A statistical ph ysics approac h to inference
problems on random net w orks: Ising and
kinetic Ising mo dels
v orgelegt v on
Ludo vica Bac hsc hmid Romano
geb oren in Milano, Italien
v on der F akult¨ at IV - Elektrotec hnik und Informatik
der T ec hnisc hen Univ ersit¨ at Berlin
zur Erlangung des ak ademisc hen Grades
Doktor der Naturwissensc haften
- Dr.-rer.-nat. -
genehmigte Dissertation
Promotionsaussc h uss:
V orsitzender: Prof. Dr. Benjamin Blank ertz
Gutac h ter: Prof. Dr. Manfred Opp er
Gutac h ter: Prof. Dr. Johannes Berg
Gutac h ter: Prof. Dr. Da vid Saad
T ag der wissensc haftlic hen Aussprac he: 15. Dezem b er 2017
Berlin, 2018

Abstract
Recen t adv ances in measuremen t tec hnologies ha v e resulted in the a v ailabil-
it y of large datasets from a v ariet y of fields spanning the natural and so cial
sciences. This p osed the c hallenge to dev elop new statistical to ols to extract
relev an t information from the data. A paradigmatic mo del that has b een suc-
cessfully applied to analyze large datasets is the Ising mo del of binary spins
in teracting through pairwise connections. In this thesis, w e use metho ds of
statistical ph ysics to tac kle sev eral op en problems related to mo delling the
sto c hastic dynamics of the Ising mo del and reconstructing the unkno wn net-
w ork of in teractions from data. First, w e deriv e a no v el mean-field solution
to the discrete time parallel dynamics of the Ising mo del, based on a w eak
coupling expansion of the log-generating function with constrained first and
second order momen ts o v er time, the result of whic h outp erforms other mean
field tec hniques in predicting single site magnetization. Next, for b oth the
equilibrium and kinetic mo dels, w e analyze the in v erse problem of learning
the couplings b et w een the v ariables based on a set of observ ations on spin
configurations. Using the ca vit y and replica metho ds of statistical ph ysics,
w e compare the p erformance of differen t inference algorithms, b y analytical
computation of the estimation error as a function of the size of the dataset,
and study its deviation from asymptotic optimalit y . W e also deriv e optimal
algorithms for learning the couplings. Finally , w e consider the case where
a subset of the spin tra jectories is observ ed while the rest are hidden. This
enabled us to mo del systems where only a finite fraction of the system is exp er-
imen tally accessible, but allo w ed the hidden v ariables to affect the dynamics
of the observ ed v ariables. A cen tral question is the prediction of the hidden
spin state when the couplings are kno wn. F or the a v erage case scenario, w e
in v estigate the theoretically optimal p erformance for predicting hidden spins
b y computing the error of the Ba y es optimal predictor. W e also deriv e a mean-
field formalism to accurately estimate the single-site magnetisation of hidden
spins for single instances of the net w ork.
iii

Zusammenfassung
Die v erf ¨ ugbare Datenmenge in Gebieten der Natur- und Sozialwissensc haften
w¨ ac hst stetig durc h den tec hnisc hen F ortsc hritt b ei Metho den zur Datenerhe-
bung. Dieser Zu w ac hs ist eine Herausforderung f ¨ ur Algorithmen, die Daten
auf relev an ten Informationen reduziert. Ein paradigmatisc hes Mo dell, das
mit Erfolg auf groen Datenmengen angew endet wurde, ist das “Ising Mo d-
ell” f ¨ ur bin¨ are Spins, w elc he durc h paarw eise W ec hselb ezieh ungen v oneinan-
der abh¨ angig sind. In dieser Arb eit v erw enden wir Metho dik der statistis-
c hen Ph ysik zur L¨ osung v ersc hiedener offener Probleme, v erbunden mit der
Mo dellierung sto c hastisc her Dynamik und der Rek onstruktion un b ek ann ter
Netzw erk e durc h In teraktionen, w elc he in Daten b eobac h tet w erden. Als er-
stes leiten wir eine neue “Mean-field” L¨ osung her f ¨ ur die zeitdiskrete par-
allele Dynamik des Ising Mo dells. Hierzu en t wic k eln wir die logarithmisc he
Momen t generierende F unktion in den sc h w ac hen Kopplungen b edingt auf
ersten und zw eiten Momen ten an jedem Zeitpunkt. W eiterhin un tersuc hen
wir das in v erse Ising Problem f ¨ ur so w ohl das Gleic hgewic h ts- als auc h das
kinetisc he Mo dell. Das heit, wir analysieren das Lernen der Kopplungen zwis-
c hen V ariablen gegeb en ein Set v on Beobac h tungen. Durc h den Gebrauc h
v on “Ca vit y-” und “Replik ametho den” aus der statistisc hen Ph ysik v ergle-
ic hen wir v ersc hieden Inferenzalgorithmen durc h analytisc he Berec hn ung des
Sc h¨ atzfehlers abh¨ angig v on der Menge der v erf ¨ ugbaren Daten und un tersuc hen
die Ab w eic h ungen dieser v on der optimalen Asymptotik. Auerdem leiten wir
optimale Algorithmen zum Lernen der Kopplungen her. Am Ende b etrac h ten
wir den F all, in dem ein T eil der Spin tra jektorien b ek ann t ist, w¨ ahrend ein
T eil nic h t b eobac h tet wird. Dies erlaubt uns Systeme zu mo dellieren, die
uns exp erimen tell n ur un v ollst¨ andig zug¨ anglic h sind, un ter Ber ¨ uc ksic h tigung,
dass die un b eobac h teten V ariablen die b eobac h teten Dynamik en b eeinflussen
k¨ onnen. V on zen traler Bedeutung ist die V orhersage des Zustands der nic h t
b eobac h teten Spins, w enn die Kopplungen b ek ann t sind. F ¨ ur den gemittel-
ten F all un tersuc hen wir das theoretisc h optimale Ergebnis zur V orhersage der
nic h t b eobac h teten Spins durc h b erec hnen des F ehlers eines Ba y es optimalen
Pr¨ adiktors. Wir leiten einen “Mean-field” F ormalism us f ¨ ur einzelne Instanzen
v on Netzw erk en her, um die marginale Magnetisierung der nic h t b eobac h teten
Spins pr¨ azise zu sc h¨ atzen.
v

Akno wledgements
I w ould lik e to express m y gratitude to Prof. Manfred Opp er for his supp ortiv e
and patien t sup ervision and for the coun tless things that I learned from him
through our discussions. His insigh ts and con tagious en th usiasm for researc h
ha v e b een a constan t source of motiv ation ov er the y ears.
I gratefully ac kno wledge the Marie Curie Initial T raining Net w ork NET ADIS
for their funding and all the professors and studen ts in v olv ed in the pro ject:
from lectures to informal c hats at sc ho ols and conferences, the pro ject pro vided
extremely enric hing exp eriences b oth from a scien tific and a p ersonal p oin t of
view. I wish to thank Prof. P eter Sollic h for co ordinating the pro ject and for
his atten tiv e guidance during m y secondmen t at King’s College London; Prof.
Y asser Roudi for hosting me at the Ka vli Institute for Neuroscience at NTNU,
for his incisiv e suggestions and encouragemen t; Prof. Andrea P agnani for the
inspiring discussions I had with him. P ascale Searle pro vided in v aluable help
in managing the pro ject. Sp ecial thanks to Barbara and Claudia for their
fruitful co op eration and for w armly w elcoming me in London and T rondheim
resp ectiv ely; to Silvia, Barbara, and Carla for all the stim ulating con v ersations,
whic h b egan in discussions of ph ysics and ended in a precious friendship.
I w ould also lik e to thank the mem b ers of m y defense committee, Prof.
Johannes Berg and Prof. Da vid Saad, for carefully examining m y thesis and
pro viding constructiv e commen ts.
I am grateful to all the mem b ers of the KI group for creating an enjo y able
w ork en vironmen t – esp ecially to m y office mates Christian and Noa for making
ro om 4.014 a fun and relaxed space, to Philipp for sharing his passion for m usic,
and to Cordula for helping me with all the administrativ e tasks.
Last but not least, I w ould lik e to extend m y gratitude to m y family for
giving me unconditional lo v e and supp ort and to all m y friends scattered o v er
the glob e for making m y do ctoral y ears so meaningful.
vii

Contents
1 Intro duction 1
1 . 1 M o t i v a t i o n ............................. 1
1.2 The Ising mo del and the kinetic Ising mo del . . . . . . . . . . 3
1.2.1 The Ising mo del . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Explaining correlations in neural spik e trains . . . . . . 5
1.2.3 Glaub er dynamics . . . . . . . . . . . . . . . . . . . . . 6
1.2.4 Inferring effectiv e connectivities in neuronal net w orks . 8
2 Thesis Outline 11
3 Mean field app roaches to dynamics on random net w o rks: the
kinetic Ising mo del 15
3 . 1 I n t r o d u c t i o n ............................ 1 5
3 . 2 P a p e r 1 . .............................. 1 8
3 . 3 D i s c u s s i o n ............................. 5 3
3 . 4 C o n c l u s i o n s ............................ 5 4
App endices 59
App endix 3.A T AP equations for the SK mo del . . . . . . . . . . . 59
3.A.1 The ca vit y approac h . . . . . . . . . . . . . . . . . . . 59
3.A.2 Plefk a’s expansion . . . . . . . . . . . . . . . . . . . . 61
App endix 3.B Mean field approac hes to the kinetic Ising mo del:
p r e v i o u s r e s u l t s .......................... 6 3
App endix 3.C Algorithm with a v eraged momen ts . . . . . . . . . . 67
4 Lea rning in kinetic Ising mo dels 69
4 . 1 I n t r o d u c t i o n ............................ 6 9
4 . 2 P a p e r 2 . .............................. 7 1
4 . 3 F u r t h e r r e s u l t s .......................... 9 2
4.3.1 The optimal linear estimator . . . . . . . . . . . . . . . 92
4.3.2 Ba y esian inference . . . . . . . . . . . . . . . . . . . . 93
4.3.3 Appro ximating the p osterior b y ca vit y argumen ts . . . 95
4.3.4 A simple exp ectation propagation algorithm . . . . . . 96
4.3.5 Av erage case: a replica analysis . . . . . . . . . . . . . 99
ix

Con ten ts
4 . 3 . 6 R e s u l t s ........................... 1 0 1
4 . 4 C o n c l u s i o n s ............................ 1 0 2
App endices 105
App endix 4.A The replica metho d: from spin glasses to neural net-
w o r k s ................................ 1 0 5
4.A.1 Spin glasses and the replica tric k . . . . . . . . . . . . 105
4.A.2 Statistical mec hanics of learning: general setup . . . . 108
App endix 4.B Maxim um lik eliho o d estimator . . . . . . . . . . . . 112
App endix 4.C Mean field estimators for the stationary state . . . . 113
App endix 4.D Mean field estimators for the transien t dynamics . . 114
App endix 4.E Exp ectation Propagation algorithm: generating func-
tion of the momen ts . . . . . . . . . . . . . . . . . . . . . . . 115
App endix 4.F Fixed p oin t of the Exp ectation Propagation algorithm 116
App endix 4.G Complete Exp ectation Propagation . . . . . . . . . 117
App endix 4.H Details of the replica calculation . . . . . . . . . . . 118
5 Lea rning Curves fo r the inverse Ising p roblem 121
5 . 1 I n t r o d u c t i o n ............................ 1 2 1
5 . 2 P a p e r 3 . .............................. 1 2 3
5.3 F urther results . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5 . 4 C o n c l u s i o n s ............................ 1 5 3
App endices 157
App endix 5.A The lik eliho o d and pseudo-likelih o o d functions . . . 157
6 Lea rning and inference in p resence of hidden units 159
6 . 1 I n t r o d u c t i o n ............................ 1 5 9
6 . 2 P a p e r 4 . .............................. 1 6 1
6 . 3 F u r t h e r r e s u l t s .......................... 1 7 8
6.3.1 Inferring hidden states in a kinetic Ising mo del via the
extended Plefk a expansion . . . . . . . . . . . . . . . . 178
6.3.2 Net w ork reconstruction . . . . . . . . . . . . . . . . . . 183
6 . 4 C o n c l u s i o n s ............................ 1 8 7
App endices 191
App endix 6.A Details of the extended Plefk a expansion . . . . . . 191
App endix 6.B Sparse observ ations . . . . . . . . . . . . . . . . . . 196
7 Summa ry and outlo ok 199
x

1 Intro duction
1.1 Motivation
Due to recen t tec hnological adv ances in data acquisition, the last t w o decades
ha v e witnessed a rapid increase in the amoun t and ric hness of data that can
b e collected in man y fields of natural sciences, finance, so cial sciences, and
comm unication net w orks. This has shifted the fo cus from the analysis of sin-
gle system comp onen ts to the attempt to understand the system as a whole.
Remark able examples can b e found in biology , where the adv en t of m ulti-
comp onen t recordings op ened up en tire new lines of researc h, suc h as genomics,
transcriptomics, proteomics, metab olomics, and the analysis of sim ultaneous
measuremen ts of man y neurons. As common feature, these large datasets en-
co de the activit y of large systems of man y in teracting units, where the direct
connections b et w een them are not directly measurable.
T o understand ho w the system op erates, it is crucial to reconstruct suc h
underlying net w orks of in teractions. The inverse pr oblem of reconstructing
the net w ork starting from the empirical kno wledge of certain observ ables, suc h
as a v erages and correlations, has raised a lot of in terest within the statistics,
mac hine-learning, and statistical ph ysics comm unities. The main c hallenge
is to disen tangle direct connections from mere correlations that could emerge
from indirect influence of in termediate comp onen ts. The task is ev en more
difficult since the system is t ypically not en tirely accessible, and the data are
noisy .
In v erse problems generally ha v e no unique solution but can b e tac kled in
a probabilistic form ulation, where the system is mo delled b y a parametrised
distribution; parameters are then inferred, either maximising the lik eliho o d
of the data or using Ba y esian estimators (where further a priori information
on the parameters is incorp orated through a prior). Ho w ev er, exact inference
requires the computation of high dimensional in tegrals and is in tractable for
most mo dels of in terest; hence, a lot of effort has b een made to deriv e efficien t
appro ximate tec hniques for this task.
The field of statistical mec hanics, whose fo cus is to study large systems of in-
teracting particles and to un v eil the relations b et w een microscopic in teractions
and macroscopic observ ables, pro vides a whole series of tec hniques that can b e
1

1 In tro duction
used b oth to construct inference algorithms based on suitable appro ximation
sc hemes and to theoretically assess the algorithms’ p erformance.
A mo del of statistical ph ysics widely used for net w ork reconstruction is the
Ising mo del, describing an equilibrium system of binary v ariables (spins) in ter-
acting via pairwise connections. Despite b eing an ob vious o v er-simplification
of real-w orld systems, it can b e used v ery effectiv ely to disen tangle direct in-
teractions among the v ariables from spurious coupling effects. Ho w ev er, the
assumption that the system is at equilibrium -whic h requires that the connec-
tions are symmetric- is unrealistic for man y applications.
More recen tly , a simple extension to dynamics of the Ising mo del (here-
after denoted as the kinetic Ising mo del) has b een emplo y ed for reconstructing
biological, financial, and gene regulatory net w orks, where the equilibrium as-
sumption is relaxed. Moreo v er, in the case of time-series data, the temp oral
structure enco ded in the dataset can b e exploited.
While the equilibrium Ising mo del has b een extensiv ely studied in the last
decades and a wide b o dy of literature has b een dev oted to dev elop appro ximate
algorithms for the in v erse Ising problem, the atten tion to its out-of-equilibrium
dynamics is more recen t, and ev en the problem of relating the system param-
eters to the time ev olution of the observ ables is not y et fully solv ed.
In this con text, the con tribution of this thesis is to w ards b oth a theoretical
analysis and the construction of new inference algorithms. First, for the kinetic
Ising mo del, w e will study the forw ard problem of predicting the time ev olution
of the single spin magnetization for fixed connections b et w een the spins. W e
will fo cus on densely and w eakly connected net w orks, for whic h an exact mean-
field theory can b e form ulated in the thermo dynamic limit of a large system.
In con trast to the equilibrium case, the v ariables of in terest are not single
spins but en tire spin tra jectories, and the goal is to compute the marginal
distribution of single-spin tra jectories. The exact mean-field solution has so far
b een found only in the case of fully asymmetric in teractions [MS11], in whic h
t w o-times correlations are negligible. In the case of generic degree of symmetry
of the couplings, a recen t approac h [MS14] whic h incorp orates the effect of
time correlations has impro v ed on the prediction of single site magnetizations;
still, it w as not clear ho w the exact mean-field solution w ould lo ok lik e in the
thermo dynamic limit. W e will deriv e a no v el appro ximate tec hnique of the
mean-field t yp e to tac kle the problem in the limit of an infinitely large system.
The second fo cus of the dissertation concerns the theoretical p erformance
of inference algorithms, whic h estimate the couplings b et w een the spins based
on a set of observ ations of s pin configurations. V arious algorithms based on
appro ximate sc hemes ha v e b een prop osed for the in v erse Ising problem, and
more recen tly for the in v erse kinetic Ising problem, with the goal of pro viding
2

1.2 The Ising mo del and the kinetic Ising mo del
computationally efficien t inference to ols for large net w orks [NZB17a]. While
their p erformance has b een only tested on sp ecific instances of the problem, it
is imp ortan t to assess their statistical efficiency in a unified theoretical setting.
Both for the kinetic and for the equilibrium case, w e analytically compute the
t ypical p erformance of v arious estimators of the couplings and pro vide new
algorithmic implemen tations of the most efficien t ones.
As a last p oin t, w e observ e that in most biological net w orks, often only
a small fraction of the system is exp erimen tally accessible. V ariables whose
activit y is recorded will also in teract with v ariables not directly observ able
or detectable, usually referred to as hidden v ariables. Hence, recen t w orks
[TH13, DR13, Hua15, BHTR15, R T15, DB16] ha v e in tro duced a mo del where
a fraction of the spin tra jectories are observ ed and a fraction are hidden.
Net w ork reconstruction is m uc h harder in this scenario, and no satisfactory
solution has b een found in dense net w orks if the hidden v ariables are connected
among themselv es. Exact learning rules imply the summation o v er all p ossible
configuration of hidden spins, whic h is in tractable for large systems; one can
resort to learning rules that are based on an appro ximate estimation of hidden
no des at fixed couplings, but the accuracy of inferring the state of the unob-
serv ed v ariables for giv en system parameters w as not clear. In the last part
of the dissertation, w e address this problem b y in v estigating the theoretically
optimal p erformance for predicting the hidden spins. W e will also in tro duce
a no v el tec hnique to predict the single site magnetization of hidden spins for
single instances of the net w ork.
1.2 The Ising mo del and the kinetic Ising mo del
After in tro ducing the Ising mo del and its simplest generalization to dynamics,
w e will sho w ho w they can b e used to mo del the dep endencies of spik es recorded
from ensem bles of neurons. Other application domains include determining the
3D structure of proteins [WWS + 09], analyzing gene expression data from gene
regulatory net w orks [LBC + 06], inferring the fitness landscap e in a ev olutionary
biology w eb of ecological in teraction b et w een sp ecies [BCG + 12]. A recen t
review of those and other metho ds can b e found in [NZB17a].
1.2.1 The Ising mo del
The Ising mo del w as in tro duced to study the macroscopic prop erties of mag-
netic materials [Hua87, LL69], man y-b o dy systems comp osed of molecules with
a magnetic moment - a v ector whic h tends to align with the magnetic field act-
ing on the molecule. Magnetic momen ts of individual molecules are describ ed
3

1 In tro duction
b y binary v ariables, σ i = ± 1 (or Ising spins), lo calized at the v ertices of a lat-
tice. T o eac h pair of v ariables at sites i, j w e assign an in teraction energy with
v alue − J ij if the spins σ i and σ j are p oin ting in the same direction ( σ i = σ j )
and with v alue J ij otherwise ( σ i = − σ j ). In some cases, eac h site i also has
its o wn energy − σ i h i , due to the presence of a (lo cal) external field h i . The
energy of the man y-particle system, or Hamiltonian, is
H = − X
i,j ∈ B
J ij σ i σ j −
N
X
i = i
h i σ i , (1.1)
where N is the total n um b er of spins. The c hoice of the set of b onds B
dep ends on the problem one is in terested in. W e will consider fully connected
net w orks, where eac h spin is directly in teracting with all the other spins in
the net w ork: i = 1 , . . . , N and j = 1 , s . . . , i − 1. In the canonical ensem ble,
the probabilit y distribution of the v ariables σ = { σ 1 . . . σ N } is the Boltzmann-
Gibbs distribution
P ( σ ) = e − β H
Z , (1.2)
where β = 1 /T is the in v erse temp erature and the normalization factor Z ,
Z = X
σ 1 = ± 1 X
σ 2 = ± 1
... X
σ N = ± 1
e − β H , (1.3)
is referred to as the partition function. In the follo wing, w e will use either the
notation P σ or T r σ to denote the sum o v er all p ossible spin configurations.
The distribution (1.2) can b e also in terpreted from an information theoretic
p oin t of view (see, for instance, [Ja y57]). Imagine w e w an t to describ e the
probabilit y distribution P ( σ ) of a set of binary v ariables σ . T o mo del the
system b y making the minim um p ossible assumptions b ey ond what w e can
directly measure from the system itself, w e can use the maxim um en trop y
principle. Indeed, the Shannon en trop y , defined as
S [ P ] = − X
σ
P ( σ ) log P ( σ ) , (1.4)
quan tifies the uncertain t y of the set of random v ariables σ : the larger the
en trop y , the less a priori information one has on the v alue of the v ariables.
Hence, making the minim um assumptions on the form of P ( σ ) corresp onds
to finding the distribution that maximizes the Shannon en trop y . If something
is kno wn ab out the statistics of the v ariables, the maximization has to b e
p erformed under the corresp onding constrain ts. Let us assume that w e can
compute sample a v erages of σ i and σ i σ j in the data; the maxim um en trop y
4

1.2 The Ising mo del and the kinetic Ising mo del
distribution sub ject to these constrain ts is (1.2), where { h i , J ij } are the La-
grange m ultipliers that ha v e to b e c hosen so that the a vera ges {h σ i i , h σ i σ j i}
with resp ect to (1.2) agree with exp erimen ts. In other w ords, the distribution
of the Ising mo del can b e seen as the less biased distribution that repro duces
the observ ed a v erages and pairwise correlations b et w een the v ariables.
While the forw ard Ising problem consists in predicting system observ ables
– suc h as spin magnetizations and correlations – giv en a complete description
of the system, in the in v erse Ising problem w e can measure magnetizations
and correlations from a system whose parameters are unkno wn; the goal is to
infer the parameters (i.e., couplings and lo cal fields) from the data. W e will
discuss metho ds to p erform this inference task in Chapters 4 and 5. In the
follo wing section, w e pro vide an o v erview of ho w the in v erse Ising problem has
b een applied to the field of computational neuroscience.
1.2.2 Explaining co rrelations in neural spik e trains
A ma jor c hallenge in neuroscience is to understand ho w neurons pro cess infor-
mation through collectiv e in teractions. Suc h understanding has b een limited
b y the p ossibilit y to exp erimen tally access only a tin y fraction of the exp o-
nen tial n um b er of p ossible activit y patterns. Recen tly , m ulti-electro de arra ys
tec hniques ha v e allo w ed to sim ultaneously record the activit y of h undreds of
neurons, and the spatio-temp oral resolution with whic h these recordings can
b e done is rapidly increasing. Although net w orks remain dramatically under-
sampled, there is no w the p ossibilit y to address questions that w ere previously
out of reac h.
One cen tral op en question is the origin of the m ulti-neuron firing patterns
observ ed in exp erimen ts [SBIB06], whic h seemed to b e in con trast with the
w eak measured correlations b et w een pairs of neurons. Recen t w orks [SBIB06,
CLM09, SF G + 06] ha v e used maxim um en trop y principles to explain ho w w eak
correlations among elemen ts can ha v e a strong effect on the state of the p op-
ulation as a whole.
F or example, the authors of [SBIB06] analyze sim ultaneous recordings from
40 neurons in the salamander retina. Time is divided in ∆ τ = 20 ms time
steps, and the activit y of eac h cell in a giv en time step is represen ted b y a
spin, with v alue σ i = 1 if the neuron is spiking, σ i = 0 if it is silen t. The
considered data is the set of observ ed sim ultaneous (i.e. within a single time
bin) spik e patterns, without regard to their temp oral order. The authors use
the Ising mo del as the minimal mo del that incorp orates pairwise correlations
and sho w that it can accurately predict the com binatorial patterns of spiking
and silence in retinal ganglion cells; in con trast, mo dels of indep enden t neurons
drastically fail. The comparison of theory and exp erimen t is done for groups
5

1 In tro duction
of N = 10 neurons, whic h are small enough that the full distribution P ( σ ) of
a spin configuration can b e sampled exp erimen tally . The external fields and
pairwise symmetric couplings are inferred b y maximizing the lik eliho o d of the
data. Here, the couplings describ e the in teractions b et w een neurons within
the terms of the fitted mo del and are referred to as effectiv e or functional
connections. Hence, inferring the v alues of the couplings starting from the data
allo ws to reconstruct the net w ork of causal in teractions b et w een the neurons.
This minimal mo del has b een generalized in differen t w a ys, for example b y
adding a stim ulus-dep enden t field acting on the spins [GA TSS13], or including
higher order in teractions [GSS11].
Ho w ev er, an eviden t limitation is that it do es not address the temp oral ev o-
lution of correlated states. In terestingly , the authors of [TJH + 08] p oin t out
that, if correlated states o ccurred in a temp orally indep enden t manner, con-
catenating the states sampled from the Ising mo del should giv e a reasonable
estimate of the lengths of observ ed m ulti-neuron firing patterns. Ho w ev er, they
observ ed that sequences of correlated states w ere significan tly longer than pre-
dicted b y concatenating states from the mo del. This suggested that temp oral
dep endencies are a common feature of cortical net w ork activit y , and should b e
considered in the mo dels.
Moreo v er, it w as sho wn [TRMH13] that correlations in the statistics of neu-
ral spik e trains could arise b oth as the effect of in teraction b et w een neurons
or b y sharing a common non-stationary input, where no in teraction among
neurons is presen t (see also [TMM + 14]).
Hence, a deep er insigh t could b e ac hiev ed through dynamical mo dels. Time-
v arying inputs and t w o-times correlations can b e tak en in to accoun t b y a simple
generalization to dynamics of the Ising mo del, i.e. the kinetic Ising mo del with
Glaub er up date rule.
1.2.3 Glaub er dynamics
Let us consider a set of Ising spins in teracting trough couplings J ij and with
a (time-dep enden t) external field. They also in teract with an external agency
(e.g., a heat reserv oir) whic h causes them to c hange their states randomly
with time. The noise in tro duced b y such external agency is parametrized
b y the in v erse temp erature β . Eac h spin σ i ( t ) is a sto c hastic function of
time and mak es random transitions b et w een the v alues ± 1, according to the
v alue of the neigh b oring spins. In particular, the lo cal transition probabilit y
w i [ σ i ( t + ∆ t ) |{ σ j ∈ ∂ i ( t ) } ] that a site i at time t + ∆ t has spin σ i ( t + ∆ t ), giv en
6

1.2 The Ising mo del and the kinetic Ising mo del
the v alue of its neighbors spins σ j ∈ ∂ i ( t ) at time t , is [Gla63]:
w i [ σ i ( t + ∆ t ) |{ σ j ∈ ∂ i ( t ) } ] = e β σ i ( t +∆ t ) θ i ( t )
2 cosh β θ i ( t ) ,
θ i ( t ) = X
j ∈ ∂ i
J ij σ ( t ) j + h i ( t ) . (1.5)
Note that - con trary to the equilibrium case - w e are not in tro ducing an y energy
function; no w the net w ork of couplings is directed, and in general J ij 6 = J j i .
Self-in teractions J ii migh t b e presen t or not: in the follo wing sections of the
thesis, w e will fo cus on cases where suc h in teractions are absen t. The param-
eter β quan tifies the randomness of the dynamics: for β → 0 the dynamics
is completely random, for β → ∞ it is deterministic. V arious up date rules
can b e defined for this dynamics. One c hoice is to up dated sim ultaneously all
the spins at discrete time steps. Suc h p ar al lel (or sinc hronous) dynamics is
defined b y the Mark o v c hain
P ( σ ( t + 1)) = X
σ t
W [ σ ( t + 1); σ ( t )] P ( σ ( t )) , (1.6)
with transition probabilit y
W [ σ ( t + 1); σ ( t )] = Y
i
w i [ σ i ( t + 1) |{ σ j ∈ ∂ i ( t ) } ] . (1.7)
In case of symmetric in teractions J ij = J j i and stationary external fields
h i ( t ) = h i , the dynamics ob eys detailed balance and the equilibrium distri-
bution can b e written in the Boltzmann form P eq ( σ ) ∼ e β H ( σ ) , where the H
is the P eretto pseudo-Hamiltonian [P er84] (i.e., an Hamiltonian dep enden t on
the in v erse-temp erature):
H ( σ ) = − X
i
h i σ i − 1
β X
i
log 2 cosh[ β θ i ( σ )] . (1.8)
A differen t dynamics is defined b y the se quential (async hronous) up date
rule, where at eac h time step only one randomly c hosen spin is up dated; the
duration of eac h up date is 1 / N , so that - on a v erage - all spins ha v e b een
up dated once on a time scale O ( N 0 ). A sequen tial dynamics with discrete time
can b e defined b y the Mark o v c hain (1.6) with transition probabilit y [Co o01]
W [ σ ( t + 1); σ ( t )] = 1
N X
i " Y
j 6 = i
δ σ j ( t +1) , σ j ( t ) # w i [ σ i ( t + 1) |{ σ j ∈ ∂ i ( t ) } ] . (1.9)
7

1 In tro duction
In the con tin uous limit N → ∞ , the pro cess ob eys the master equation
d
dt P ( σ ( t )) = X
i
[ P ( F i σ ( t )) w i ( F i σ ( t )) − P ( σ ( t )) w i ( σ ( t ))] , (1.10)
where no w
w i ( σ ( t )) = 1
2 [1 − σ i tanh β θ i ( t )] , (1.11)
and where F i is the flip op erator: F i σ = { σ 1 ,..., − σ i , . . . , σ N } . In case of
symmetric in teractions and stationary external fields, the dynamics con v erges
to the equilibrium distribution P eq ( σ ) ∼ e β H ( σ ) , where H is the Hamiltonian
of the Ising mo del (1.1).
1.2.4 Inferring effective connectivities in neuronal net w o rks
The Glaub er parallel dynamics for an Ising system has b een used to mo del
net w orks of neurons that spik e at a time-v arying rate whic h dep ends on earlier
spik es and on external co v ariates (suc h as a stim uli).
Spik e trains recorded from N neurons are divided in to small time bins, and a
binary v ariable σ i ( t ) is assigned to eac h neuron i at eac h time bin t , with v alue
1 if the neuron has emitted one or more spik es in the time bin, − 1 otherwise.
Tw o recen t studies [RH11b, CF G + 15] sim ulated biologically realistic corti-
cal mo dels and sho w ed that functional connections inferred using the kinetic
Ising mo del can b e successfully used for net w ork reconstruction, i.e. to dis-
tinguish connected vs disconnected pairs of neurons. In addition, the authors
of [CF G + 15] prop ose a metho d to o v ercome the limitations of a probabilistic
dynamics with a single arbitrary time-step and to correct for the bias in-
tro duced b y the arbitrary c hoice of the time-bin used to binarize the spik e
trains. A dynamic mo del - con trary to an equilibrium one - allo ws us to infer
non-symmetric in teractions, and real synaptic in teraction are in general not
symmetric. Y et, the exact relation b et w een the inferred functional connec-
tions and the real synaptic connections is non-trivial and could b e understo o d
only analysing recordings from circuits where the actual ph ysiological synapses
b et w een neurons are kno wn 1 .
Ho w ev er, some features of the inferred connections can b e robust to c hanges
in the mo del and giv e precious insigh ts on the prop erties of the real system.
1 So far, only few w orks [GK G + 13, VIS15, LCRP14] ha v e v alidated connectivity estimates
with some form of ’ground truth’; they sho w ed that approaches based on Generalized
Linear Mo dels (GLMs) w ere successful in inferring the true connectivit y of the circuit,
while linear mo dels and mo del-free approac hes failed; this encourages for the choice of
GLM-based approac hes to estimate synaptic connectivit y (note that the kinetic Ising
mo del can b e seen as a simplified GLM with a limited temp oral memory).
8

1.2 The Ising mo del and the kinetic Ising mo del
A relev an t example is the work b y Dunn and collab orators [DMR15]. They
analyse sim ultaneous recordings from tens of grid cells in t w o rats freely mo ving
in a 2D en vironmen t. Grid cells are neurons that presen t a particular spatial
selectivit y , suc h that the p ositions in real space where one particular cell is
firing form an hexagonal grid; the relativ e p osition of the grids of t w o distinct
cells is called their relativ e phase. Fitting a kinetic Ising mo del with parallel
up date rule to the data, the authors find a systematic dep endence of the
couplings b et w een t w o cells and their relativ e phase: cells with nearb y phases
ha v e p ositiv e functional connection strengths, while those further apart ha v e
negativ e ones. The authors explain a w a y v arious sources of correlations that
could lead to spurious connections, suc h as the o v erlap of the firing fields, and
head directional input. The result is relev an t as, since attractor mo dels of
grid cells rely hea vily on this t yp e of effectiv e connectivit y , this w ork pro vides
supp ort for the idea of attractor dynamics in the grid cell assem bly .
9

2 Thesis Outline
This dissertation is written in the form of a thesis b y publication, where I col-
lect the pap ers that I ha v e co-authored in separate c hapters. In eac h c hapter,
the pap er is follo w ed b y unpublished results and b y an app endix that briefly
reviews the fundamen tal metho ds used in the pap er.
Chapter 2 considers the forw ard problem of predicting the time ev olu-
tion of system observ ables in the kinetic Ising mo del, assuming that the pa-
rameters are kno wn. With the aim of analysing the system in a mean-field
framew ork, w e will in tro duce a no v el tec hnique referred to as the extended
Plefk a expansion. It is an extension to dynamics of the Plefk a expansion for
the Sherrington–Kirkpatric k mo del, where the no v elt y lies in constraining not
only the first momen ts in the expansion, but also all marginal second momen ts.
W e conjecture that it pro vides the exact mean field solution to the forw ard
problem in the thermo dynamic limit of infinitely man y particles, when the
couplings are w eak and long-ranged.
Publication included at page 17 (Publisher v ersion): Bac hsc hmid-Romano,
Ludo vica, et al. V ariational p erturb ation and extende d Plefka appr o aches to
dynamics on r andom networks: the c ase of the kinetic Ising mo del. Journal of
Ph ysics A: Mathematical and Theoretical 49.43 (2016): 434003.
doi:10.1088/1751-8113/49/43/434003
In Chapter 3 , w e analytically study the p erformance of t w o algorithms
for learning the couplings in the kinetic Ising mo del, fo cussing on the case of
asymmetric couplings. The first one is based on the exact mean-field solution
for the asymmetric mo del deriv ed in [MS11]; the second is a Ba y esian estimator
of the couplings, where w e appro ximate the p osterior means - whose exact
computation is in tractable- using the ca vit y metho d of statistical ph ysics. W e
compute the estimation error of these metho ds, as a function of the length of
observ ed tra jectories. The theoretical setting for our analysis is offered b y the
statistical mec hanics of in v erse problems, where the phase space consists of the
couplings to b e inferred, while the spin v alues are treated as fixed observ ations;
the replica metho d of spin glasses is used to compute the a v erage error of a
giv en estimator of the couplings in the limit of large systems, where the ratio
α = M / N remains finite - M b eing the size of the dataset and N the size of the
system. The main c hallenge will b e to treat analytically the distribution of the
spin observ ations, but the analysis will b e simplified b y the fact that t w o-times
11

2 Thesis Outline
correlations deca y after one time step for asymmetric net w orks. Ho w ev er, the
equal-time correlation matrix will pla y a ma jor role in determining the sp eed
of learning and w e will compute its statistics in order to get an explicit result
for the estimation error. W e also design an efficien t algorithm to n umerically
implemen t the optimal Ba y es estimator.
Publication included at page 68 (Publisher v ersion): Bac hsc hmid-Romano,
Ludo vica, and Manfred Opp er. L e arning of c ouplings for r andom asymmet-
ric kinetic Ising mo dels r evisite d: r andom c orr elation matric es and le arning
curves. Journal of Statistical Mec hanics: Theory and Exp erimen t 2015.9
(2015): P09016.
doi:10.1088/1742-5468/2015/09/P09016
The formalism of Chapter 3 is then extended in Chapter 4 to analyse the
error of learning the couplings in an equilibrium mo del. The distribution of
the data is ev en more difficult to treat, due to the presence of the normalising
partition function, and w e will use a com bination of the ca vit y and replica
metho ds of spin glasses to carry out the calculation and include the effect
of correlations. W e study the p erformance of algorithms based on the min-
imisation of a lo cal cost function, fo cussing on the pseudo-lik eliho o d and the
mean-field estimators. Surprisingly , w e will find that a simple quadratic cost
function is the one that ac hiev es minimal error, and the explicit estimator
asso ciated with it can b e en tirely constructed from data.
Publication included at page 120 (Publisher v ersion): Bac hsc hmid-Romano,
Ludo vica, and Manfred Opp er. A statistic al physics appr o ach to le arning
curves for the inverse Ising pr oblem. Journal of Statistical Mec hanics: Theory
and Exp erimen t 2017.6 (2017): 063406.
doi:10.1088/1742-5468/aa727d
Chapter 5 treats an extension of the kinetic Ising mo del, where a fraction of
the tra jectories is observ ed and a fraction is hidden. Using the replica metho d,
w e analytically compute the a v erage error of the Ba y es optimal estimator of
hidden spins, whic h is obtained from the p osterior distribution of unobserv ed
spins giv en the observ ed ones. W e then turn to the study of single instances
of the net w ork. Using the extended Plefk a expansion, w e deriv e a set of mean-
field equations c haracterising the dynamics of the hidden-spin v ariables, and
w e can accurately estimate the single-site magnetisation of hidden spins. In
the end, w e discuss the applicabilit y of our result as building blo c k for an
algorithm aimed at reconstructing the net w ork connections.
Publication included at page 158 (Publisher v ersion): Bac hsc hmid-Romano,
Ludo vica, and Manfred Opp er. Inferring hidden states in a r andom kinetic
Ising mo del: r eplic a analysis. Journal of Statistical Mec hanics: Theory and
Exp erimen t 2014.6 (2014): P06013.
12

doi:10.1088/1742-5468/2014/06/P06013
In the sixth and last c hapter, w e summarise the conclusions of the single
c hapters and indicate future researc h directions.
13

3 Mean field app roaches to
dynamics on random net w o rks:
the kinetic Ising mo del
3.1 Intro duction
Mean-field metho ds of statistical ph ysics allo w for a tractable description of
complex systems of man y in teracting v ariables. Based on the assumption that
the fluctuations around the a v erage v alue of the order parameters are small,
they pro vide a solution in whic h eac h v ariable is sub ject to an effectiv e lo cal
field, considering the in teractions with other degrees of freedom, on a v erage.
Man y highly non-trivial mean-field appro ximations w ere used to deriv e the
main equilibrium prop erties of the Ising spin glass mo del [MPV87] 1 , and made
it p ossible to in tro duce efficien t algorithms for statistical inference and opti-
misation [MM09]. In recen t y ears, a gro wing in terest has b een dedicated to
extending suc h mean-field tec hniques to the dynamic coun terpart of the Ising
mo del.
V arious kinds of dynamics can b e defined for the Ising mo del. W e are
in terested in studying the Glaub er dynamics with parallel up date rule, for
systems where the matrix of couplings is fixed and can ha v e an y degree of
symmetry , b earing in mind the applicabilit y of this framew ork to mo del the
dynamics in biological net w orks, based on time-series data.
Ho w ev er, initial mean field approac hes to the dynamics of spin systems w ere
dev elop ed for the disorder a v eraged case scenario. Soft spin mo dels w ere the
first to b e considered. If the connections b et w een the spins are symmetric,
the dynamics of the net w ork can b e describ ed as a relaxation of a global en-
ergy function to w ards lo cal minima. A relaxation dynamic of the Langevin
t yp e has b een extensiv ely analysed (for a review, see [BCKM98, Cug03]), as it
pro vides a framew ork for studying off-equilibrium b eha viours (suc h as the phe-
nomenon of ageing in glassy systems) and un v eiled strong analogies b et w een
disordered systems and other t yp es of glasses where disorder is absen t, suc h
as structural fragile glasses [P ar06]. If the connections are not symmetric, an
1 F or a brief introduction to spin glasses and disorder av erages, see section 4.A.1.
15

3 Dynamics on random net w orks
energy function cannot b e defined; ho w ev er, a Langevin equation formalism
w as dev elop ed in [CS87], where a set of lo cal self-consisten t equations for the
spin correlations and resp onse functions is deriv ed.
F or discrete spins, the discon tin uous nature of the v ariables rules out an
approac h based on Langevin equations. Instead, the sto c hasticit y of the dy-
namics can b e form ulated in a probabilistic setting, t ypically in terms of a
Glaub er rule [Gla63] that - giv en the v alue of the spin v ariable at the curren t
time - sp ecifies the probabilit y of observing a giv en spin configuration at a
follo wing time. Time can b e considered either as a discrete or a con tin uous
v ariable, and the spin v alues can b e up dated all at the same time or sequen-
tially one b y one, giving rise to div erse t yp es of dynamics. F or spin glass
mo dels with hard spins, a path in tegral formalism to describ e suc h Glaub er
dynamics with con tin uous time w as in tro duced b y Sommers [Som87]. Crisan ti
and Somp olinsky [CS88] observ ed that the mean-field equations for a net w ork
with partially asymmetric couplings are quite difficult to solv e, but they sim-
plify remark ably in the particular case of a net w ork with fully asymmetric
couplings: t w o-times correlation deca y to zero, and - in the N large limits -
the lo cal fields can b e replaced b y a time-dep enden t Gaussian random field.
An alternativ e approac h w as prop osed b y Co olen and collab orators [CS93,
CS94, CLS96]. Their dynamical replica analysis is based on a generating func-
tional formalism and deriv es deterministic flo w equations for macroscopic state
v ariables. Suc h w orks w ere follo w ed b y [NY96], where the Glaub er dynamics
of the SK mo del is studied at high temp eratures. The authors explicitly com-
pute the microscopic probabilit y distribution of the spin configuration as a
function of time, via a high-temp erature expansion.
The role of the degree of symmetry on the transien t dynamics of a system
at zero temp erature w as analysed in [EO94]. A com bination of dynamical
functional metho ds and Mon te Carlo sim ulations allo ws us to iden tify – b y
v arying the a v erage symmetry of the couplings – a transition from ergo dic-
dynamics to a phase where a finite fraction of spins freezes 2 .
Mo dels with fixed quenc hed disorder w ere studied more recen tly . Dynamical
T AP equations ha v e b een first deriv ed for the spherical p-spin mo del in [Bir99].
This w ork also analyses the conditions under whic h the dynamics is a relax-
ation in the T AP free-energy landscap e. Analogous T AP equations w ere also
recen tly found in [BSO16], where the mo del is extended to comprise generic
con tin uous v ariables and nonlinear in teraction terms; the solution is found b y
a generating functional approac h closely related to the one that w e will discuss
in P ap er 1.
2 A review of those and other metho ds used for b oth soft and hard spin mo dels can b e
found in [HKP91, Co o01].
16

3.1 In tro duction
F or Ising spins, b y information geometric argumen ts, Kapp en and Span-
jers [KS00] argued that asymmetric net w orks at the stationary state ob ey the
same T AP equations (3.6) v alid for the equilibrium mo del. This result is a
go o d appro ximation for w eak couplings, but do es not pro vide the exact mean
field description, as w as later pro v ed b y Mezard and Sak ellariou [MS11]. The
authors, follo wing the approac h of [CS88], observ e that asymmetric net w orks
exhibit small correlations among spins at v arious times. A cen tral limit the-
orem argumen t sho ws that effectiv e fields are Gaussian distributed, and the
resulting mean-field description of the dynamics is exact in the thermo dynamic
limit for w eak and long-range couplings 3 . The whole transien t dynamics for
net w orks with an arbitrary degree of symmetry w as first studied in [RH11a],
where, using a generating functional approac h, the authors deriv e T AP-lik e
equations via a small couplings expansion. Another deriv ation of these equa-
tions using information geometry w as rep orted in [AM12]. Ho w ev er, in the
limit of an asymmetric net w ork, their result do es not agree with the exact
one of [MS11]. Saad and Mahmoudi extended the w ork of [MS11] to the
case of couplings with arbitrary symmetry [MS14]. The authors still consider
the effectiv e fields as Gaussian distributed but in tro duce non-zero co v ariance
b et w een spins at differen t times and pro vide recursiv e equations to compute
correlations at all times. The result impro v es on the other metho ds and reco v er
the exact theory for asymmetric net w orks; ho w ev er, for a arbitrary degree of
symmetry , the exactness of this metho d in the limit N → ∞ remains an op en
question.
P ap er 1 in tro duces no v el approac hes to the problem. First, t w o metho ds
for deriving a naiv e mean-field equation in the static case are extended to the
kinetic case: a v ariational approac h based on minimising the Kullbac k-Leibler
div ergence b et w een the true distribution of spins and the distribution of in-
dep enden t tra jectories, and a saddle-p oin t appro ximation to the generating
functional. Then, t w o no v el appro ximations are presen ted. In a v ariational
p erturbativ e appro ximation, the action in the path in tegral represen tation of
the generating functional is expanded around a quadratic function in the fields
and conjugate fields; the latter function dep ends on v ariational parameters
that are optimised to obtain minim um sensitivit y of the appro ximating func-
tional to the v ariational parameters. In other w ords, the generating functional
of the dynamics is appro ximated b y a Gaussian distribution, and its param-
eters are later optimised. The result will strongly dep end on the constrain ts
imp osed on the parameters of suc h Gaussian distribution, in particular on its
co v ariance matrix. Finally , w e presen t an extension of the Plefk a expansion
for dynamics in tro duced in [RH11a].
3 The result is not consisten t with the one of [KS00]).
17

3 Dynamics on random net w orks
Plefk a’s expansion [Ple82] w as originally p erformed to deriv e mean-field
(T AP) equations for the equilibrium Sherrington-Kirkpatric k mo del, b y ex-
panding the free energy at fixed magnetisation (Gibbs p oten tial) in p o w ers
of the in teraction strength. In the kinetic case, the v ariables of in terest are
no longer single spins but en tire spin tra jectories, and a path in tegral formal-
ism is in tro duced to compute a v erages o v er tra jectories; a Gibbs free energy
cannot b e defined in the out-of-equilibrium scenario, but the logarithm of the
partition function at fixed momen ts o v er time will pro vide the analogous func-
tion to b e expanded. While in the first generalisation to dynamics [RH11a] of
Plefk a’s expansion only marginal first momen ts o v er time are fixed, in P ap er
1 w e will sho w that also the second order momen ts m ust b e considered for a
correct analysis. The result will outp erform other curren t metho ds in predict-
ing the time ev olution of the single-spin magnetisations, and w e conjecture
that it pro vides the correct mean field solution to the forw ard problem in the
thermo dynamic limit of infinitely man y particles when the couplings are w eak
and long-ranged. In the discussion section, w e will further compare among
differen t metho ds.
An in tro duction to Plefk a’s expansion (and to the ca vit y metho d) is giv en in
section 3.A, where w e deriv e the mean field (T AP) equations for the equilib-
rium Sherrington-Kirkpatric k mo del. Section 3.B reviews previous approac hes
to the transien t dynamics of an Ising mo del with parallel Glaub er up date rule.
3.2 P ap er 1.
Author’s con tribution : I p erformed the analytical and n umerical calcula-
tions relativ e to: Section 5 (Extended Plefk a expansion); App endix D (Details
on the extended Plefk a expansion); App endix E (The Y uleW alk er equations).
I wrote the relativ e sections in the pap er. I con tributed to writing section 6
(Numerical results), Section 7 (Summary and Conclusions) and to the prepa-
ration of the figures.
18

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Variational perturbation and extended
Plefka approaches to dynamics on random
networks: the case of the kinetic Ising
model
L Bachschmid-Romano
1 , 4
, C Battistin
2 , 4
, M Opper
1
and
Y Roudi
2 , 3
1
Department of Arti fi cial Intelligence, Technische Universität Berlin, Marchstraße 23,
Berlin, D-10587, Germany
2
Kavli Institute for Systems Neuroscience and Centre for Neural Computation, NTNU,
Trondheim, Norway
3
Institute for Advanced Study, Princeton, USA
E-mail: [email protected] and [email protected]
Received 19 April 2016, revised 22 July 2016
Accepted for publication 3 August 2016
Published 3 October 2016
Abstract
We describe and analyze some novel approaches for studying the dynamics of
Ising spin glass models. We fi rst brie fl y consider the variational approach
based on minimizing the Kullback – Leibler divergence between independent
trajectories and the real ones and note that this approach only coincides with
the mean fi eld equations from the saddle point approximation to the generating
functional when the dynamics is de fi ned through a logistic link function,
which is the case for the kinetic Ising model with parallel update. We then
spend the rest of the paper developing two ways of going beyond the saddle
point approximation to the generating functional. In the fi rst one, we develop a
variational perturbative approximation to the generating functional by
expanding the action around a quadratic function of the local fi elds and
conjugate local fi elds whose parameters are optimized. We derive analytical
expressions for the optimal parameters and show that when the optimization is
suitably restricted, we recover the mean fi eld equations that are exact for the
fully asymmetric random couplings ( Mézard and Sakellariou 2011 J. Stat.
Mech. 2011 L07001 ) . However, without this restriction the results are dif-
ferent. We also describe an extended Plefka expansion in which in addition to
the magnetization, we also fi x the correlation and response functions. Finally,
we numerically study the performance of these approximations for
Jour nal of Ph ysics A: Mathematical and Theoretical
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 ( 33pp ) doi:10.1088 / 1751-8113 / 49 / 43 / 434003
4
These authors contributed equally to this work.
1751- 8 113 / 16 / 434003 + 33$ 33. 00 © 201 6 IOP Pu bli shin g Ltd Printe d in the UK 1

Sherrington – Kirkpatrick type couplings for various coupling strengths and the
degrees of coupling symmetry, for both temporally constant but random, as
well as time varying external fi elds. We show that the dynamical equations
derived from the extended Plefka expansion outperform the others in all
regimes, although it is computationally more demanding. The unconstrained
variational approach does not perform well in the small coupling regime, while
it approaches dynamical TAP equations of ( Roudi and Hertz 2011 J. Stat.
Mech. 2011 P03031 ) for strong couplings.
Keywords: random graphs, non-equilibrium processes, spin glasses, varia-
tional methods, perturbational methods
( Some fi gures may appear in colour only in the online journal )
1. Introduction
The kinetic Ising spin glass model is a prototypical model for studying the dynamics of
disordered systems. Previous work on this topic focused both on studying the average — over
couplings — behavior of various order parameters, such as magnetizations, correlations and
response functions, and in more recent years, developing approximate methods for relating
the dynamics of a given realization of the model to its parameters. The latter line of work has
received a lot of attention in recent years, in part, because of the applications it has on
developing approximate inference methods for point processes which in turn are receiving
particular attention due to the on going improvements in data acquisition techniques in
various disciplines in life sciences.
Most of the early work on the topic dealt with systems with symmetric interactions, until
Crisanti and Sompolinsky [ 3 ] studied the disorder averaged dynamics of Ising models with
various degrees of symmetry and Kappen and Spanjers [ 4 ] derived naive mean fi eld and TAP
equations for the stationary state of the Ising model for arbitrary couplings, in both cases
considering Glauber dynamics. Roudi and Hertz [ 2 ] derived dynamical TAP equations
( hereafter denoted by RH-TAP ) for both discrete time parallel and continuous time Glauber
dynamics using Plefka ’ s method [ 5 ] , originally used for studying equilibrium spin glass
models, extended to dynamics. This was followed by [ 6 ] who reported another derivation of
these equations using information geometry following the approach of [ 4 ] . Mezard and
Sakellariou [ 1 ] developed a mean fi eld method ( hereafter denoted by MS-MF ) which is exact
for large networks with independent random couplings; see also [ 7 ] . Two schemes for
improving the existing mean- fi eld description were proposed in [ 8 ] an elegant generalized
mean fi eld methods was followed in [ 9 ] .
In the current paper we follow up on these efforts and report some new results on the
dynamics of kinetic Ising model with parallel dynamics. We fi rst look at the relationship
between the saddle point approximation to the path integral representation of the dynamics
and the simplest variational approach based on minimizing the Kullback – Leibler ( KL )
divergence between the true distribution of the spin trajectories and a factorized distribution.
Although for the standard kinetic Ising model the two methods yield the same equations of
motion, we see that this is not in general the case when the probability of spin con fi gurations
at a given time given those of the previous time is not a logistic function of the fi elds. After
this, we consider two approaches for going beyond the saddle point solution of the path
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
2

integral representation of the dynamics of the standard kinetic Ising model with parallel
dynamics ( de fi ned in more detail in the following sections ) .
In one of these approaches, which we refer to as gaussian average variational method, we
perform a Taylor expansion of the action in the path integral representation of the generating
functional around a quadratic function of the fi elds and conjugate fi elds. As described in
described in detail in section 4 , we then choose the parameters of this function such that the
resulting functional minimally depends on these parameters. We derive analytical expressions
for these optimal solutions and show that for a fully asymmetric network under a further
assumption about the interaction between the fi elds and the conjugate fi elds, we can recover
the equations of motion for the magnetization identical to MS-MF equations [ 1 ] . Without this
assumption we observe that the resulting equations are different from MS-MF. In the second
approach, we go beyond the saddle point by performing an extended Plefka expansion. The
standard Plefka expansion for the equilibrium model involves performing a small coupling
approximation of the free energy at fi xed magnetization and is the approach that was ori-
ginally taken in [ 2 ] . As we show here, however, similar to the soft spin models [ 10 , 11 ] ,a
better description of the dynamics can be achieved by not only fi xing the magnetizations but
also pairwise correlation and response functions while expanding around the uncoupled
model.
2. The dynamical model
We consider the synchronous dynamics of N interacting binary spins in the time window
T 0,

[]

de fi ned by

=+
=
-
PP t t P ss s s 10 , 1
T
t
T
0:
0
1
() ( ( ) ∣ ( ) ) ( ( ) ) ( )
in which

+=
= +
Pt t f H t ss 1, 2
i
N
st i
1
1
i
( ( ) ∣ () ) ( () ) ( )
()
where
å
=+
=
Ht h t J s t 3
ii
j
N
ij j
1
() () () ( )
is the total fi eld acting on spin i at time t composed of the external fi eld h
i
and the fi elds felt
from other spins in the system. The function
+
fH t
st i
1
i (( ) )
()
is a generic transfer function or
conditional probability of the state of the spin i at time
+ t 1

(

)
given the fi eld at time t . Our
goal will be to calculate the mean magnetizations of the spins.
The generating functional of the distribution
P s
T 0:
()
, expressed as a path integral is


 ò
y p
==
yy -

Z

D hJ ,, e 1
22 e, 4
P NN T
L sh i, ,
[] ⟨ ⟩ () ()
[]
where ... P

⟨

⟩ denotes averaging with respect to the history of trajectories de fi ned by ( 1 ) and ( 2 ) ,
and
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
3

⎡
⎣ ⎤
⎦

åå
åå
y º- -
+-
å
y
==
-
==
-
Lg t g t h t
fg t
h ,, i
lnTr 1 e 5
i
N
t
T
ii i
i
N
t
T
st st i
st t J g t
10
1
10
i
i i
i i j ji j
[ ] ˆ () ( () () )
(( ) ) ( )
() ()
() () ˆ ()
having set = gT 0
i
ˆ () and -= fg 11
si 0
i (( ) )
() . Notice that we assumed the initial state
s 0 ()
to
be uniformly distributed, manifested in the factor
12
N
in ( 4 ) , and that we refer to the two
auxiliary variables with the compact notation



º ==
gt gt ,
ii
iN tT 1 ... , 0 ...
{( ) ˆ () }
and
 = 

D

gt gt dd
it ii
,
() ˆ ()
.
The magnetization of spin i at time t can then be obtained as the fi rst derivative of the log-
generating functional:
y
y
=- ¶
¶
y 

m

t Z
t
hJ
i lim ln , , .6
i
i
0
() ([ ] )
() ()
Let us make a brief note on how the the integral representation of the generating func-
tional in ( 4 ) and ( 5 ) has been derived. This is done by fi rst replacing H
i
( t ) in ( 2 ) by g
i
( t ) and
integrating over all g
i
( t ) while enforcing that at each time step and for each spin = gt Ht
i i
() (

)

by inserting δ -functions,

d

- gt Ht
i i
[( ) ( ) ]
, in the integral. One then writes this delta function
in its integral representation
ò

d

p
-= - gt Ht gt gt gt Ht
d
2 exp i 7
i i i
ii i
[( ) ( ) ] ˆ( ) {ˆ ( ) [ ( ) ( ) ] } ( )
which is how the gt
i
ˆ (

)

appear in the equations. This rewriting of the generating functional
constitutes the fi rst steps in the Martin – Siggia – Rose – De Domenicis – Peliti formalism [ 12 , 13 ]
once it is adapted for hard spins. For more details about this approach and a pedagogical
review on its application to soft and hard spin dynamics see [ 14 , 15 ] .
A logistic transfer function f in ( 2 ) , such that
=
+
fH t
st i
1
i
(( ) )
()
++ st H t 11 t a n h
ii
1
2
(( ) ( ) )
, yielding the following probability distribution over spin paths

=
==
- +
Ht
s

P

1
2
e
2 cosh ,8
T
N i
N
t
T st Ht
i
0:
10
1 1
ii
() (( ) ) ()
() ( )
corresponds to the standard kinetic Ising model with parallel update studied in previous
work [ 1 , 2 , 6 , 9 ] .
This path integral representation in ( 4 ) allows us to explicitly perform the trace over the
spins in the generating functional of ( 4 ) and ( 5 ) yielding
⎡
⎣
⎢ ⎤
⎦
⎥

åå å
åå
åå å
y y
y
º- - + -
--
-- + +
==
-
==
-
== =
-
Ll c g t t J g t
gt gt h t
lc g T T lc g t
h ,, 1 i i
i
1i , 9
i
N
t
T
i i
l
li l
i
N
t
T
ii i
i
N
i i
i
N
t
T
i
10
1
10
1
11 0
1
[] ( ) ( ) ˆ ( )
ˆ () ( () () )
[( ) ( ) ] [( ) ] ( )
where we have set
-= g 10
i
()
" i
and
º lc x x log cosh . 10 [] () ( )
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
4

3. Mean fi eld
As a prologue to our more important results in the following sections, in this section we
review the derivation of mean fi eld equations for the dynamical model in ( 2 ) using two
approaches. These are the saddle point approximation to the path integral representation of the
generating functional in ( 4 ) , and the minimization of the KL distance between the true
distribution

P

in ( 1 ) and a factorized one. Despite being formally different methods, in the
literature they are both often referred to as mean fi eld and it is indeed well know that for the
speci fi c case of the equilibrium Ising model, they lead to the very same set of equations,
known as na ï ve mean fi eld equations [ 16 ] . Throughout this section the transfer function f in
( 2 ) is considered a generic function of the fi eld H
i
( t ) . Only towards the end of this section we
are going to consider f as a logistics function of the kinetic Ising model.
3.1. Saddle point mean field
In the equilibrium case, one way to derive the naïve mean fi eld equations is as the equations
describing the saddle point approximation to a path integral representation of the free energy,
while the TAP equations are those derived by calculating the gaussian integral around the
saddle point [ 17 ] . ( Another way is by means of Plefka expansion, which at this point we do
not discuss but will get back to later on ) . Let us consider this saddle point approach for the
kinetic model in ( 2 ) and the corresponding generating functional ( 4 ) .D e fi ning a complex
measure q as
⎡
⎣ ⎤
⎦
y y
-=
-
-
å
y -

q

st gt t t fg t
Fg t t t
g g
1, , 1e
1, , ,1 1
i i i
st i
st t J g t
it i i
i
i
i i j ji j
(( ) ∣ ( ) ˆ () () ) (( ) )
(( ) ˆ () () ) ()
()
() () ˆ ()
where
y - Fg t t t g 1, ,
it i i
(( ) ˆ () () )
is the normalization constant, the saddle point equations for
the generating functional of ( 4 ) , namely the stationary points of the function
 y

L

h ,, [

]

,i n
( 5 ) , gt
i
SP
ˆ (

)

and gt
i
SP (

)

, read
= ¢
y
+
+ ++ +
gt
fg t
fg t a i, 1 2
i
st i
st i qs t g t t t g
SP 1
SP
1
SP
1, 1 , 1
i
i i i i
SP SP
ˆ( ) (( ) )
(( ) ) ()
()
() (( ) ∣ ( )
ˆ () () )
å
=+ y -
gt h t J s t b ,1 2
i i
j
ij j q s t g t t t g
SP 1, ,
j j j
SP SP
() () ⟨ () ⟩ ( )
(( ) ∣ ( )
ˆ () () )
where we have de fi ned ¢ º ¶
¶
fy
x
fy
y
x
() ()
. Notice that in the limit y  0 ,
=

g

0
SP
ˆ
is a self-
consistent solution of the previous saddle point equation ( 12 a ) , while ( 12 b ) turns into
å
=+
-
gt h t J s t .1 3
i i
j
ij j f g t
SP 1
s j t j
SP
() () ⟨ () ⟩ ( )
(( ) )
()
The approximate log generating functional  yy -+  ZL hJ h ln , , , , const.
SP
[] [ ] allows
us to estimate the magnetizations using ( 6 ) and ( 13 ) as
= å
-+ -

m

ts t .1 4
ii
fh t J m t 11
s i t i j ij j
()
() ⟨ () ⟩ ( )
() ()
()
These are the saddle point mean fi eld equations for a general function f . Note that the
marginal here yields the same expression as the conditional probability in ( 2 ) , namely
+
fH t
st i
1
i (( ) )
()
except that in ( 14 ) , the fl uctuating fi eld H
i
( t ) has been replaced by an effective
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
5

( mean ) fi eld
=+
å

H

th t J m t
i i j ij j
eff
() () ()
, in analogy with the physical intuition behind the
original formulation of the mean fi eld theory by Weiss [ 18 ] .
3.2. Mean field from KL distance
A second way of deriving mean fi eld equations, usually employed in the machine learning
community, is based on a variational approximation. Within this framework, one approx-
imates the model distribution
P s
T 0:
()
with a Markovian process

Q

s T 0:
()
that factorizes over
the spin trajectories [ 19 ] . In other words, assuming

=+
=
-
QQ t t Q ss s s 10 , 1 5
T
t
T
0:
0
1
() ( ( ) ∣ ( ) ) ( ( ) ) ( )
where

+= +
=
Q t t Q st st ss 11 , 1 6
j
N
jj
1
( ( )∣ ( )) ( ( )∣ ( )) ( )
one minimizes the KL divergence, 

D

QP ss
TT
KL 0: 0:
[( ) ( ) ]
, between the approximate
distribution

Q

s T 0:
()
and the model
P s
T 0:
()
. In the case of the model de fi ned in ( 2 ) and an
approximate distribution satisfying ( 15 ) and ( 16 ) , the KL-divergence can be rewritten as
º 

D

QP Q Q
P a ss s s
s
Tr ln , 17
TT T T
T
s
KL 0: 0: 0: 0:
0:
T 0:
[( ) ( ) ] ( ) ()
() ()
å
å
åå
=+
+
+
=+
+
+
++ +
+
+
¹
+
Qt Qt t Qt t
Pt t
Q st Q st st Qs t s t
us t s t
Q st Q st st Q st st b
ss s
ss
ss
Tr Tr 1 ln 1
1
Tr Tr 1 ln 1
1
Tr Tr 1 ln 1 , 17
t
tt
t
st j st j j
jj
jt j j
ti j
st i st i i i i
ss 1
1
1
jj
ii
(( ) ) (( ) ∣( ) ) (( ) ∣( ) )
(( ) ∣( ) )
(( ) ) (( ) ∣ ( ) ) (( ) ∣( ) )
(( ) ∣( ) )
( ( )) ( ( )∣ ( )) ( ( )∣ ( )) ( )
() ( )
() ( )
() ( )
where the fi rst line is just the de fi nition of the KL-divergence, in the second line we have
exploited the Markovian property of

P

and

Q

and assumed
=

P

Q ss 00 (( ) ) (( ) )
, while in the
last line we have use the factorizability of

Q

over spin trajectories. Notice that the last equality
is valid for any choice of j and that we have de fi ned u
jt
as
+º + +
+
us t s t Q t t P t t ss s s 1 exp Tr 1 , ln 1 18
jt j j t t j j ss 1,
jj
( ( )∣ ( )) { ( ( ) ( )) ( ( )∣ ( )) } ( )
{( ) ( ) } ⧹ ⧹
⧹⧹
and t s j ()
⧹ denotes all components of
t s (

)

apart from j . Observe that thanks to the Markovian
property of the two distributions

P

and

Q

we were able to reduce the average over a NT
dimensional space to a sum of T averages over N

2

dimensional spaces.
In order to determine the variational mean fi eld equations, one has to minimize the KL-
divergence in the space of marginals

Q

st
j
(( ) )
and transition probabilities +

Q

st st 1
jj
(( ) ∣( ) )
.
Given that these are not independent, we enforce the constraints:
+= + Q st Q st st Q st 1T r 1 1 9
js t j j j
j
(( ) ) (( ) ∣( ) ) (( ) ) ( )
()
using Lagrange multipliers

l

st
j
(( )

)

, ultimately optimizing the following cost function:

å l
º
-- - -
-
 DQ P
st Q st Q st st Q st
ss
Tr Tr 1 1 . 20
TT
jt
st j j st j j j
KL 0: 0:
,
1
jj
[( ) ( ) ]
( ( )){ ( ( )) ( ( )∣ ( )) ( ( )) } ( )
() ( )
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
6

The stationary points of



in ( 20 ) are the zeros of the functional derivatives
 d
d l
l
=+ +
+ -
++ +
+
+
Qs t Qs t s t Qs t s t
us t s t st
st Q st st a
Tr 1 ln 1
1
Tr 1 1 , 21
j
st j j
jj
jt j j
j
st j j j
1
1
j
j
(( ) ) (( ) ∣( ) ) (( ) ∣( ) )
(( ) ∣( ) ) (( ) )
(( ) ) (( ) ∣( ) ) ( )
()
()
⎧
⎨
⎩
⎫
⎬
⎭
 d
d l
+ = +
+ ++ +
Qs t s t Qs t Qs t s t
us t s t st b
1 ln 1
1 11 2 1
jj
j
jj
jt j j
j
(( ) ∣( ) ) (( ) ) (( ) ∣( ) )
(( ) ∣( ) ) (( ) ) ( )
that can be reduced to the relation:
+= +
+
+
Qs t s t us t s t
us t s t
1 1
Tr 1 .2 2
jj
jt j j
st j t j j 1
j
(( ) ∣( ) ) (( ) ∣( ) )
(( ) ∣( ) ) ()
()
It is worth emphasizing that this solution is valid for any Markov chain

P

and any
approximate Markov distribution

Q

that factorizes over the spin trajectories. From now on we
will require the spins at time t to be conditionally independent under the model distribution, as
in ( 2 ) . This assumption and a little algebra allow us to simplify ( 22 ) as follows:
å
å
+= +
+
+
+ +
Qs t s t
Qt f h t J s t
Qt f h t J s t
s
s
1
exp Tr ln
Tr exp Tr ln
,2 3
jj
tj st j l jl l
st t j st j l jl l
s
s
1
1 1
j j
jj j
{}
{}
()
()
[( ) ∣( ) ]
[ () ] () ()
[ () ] () ()
()
() ⧹ ()
() ( ) ⧹ ()
⧹
⧹
where we imposed the normalizability to

Q

.
If there are no self-couplings in the model distribution

P

, the right-hand side of ( 23 ) will
not depend on s
j
( t ) and consequently the solution for the joint distribution

Q

s T 0:
()
will
factorize in time. The spin independent 1st order Markov chain

Q

that best approximates the
model

P

de fi ned in ( 2 ) with = J 0
jj , is actually a 0th order Markov chain. Additionally the
absence of self-interactions in

P

makes ( 23 ) an explicit relation between the marginal of spin j
at time
+ t 1
and the marginals of all spins but j at the previous time step t . Since we are
dealing with a system of binary units, marginals are fully determined by their fi rst moments,
thus the marginal of spin j at time
+ t 1
,i n ( 23 ) , becomes a function of the magnetizations at
time t . Taking one step further one can easily verify that the fi rst moments of ( 23 ) equal the
naïve mean fi eld magnetizations of ( 14 ) if the transition probability
+
fH t
st i
1
i (( ) )
()
belongs to
the exponential family with the fi eld H
i
( t ) as natural parameter
= +
+
+
fH t as t H t
as t H t
exp 1
Tr exp 1 ,2 4
st i
ii
st i i
1
i
i
(( ) ) [( ( ) ) ( ) ]
[( ( ) ) ( ) ] ()
() ()
where

a

(·)
is a generic function of the state
+

s

t 1
i
(

)

. For the kinetic Ising model

a

(·)
is the
identity function and the equations for the magnetizations read:
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
å
=- + -

m

th t J m t tanh 1 1 , 25
ii
j
ij j
( ) () () ( )
equivalent to ( 14 ) and know as the dynamical Naïve mean fi eld equations [ 2 ] .
4. Gaussian average method
What we have shown so far is that the saddle point approximation to the generating functional
for the kinetic Ising model and the one based on the KL divergence match each other,
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
7

although this is not the case for non-logistics transfer functions. In this section, we study an
improvement over the saddle point approximation. Our approach is to fi nd the optimal
gaussian distribution for approximating the generating functional perturbatively, and then
using the resulting approximation to calculate the magnetizations. This can be thought as an
extension to complex measures of a standard variational method: it was taken by Müschlegel
and Zittartz [ 20 ] for the equilibrium Ising model, while a general framework is set in [ 21 ] .W e
describe this approach in detail in this section.
4.1. Optimization
We consider the fi rst order Taylor expansion of the log-generating functional de fi ned in ( 4 )
and ( 5 ) around a gaussian integral:



ò
ò
ò
y p -- +
- ++  ZD
DL L
D NT N hJ ln , , ln ln 2 ln 2 , 26
s
([ ] ) ˜ ˜ ()
˜ ()
where we have de fi ned the complex gaussian measure
 = -

D

Da e, 2 7
L s ˜ ()


hh =- - Lb S
1
2 27
s
( ¯ )( ¯ )( )
parametrized by the interaction matrix
S
and the mean
h
¯
. Here we split the vectors
h
¯
into
hh
=
-
tt ,
t
T
0
1

{

() ˆ () } and h t () into
h
=
t
i i
N
1

{

() }
similar to



. From now on we will use the form of
the action L in ( 9 ) since we are going to focus on the standard parallel update kinetic Ising
model.
The choice of a quadratic form for L
s
allows us to easily calculate many of the terms in
( 26 ) , simplifying the expression for the log-generating functional as
⎤
⎦
⎥
⎡
⎣
⎢ ⎤
⎦
⎥




ò
ò
ò
ò
åå
åå å
åå
åå å
å
å
hh
h p hy
h
p yh
p hy
p h
-= - - -
+- ¢
¢ -+ -+
- ¢ -
-¢ -
¢ -
-¢
¢ -+ -+
+¢
¢ ++
++ +
-
=
-
ZN T t t
th t D l cg t t t
Jg t J t
Dl c J g J
Dl c g T T T
Dl c g t t N
SS
S
S
S
S
ln 1
2 ln det i i
i det
2 11 i
ii
det
2 i0 i 0 i 0
det
2 11 i
det
2 ln 2 ,
28
it
Nt i Nt N i
it
ii
it
i i NT it
T
i i i
l
li l
l
li l
NT i
i
l
li l
l
li l
NT i
i i i
NT it
i i
,
2; 2
1
,
,1
1
,
() ˆ ()
ˆ () () () ˜ [( ) ( ) ( )
ˆ( ) ˆ ()
() ˜ () ˆ () ˆ ()
() ˜ [( ) ( ) ( ) ]
() ˜ [( ) ( ) ]
()
where we have replaced ( 9 ) in ( 26 ) and we have performed the change of variables



 h ¢= - ¯ . Notice that when not stated otherwise the sum over t runs from t  =  0t o
=- tT 1
. From now on we will just drop off the superscript ′ from variables



.
If all measures were real probability measures, the fi rst order approximation on the right-
hand side of ( 26 ) would be an upper bound to the free energy

- Z

ln . In this case a mini-
mization of the bound with respect to the variational parameters would be the obvious choice
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
8

for optimizing the approximation. Since integrations in our case are over complex measures
this argument cannot be applied. Instead, we base our optimization on the idea of the var-
iational perturbation method [ 22 ] : if the Taylor series expansion of the log generating
functional ( 4 ) – ( 9 ) would be continued to in fi nite order it would represent the functional and
the resulting series would be entirely independent of the parameters of the gaussian measure
( 27 a ) . On the other hand, the truncated series ( 26 ) inherits a dependence on the variational
parameters
h
,
h
ˆ
,
S
. Hence, one would expect that the truncation represents the most sensible
approximation if it depends the least on these parameters. One should therefore choose their
optimal values such that the approximation to

Z

ln is the most insensitive to variations of
these parameters. This simply corresponds to computing the stationary values of the log
generating functional in the
h
,
h
ˆ
,
S
space. This requirement of minimum sensitivity to the
variational parameters was introduced in [ 23 ] as an approximation protocol.
Using the logic in the previous paragraph and setting the fi rst derivative of the expression
for

- Z

ln in ( 28 ) with respect to

h

t
j
ˆ( )
to zero, one gets the equation for stationary

h

t
j
()
, the
fi rst moment of the gaussian form for g
j
( t ) :
å

h

m =+ th t J t ,2 9
i i
j
ij i
() () () ( )
where we have de fi ned for =¼ - tT 1, , 1 :
⎡
⎣
⎢ ⎤
⎦
⎥

ò å
m p hy h =- + - + - + tD g t t t J g t t
S det
2 tanh 1 1 i i ,
30
i NT ii i
l
li ll
() () ˜ () () ( ) ( ˆ ( )
ˆ () )
()
while for t  =  0 and t  =  T we have respectively:
⎡
⎣
⎢ ⎤
⎦
⎥

ò å

m

p yh =- + DJ g a
S
0 det
2 tanh i 0 i 0 0 , 31
i NT i
l
li ll
() () ˜ () ( ˆ () ˆ () ) ( )

ò

m

p hy =- + - + TD g T T T b
S det
2 tanh 1 1 i . 31
i NT ii i
() () ˜ [( ) ( ) ( ) ] ( )
Solving
h

-

¶¶ = Zt ln 0
i
()
gives:

ò

h

m p h =+ - + tt D g t t
S
i1 i
det
2 tanh . 32
ii NT ii
ˆ () ( ) () ˜ [( ) ( ) ] ( )
Looking for the stationary points of ( 26 ) with respect to -
S 1 corresponds to solving the
following set of equations:
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
9

⎪
⎪
⎤
⎦
⎥
⎫
⎬
⎭


ò
ò
å
å
å
dd
p h
p h
yh
¶
¶¢
=- ¢ -
+¶ +
-¶ +
++ - + + + =
- +¢
¢
¢
Z
St t St t
Dg s s
Dg s s
sJ g s s
S
S
ln
,
1
2 ,i
det
2
1
2 l n2c o s h
det
2
1
2 l n2c o s h
i1 i 1 1 0 , 3 3
ij
ji ji N tt
NT ms
it jt mm
NT ms
it jt mm
m
l
lm ll
1
12
,
,
12
,
,
{}
{
() ()
() ˜ [( ) ( ) ]
() ˜ [( ) ( )
() ( ˆ ()
ˆ () ) ( )
where we have de fi ned 

¶

º
¢ ¶
¶¶ ¢
it jt tt
, ij
2
() ( )
.
4.2. Equations for the magnetizations
In the previous subsection we derived expressions for the parameters of the gaussian used
for perturbative approximation of the log-generating functional at fi xed ψ . Now we
want to derive an expression for the magnetizations using ( 6 ) . We will fi rst perform
the derivative of ( 28 ) with respect to ψ ; notice that even
h
,
h
ˆ
and
S
are ψ dependent, such
that ( 6 ) reads:
åå
å
y
h
yh
h
yh
y
=- ¶
¶ + ¶¢
¶
¶
¶¢
+ ¶¢
¶
¶
¶¢
+ ¶¢ 
¶
¶
¶¢ 
y  ¢¢
¢
mt Z
t
t
t
Z
t
t
t
Z
t
St t
t
Z
St t
i lim ln ln ln
, ln
, .3 4
i
ij t
j
i j jt
j
i j
ljt t
lj
i lj
0 ,,
,, ,
,
,
() ()
()
() ( )
ˆ ()
() ˆ ()
()
() ( ) ()
However, since in our optimization scheme we looked for the stationary values of

Z

ln
with respect to the variational parameters,

y ¶

¶ Z ln will only consist of its explicit derivative
with respect to ψ , leading to:
m =
y 

m

tt lim 35
i i
0
() () ( )
for all t = 0, K , T and

m

t
i
(

)

has been de fi ned in ( 30 ) – ( 31 b ) .
4.3. The optimized values of the parameters
In principle, one needs to solve the full set of equations ( 29 ) – ( 33 ) and take the limit of y  0
to calculate the magnetization in ( 35 ) . This is obviously a very dif fi cult task to do analytically
given the high dimensional integrals that appear in ( 30 ) – ( 33 ) and that the equations have
to be solved simultaneously. The solutions, however, can be very much simpli fi ed if we
assume
h =
y  t lim 0 36
i
0 ˆ () ( )
"" it ,
. With ( 36 ) , which we will justify in section 4.4 below, the optimal interaction matrix
S
in ( 33 ) in the limit y  0 assumes the following block tridiagonal structure:
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
10

⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
=




 

S

SS
SSS
SSS
SSS
0 , 0 0 , 1 000
1, 0 1, 1 1, 2 0 0
0 2 ,1 2 ,2 2 ,3 0
0 0 3, 2 3, 3 3, 4
,3 7
() ()
() () ()
() () ()
() () ()
()
where
⎡
⎣
⎢ ⎤
⎦
⎥ ⎡
⎣
⎢ ⎤
⎦
⎥

 g
l
= -
- += +
tt t tt tt

S

S , 0i
i ,, 1 0, 1
00
,3 8 () () () () ()
the blocks
¢ tt S , (

)

are of size ´ NN

2

2 ,

+= + tt t t SS ,1 1 , () ( )
and
å

g

=- = ¼ - tJ J m t t T a 10 , , 1 , 3 9
ij
k
ik jk k 2
() ( () ) ( )
l += - + = ¼ - tt J m t t T b , 1 i 1 1 0, , 2. 39
ij ji i 2
() ( ( ) ) ( )
Observe that the matrix
S
in ( 37 ) is a symmetric complex matrix ( not Hermitian ) , whose
Hermitian part is positive symmetric. ( Recall that the Hermitian part of a matrix
S
is de fi ned
as + SS

2 (

)
† . ) This is consistent with its derivation given that — as pointed out in [ 24 ] — the
gaussian integral



ò
D
˜
converges only if the Hermitian part of
S
is a positive symmetric
matrix.
In ( 39 a ) and ( 39 b ) we implicitly state that
= S det 1
: as a matter of fact it can be proven
to be a mere consequence of the block structure of the matrix
S
, as shown in appendix A .
Since

ò
p = D S 2d e t
NT
˜ ()
this means that the gaussian integral and the model log gen-
erating functional match in the limit y  0 .
Finally we can substitute the optimal values of the variational parameters in ( 35 ) and
exploit ( 32 ) to get:

ò å
p
=- + - + -

m

tD g t h t J m t
1
2 tanh 1 1 1 40
i NT i i
j
ij j
() () ˜ [( ) ( ) ( ) ] ( )
for =¼ - tT 1, , 1 .
We are now left to evaluate a multidimensional integral in ( 40 ) . In fact the integration in
( 40 ) can be reduced to a one-dimensional integral marginalizing the multivariate gaussian
distribution, yielding
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
ò
å
p s =+ - + -
-

m

t x xt h t J m t
de
2 tanh 1 1 , 41
i
x
i
j
ij j
2
2
() () ( ) ( ) ( )
s =
- -+ -+
t S ,4 2
Nt i Nt i
1 21 , 21
() ( ) ( )
() ()
where the integral is now over = - -+ -+
xg t S
i Nt i Nt i
1 21 , 21
() ( ) () ()
, a normally distributed,
zero mean unit variance, random variable.
For performing the one-dimensional integral in ( 41 ) , we need to compute the entries of
the inverse of matrix
S
. In appendix B we demonstrate that, given
S
as de fi ned in ( 37 ) – ( 39 b ) ,
the entries of -
S 1 in which we are interested in can be calculated recursively as

gg g l g l == - - - -
++
-
tt t t t t t t

S

where 1 , 1 1 , . 43
Nt i Nt i ii
2, 2
1
˜ () ˜ () () ( ) ˜ () ( ) ( )
As we show in appendix B ,

g

t
ii
˜ () can only take positive values and therefore the integral
in ( 41 ) is physically well-de fi ned.
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
11

Recalling the de fi nitions of the matrices

g

and l , one can verify that the magnetizations
in ( 41 ) only depend on the past magnetizations
¢ mt
j
()
with ¢< tt
, j = 1, K , N . Since this
dependence goes back to ¢= t 0 it is natural to wonder if the error in estimating the past
magnetizations would accumulate impairing the inference process. We notice ( not included in
section 6 ) that for the gaussian average method knowledge of the history of the experimental
magnetization — knowing

g

- t 1
ii
˜ ()
when computing m
i
( t ) with ( 41 ) — does not affects the
reconstruction signi fi cantly. Whether we are using experimental magnetizations or approx-
imate ones in ( 43 ) , we observe that

g

- t 1
ii
˜ ()
grows exponentially with time for strong
couplings while it converges to a fi nite value for weak couplings. This behavior can be
understood by studying the stability of the map ( 43 ) of

g

- t 1
˜ ()
into

g

t
˜ () that de fi nes a
dynamical system, as we do in appendix C . Averaging over the disorder one realizes that this
dynamical system is chaotic for couplings strength above a certain critical value. Its critical
value depends on the degree of symmetry of the connectivity and on the presence of an
external fi eld.
4.4. The solution lim ψ - 0
b
η i ð t Þ¼ 0
In principle, the value of limit of y  0 of

h

t
i
ˆ( ) that satisfy the optimality equations, may be
non-zero. In this section, we justify the choice of
h =
y 
t lim 0
i
0
ˆ( )
that we made in the
previous section. We fi rst note that zero is a good candidate for the optimal value of

h

= tg t
i i L

s

ˆ( ) ⟨
ˆ () ⟩ — here ¼ L s

⟨

⟩ indicates the average under the complex measure
-
e
L

s

in ( 27 a )
and ( 27 b ) — since
=
y  gt lim 0 , 44
i L
0 ⟨ˆ ( ) ⟩ ( )
where L for the kinetic Ising model has been de fi ned in ( 9 ) and
¼
L

⟨

⟩
indicates the average
under the complex measure
-
e
L
. This choice for the mean in the
g s
ˆ
can be justi fi ed by
analogy with the mean in the g s: the stationary value for latter is also the saddle point value of
the kinetic Ising generating functional, while the saddle point in the
g s
ˆ
is conventionally set
to zero.
Furthermore, we can show that
h =
y 
t lim 0
i
0
ˆ( )
yields a consistent solution. To do this
we fi rst note that by inverting the matrix
S
, as shown in appendix B , two point correlation
functions - ¢
gt g t 1
ij L
s

⟨

()
ˆ () ⟩ and ¢
gt gt
ijL
s

⟨

ˆ () ˆ () ⟩ are both zero, where notation ¼ ¢
L s

⟨

⟩ indicates
averages under the gaussian measure - ¢
e L s , with




¢ =

L

S
s
1
2 . Consequently, we have
⎡
⎣
⎢ ⎤
⎦
⎥
⎡
⎣
⎢ ⎤
⎦
⎥
⎛
⎝
⎜ ⎞
⎠
⎟
å
åå
åå
å
mh
h
h
dh
h
=- + - -
=- + - -
=- - -
=- -
=- + -
¢
¢
¢
¢
¢
y 
=
¥ -
-
--
- -
tg t t J g t
ag t t J g t
bt g t J g t
bt g t
gt t
lim tanh 1 1 i
11 i
11 i
11
tanh 1 1 . 45
ii i
l
li l
L
n
n ii
l
li l
n
L
nk l
nk l i kl i l
l
li l
nk
L
nk l
nk l k n i kl i l L
ii
L
0
1
21
,,
,,
21
,,
,, , 2 1
s
s
s
s
s
( ) () () ˆ ( )
() () ˆ ( )
(( ) ) ( ) ˆ ( )
(( ) ) ⟨ ( ) ⟩
⟨ [ () () ] ⟩ ( )
Now, note that the previous equality corresponds to setting

h

= t 0
i
ˆ( )
in ( 32 ) .
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
12

4.5. The fully asymmetr ic limit
In [ 1 ] Mezard and Sakellariou derive equations for the magnetizations that are exact for fully
asymmetric couplings:
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
ò
å
p g =- + - + -
-

m

t x ht J m t x t
d
2 et a n h 1 1 1 , 4 6
i x i
j
ij j ii
2
2
( ) () () () ( )
where

g

t
ii
()
has been de fi ned in ( 39 a ) .
In section 4.1 all entries of
S
were free to be optimized. However, we could have
assumed that the blocks corresponding to l - tt 1, ()
are set to zero a priori . By looking at
( 41 ) and ( 43 ) one easily realizes that with this constraint our optimization would have lead to
( 46 ) , which is exact in the fully asymmetric limit. Notice that this prescription on l would not
affect the optimal value of any other variational parameter, since we optimized

Z

ln inde-
pendently with respect of
h
,
h
ˆ
or
¢ tt S ,
ij
()
.
5. Extended Plefka expansion
As mentioned in the Introduction, two particularly powerful approaches to studying dis-
ordered systems both in machine learning and statistical physics community are variational
and weak coupling expansions. In the previous sections we reported some results regarding
the variational approach. In this section we aim at developing a comprehensive weak coupling
expansion for the disordered spin systems.
Weak coupling expansions in fi eld theory and statistical physics of disordered systems
take several forms. One of the most powerful amongst these, which has proven to be parti-
cularly useful for studying the equilibrium properties of glassy systems, is the Plefka
expansion. The Plefka expansion was originally performed for the equilibrium Sherrington –
Kirckpatrick model by expanding the Gibbs free energy at fi xed magnetization, enforced via a
Legendre transform, around the free energy of an uncoupled system. To the fi rst order in J it
yields the naive mean fi eld results while to the second order the TAP equations are recovered.
Although higher order terms vanish for the SK model, they can in general be computed [ 25 ] .
In performing the Plefka expansion for the equilibrium model with binary spins it is
suf fi cient to fi x the magnetization and this has been the line taken by Roudi and Hertz in
deriving Plefka expansion and dynamical TAP equations for the kinetic Ising model. How-
ever, in contrast to the equilibrium case, for the dynamics the magnetization is not the only
relevant order parameter. Including other observables in the Plefka expansion, namely the
correlation and response functions, is what we do in this section. As we will show with
numerical results in the next section, this will lead to a signi fi cant improvement for predicting
the dynamics of the system.
Instead of the generating functional in ( 4 ) and ( 5 ) , let us now consider the following
functional:

ò
yy
p
=
aa
hh

Z

DL CBR CBR s ,, , , 1
22 Tr exp , , , , , , , 47
NN T s
[ ˆ ˆˆ
] () ([ ˆ ˆˆ ]) ( )
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
13

with
⎪
⎪
⎧
⎨
⎩
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎫
⎬
⎭
 åå
å
åå
y a
y
=- + +
-- +
+¢ ¢
+¢ ¢ - ¢ ¢
a
¢
¢¢
h Lg t g t J s t s t g t
gt h t gt t s t
Ct t s t s t
Btt g t g t R t t g t st
CBR s ,, , , , , i 1
l n2c o s h i
1
2 ,
1
2 ,i , , 4 8
it
ii
j
ij j i i
i i i i i
t
ii i
t
i ii
t
i i i
,
[ ˆ ˆˆ ] ˆ () () () ( ) ()
() () ˆ () () ()
ˆ () ( ) ( )
ˆ () ˆ ( ) ˆ ( ) ˆ () ˆ ( ) ( ) ( )
where we have introduced the parameter α to control the interaction strength. The
introduction of the new auxiliary fi elds

CB

,
ˆ ˆ and

R

ˆ in the action ( 48 ) is related to the averages
of the observables that we want to constrain when performing the Legendre transform. In
particular, here we decide to fi x all marginal fi rst and second moments over time. One can
fi nd the moments and the physical meaning of these auxiliary fi elds by fi rst derivatives of the
generating functional with respect to the fi elds as follows:
y
-=
¶
¶ =-
= ¶
¶ =
¢= ¶
¶¢
=¢ ¢ ¹
¢= ¶
¶¢
=¢
¢= ¶
¶¢
=- ¢
a a
a a
a a
a a
a a
mt Z
ht gt
mt Z
t st
Ct t Z
Ct t st st t t
Btt Z
Btt gt gt
Rt t Z
Rt t gt s t
i ln i
ln
, ln
, for
1
2 , ln
,
1
2
, ln
, i, 4 9
i
i
i
i
i
i
i
i
ii
i
i
ii
i
i
i i
ˆ( ) () ⟨ˆ ( ) ⟩
() () ⟨( ) ⟩
() ˆ () ⟨( ) ( ) ⟩
() ˆ () ⟨ˆ ( ) ˆ ( ) ⟩
() ˆ () ⟨ˆ ( ) ( ) ⟩ ( )
where
a


⟨

⟩
denotes averaging over the distribution de fi ned by the measure inside the
functional ( 47 ) . Namely, for any function
F s ()
of the trajectory of spins
s
we de fi ne:


ò
ò
y
y
=
a
a
a
h
h
F DF L
DL
sC B R s
CBR s
Tr exp , , , , , ,
T r e x p , ,, ,, , .5 0
s
s
⟨⟩ () ( [ ˆ ˆˆ ])
([ ˆ ˆˆ ]) ()
The moments of the original dynamical system ( 2 ) can be found by setting the auxiliary fi elds
to zero and a = 1 at the end of the calculation. Note that
=

C

tt ,

1

()
and its conjugate
fi eld =

C

tt ,0
ˆ () .
The Legendre transform of

Z

ln is given by
åå
åå å
y y G= - +
-¢ ¢ - ¢ ¢ - ¢ ¢
aa
¢¢ ¢
h Z t mt ht mt
C t tC t t B t tB t t R t t R t t
mmCBR CBR , , ,, l n , , ,, i
1
2 ,, 1
2 ,, , , , 5 1
it
i i
it
ii
itt
ii
itt
ii
itt
ii
[ ˆ ][
ˆ ˆˆ
] () () ()ˆ ()
ˆ () () ˆ () () ˆ () () ( )
where the fi elds y h CBR ,, , ,
ˆ ˆˆ
in the above equation are to be considered as functions of the
moments, and dependent on the α parameter, according to the following set of equations:
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
14

y
¶G
¶ =-
¶G
¶ =
¶G
¶¢
=- ¢ ¢ ¹
¶G
¶¢
=- ¢
¶G
¶¢
=- ¢
a a
a a
a a
a a
a a
mt t
mt ht
Ct t Ct t t t
Btt Bt t
Rt t Rt t
mmCBR
mmCBR
mmCBR
mmCBR
mmCBR
,, , ,
i, , , ,
, ,, , , , f o r
,
1
2 ,, , , ,
, ,, , , , . 5 2
i i
i
i
i
i
i
i
i
i
() [ ˆ ]( )
ˆ( ) [ ˆ ]( )
() ˆ [ ˆ ]( )
() ˆ [ ˆ ]( )
() ˆ [ ˆ ]( ) ( )
We now perform a second order expansion of

G

a
around
a = 0
and consider the set of
equations ( 52 ) within the expansion; the details of the calculation are reported in appendix D .
Setting the auxiliary fi elds to zero, we can extract the value of the fi elds
y h CBR ,, , ,
00 0 00
ˆ ˆˆ
as functions of the correct ( within the expansion ) marginal fi rst and second moments. Those
fi elds thus represent effective external fi elds which have to be applied to the model without
interactions (
a = 0
) to obtain the same moments as the interacting model. Hence, we may
consider y h Z CBR ,, , ,
0 00 0 00
[ ˆ ˆˆ

]

as the generating functional for the true marginal distribu-
tions, giving us an effective non-interacting description of the true interacting dynamics. The
explicit calculation ( appendix D ) yields = 
Zh Z h
i i i
0 0
[] [

]

, where
⎛
⎝
⎜ ⎞
⎠
⎟
⎤
⎦
⎥
⎥
ò 
åå
d
f
µ
-- - ¢ ¢ - ¢ -
f
+
¢=
-
Z gt gt
tJ m t J J R t t s t m t h t
g dT r e
2 cosh
,5 3
i i s
t
st g t
i t
i
i
j
ij j
t
t
ij ji j i i i
0 1
0
1
i
i i
i
() [( )
() () ( ) [ ( ) ( ) ] () ( )
() ( )
and where

f

t
i () is a gaussian random variables, drawn independetly for each i , with zero
mean and covariance
å
ff ¢= ¢ - ¢ t t J C t t mt mt ,. 5 4
ii
j
ij jj j
2
⟨ ( ) ( )⟩ [ ( ) ( ) ( )] ( )
This corresponds to a stochastic equation for a single spin, where each spin i is subjected to an
effective fi eld
⎛
⎝
⎜ ⎞
⎠
⎟
åå
f =+ - ¢ ¢ - ¢ +
¢=
-
gt t J m t J J R t t s t m t h t ,. 5 5
i i
j
ij j
t
t
ij ji j i i i
0
1
() () () ( ) [ ( ) ( ) ] () ( )
The effective fi eld in ( 55 ) is composed of a coloured gaussian noise ( f ) , a naive mean fi eld
( the second term ) , a retarded interaction with the past values of the spins ( third term ) and
fi nally the external fi eld ( h
i
( t )) .
The retarded interactions and the noise covariance have to be computed as averages from
the entire ensemble of independent spins. Luckily, this can be done in a causal fashion, i.e. the
spin dynamics depends only on past spin history. However, this can not be done analytically,
although one may proceed again with a perturbation expansions in order to get equation of
motions for one and two time functions. The fact that the external noise is gaussian should be
helpful. As an alternative, we have resorted to numerical simulations, where the necessary
averages are estimated from a large number N
T
of samples of trajectories. Sample averages
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
15

will be denoted by overbars; namely, for any function F s
kk
()
of the

k

th
trajectory of spins s k
we de fi ne the following average:
å
=
=
F N F
1 .5 6
T k
N
k
1
T
()
In order to compute the retarded interaction
¢

R

tt ,
i
()
, we recall that given a vector

f

with
gaussian distributed components

f

t (

)

, with zero mean and covariance matrix
 ff ¢= ¢ tt t t ,

⟨

() ( ) ⟩ ( )
, and given a function f F (

)

of the vector

f

, the following relation
holds

å
ff f f
=¢
¶
¶¢
¢
Ft t t t F ,, 5 7
t
⟨( )( ) ⟩ ( ) () () ( )
as can be shown using integration by parts. By considering the function
ff =º Fs t s t ; () ( ) ( )
and using ( D.19 ) one fi nds the following equation relating the
response and correlation functions:
åå
ft t t ¢= ¢ - ¢
f
t =
-
st t R t J C t m m t ,, . 5 8
i i
t
i
j
ij jj j
1
1
2
i
⟨ ( ) ( )⟩ ( ) [ ( ) ( ) ( )] ( )
The algorithm can be described as follows.
• Initial condition: set
== =

s

iN kN 0 1 , 1 ... , 1 ... .
i
k T
()
• For
= tT 1 ...
:
( i ) Draw the spins at time t from
= - ==
-

p

st gt iN kN
e
2c o s h 1 , fo r 1 ... , 1 ... ,
i
k st gt
i
k T
1
i k
i
k
(( ) ) ()
() ( )
using the fi elds
- gt 1
i
k
()
calculated at the previous time step.
( ii ) Compute the sample averages
¢= - ¢ ¢ = - =

C

tt g t st t t i N , ta nh 1 , fo r 1 ... 1 , 1 ... .
i i i
() [ ( ) ] ( )
( iii ) Draw the noise variables

f

== ti N k N for 1 ... , 1 ...

,

i
k T
() from the conditional
probability ff f - pt t 0 ... 1 ,
i
k
i
k
i
k
(( ) ∣( ) ( ) )
which can be computed using the Yule
Walker equations ( appendix E ) .
( iv ) Compute the sample averages that will be needed in ( v ) :
ff ¢= - ¢ ¢ = - =

s

tt g t t t t i N ta nh 1 , for 1 ... 1 , 1 ... .
i i i i
( ) () [ ( ) ] ()
( v ) Compute ¢¢ = -

R

tt t t , , for 1 ...

1

i
() using ( 58 ) by solving the system of linear
equations:
åå
ft t t ¢= ¢- ¢
t =
-

s

tt R t J C t mm t ,, .
i i
t
i
j
ij jj j
1
1
2
() ( ) ( ) [ ( ) ( ) ( ) ]
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
16

( vi ) Compute the fi elds
⎛
⎝
⎜ ⎞
⎠
⎟
åå
f =+ - ¢ ¢ - ¢ +
==
¢=
-
gt t J m t J J R t t st m t h t
iN k N
,,
fo r 1 ... , 1 ... .
i
k
i
k
j
ij j
t
t
ij ji j i
k ii
T
0
1
() () () ( ) [ ( ) ( ) ] ()
( vii ) Compute the magnetizations at time
+ t 1
:
+= =

m

tg t i N 1 ta nh , for 1 ... .
i i
() [ ( ) ]
To conclude this section, let us point out that the mean fi eld result ( 46 ) , which is exact for
asymmetric networks in the thermodynamic limit for gaussian couplings with variance N 1 ,
can be obtained in two ways. One either considers the result ( 54 ) and neglects the term JJ
ij ji
for an asymmetric network in the limit of large N , or one works with a simpli fi ed Plefka
expansion where all two-time moments for different times are excluded from the beginning.
Hence, from the second moments, one keeps only

B

tt , ()
in the expansion.
6. Numerical results
In the previous sections we studied analytically two approaches to improve on the saddle
point approximation to the generating functional of the kinetic Ising model with synchronous
update. In section 4.5 we have argued that the constrained gaussian average optimization
leads to the mean fi eld ( MS-MF ) equations of [ 1 ] , whose performances was studied in [ 26 ] .
One could wonder how this compares to the unconstrained gaussian average method, and so
we iterated ( 41 ) and ( 43 ) to reconstruct the entire dynamics of magnetizations. In order to
estimate the magnetizations for the extended Plefka expansion described in the previous
section we designed the algorithm explained in section 5 . Thus we can evaluate numerically
the goodness of the two approximations in terms of magnetizations and compare them with
existing algorithms. Speci fi cally we investigate how they perform with respect to three mean
fi eld methods, namely Naive mean fi eld, dynamical TAP ( RH-TAP ) equations of [ 2 ] and MS-
MF equations of [ 1 ] . To recapitulate, Naïve mean fi eld and TAP equations can be obtained
via perturbative expansion in the magnitude of the couplings of the Legendre transform of the
log generating functional at fi xed magnetizations [ 2 ] , without making any restriction on
symmetry and distribution of the couplings. The fi rst order expansion gives Naïve Mean
Field, while second order terms lead to RH-TAP. MS-MF equations can be derived via central
limit theorem arguments exploiting the fact that the couplings are independent identically
distributed random variables with variance that scales as N 1 [ 1 ] , without making any
assumption on the couplings strength.
RH-TAP magnetizations under the kinetic Ising model with synchronous update are:
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
å g =- + - - -

m

th t J m t m t t tanh 1 1 1 , 59
ii
j
ij j i ii
( ) () () ( ) () ( )
where

g

- t 1
ii
()
has been de fi ned in ( 39 a ) . MS-MF equations correspond to ( 46 ) and Naïve
mean fi eld to ( 25 ) .
In order to test the performances of our methods as a function of couplings asymmetry
and strength we chose our couplings, following Crisanti and Sompolinsky [ 3 ] :
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
17

Figure 1. Mean squared error of Näive mean fi eld ( blue ) , RH-TAP ( green ) , MS-MF
( red ) , unconstrained gaussian average approach ( light blue ) and extended Plefka
( magenta ) for predicting entire dynamics of the magnetizations. The mean squared
error is plotted as a function of the couplings strength g for a system of 50 spins. We
have used 100 time steps and 50 000 repeats to calculate the experimental
magnetizations and have averaged the errors over 10 realizations of the couplings.
The error bars are standard deviations over these realizations. The number of sample
trajectories used in the algorithm for the extended Plefka method is N
T
= 50 000. The
different panels correspond to different values of the asymmetry parameter
= k 0, 0 .5, 1 from top to bottom. Left: stationary external fi eld drawn independently
for each spin from a normal distribution ( zero mean, standard deviation 0.5 ) . Right:
sinusoidal external fi eld with amplitude 0.5.
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
18

=+

J

Jk J ,6 0
ij ij ij
sym a ntisym
()
where
= JJ
ij ji
sym sym
and
=- JJ
ij ji
antisym antisym , while k is the parameter that controls the
asymmetry, interpolating between the fully asymmetric = k 1

(

) and the fully symmetric
= k 0

(

) distributions. We draw all the couplings J ij
sym and
J
ij
antisym
independently from a
distribution with zero mean and variance:
Figure 2. Mean squared error of Naive mean fi eld ( blue ) , RH-TAP ( green ) , MS-MF
( red ) , unconstrained gaussian average approach ( light blue ) and extended Plefka
( magenta ) for predicting entire dynamics of magnetizations. The mean squared error is
plotted as a function of the system size N . We have used 100 time steps and 50 000
repeats to calculate the experimental magnetizations and have averaged the errors over
10 realizations of the couplings. The error bars are standard deviations over these
realizations. The number of sample trajectories used in the algorithm for the extended
Plefka method is N
T
= 50 000 for
=

N

25 , 75, 100
, while N
T
= 100 000 for N  =  200.
Stationary external fi eld drawn independently for each spin from a normal distribution
( zero mean, standard deviation 0.5 ) . ( A ) : symmetric case k  =  0, couplings strength
g  =  1. ( B ) : fully asymmetric case k  =  1, g  =  1. ( C ) : k  =  0, = g 0.1; unconstrained
gaussian average, MS-MF and RH-TAP curves overlap. ( D ) : k  =  0, g  =  2. Notice the
different scale on the y -axis in ( C ) with respect to the other panels.
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
19

==
+
JJ g
Nk 1 ,6 1
ij ij
sym 2 an tisym 2
2
2
⟨( ) ⟩ ⟨( ) ⟩ () ()
where g controls the strength of the couplings.
We initialize the algorithms with the same initial condition and then we iterate them for
reconstructing the whole dynamics of magnetizations. We compare the predicted magneti-
zations with the experimental ones computing the mean square errors:
åå
=-
==
TN mt m t

M

SE 11 ,6 2
i
N
t
T
i i
J
11
exp 2
reali zations
(( ) ( ) ) ( )
where
mt
i
exp
(

)

are obtained by sampling the kinetic Ising model distribution of ( 8 ) .
The results are shown in fi gure 1 . From these plots it is clear that, apart from Naïve Mean
fi eld, all methods that we considered are compatible in the high temperature limit. At lower
temperatures the extended Plefka expansion is superior independently of the external fi eld.
Note, however, that for fully asymmetric couplings and sinusoidal external fi eld the MS-MF
method is performing slightly better than the extended Plefka approximation. This is likely
due to the fi nite size effects, since the two approaches are equivalent for asymmetric networks
with large N , as explained in section 5 . Regardless the degree of symmetry of the couplings
and the external fi eld RH-TAP systematically improves on the unconstrained gaussian
average approach, which fails at intermediate temperatures. The fact that the reconstruction is
noisier with respect to [ 26 ] is due to error propagation during the dynamics.
The scaling of the MSE errors with N is shown in fi gure 2 . Numerical simulations show
that the error of the extended Plefka method decays with the system size N for every value of
the parameters g and k , while the errors of the RH-TAP and MS-MF approximations decrease
with N only in the range of the parameters for which the the approximations were developed,
which corresponds respectively to a symmetric network with small couplings and to an
asymmetric network. The error computed using Naïve mean fi eld and unconstrained gaussian
average approximations shows no scaling with N . This seems to suggest that the extended
Plefka expansion provides an accurate mean fi eld description of the dynamics. Notice
however that evaluating the local moments with grater accuracy requires considering the
whole history of the single spin trajectory and that the complexity of the algorithm described
in section 5 scales with the degrees of freedom as + TN T N N
T
2
. To speed up the algorithm
one could argue that, when the couplings J
ij
scale as N 1 , the two sums
åå
¢¢ JC t t J J R t t ,; ,
j
ij j
j
ij ji j
2
() ()
appearing in ( 54 ) and ( 55 ) can be replaced by their self-averaging value
åå
¢ -
+ ¢ gC t t g k
k Rt t ,; 1
1 ,,
j
j
j
j
22
2
2
() ()
where we considered the distribution ( 61 ) for the couplings. This would allow us to write a
self-averaging version of ( 58 ) and the computational cost of the algorithm would reduce to
+ TT N N
T
2 . We postpone this analysis to future work.
7. Summary and discussion
In this paper we studied new approximations for predicting the dynamics of the kinetic Ising
model with arbitrary couplings. First we distinguished between the variational and fi eld
theoretical approaches to the Naïve mean fi eld theory for a generic Markov chain and pointed
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
20

out that the two do not coincide unless the transfer function is logistic, as is the case for the
kinetic Ising model and there are no self-interactions in the system. ( For an overview of the
approaches to Naïve mean fi eld theory for the equilibrium case see [ 16 ] . ) For the speci fi c case
of the kinetic Ising model with discrete time parallel updates, we then proposed two
approximations based on generating functional integral technique: the gaussian average
Variational method and the extended Plefka expansion. In the gaussian average variational
method we expand the generating functional of the process to fi rst order around a high
dimensional complex gaussian integral and optimize the resulting expression. An uncon-
strained optimization of the parameters of this gaussian function, in which we assume no
structure for the covariance matrix, provides equations of motion which, as our numerical
analysis indicates, perform at the naive mean fi eld level for small couplings while they get
close to RH-TAP equations [ 2 ] for larger couplings. On the other hand making suitable
assumptions on the covariance matrix, allows us to recover the MS-MF equations [ 1 ] , known
to be exact for fully asymmetric connectivities in the thermodynamic limit.
Although we numerically compared the dynamics of magnetizations predicted from our
dynamical equations with those of simulating the system, we did not study the relaxation
dynamics that our dynamical equations predict for such systems analytically. Such an analysis
has been performed in the case of the p -spin spherical spin glass model in [ 10 ] where it is
shown that the long term dynamics of the dynamical TAP equations for this system can be
seen as descending through the free energy landscape. For symmetric couplings and constant
external fi elds, the synchronous update model that we have considered here in the long time
will equilibrate to a Boltzmann distribution determined by the Peretto ’ s Hamiltonian [ 27 ] .
The replica analysis for this model has not been performed and, therefore, we cannot make
any statements as to what degree our extended Plefka and variational equations will be in
agreement with such analysis. However, we would like to note that for the asynchronous
update Glauber dynamics, once stationary magnetizations are assumed, the RH-TAP
equations will coincide with the standard static TAP equations [ 28 ] . As noted before the
equations derived here using the extended Plefka expansion are generalization of the RH-TAP
equations [ 2 ] and reduce to those if correlations and response functions are not taken into
account. Static TAP equations — which can be derived as the stationary limit of RH-TAP
equations — in turn, describe the multitude of local minima observed in the low temperature
phase of the SK model, and whose consistency with the replica approach has been formally
established [ 29 ] . The situation regarding the variational method is less clear, for one reason
because, besides that little is known of the low temperature properties of the Peretto ’ s
Hamiltonian [ 30 ] , the resulting dynamical equations can change with the ansatz chosen for
the covariance matrix of the fi elds and conjugate fi elds. If no ansatz is assumed, our numerical
results show that at low temperatures ( strong couplings ) , the error in predicting the magne-
tizations approaches those of the RH-TAP equations, which as stated before lead to static
TAP equations in the stationary state. We will leave it to future studies to explore this
similarity and the relaxation dynamics predicted by the variational approach in more detail
and analytically.
In the extended Plefka approach, by expanding the log generating functional in the
coupling strength, while fi xing fi rst and second order moments over time, we approximate the
true interacting dynamics by an effective single site dynamics. Namely, within the approxi-
mated description, each spin is subjected to an effective local fi eld ( 55 ) that contains a
retarded interaction with its own past values and a coloured gaussian noise. The main dif-
ference with other mean fi eld techniques is that the whole history of the single spin trajectory
is taken into account in the equation for local order parameters. Numerical simulations show
that considering this term leads to greater accuracy in predicting local magnetizations for all
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
21

values of couplings strength, coupling asymmetry and different choices of external fi elds. We
fi nd that this memory term is stronger for larger degree of symmetry of the network, and
negligible when the couplings are uncorrelated: in this case the MS-MF approximation is
retrieved.
The methods proposed in this paper are quite general in their scope and in theory can be
used for studying the dynamics of other kinetic models. In particular, we fi nd it interesting to
see how these approximations perform for point process models from the generalized linear
model family, from which the kinetic Ising model is just one simple example. Furthermore,
these methods can also be applied for inverse problems: inferring the interactions and fi elds
given spin trajectories [ 31 ] . In particular, given the fact that inference and learning in the
presence of hidden nodes can be casted in a functional integral language [ 32 ] , our methods
can naturally lend themselves to developing novel approximations in this case for point
processes. In fact, very recently, the extended Plefka approach has been used for learning and
inference of the continuous variables in the presence of hidden nodes [ 33 ] .
Acknowledgments
This work has been partially supported by the Marie Curie Initial Training Network
NETADIS ( FP7, grant 290038 ) . YR and CB also acknowledge fundings from the Kavli
Foundation and the Norwegian Research Council Centre of Excellence scheme. YR is also
grateful to the Starr foundation for fi nancing his membership at the IAS.
Appendix A. Determinant of S
As was mentioned in section 4.1 of the main text, in this appendix we demonstrate that the
determinant of the matrix
S
that appears in ( 37 ) and ( 38 ) equals one. We are going to prove it
irrespectively to the speci fi c details of matrices

g

and l in ( 38 ) . Consider a complex matrix
S
with the block structure de fi ned in ( 37 ) , where l + tt ,1 ()
and

g

t () are generic complex
square matrices of order N .
In order to compute its determinant partition the matrix
S
as follows:
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
=




  

S

SS
SS S
SSS
SSS
0 , 0 0 , 1 000
1, 0 1, 1 1, 2 0 0
0 2 ,1 2 ,2 2 ,3 0
0 0 3, 2 3, 3 3, 4
.A . 1
() ()
() () ()
() () ()
() () ()
()
The determinant of this partitioned matrix can be formulated in terms of its blocks
through the properties of Shur complements. Indeed for a generic matrix M :
⎡
⎣
⎢ ⎤
⎦
⎥
== -
-
M AB
CD MA D C A B det det det . A.2
1
[] ( )
Since the square matrix S 0, 0 ()
is invertible, as can be easily checked in ( 38 ) and ( A.2 )
can be used to express the determinant of
S
as:
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
22

⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎤
⎦
⎥
⎥
⎥
⎥
⎥
=-
-

    
SS S
S
S S
det det 0 , 0 det
1, 0
0 0,0 1 , 0 0 ... ,A . 3
S
01
0
()
()
()
() () ()
⧹
˜ ⧹
where we have denoted with
S
0 ⧹
the bottom right matrix in the partition ( A.1 ) .
Notice that second term in S
0
˜
⧹
— the Shur complement of S 0, 0 ()
— has the form:
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
=
-





 
S
S S
S
1, 0
0 0,0 1 , 0 0 ...
1 , 1 0000
0 0000
0 0000
0 0000
A.4
1
()
()
() ()
ˆ ()
()
with
⎡
⎣
⎢ ⎤
⎦
⎥


 r rl g l = -
- =

S

1, 1 0i
i1
,1 0 , 1 0 0 , 1 A . 5
ˆ () () () ( ) () ( ) ( )
such that the matrix S
0
˜
⧹
in ( A.3 ) turns out having the same block form as
S
0 ⧹
,
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
=




  

S

SS
SS S
SSS
SSS
1 , 1 1 , 2 000
2, 1 2, 2 2, 3 0 0
0 3 ,2 3 ,3 3 ,4 0
0 0 4, 3 4, 4 4, 5
,A . 6
0
˜
˜ () ()
() () ()
() () ()
() () ()
()
⧹
⎡
⎣
⎢ ⎤
⎦
⎥

 g
= -
-
tt t

S

, 0i
i A.7
˜ () ˜ () ()
and


g

gl g l =- - - - tt t t t t t 1, 1 1, . A .8
˜ () () ( ) ˜ () ( ) ( )
As a consequence S
0
˜
⧹
is a block tridiagonal matrix, just like
S
, and in order to compute
its determinant one can apply ( A.3 ) again, to express S det
0
˜
⧹
as a function of the determinant
of
S 1, 1
˜ (

)

. By repeatedly applying ( A.3 ) to the Shur complements S t
˜ ⧹ of tt S ,
˜ ()
, one shows
that the determinant of
S
can be factorized into determinants of tt S ,
˜ ()
s. As proven for t  =  0
these matrices tt S ,
˜ ()
preserve the structure of
tt S , ()
and therefore their determinants are 1.
Finally:

== tt SS det det , 1. A.9
t
˜ () ( )
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
23

Appendix B. Inverse of S
In sections 4.2 and 4.4 we relate the optimal values of the variationl parameters and
the magnetizations to the elements of the covariance matrix -
S 1 in the framework of
the gaussian average method. In this appendix we derive expressions for these elements,
namely the correlations between fi eld and conjugate fi elds
=
++
-

¢

gt S
Nt i Nt i iL
2, 2
1 2
s
⟨( ) ⟩
,
=
++ ++
- ¢
gt gt S Nt N i Nt N j ijL
2, 2
1
s
⟨ ˆ () ˆ () ⟩ and =+
++ + +
- ¢
gt gt S 1
Nt i N t N j ij L
2, 2 1
1
s
⟨( )
ˆ () ⟩
() — where ¼ ¢
L s

⟨

⟩
indicates averages under the gaussian measure - ¢
e L s , with




¢ =

L

S
s
1
2 .
V ariance 〈 g i ð t Þ 2 〉 L ′
s
Here we close the set of equation ( 41 ) for the magnetizations with equations for the variances
++
-
S Nt i Nt

i

2, 2
1 in terms of the interaction matrix
S
, whose entries are linked to the magnetizations
through ( 37 ) – ( 39 b ) .
Recall that the inverse of the non-singular matrix
S
can be computed as [ 34 ]
= -
- +

S

S
S
1d e t
det ,B . 1
ij
ij ji
1 () ( [ ] )
() ()
where S ji

[

] is the ji minor of the matrix
S
, obtained removing the j th row and the i th column
from the matrix itself. In case of the matrix de fi ned by ( 37 ) , whose determinant equals 1, the
problem of inverting the matrix corresponds to computing the determinant of these minors.
We now aim to calculate the determinant of
++
S
Nt i Nt i 2, 2

[

]
following the derivation of the
determinant of
S
.
As in appendix A , we start with factorizing out the determinant of the diagonal blocks
¢¢ tt S , ()
up to -- tt S 1, 1 ()
, according to ( A.3 ) . Given that these all equal 1, we can rewrite
the determinant of
++
S
Nt i Nt i 2, 2

[

]
as:
=
++
Ii t S det det , , B.2
Nt i Nt i 2, 2
([ ] ) ( ( ) ) ( )
where
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
º-
-
-- -
--


I

it
tt
tt tt
S
S
S S
,
,1
0 1, 1 1 , 0 ... B.3
t ii
i
i
11
()
() [ ]
[( ) ] ˜ ()
[( ) ] ()
⧹
⧹
⧹
and we have de fi ned tt S ,
˜ ()
in ( A.8 ) . - tt S ,1

i
[

() ]
⧹ and
- tt S ,1
i

[

() ]
⧹ have been obtained
removing respectively the i th row and the i th column from - tt S ,1 ()
. -
S t i

i

1

[

]
⧹ is instead the
ii minor of
-
S t 1 ⧹
de fi ned analogously as
S
0 ⧹
in appendix A :
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
=
+
++ + + +
++ ++ ++
++ ++
-




  
tt tt
tt tt tt
tt tt tt
tt tt

S

SS
SS S
SSS
SS
,, 10 0
1, 1, 1 1, 2 0
0 2 ,1 2 ,2 2 ,3
00 3 , 2 3 , 3
.B . 4
t 1
() ( )
( ) () ()
() () ()
() ()
()
⧹
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
24

One can easily see that

I

it , ()
in ( B.2 ) preserves the block form of -
S t i

i

1

[

]
⧹ , namely
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
=
+
+


I

it
tt tt
tt
SS
S
S
,
,, 1 0 0 0
1,
0
...
B.5
ii i
i
t
()
[ ˜ () ] [ ( ) ]
[( ) ] ()
⧹
⧹
⧹
and therefore one can apply the formula in ( A.3 ) once more to factorize the determinant in
( B.2 ) into a product of two determinants as follows:
=+
++
tt I it SS det det , det , 1 , B.6
Nt i Nt i ii 2, 2
([ ] ) ([ ˜ () ] ) ( ( ) ) ( )
where
+

I

it ,1 ()
has been de fi ned in ( B.3 ) .
With a bit of algebra it is possible to show that the matrix
+

I

it ,1 ()
in ( B.6 ) , has the
very same structure as S

t

⧹
and consequently of
S
. Thus the second factor in the above
equation is 1 ( appendix A ) and what ’ s left is to compute the determinant of the ii minor of the
matrix tt S ,
˜ ()
.
Given the structure of tt S , ii

[

˜ () ]
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥

 g
º -
-
tt t
S , 0i
i ,B . 7
ii
i
i

[

˜ () ] ˜ () ()
⧹
⧹
where

g

t
˜ () has been de fi ned in ( A.8 ) , its determinant reduces to:
 gg
gg
g
=-
=-
=
--
--
tt t t
tt
t
S det , 1 det det i i
1 det det
.B . 8
ii N i i
N ii
ii
11
22 1
([ ˜ () ] )( ) (
˜ () ) ( ˜ () )
() (
˜ () ) ( [
˜ () ] )
˜ () ( )
⧹ ⧹
Finally we will check that the diagonal elements of -
S 1 we ’ ve just obtained are well
de fi ned variances by proving that they can take only positive values. In order to do that we
will show that the matrix

g

t
˜ () is positive de fi nite.
By substituting

g

and l , using respectively ( 39 a ) and ( 39 b ) ,i n ( 43 ) one can express

g

t
˜ ()
in terms of

g

- t 1
˜ ()
, the matrix of the couplings

J

and the matrix
d º- Mt m t 1
ij ij i 2
() ( () )
( m are the magnetizations ) as


g

g =+ - tt t t t t JM JM JM M J 1. B . 9
22
˜ () ( () ) ( () ) () ˜ () ( ) ( )
The fi rst matrix on the right-hand side of ( B.9 ) is positive de fi nite. Since the sum of two
positive de fi nite matrices is positive de fi nite, it is left to show that the second term on the
right-hand side of ( B.9 ) is positive de fi nite. We will prove it by induction. First of all given
that

g

g = 00
˜ () ()
, from the de fi nition of

g

in ( 39 a ) , we know that

g

0
˜ (

)

is positive de fi nite.
Then we assume that

g

- t 1
˜ ()
is positive de fi nite and we prove that 
g - tt t

J

MM J 1
22
() ˜ () ( )
is positive de fi nite. If

g

- t 1
˜ ()
is positive de fi nite, it exist a matrix A such that


g

-= tA A 1
˜ () . Exploiting the latter one can rewrite:

g -= tt t t A t A JM M J JM JM 1B . 1 0
22 2 2
() ˜ ( ) ( ) (( ) ) (( ) ) ()
proving that the second term on the right-hand side of ( B.9 ) is positive de fi nite. Consequently

g

t
˜ () is a positive de fi nite matrix and its diagonal entries take only positive values.
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
25

Correlations 〈 b
g i ð t Þ b
g j ð t Þ 〉 L ′
s
Here we will prove that the two point correlation function between conjugate fi elds
++ ++
-
S Nt N i Nt N j 2, 2
1 is zero, as claimed in 4.4 , where it enters the proof of consistency of the
optimal parameter

h

= 0 ˆ
.
Similarly to the previous subsection we will use ( B.1 ) to invert the matrix S and compute
the determinant of the minor through Shur ’ s complement formula ( A.2 ) :
= -
=-
++ ++
- ++ + ++ ++
+ Y ijt
S S
S
1d e t
det
1d e t , , B . 1 1
Nt N i Nt N j
Nt N i j Nt N i Nt N j
ij
2, 2
1
42 2, 2
() ( [ ] )
()
() ( ( ) ) ( )
with
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
=-
-
´- - -
- ++
+
- +

Y ijt
tt
tt tt
S
S
S S
,,
,1
0
1, 1 1 , 0 ... B.12
t Ni Nj
Ni
Nj
1 ,
1
()
() [ ]
[( ) ]
˜ ()
[( ) ] ()
⧹
⧹
⧹
and we have de fi ned tt S ,
˜ ()
in ( A.7 ) and
-
S t 1 ⧹
in ( B.4 ) .

Y

ijt ,, ()
in ( B.12 ) preserves the
block form of
- ++
S
t Ni Nj
1 ,

[

]
⧹
, namely
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
=
+
+
++ +
+

Y ijt
tt tt
tt
SS
S
S
,,
,, 1 0 0 0
1,
0
...
.B . 1 3
Ni Nj Ni
Nj
t
,
()
[ ˜ () ] [ ( ) ]
[( ) ] ()
⧹
⧹
⧹
Conversely to the previous section we cannot express the determinant of

Y

ijt ,, ()
in
terms of the Shur ’ s complement of ++
tt S , Ni Nj ,

[

˜ () ] , since the latter is a singular matrix. One
has instead to resort to the Shur ’ s complement of the matrix S

t

⧹
that we know is invertible and
its determinant is 1, having the same structure of the matrix
S
( appendix A ) :
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎞
⎠
⎟
⎟
=
- + +
++
+ -
+

Y ijt tt
tt
tt
SS
S S
S
det , , det det ,
, 1 0 ...
1,
0 .B . 1 4
t Ni Nj
Ni t
Nj
,
1
()
() ( [
˜ () ]
[( ) ] []
[( ) ]
()
⧹
⧹ ⧹
⧹
Just like ++
tt S , Ni Nj ,

[

˜ () ] , the matrix whose determinant is the second factor on the right-
hand side of ( B.14 ) is singular: as can be easily checked its i th column is null, regardless of
the elements of -
S
t 1

[

]
⧹ . This completes the proof that =
++ ++

S

0
Nt N i Nt N j 2, 2 for all
=¼ ij N ,1 , ,
and =¼ - tT 0, , 1 .
Correlations 〈 g i ð t Þ b
g j ð t + 1 Þ 〉 L ′
s
Here we will prove that the two point correlation function between conjugate fi elds
=
++ + +
-
S 0
Nt i N t N j 2, 2 1
1 () is zero, as claimed in 4.4 , where it enters the proof of consistency
of the optimal parameter

h

= 0 ˆ
. The derivation is very similar to the one for
=
++ ++
-
S 0
Nt N i Nt N j 2, 2
1 in the previous subsection.
We will use ( B.1 ) to invert the matrix S and compute the determinant of the minor
through Shur ’ s complement formula ( A.2 ) :
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
26

= -
=-
++ + +
- ++ + ++ + +
++ Z ijt
S S
S
1d e t
det
1d e t , , B . 1 5
Nt i N t N j
Nt N i j Nt i N t N j
Ni j
2, 2 1
1
43 2, 2 1
3
() ( [ ] )
()
() ( ( ) ) ( )
()
()
with
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
º-
-
´- - -
- +
- +

Z ijt
tt
tt tt
S
S
S S
,,
,1
0
1, 1 1 , 0 ... B.16
t iN j
i
Nj
1 ,3
1
()
() [ ]
[( ) ]
˜ ()
[( ) ] ()
⧹
⧹
⧹
and we have de fi ned tt S ,
˜ ()
in ( A.8 ) and
-
S t 1 ⧹
in ( B.4 ) .
Z ijt ,, ()
in ( B.16 ) preserves the
block form of
- +
S
t iN j
1 ,3

[

]
⧹
, namely
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
=
+
++ + + +
++
+
+
+



Z

ijt
tt tt
tt tt tt
tt
SS
SS S
S
S
,,
,, 1 0 0
1, 1, 1 1, 2 0
02 , 1
... 0
...
.B . 1 7
ii N j
Nj
t
,
1
()
[ ˜ () ] [ ( ) ]
( ) () ()
() ()
⧹
⧹
⧹
Analogously to the previous section we will now express the determinant of
Z ijt ,, ()
using the Shur ’ s complement of the matrix
+
S t 1 ⧹
that we know is invertible and its deter-
minant is 1, just like the matrix
S
, as shown in appendix A :
⎜⎟
⎛
⎝
⎜
⎜
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎛
⎝
⎞
⎠
⎛
⎝
⎜ ⎞
⎠
⎟
⎞
⎠
⎟
⎟
= +
++ +
- ++
++
+ +
+
+-


Z ijt tt tt
tt tt
tt
tt
S SS
SS
S S S
det , , det det ,, 1
1, 1, 1
00
1, 2 0
02 , 1
00
... ...
.B . 1 8
t ii N j
Nj
t
1 ,
11
() [ ˜ () ] [ ( ) ]
() ( )
()
[] ()
()
⧹ ⧹
⧹
⧹
The structure of
+-
S
t 11

[

]
⧹
re fl ects -
S 1 structure:
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
=
W+ + W+ + W+ +
W+ + W+ +
W+ +
+-


 
 
tt tt tt
tt tt
tt
S
2, 2 2, 3 2, 3
3, 2 1, 1
4, 2
B.19
t 11

[

]
() () ()
() ()
() ()
⧹
with
⎡
⎣
⎢ ⎤
⎦
⎥
g
W+ + = +D
G
tt t
2, 2 2
0 ,B . 2 0 ()
˜ () ()
where Δ and Γ are matrices of order N

2

. The block form of
W+ + tt 2, 2 ()
follows that of
the diagonal blocks of -
S 1 , that was proven to be such in the previous sections of this
appendix.
Using ( B.19 ) one can check that the matrix whose determinant is the second factor on the
right-hand side of ( B.18 ) is singular: its ( N + j ) -th row is null. This completes the proof that
=
++ + +
-
S 0
Nt i N t N j 2, 2 1
1 () for all
=¼ ij N ,1 , ,
and =¼ - tT 0, , 1 .
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
27

Appendix C. Gaussian average method: variance
In this appendix we study the stability of the dynamical system for the matrix

g

˜
de fi ned in the
main text by ( 43 ) . In order to do that we fi rst average ( 43 ) over the distribution of the
couplings introduced in section 6 through ( 60 ) and ( 61 ) . Consider then entries of

g

1
˜ (

)

:
gg
gg
=+ =
=+ =
gg k
gg k
1 1 0 for 0
1 2 1 8 0 for 1 , C.1
ij lm
ij lm
22
22
˜ () ( ˜ () )
˜ () ( ˜ () ) ( )
Figure C1. The mean variance in the gaussian integral for the magnetizations versus
time of reconstruction, when the experimental history of magnetization is known.
Green: the gaussian average method; blue: the variance in [ 1 ] . N  =  20, single
realization of the couplings with g = 0.4 ( A ) , g = 0.6 ( B ) , g = 0.9 ( C ) and g = 1.1 ( D ) .
The experimental magnetizations are computed using 10
4
samples of the dynamics.
Zero external fi eld. Top: asymmetry parameter k  =  0. Bottom: asymmetry para-
meter k  =  1.
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
28

where the overbar indicates the average over the disorder, k is the parameter controlling the
asymmetry of the couplings, while g is the coupling strength. Notice the different factors in
( C.1 ) due to the correlations between the couplings. If we consider the univariate analogous to
( C.1 ) :
=+ - =
=+ - =
xt g g xt k
xt g g xt k
11 f o r 0
21 8 1 f o r 1 C . 2
22
22
() ( ( ) )
() ( ( ) ) ( )
one can easily check that this dynamical system is characterized by a critical value for g , g
0
that discriminates between different stability classes for the system. Below g
0
x in ( C.2 )
converges to a fi nite value, while it grows exponentially in time for
> gg
0
. The critical value
for this chaotic behavior is respectively
= g

1

0
for fully asymmetric couplings ( k = 1 ) and
<< g

0

.7 0.8
0 for fully symmetric ones ( k = 0 ) .
We got numerical evidence to support our intuition. We found that the variance in the
gaussian average method undergoes a chaotic behavior when the couplings strength reaches a
certain critical value, ~ g 0.5
0 for symmetric connectivities and
~ g

1

0
for fully asymmetric
ones. This value depends on the degree of symmetry of the couplings, on the presence of the
external fi eld and there are small fl uctuation across different realizations of the couplings, but
the phenomenon is qualitatively conserved. Figure C1 shows single realizations of the
couplings below and above critical values and compares the mean of the gaussian average
variances g
å t
N i ii
1 ˜ (

)

with the mean of the MS-MF variances ( mean of

g

t
ii
()
( 39 a ) in our
notation ) in the gaussian integral for fully asymmetric couplings [ 1 ] .
Appendix D. Details on the extended Plefka expansion
We rewrite the functional

G

a
( 51 ) as

ò p

G

=X - -
aa
NT N mmCBR s ln d Tr exp , , , , , , ln 2 ln 2 , D.1
s
([ ˆ ]) ( )
where
⎪
⎪
⎧
⎨
⎩
⎫
⎬
⎭
 åå
åå
å
a
y
X= - + +
-- - + -
+¢ ¢ - ¢ + ¢ ¢ - ¢
-¢ ¢ - ¢
a
¢¢
¢
gt gt J s t s t gt
gt h t gt m t t st m t
C t ts t s t C t t B t tg t g t B t t
Rt t g t s t Rt t
mmCBR s ,, , , , , i 1
l n2c o s h i
1
2 ,,
1
2 ,,
i, i , , D . 2
it
ii
j
ij j i i
i i i i i ii
t
ii i i
t
i ii i
t
i i ii
,
[ ˆ ] ˆ () [ () () ] ( ) ()
() () [ ˆ () ˆ () ] () [ () () ]
ˆ () [ ( ) ( ) () ] ˆ () [ ˆ ( ) ˆ ( ) () ]
ˆ () [ ˆ ( ) ( ) () ] ( )
and proceed with the perturbation expansion of

G

a
around
a = 0
up to the second order:
a a

G

=G + G + G
a 2 ,D . 3
01
2 2 ()
() () ()
where
a

G

=¶ G ¶
aa =
kk k

0

∣
()
. At the end of the calculation we will set a = 1 . The fi rst term in
the expansion is given by
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
29

åå
åå å
y y G= - +
-¢ ¢ - ¢ ¢ - ¢ ¢
¢¢ ¢
h Z t mt h t mt
C t tC t t B t tB t t R t t R t t
mmCBR C B R ,, , , l n , , , , i
1
2 ,, 1
2 ,, , , ,
D.4
it
i i
it
i i
itt
i i
itt
i i
itt
i i
0 0 00 0 00 00
0 00
[ ˆ ][ ˆ ˆˆ
] () () ()ˆ ()
ˆ () () ˆ () () ˆ () ()
()
()
where
⎫
⎬
⎭

ò 
å
åå
y p
y
=+ +
-- + + ¢ ¢
+¢ ¢ - ¢ ¢
¢
¢¢
h Zg t g t s t g t
gt gt h t t s t C t t s t st
Btt g t g t R t t g t st
CBR ,, , , 1
22 dT r e x p i 1
ln cosh i 1
2 ,
1
2 ,i , D . 5
NN T s
it
ii i i
ii
i i i
t
ii i
t
i ii
t
i i i
0
[ ˆ ˆˆ
] () { ˆ () () ( ) ()
() ˆ () () () () ˆ () ( ) ( )
ˆ () ˆ ( ) ˆ ( ) ˆ () ˆ ( ) ( ) ( )
and
y h CBR ,, , ,
00 0 00
ˆ ˆˆ
are the fi elds for which the set of equations ( 52 ) is satis fi ed for

G

=G
a 0 ()
for a given value of

m

mCBR ,, , ,
ˆ
. We compute

G

1 ()
as follows:
aa

G

= ¶G
¶ = ¶X
¶
a
a
a
a
= =
.D . 6
1
0 0
()
()
Using ( D.2 ) one fi nds:
åå å
åå
åå
åå
aa
y
a
a
a
a
¶X
¶ =- - ¶
¶ -+
¶
¶ -
+ ¶¢
¶ ¢- ¢
+ ¶¢
¶ ¢- ¢
- ¶¢
¶ ¢- ¢
a
¢¢
¢¢
¢¢
Jg t s t ht gt m t t st m t
Ct t st st C t t
Btt gt gt B t t
Rt t gt s t i R t t
ii
1
2
, ,
1
2
, ,
i , ,.
D.7
ijt
ij i j
it
i
i i
it
i ii
itt t
i ii i
itt t
i
ii i
itt t
i
i ii
ˆ( ) ( ) () [ˆ ( ) ˆ ( ) ] () [( ) ( ) ]
ˆ ()
[( ) ( ) ( ) ]
ˆ ()
[ˆ ( ) ˆ ( ) ( ) ]
ˆ ()
[ˆ ( ) ( ) ( ) ]
()
When computing the average a ¶X ¶
aa

⟨

⟩ as de fi ned in ( 50 ) , all the terms on the right-hand
side of ( D.7 ) except for the fi rst one vanish because of the set of equations ( 49 ) . Moreover at
α  =  0 the spins are decoupled and the averages are trivial:
åå

G

=- =- Jg t s t J m t m t ii . D . 8
ijt
ij i j
ijt
ij i j
1 0
⟨ ˆ () () ⟩ ˆ () () ( )
()
For the second derivative of

G

a
with respect to α we have
⎛
⎝
⎜ ⎞
⎠
⎟ ⎛
⎝
⎜ ⎞
⎠
⎟
aa a a
¶G
¶ = ¶X
¶ + ¶X
¶ - ¶X
¶
aa
a
a
a
a
a
.D . 9
2
2
2
2
2 2
()
Using ( D.7 ) and the set of equations ( 49 ) , it is easy to show that the fi rst term on the right-
hand side of the above equation is zero. One thus fi nds
⎛
⎝
⎜ ⎞
⎠
⎟
aa

G

= ¶X
¶ - ¶X
¶
aa
a =
,D . 1 0
2
2
0
()
()
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
30

which can be computed using ( D.7 ) and the following Maxwell equations:
å
å
y
aa
aa
¶
¶ =- ¶
¶
¶G
¶ =
¶
¶ = ¶
¶
¶G
¶ =-
a
a
a
a
a
a
= =
==
t
mt Jm t
ht
mt Jm t
i
ii . D . 1 1
i
i j
ji j
i
i j
ij j
0 0
00
()
() ˆ( )
()
ˆ( ) () ( )
Note that the derivatives of the two-time conjugate fi elds with respect to α are zero, e.g.
aa
¶¢
¶ = ¶
¶¢
¶G
¶ =
a
Ct t
Ct t
,
, 0. D.12
i
ˆ ()
() ()
We fi nally obtain
å dd d d

G

=- ¢ ¢
¢¢ ¢ ¢ ¢¢ ¢
gt J s t g t J s t ,D . 1 3
i j ijt t
i ij j i ij j
2 0
⟨ ˆ () () ˆ ( ) ( ) ⟩ ( )
()
where we de fi ned

d

=- st st m t
ii i
() () ()
and

d

=- gt gt m t
ii i
ˆ () ˆ () ˆ ()
. Since the averages are
taken at
a = 0
spins at different sites are decoupled and the only non-vanishing terms in
( D.13 ) correspond to the case ¢= ¢= ii j

j

, and ¢= ¢= ij j i , :
å
dd d d
dd d
G= - ¢ ¢
+¢ ¢
¢
J g tg t s ts t
JJ g t s t g t s t ,D . 1 4
ijtt
ij ii jj
ij ji i i j j
2 2 00
00
[ ⟨ ˆ ( ) ˆ () ⟩ ⟨ ( ) () ⟩
⟨ˆ ( ) ( ) ⟩ ⟨ ˆ ( ) ( ) ⟩ ] ( )
()
which can be written in terms of the moments as follows
å
G= - ¢ - ¢ ¢ - ¢
+¢ - ¢ ¢ - ¢
¢
J B t t mt mt Ct t m t m t
J J R t t mt mt R t t mtmt
,,
i, i , . D . 1 5
ijtt
ij ii i jj j
ij ji i i i j j j
2 2
[ ( ( ) ˆ ()ˆ ( ) ) ( ( ) () ( ) )
( ( ) ˆ ( ) ( ))( ( ) ˆ ( ) ( ) )] ( )
()
Inserting ( D.4 ) , ( D.8 ) and ( D.15 ) in ( D.3 ) we fi nd the explicit expression of the functional

G

a
expanded up to the second order. Considering the set of equations ( 52 ) within the second
order expansion and setting the auxiliary fi elds to zero, we can extract the value of the fi elds
y h CBR ,, , ,
00 0 00
ˆ ˆˆ
as functions of the correct ( within the expansion ) marginal fi rst and
second moments:
åå
å
å
y =
=+ - ¢ ¢
¢=
¢= - ¢ - ¢
¢= ¢
¢
t
ht h t J m t J J R t t m t
Ct t
Bt t J C t t m t m t
Rt t J J R t t
0
,
,0
,,
,, . D . 1 6
i
i i
j
ij j
jt
ij ji j i
i
i
j
ji jj j
i
j
ij ji j
0
0
0
0 2
0
()
() () () ( ) ( )
ˆ ()
ˆ () ( () ( ) ( ) )
ˆ () ( ) ( )
From general results for generating functional analysis of spin systems [ 14 ] it can be shown
that = m 0
ˆ , B  =  0 and ¢

R

tt , ()
has the meaning of a local response function and is non-
vanishing only for >

¢

tt
. To get an explicit expression for Z
0
we insert ( D.16 ) in ( D.5 ) .I t
yields
= 
Zh Z h
i i
0 0
[] [

]

, where
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
31

⎪
⎪
⎫
⎬
⎭

ò 
åå
å
p
=+ - -
+- - ¢
-¢ ¢ - ¢ ¢ - ¢
¢
¢
Z h st gt gt gt h t
gt gt gt J m t gt J C t t
mt mt g t g t J J Rt t s t m t
1
22 d Tr exp 1 ln cosh i
ii 1
2 ,
i, .
D.17
i i NN T s
t
i ii i
i
ii
j
i ij j
jt
i ij j
jj i
jt
i ij ji j i i
0
2
i
[] () { ( ) () () ˆ () ()
ˆ () () ˆ () () ˆ () [ ( )
( ) () ] ˆ () ˆ ( ) ( ) [ () () ]
()
To linearize the quadratic terms in ( D.17 ) , we introduce the gaussian random variables

f

t
i ()
,
independently for each i , with zero mean and covariance
ff ¢=
å ¢- ¢ t t J C t t mt mt ,
ii j
ij jj j
2

⟨

() ( ) ⟩ ( ( ) () ( )

)

, obtaining:
⎪
⎪
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎤
⎦
⎥
⎥
⎫
⎬
⎭

ò 
åå å
p
f
=+ - +
-+ - ¢ ¢ - ¢ +
f
¢
-
Z h s t gt gt gt gt
tJ m t J J R t t s t m t h t
1
22 d Tr exp 1 ln cosh i
,.
D.18
i i NN T s
t
i ii i i
i
j
ij j
jt
t
ij ji j i i i
0
1
i
i
[] () {( ) ( ) ( ) ˆ ( ) [ ( )
() () ( ) [ ( ) ( ) ] ()
()
From the above equation one can see that the moment
¢

R

tt ,
i
()
de fi ned in ( 49 ) can be written
as an average over the fi elds

f

t
i () as follows
f
¢= ¶
¶¢
f
Rt t st
t
,, D . 1 9
i i
i i
() ()
() ()
and can be interpreted as a response function.
Appendix E. The Yule – Walker equations
We want to generate the gaussian random fi eld

f

t
i
k
()
for given trajectory k and spin i based on
the past values of the fi eld

f

ff - t 0 , 1 ... 1
i
k
i
k
i
k
() () ( )
. Since the random variables

f

ff t 0 , 1 ...
i
k
i
k
i
k
() () (

)

are jointly gaussian distributed with zero mean, we know that the
conditional expectation

f

ff f f º-

tE t t 0 , 1 ... 1
i
k
i
k
i
k
i
k
i
k
( ) { ( ) ∣ () () ( ) }
is given by the linear
estimate
å

f

f =
=
-



ta r r ,E . 1
i
k
r
t
i
k
0
1
( ) () () ( )
which also happens to be the best mean square estimate of

f

t
i
k
()
given

f

=- rr t , 0 ...

1

i
k () .
The coef fi cients -

a

aa t 0 , 1 ,... 1 () () ( )
are such that the mean square value of the estimation
error
ff -


E

tt
i
k
i
k 2
{ [ () () ] }
is minimum. By the orthogonality principle, this condition holds
if the following set of equations is satis fi ed
å
ff f -¢ = ¢ = -
=
-

E

ta r r r r t 0, 0 .. . 1, E .2
i
k
r
t
i
k
i
k
0
1
{ [ ( ) () () ] ( ) } ( )
which can be rewritten in matrix form as
 = A ,E . 3
t ()
where
=- aa t A 0 ... 1 [( ) ( )

]

is the vector of coef fi cients,



is the correlation matrix with
elements



ff ¢= ¢ rr E r r , i
k
i
k
() { ( ) ( ) }
for
¢= -

r

rt , 0 ... 1
and



t
is the vector with elements
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
32



ff = rE t r
t i
k
i
k
() { ( ) () } for =-

r

t 0 ...

1

. Since

f

ff -^

tt r
i
k
i
k
i
k
() () ( )
for every
=-

r

t 0 ,... 1 , from ( E.1 ) we conclude that

f

ff -^

tt t
i
k
i
k
i
k
() () ()
and the error reduces to

å
ff ff f -= - = - -


E

tt E tt t J m t A 1. E . 4
i
k
i
k
i
k
i
k
i
k
j
ij j t
2 22
{ [ () () ] } { [ () () ] () } () ( )
Knowing that ff ¢=
å ¢- ¢ ¢ =

E

r r J C r r mr mr r r t , , 0 ...
i
k
i
k
j ij jj j
2
{ ( ) ( )} [ ( ) ( ) ( )] , we can
compute the coef fi cients a ( r ) from ( E.3 ) and draw the gaussian random variable

f

t
i
k
()
with
mean and covariance given, respectively, by ( E.1 ) and ( E.4 ) .
References
[ 1 ] Mézard M and Sakellariou J 2011 J. Stat. Mech. L07001
[ 2 ] Roudi Y and Hertz J 2011 J. Stat. Mech. P03031
[ 3 ] Crisanti A and Sompolinsky H 1987 Phys. Rev. A 36 4922
[ 4 ] Kappen H and Spanjers J 2000 Phys. Rev. E 61 5658
[ 5 ] Plefka T 1982 J. Phys. A: Math. Gen. 15 1971
[ 6 ] Aurell E and Mahmoudi H 2012 Phys. Rev. E 85 031119
[ 7 ] Toyoizumi T, Rad K R and Paninski L 2009 Neural Comput. 21 1203 – 43
[ 8 ] Huang H and Kabashima Y 2014 J. Stat. Mech.: Theory and Experiment 2014 P05020
[ 9 ] Mahmoudi H and Saad D 2014 J. Stat. Mech. P07001
[ 10 ] Biroli G 1999 J. Phys. A: Math. Gen. 32 8365
[ 11 ] Bravi B, Sollich P and Opper M 2015 arXiv: 1509.07066
[ 12 ] Martin P C, Siggia E and Rose H 1973 Phys. Rev. A 8 423
[ 13 ] De Dominicis C and Peliti L 1978 Phys. Rev. B 18 353
[ 14 ] Coolen A C C 2000 arXiv: cond-mat / 0006011
[ 15 ] Hertz J A, Roudi Y and Sollich P 2016 arXiv: 1604.05775
[ 16 ] Opper M and Saad D 2001 Advanced Mean Field Methods: Theory and Practice ( Cambridge,
MA: MIT Press )
[ 17 ] Kholodenko A 1990 J. Stat. Phys. 58 355 – 70
[ 18 ] Weiss P 1907 J. Phys. Theor. Appl. 6 661 – 90
[ 19 ] Bishop C M 2006 Pattern Recognition and Machine Learning ( Berlin: Springer )
[ 20 ] Müschlegel B and Zittarts H 1963 Z. Phys. 175 553 – 73
[ 21 ] Sissakian A and Solovtsov I 1992 Z. Phys. C 54 263 – 71
[ 22 ] Kleinert H 2009 Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial
Markets 5th edn ( Singapore: World Scienti fi c )
[ 23 ] Stevenson P M 1981 Phys. Rev. D 23 2916 – 44
[ 24 ] Altland A and Simons B D 2010 Condensed Matter Field Theory ( Cambridge: Cambridge
University Press )
[ 25 ] Georges A and Yedidia J S 1991 J. Phys. A: Math. Gen. 24 2173
[ 26 ] Sakellariou J, Roudi Y, Mezard M and Hertz J 2012 Phil. Mag. 92 272 – 9
[ 27 ] Peretto P 1984 Biol. Cybern. 50 51 – 62
[ 28 ] Thouless D J, Anderson P W and Palmer R G 1977 Phil. Mag. 35 593 – 601
[ 29 ] Cavagna A, Giardina I, Parisi G and Mézard M 2003 J. Phys. A: Math. Gen. 36 1175
[ 30 ] Scharnagl A, Opper M and Kinzel W 1995 J. Phys. A: Math. Gen. 28 5721
[ 31 ] Roudi Y and Hertz J 2011 Phys. Rev. Lett. 106 048702
[ 32 ] Dunn B and Roudi Y 2013 Phys. Rev. E 87 022127
[ 33 ] Bravi B and Sollich P 2016 arXiv: 1603.05538
[ 34 ] Friedberg S, Insel A and Spence L 1996 Linear Algebra 3rd edn ( Englewood Cliffs, NJ:
Prentice-Hall )
J. Phy s. A: Math. Theor . 49 ( 2016 ) 434003 L Bachschmid-Romano et al
33

3.3 Discussion
3.3 Discussion
In this section, w e presen t additional comparisons b et w een differen t algorithms
aimed at predicting single site magnetizations. The analysis of P ap er 1 rev eals
that a no v el appro ximation denoted as the Extended Plefk a Expansion out-
p erforms other existing metho ds, at the cost of k eeping memory of the spin
fluctuations at all past times. The required Mon te Carlo algorithm can b e
sp eeded up from T 2 N + T N N T to T 2 + T N N T computational time steps,
where N is the size of the system, T the n um b er of time steps and N T the
n um b er of Mon te Carlo tra jectories. If the couplings J ij ha v e v ariance ∼ 1 / N ,
w e exp ect lo cal t w o-times momen ts to b e self-a v eraging. Hence, w e can rewrite
the effectiv e single site field of equation (55) as
g i ( t ) = φ i ( t ) + X
j
( J ij m j ( t )
− g 2 1 − k 2
1 + k 2
1
N
t − 1
X
t 0 =0 X
j
R j ( t, t 0 )[ s i ( t 0 ) − m i ( t 0 )] ! + h i ( t ) .
(3.1)
and the co v ariance of the Gaussian noise of equation (54) b ecomes
h φ i ( t ) φ i ( t 0 ) i = g 2
N X
j
[ C j ( t, t 0 ) − m j ( t ) m j ( t 0 )] . (3.2)
The new v ersion of the algorithm is written in App endix 3.C. Figure 3.1 sho ws
that - for net w orks of 50 spins- the accuracy of the prediction on the magne-
tization is reduced if w e replace the lo cal second order momen ts with their
a v erage v alue, esp ecially for small couplings. Ho w ev er, the scaling of the mean
squared error with the system size (Figure 3.2) suggests that the t w o algo-
rithms p erform equally w ell for v ery large net w orks: in b oth implemen tations
of the extended Plefk a metho d, error of the predicted magnetization go es to
zero in the limit N → ∞ .
As a final comparison, w e in v estigate the p erformance of the Extended Gaus-
sian appro ximation of [MS14] (see section 3.B) and plot its error in Figure 3.1,
in case of a symmetric net w ork. W e recall that, for completely asymmetric
net w orks, this metho d agrees with the exact mean field theory of [MS11]. One
observ es that, for v ery small couplings (high temeratures), all the metho ds
approac h eac h other. F or quite large v alues of the couplings (corresp onding
to v alues of g greater than g ≈ 1 . 5), the extended Gaussian appro ximation
p erforms equally w ell as the extended Plefk a algorithm. T o b e more precise,
53

3 Dynamics on random net w orks
the extended Gaussian metho d outp erforms the extended Plefk a appro xima-
tion for N = 50; ho w ev er, w e observe that the result of the extended Plefk a
metho d decreases for increasing N , con trary to the extended Gaussian metho d
(Figure 3.2). There are in termediate v alues of the strength of the couplings
(appro ximately corresp onding to v alues 0 . 2 < g < 1 . 5 ) for whic h the p er-
formance of the extended Gaussian appro ximation is p o orer that b oth the
Extended Plefk a and the RH-T AP , where the latter metho d do es not consider
the effect of correlations and resp onses.
3.4 Conclusions
W e hav e presen ted sev eral appro ximate metho ds to study the transien t dy-
namics of an Ising mo del with the parallel up date Glaub er rule, deriving
equations for the time ev olution of single-spin magnetisations for fixed v al-
ues of the quenc hed couplings. W e considered system with couplings that are
w eak, long-ranged and are allo w ed to ha v e an y degree of symmetry . While a
lot of efforts ha v e b een dev oted to the study of the dynamics of spin glasses
with symmetric couplings (for a review see [BCKM98] ), comparativ ely less
atten tion has b een dev oted to net w orks with asymmetric couplings - for whic h
an energy function cannot b e defined. In the fully asymmetric case, correla-
tions b et w een spins at v arious times are negligible, and the fields acting on
eac h spin can b e simply describ ed in terms of effectiv e Gaussian fields [MS11]
. Ho w ev er, if the couplings ha v e a non-zero degree of symmetry , these corre-
lations p ersist also at distan t times, and studying the dynamics is less trivial.
In this c hapter, w e in tro duced a no v el approac hes to this task, denoted as the
extended Plefk a expansion. It is based on a w eak coupling expansion of the
log-generating functional, where the lo cal momen ts o v er time are constrained
via a Legendre transform. The no v elt y of our formalism relies in including not
only the first-order marginal momen ts, but also all the second-order marginal
momen ts; the latter quan tities are trivial for the equilibrium Ising mo del (since
σ 2 = 1) but are non-negligible in the dynamic case for a correct mean-field de-
scription of the system. The result is an effectiv e log-generating functional that
factorizes o v er single-site tra jectories and con tains the correct first and second
order momen ts within the appro ximation. Namely , within the appro ximate
description, eac h spin is sub jected to an effectiv e lo cal field whic h con tains
Gaussian noise - enco ding the effect of correlations with past spin v alues - and
a memory term where the resp onse function is coupled to the whole history
of spin fluctuations. The con tribution of the memory term is stronger for
larger degree of symmetry of the net w ork, and negligible when the couplings
are uncorrelated, in whic h case the exact mean field theory is retriev ed. The
54

3.4 Conclusions
0 0.5 1 1.5 2 2.5 3
g
0.0001
0.01
1
ε

Figure 3.1: Mean squared error of the predicted magnetization a v eraged o v er
spins and time: ε = h m i ( t ) exp erimen tal − m i ( t ) predicted i , where the
’exp erimen tal’ magnetization is the one w e get from sim ulating
the exact dynamics b y using 50000 Mon te Carlo rep eats. The
mean squared error is plotted as a function of the parameter g -
represen ting the strength of the couplings- for a system of N = 50
spins with fully symmetric couplings. W e ha v e used 100 time steps
and ha v e a v eraged the errors o v er 10 realizations of the couplings.
The error bars are standard deviations o v er these realizations. Dif-
feren t colors corresp ond to differen t metho ds to predict the magne-
tizations, some of them referring to P ap er 1: RH-T AP (green), MS-
MF (red), extended Gaussian a v erage (ligh t blue), extended Ple-
fk a (violet) and extended Plefk a with a v eraged momen ts (blac k).
The n um b er of sample tra jectories used in the algorithm for b oth
the extended Plefk a and extended Plefk a with a v eraged momen ts
metho ds is N T = 50000.
55

3 Dynamics on random net w orks
20 40 60 80 100
N
0
0.005
0.01
0.015
ε

Figure 3.2: Mean squared error of the Extended Plefk a metho d with a v eraged
momen ts (blac k), Extended Gaussian appro ximation (blue) and
Extended Plefk a metho d of P ap er 1 (magen ta) for predicting en tire
dynamics of magnetizations. The mean squared error is computed
for a fully symmetric net w ork with fixed coupling strength g = 1,
and plotted as a function of the system size N . W e ha v e used 100
time steps and 50,000 rep eats to calculate the exp erimen tal mag-
netizations and ha v e a v eraged the errors o v er 10 realizations of the
couplings. The error bars are standard deviations o v er these real-
izations. The n um b er of sample tra jectories used in the algorithm
for the algorithms based on the Extended Plefk a appro ximation is
N T = 50000. Stationary external field dra wn indep enden tly for
eac h spin from a normal distribution (zero mean, standard devia-
tion 0.5). The Extended Gaussian metho d sho ws no scaling with
N, a b eha viour that w e also observ ed at g = 0 . 5 and g = 3. F or
the error of the t w o algorithms based on the Extended Plefk a ap-
pro ximation, w e fitted a shifted p o w er la w to the data, to infer
the v alue of the asymptotic error for large net w orks. The fit seems
to indicate that the mean squared error go es to zero in the limit
N → ∞ in b oth cases, with exp onen t 1 . 8 ± 0 . 2 for the Extended
Plefk a metho d and 0 . 89 ± 0 . 15 for the algorithm with a v eraged
momen ts.
56

3.4 Conclusions
complex lo cal fields require a Mon te Carlo sim ulation to b e computed, whic h
mak es the metho d n umerically complex. A less complex v ersion of the Mon te
Carlo algorithm, where the second-order momen ts are replaced b y their self-
a v eraging v alue, p erforms equally w ell only for v ery large systems. The result
outp erforms the other metho ds in predicting the single site magnetization,
and the mean squared error of the predicted magnetization go es to zero with
increasing system size as a p o w er lo w. This result, together with the retriev al
of the exact equations for the fully asymmetric case, seems to indicate that
the appro ximation should b ecome exact in the thermo dynamic limit of a large
net w ork. A y ardstic k for comparison can b e the results of [EO94], where the
transien t zero-temp erature dynamics of an Ising mo del with arbitrary degree
of symmetry w as deriv ed for the disordered a v erage system. Our equation for
the magnetisation sho ws strong analogies with their result; as a next step, w e
in tend to p erform the prop er comparison b y a v eraging our mean-field solution
o v er the quenc hed couplings. The complexit y of p erforming this a v erage lies
in the nonlinear dep endence of the magnetisations on b oth the couplings and
the magnetisations at previous times, whic h in turn dep end on the couplings.
W e exp ect that the computation could b e treated using the replica trick.
The mean-field equations that w e discussed in this c hapter can b e also used
as inference to ols to compute the v alue of the couplings from data. A future
direction of the presen t w ork consists in designing a mean-field estimator for
the couplings starting from the extended Plefk a expansion; for example, one
could extend the approac h of [MS11, MS14], where a relation b et w een equal
times and one-time dela y ed correlation matrices is deriv ed, and then in v erted
to compute the couplings from correlations observ ed from data.
Along with dev eloping more accurate mean-field inference tec hniques, w e
find it imp ortan t to assess the qualit y of mean-field estimators against other
t yp es of estimators. In the next c hapter, w e carry out suc h analysis for fully
asymmetric net w orks, for whic h the exact mean-field equations are kno wn.
57

App endix
3.A T AP equations fo r the SK mo del
The T AP equations were deri v ed b y Thouless Anderson and P almer [T AP77]
in 1977 and pro vided the mean field solution to the Sherrington-Kirkpatric k
(SK) mo del of spin glasses [SK75].
H ( σ ) = − X
i<j
J ij σ i σ j − X
i
h ex
i σ i , (3.3)
where the couplings J ij are indep enden t Gaussian random v ariables with zero
mean and v ariance J 0 / N , and h ex
i are external lo cal fields. The T AP equations
consists in a set of mean field equations for the lo cal magnetization, v alid for
a giv en realization of the random couplings J ij :
m i = tanh( β ˜
h i ) + O (1 / N ) (3.4)
where
˜
h i = h ex
i + X
j
J ij m j − m i X
j
J ij χ j j ! (3.5)
= h ex
i + X
j
J ij m j − m i X
j
J 2
ij β (1 − m 2
j ) . (3.6)
The last equalit y follo ws from the definition of the susceptibilit y χ j j , whic h
represen ts the reaction of the magnetization m j to a small c hange of the field
˜
h j :
χ j j = ∂ m j
∂ ˜
h k
= β (1 − m 2
k ) . (3.7)
In the follo wing t w o sections, w e presen t t w o metho ds to deriv e these equations.
3.A.1 The cavit y app roach
The T AP equations for the lo cal magnetizations m i = h σ i i of the SK mo del
can b e deriv ed from the ca vit y [MPV87] metho d ( see [OS01, Nis01, O W01a]).
59

3 Dynamics on random net w orks
W e start b y deriving an set of appro ximate equations for the single site
marginal distribution of spins P i ( σ i ). The k ey observ ation is that the single
spin σ i dep ends on the other spins through the lo cal fields h i = P j J ij σ j , and
that the join t distribution of σ i and h i is
P ( σ i , h i ) ∝ e β σ i ( h i + h ex
i ) P ( h i \ σ i ) . (3.8)
P ( h i \ σ i ) is the distribution of the lo cal field h i in an auxiliary system of N − 1
spins, where the spin σ i has b een remo v ed b y setting J ij = 0 for j 6 = i . It is
called the ca vit y distribution, and can b e explicitly written as
P ( h i \ σ i ) ≡ X
σ \ σ i
δ ( h i − X
j
J ij σ j ) P ( σ \ σ i ) , (3.9)
where P ( σ \ σ i ) is the distribution of the spin configuration in a system where
the spin σ i has b een remo v ed. Hence, the marginal distribution of single spins
can b e found once the ca vit y distribution of lo cal field is sp ecified:
P i ( σ i ) ∝ Z dh i e β σ i ( h i + h ex
i ) P ( h i \ σ i ) . (3.10)
Let us consider the distribution (3.9). The SK mo del is a fully connected
system, and the n um b er of terms in the sum P j J ij σ j is N − 1. If all the
spins in the sum w ere indep enden t and iden tically distributed, the cen tral
limit theorem w ould tell us that the ca vit y distribution (3.9) is Gaussian. W e
assume that this is the case for the SK mo del, where correlations of differen t
sites σ j are w eak 4 :
P ( h i \ σ i ) ≈ 1
√ 2 π V i
exp  − ( h i − h h i i \ i ) 2
2 V i  , (3.11)
where h ... i \ i denotes a v erages with resp ect to the ca vit y distribution. By in-
serting this Gaussian distribution in (3.10) one finds the follo wing equation
for the single site magnetization
m i = tanh β h h i i \ i . (3.12)
W e now ha v e to ev aluate the exp ectations h h i i \ i and the v ariances V i .
The definition of full exp ectation
h h i i = X
σ i Z dh i h i P ( σ i , h i ) , (3.13)
60

3.A T AP equations for the SK mo del
together with (3.8) and the Gaussian ca vit y field (3.11) yields
h h i i = h h i i \ i + V i h σ i i . (3.14)
The v ariance of the cavit y field is defined as
V i = X
ij
J ij J ik ( h σ j σ k i \ i − h σ j i \ i h σ k i \ i ) . (3.15)
It can b e sho wn [MPV87] that, in the ab o v e equation, only diagonal terms
j = k con tribute to the sum; w e do not pro v e the v alidit y of this result, but
only men tion that it is due to the indep endence of J ij and J ik ( i 6 = k ) and to
the prop ert y called clustering
lim
N →∞
1
N 2 ( h σ j σ k i−h σ j ih σ k i ) = 0 , (3.16)
that w as sho wn to b e v alid when the temp erature is sufficiently high and there
is only one solution to (3.4-3.6). W e finally get
V i ≈ X
j
J 2
ij (1 − h σ j i 2
\ i ) ≈ X
j
J 2
ij (1 − h σ j i 2 ) = X
j
J 2
ij (1 − m 2
j ) , (3.17)
b y assuming that the a v erage in the original system and the a v erage in the
auxiliary system where one spin has b een remo v ed are appro ximately the same.
F rom (3.12), (3.14) and (3.17) one gets the T AP equations (3.6).
3.A.2 Plefk a’s expansion
Another metho d to deriv e T AP equations is the Plefk a [Ple82] expansion. It
consists in a w eak coupling expansion of the free energy at fixed magnetiza-
tions, whic h are constrained through a Lagrange transform (i.e., the Gibbs
p oten tial):
− β G α ( β , m ) = E xtr h log T r e − β H α − β X
i
h i m i ! , (3.18)
where
H α = − α X
i<j
J ij σ i σ j − X
i
σ i ( h ex
i + h i ) . (3.19)
4 F or a more detailed discussion of the v alidit y of the mentioned appro ximation, see
[MPV87].
61

3 Dynamics on random net w orks
the parameter α is in tro duced to con trol the strength of the couplings, α = 0
corresp onding to the non-in teracting system, while a α = 1 to the the SK
mo del. The auxiliary fields h i are Lagrange m ultipliers that enforce the con-
strain t
m i = h σ i i α , (3.20)
where h ... i α is the exp ectation with resp ect to the Hamiltonian (3.19). These
auxiliary fields are to b e considered as functions of the momen ts, according to
the follo wing set of equations
h i [ m ] = ∂ G α
∂ m i
, (3.21)
where w e ha v e used the condition (3.20). Note that when the auxiliary fields h i
are set to zero, the Gibbs free energy (3.18) is equiv alen t to the unconstrained
equilibrium free energy , so that the condition for the equilibrium magnetization
is ∂ G α
∂ m i
= 0 . (3.22)
The idea is to expand (3.18) in p o w ers of α and set α = 1 at the end of the
calculations to reco v er the result for the SK mo del:
G α = G 0 + ∂ G
∂ α     α =0
+ 1
2
∂ 2 G
∂ α 2     α =0
α 2 + O ( α 3 ) . (3.23)
Let us compute it term b y term. The Gibbs p oten tial of the non-in teracting
Ising system is computed to
β G 0 = X
i
h ex
i m i + 1
2 X
i
[(1 + m i ) log 1
2 (1 + m i ) + (1 − m i ) log 1
2 (1 − m i )] , (3.24)
while the first and second deriv atives giv e, resp ectiv ely ,
∂ G
∂ α = h H int i α ,
∂ 2 G
∂ α 2 = − β * H in t H in t − h H int i α − X
i
∂ h
∂ α ( σ i − m i ) !+ α
,
(3.25)
where H in t = ∂ H /∂ α is the in teracting part of the Hamiltonian. By ev aluating
the ab o v e equations at α = 0 w e find
∂ G
∂ α     α =0
= − 1
2 X
i 6 = j
J ij m i m j ,
∂ 2 G
∂ α 2     α =0
= − 1
2 β X
i 6 = j
J 2
ij (1 − m 2
i )(1 − m 2
j ) ,
(3.26)
62

3.B Mean field approac hes
where w e ha v e used the condition (3.20) and where the latter equation follo ws
from the relation:
∂ h i
∂ α = ∂
∂ m i
∂ G
∂ α     α =0
= − X
j,j 6 = i
J ij m j , (3.27)
whic h, in turn, follo ws from (3.21). Inserting (3.24) and (3.26) in (3.23) w e
get the final equation for the Gibbs p oten tial:
β G α ( β , m ) = X
i
h ex
i m i + 1
2 X
i
[(1 + m i ) log 1
2 (1 + m i ) + (1 − m i ) log 1
2 (1 − m i )]
− β α
2 X
i 6 = j
J ij m i m j −  β α
2  2 X
i 6 = j
J 2
ij (1 − m 2
i )(1 − m 2
j ) + O ( α 3 ) .
(3.28)
F or α = 1, if terms O ( α 3 ) are neglected, the T AP equations for the mag-
netization are then reco v ered b y extremization of (3.28) with resp ect to m i ,
according to (3.22). In [Ple82], Plefk a sho ws that these higher order terms in
(3.28) can b e neglected in the N → ∞ limit, as long as the system is not in
the spin glass phase.
3.B Mean field app roaches to the kinetic Ising
mo del: p revious results
Let us no w review the first generalization to dynamics of Plefk a’s expansion,
as prop osed in [RH11b]. W e consider the kinetic Ising mo del in tro duced in
section 1.2, comp osed of N Ising spins s i ( t ) in teracting through couplings J ij
and with lo cal external fields h i ( t ). It ev olv es in time according to a Glaub er
dynamics with parallel up date rule. The distribution of spin tra jectories has
the follo wing Mark o vian form
P( σ 0: T ) =
T − 1
Y
t =0
P( σ ( t + 1) | σ ( t ))P ( σ (0)) , (3.29)
where the transition probabilit y is giv en b y
P( σ ( t + 1) | σ ( t )) =
N
Y
i =1
exp σ i ( t + 1)[ h ex
i ( t ) + P N
j =1 J ij σ j ( t )]
2 cosh[ h ex
i ( t ) + P N
j =1 J ij σ j ( t )] . (3.30)
63

3 Dynamics on random net w orks
Standard field theoretical manipulations [ZJ02] lead to define the follo wing
Martin-Siggia-Rose generating functional [MSR73, DDP78]:
Z α [ ψ , h ] = Z Y
i,t
( dg i ( t ) d ˆ g i ( t )) T r σ Y
it
exp ( i ˆ g i ( t ) " h i ( t ) − α X
j
J ij σ j ( t ) #
+ σ i ( t + 1) g i ( t ) − log 2 cosh g i ( t ) − h i ( t ) ˆ g i ( t ) + ψ i ( t ) σ i ( t ) } (3.31)
where α is in tro duced to con trol the coupling strength. By deriv ativ es of the
generating functional w e find the momen ts
− i ˆ µ i ( t ) .
= ∂ log Z α [ ψ , h ]
∂ H i ( t ) = − i h ˆ g i ( t ) i α (3.32)
µ i ( t ) .
= ∂ log Z α [ ψ , h ]
∂ ψ i ( t ) = h σ ( t ) i α (3.33)
where h . . . i α denotes exp ectation under the measure inside the in tegral in
(3.31). Note that b y taking the limits ψ → 0 , h → h ex and setting α = 1 at
the end of the calculation, (3.33) reduces to the magnetization
m i ( t ) .
= h σ ( t ) i P = lim
ψ → 0 µ i ( t ) (3.34)
a v eraged o v er the distribution (3.29). In this framew ork, Hertz and Roudi de-
riv e a set of mean field equations b y extending the Plefk a expansion for the SK
mo del [Ple82] to the dynamical case. T o this end, they dra w a parallel b et w een
the logarithm of the generating functional and the Helmholtz free energy in
the equilibrium statistical mec hanics; accordingly , the Legendre transform of
log Z α corresp onds to the Gibbs free energy . While the original Plefk a expan-
sion consisted in a w eak coupling expansion of the Gibbs free energy , now the
Legendre transform of log Z α at fixed momen ts (3.32-3.33) is T a ylor expanded
in p o w ers of α , around α = 0. A t the end of the calculation one sets α = 1.
The Legendre transform of log Z with resp ect to the real and auxiliary fields
is
Γ α [ µ , ˆ
µ , ] = log Z α [ ψ , h , ] − X
it
ψ i ( t ) µ i ( t ) + ih i ( t ) ˆ µ i ( t ) (3.35)
where the fields ψ , h are functions of the momen ts (3.32-3.33) according to
the follo wing equations:
∂ Γ[ µ , ˆ
µ ]
∂ m i ( t ) = − ψ i ( t )
∂ Γ[ µ , ˆ
µ ]
∂ ˆ m i ( t ) = ih i ( t ) .
(3.36)
64

3.B Mean field approac hes
By expanding (3.35) in p o w ers of α and considering the equations (3.36) within
expansion, one finds the follo wing ’naiv e’ mean field equation
m i ( t + 1) = tanh " h ex
i ( t ) + X
j
J ij m j ( t ) # (3.37)
at the first order; a second order expansion yields T AP-lik e equations,
m i ( t + 1) = tanh " h ex
i ( t ) + X
j
J ij m j ( t ) − m i ( t + 1) J 2
ij [1 − m j ( t ) 2 ] # , (3.38)
that should b e solv ed self-consistenly for m i ( t + 1) at eac h time step. In this
form ulation, the generating functional (3.31) is defined in terms of fields that
act linearly on the degrees of freedom and the Legendre transform (3.35) is
p erformed b y fixing the first order statistics o v er time. In P ap er 1 w e will
observ e that, in con trast to the equilibrium case, the mean field description is
considerably impro v ed if w e tak e in to accoun t also the second order statistics
-namely correlations and resp onses. In particular, w e will in tro duce these
quadratic terms in the generating functional, so that the second order momen ts
can b e easily found b y first deriv atives of the generating functional.
In the case of a completely asymmetric net w ork, where the couplings ha v e
v ariance scaling as 1 / N , the correlations b et w een spins at differen t times is
small [CS88] and the cen tral limit theorem can b e used to describ e the statistics
of lo cal fields. This the approac h follo w ed in [MS11], where the field acting on
eac h spin at site i and time-step t :
X
j
J ij σ j ( t − 1)
is treated as a Gaussian distributed field with mean
g i ( t − 1) = X
j
J ij m j ( t − 1) (3.39)
and v ariance
∆( t − 1) = X
j
(1 − m j ( t − 1) 2 ) . (3.40)
F rom the definition of the dynamics (3.29), one retriev es the follo wing equation
for the magnetization:
m i ( t ) = Z D x tanh h g i ( t − 1) + h ex
i ( t − 1) + x p ∆ i ( t − 1) i , (3.41)
65

3 Dynamics on random net w orks
where the in tegral is o v er the Gaussian noise D x = dx e − x 2 / 2 / (2 √ π ).
A comparison b et w een the appro ximations (3.38) and (3.41) is presen ted
in [SRMH12], where the limit of v alidit y of the t w o metho ds are compared.
An extension of the latter metho d to the case of arbitrary degree of symmetry
is deriv ed in [MS14]. The effectiv e lo cal fields are still assumed to b e Gaussian
distributed, but a non-zero co v ariance b et w een spins at differen t times is also
considered. The resulting magnetization is
m i ( t ) = Z D x tanh h g i ( t − 1) + h ex
i ( t − 1) + x p V ii ( t − 1 , t − 1) i , (3.42)
where g i ( t ) w as defined in (3.39), and
V ij ( t, s ) = h δ g i ( t ) δ g j ( s ) i (3.43)
is the co v ariance of the field fluctuations δ g i ( t ) = P j J ij δ σ j ( t − 1),where δ s i ( t −
1) = s i ( t ) − m i ( t ). Starting from ca vit y argumen ts, the authors analytically
deriv e a set of recursiv e equations for calculating co v ariances at differen t times
in terms of co v ariances at previous times. F or time indices s ≤ t the result is
V ( t, s ) = J > A ( t − 1) V ( t − 1 , s ) + J > C ( t − 1 , s − 1) J , (3.44)
where C ij ( t, s ) = δ ij h δ s i ( t ) δ s i ( s ) i is the auto-co v ariance function and
A ij ( t ) = δ ij Z D x h 1 − tanh 2  g i ( t ) + h ex
i ( t ) + x p V ii ( t, t ) i .
Note that to compute the magnetization (3.42) one needs V ii ( t − 1 , t − 1), whic h
is obtained from (3.44) in terms of past v alues of V and m , and in terms of
the auto-co v ariance function at t w o successiv e time steps; the latter quan tit y
is compute to b e
C ii ( t − 1 , t ) = Z dh i ( t ) Z dh i ( t − 1) p ( h i ( t ) , h i ( t − 1))
tanh [ h i ( t ) + H i ( t )] tanh [ h i ( t − 1) + h ex
i ( t − 1)] ,
(3.45)
where p ( h i ( t ) , h i ( t − 1)) is a biv ariate Gaussian distribution with mean v ector
( g i ( t ) , g i ( t − 1)) >
and co v ariance matrix giv en b y
 V ii ( t, t ) V ii ( t, t − 1) p V ii ( t, t ) V ii ( t − 1 , t − 1)
V ii ( t, t − 1) p V ii ( t, t ) V ii ( t − 1 , t − 1) V ii ( t − 1 , t − 1)  .
66

3.C Algorithm with a v eraged momen ts
3.C Algo rithm with averaged moments
Keeping the same notation, the algorithm of (P ap er 1, section 5) is mo dified
as follo ws.
• Initial condition: set s k
i (0) = 1 , i = 1 ...N , k = 1 ...N T .
• F or t = 1 ...T :
1. Dra w the spins at time t from
p ( s k
i ( t )) = e s k
i ( t ) g k
i ( t − 1)
2 cosh g k
i ( t − 1) , for i = 1 ...N , k = 1 ...N T ,
using the fields g k
i ( t − 1) calculated at the previous time step.
2. Compute correlations
C i ( t, t 0 ) = tanh[ g i ( t − 1)] s i ( t 0 ) , for t 0 = 1 ...t − 1 , i = 1 ...N ,
and their a v eraged v alue h C ( t, t 0 ) i .
= 1
N P i C i ( t, t 0 ).
3. Dra w the Gaussian random v ariable φ k
i ( t ) with mean
b
φ k
i ( t ) =
t − 1
X
r =0
a ( r ) φ k
i ( r ) , (3.46)
and co v ariance
E { [ φ k
i ( t ) − b
φ k
i ( t )] 2 } = 1 − g 2
N X
j
m 2
j ( t ) −
t − 1
X
r =0
a ( r ) h C ( t, r ) i , (3.47)
where the co efficien t { a (0) . . . a ( t − 1) } are computed from
t − 1
X
r =0
a ( r ) h C ( r , t 0 ) i = h C ( t, t 0 ) i , t 0 = 0 . . . t − 1 .
4. Compute the sample a v erages that will b e needed in (5):
s i ( t ) φ i ( t 0 ) = tanh[ g i ( t − 1)] φ i ( t 0 ) , for t 0 = 1 ...t − 1 , i = 1 ...N .
5. Compute h R ( t, t 0 ) i .
= 1
N P i R i ( t, t 0 ) , for t 0 = 1 ...t − 1 b y solving the
system of linear equations:
1
N X
i
s i ( t ) φ i ( t 0 ) =
t − 1
X
τ =1 h R ( t, τ ) i g 2 [ h C ( τ , t 0 ) i − 1
N X
j
m j ( τ ) m j ( t 0 )] .
67

3 Dynamics on random net w orks
6. Compute the fields
g k
i ( t ) = φ k
i ( t ) + X
j
( J ij m j ( t )
− g 2 1 − k 2
1 + k 2
t − 1
X
t 0 =0
J ij J j i h R ( t, t 0 ) i [ s k
i ( t 0 ) − m i ( t 0 )] ! + h i ( t ) ,
for i = 1 ...N , k = 1 ...N T .
7. Compute the magnetizations at time t + 1:
m i ( t + 1) = tanh[ g i ( t )] , for i = 1 ...N .
68

4 Lea rning in kinetic Ising mo dels
4.1 Intro duction
In Chapter 3 , w e discussed mean-field approac hes to the forw ard problem
for the kinetic Ising mo del. Giv en a sp ecific set of mo del parameters, w e
describ ed the time ev olution of system observ ables, suc h as magnetisations
and correlations.
Here, w e fo cus on the in v erse problem: based on a set of measuremen ts from
the system, w e w an t to infer the mo del parameters (i.e., couplings b et w een the
spins and external fields). The amoun t of information enco ded in the data will
affect the qualit y of parameter estimation and it is imp ortan t to quan tify ho w
the p erformance of the inference algorithm dep ends on the size of the data
set. This question is particularly relev an t in the con text of the new high-
throughput data collection tec hniques, where the n um b er of v ariables that
can b e sim ultaneously recorded is almost as large as the n um b er of p ossible
trials [A G16].
In this Chapter, w e assume to ha v e access to time series data of length T for a
system of N spins, sp ecifying the v alue of eac h spin at successiv e time p oin ts. A
widely used estimator for the parameters is the maxim um lik eliho o d estimator,
whic h con v erges in probabilit y to the true v alue of the parameters when the
size of the dataset (rescaled b y the size of the system) tends to infinit y , with the
lo w est p ossible asymptotic mean squared error [Cra16]. The lik eliho o d can b e
computed in p olynomial time in T and N , whic h mak es the computation m uc h
faster with resp ect to the equilibrium case (see section 5.A). Still, maxim um
lik eliho o d conditions m ust b e computed at ev ery step of the iteration, based
on the curren t v alue of the parameters, and the iteration can tak e a long time
to con v erge, also dep ending on the c hoice of initial conditions and learning
rate. A m uc h faster metho d is pro vided b y appro ximate tec hniques, suc h as
the mean-field metho ds discussed in c hapter 3.
The first fo cus of this c hapter is to analyse the theoretical p erformance of
estimators based on a mean-field appro ximation. In a mean-field framew ork,
inference in kinetic Ising mo dels w as initially studied based on data from their
non-equilibrium steady state. The T AP equations (3.6) for the magnetisa-
tion deriv ed at equilibrium for the SK mo del w ere argued to b e v alid for the
69

4 Learning in kinetic Ising mo dels
async hronously [KS00] and sync hronously [RH11b] up dated Glaub er dynam-
ics. Based on those equations, a linear relation b et w een the one-time-dela y ed
and equal-time correlation matrix, resp ectiv ely denoted b y D and C , can b e
found:
J = AD C − 1 , (4.1)
where the matrix A enco des for the details of the considered appro ximation
(App endix 4.C). If the true correlation matrices are replaced b y the empirical
ones, this relation pro vides a linear estimator for the couplings, whic h can
b e simply computed via a matrix in v ersion. More recen tly , these results ha v e
b een extended to transien t dynamics. In c hapter 3, w e sa w that the mean-field
description is particularly simple in the case of an asymmetric net w ork, where
a linear relation b et w een one-time-dela y ed and equal-time correlation matrices
pro vides the exact solution in the thermo dynamic limit. This relation can b e
easily in v erted to infer the couplings in the same form as (4.1), where no w the
correlation matrices dep end on time (see section 4.C). Ideally , correlations are
computed from m ultiple trials of spin tra jectories. Ho w ev er, it is hard to ha v e
access to suc h data, and a v erages o v er trials are replaced b y a v erages o v er
time [MS11]. Hence, the qualit y of the estimator will dep end on the length of
the observ ed spin tra jectories.
In this framew ork, w e aim to compute the error asso ciated with the linear
mean-field estimator and study ho w it scales with the length of the observ ed
tra jectories.
The theoretical framew ork for our analysis is giv en b y the statistical me-
c hanics of learning [EVdB01], where the phase space consists of the couplings
to b e inferred, while the spin v alues are considered to b e fixed observ ations.
W e w ork in the so called studen t-teac her scenario, where the data are gener-
ated indep enden tly from a teac her net w ork and a learning algorithm adapts
the couplings of a studen t net w ork as estimator for the teac her. The error as-
so ciated to the algorithm is giv en b y the a v erage mean squared error b et w een
the teac her coupling v ector and the studen t one, where a v erages are computed
b y using the replica metho d of statistical ph ysics [MPV87, Nis 0 1] 1 .
In this c hapter, w e extend the replica formalism used for learning of p ercep-
trons to the kinetic Ising mo del. In the large N limit, t w o-times correlations
can b e neglected in asymmetric net w orks, for whic h the memory of the system
1 Replica calculations ha v e b een widely applied to problems related to learning in feed-
forw ard neural net w orks [WRB93, SST92, OK96], follo wing the seminal w ork of Gard-
ner [Gar87, Gar88], who exploited it to compute the critical capacity of the p erceptron
with con tin uous synaptic w eights; more recen t applications also include comm unication
theory [GV05], compressed sensing [GBS09, R GF09, GS10, KMS + 12],matrix factoriza-
tion [KKM + 16] and and high-dimensional regression [A G16].
70

4.2 P ap er 2
is lost after one time step (see, e.g., [CS87]). This allo ws us to use a cen-
tral limit theorem argumen t to treat the probabilit y distribution underlying
the Mark o vian dynamics as the distribution of T indep enden t p erceptrons,
in the thermo dynamic limit of a large system. In eac h p erceptron (corre-
sp onding to eac h time step), the inputs are not indep enden t but spatially
correlated through the equal-time correlation matrix. Surprisingly , w e find
that the equal-time spatial correlation matrix has a non-negligible influence
on the estimation error. W e compute the statistics of this random correlation
matrix and obtain an explicit result for the estimation error as a function of
the gro wing length of observ ed tra jectories.
In section 4.3, w e study the p erformance of other t w o appro ximate algo-
rithms. First, within the class of estimators that minimize a lo cal cost function
whic h is quadratic in the couplings, w e consider the optimal one, minimising
the mean square error of estimated parameters. Then, w e turn to a Ba y esian
probabilistic form ulation. In tro ducing a prior distribution, the Ba y es optimal
estimator of the parameters is giv en b y their p osterior exp ectation. Since
computing p osterior a v erages exactly is in tractable, w e prop ose an analytic
appro ximation to the p osterior exp ectations based on ca vit y argumen ts and
design an efficien t algorithm to n umerically implemen t our solution. Finally ,
w e use an analogous formalism to the one dev elop ed in P ap er 2 to compute
the error of the Ba y es optimal estimator and compare it with the mean-field
and linear optimal estimators.
An in tro duction to the replica metho d, applied to the ph ysics of spin glasses
and to the problem of learning in neural net works is presen ted in section 4.A.1;
the general framew ork for the statistical mec hanics of learning is also briefly
explained. The deriv ation of the maxim um lik eliho o d estimator and of the
linear mean field estimator - b oth for the stationary and for the transien t
dynamics - is giv en in sections 4.B, 4.C, 4.D, resp ectiv ely .
4.2 P ap er 2.
Author’s con tribution : I p erformed the analytical and n umerical calcula-
tions, prepared the figures and con tributed to writing the pap er.
71

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Variational perturbation and extended Plefka approaches to dynamics on random networks: the case of
the kinetic Ising model
L Bachschmid-Romano, C Battistin, M Opper et al.
L1 regularization for reconstruction of a non-equilibrium ising model
Hong Li Zeng, John Hertz and Yasser Roudi
Belief propagation and replicas for inference and learning in a kinetic Ising model with hidden
spins
C Battistin, J Hertz, J Tyrcha et al.
A statistical physics approach for the analysis of machine learning algorithms on realdata
Dörthe Malzahn and Manfred Opper
Dynamics in the Griffiths phase of the diluted Ising ferromagnet
A Mozeika and A C C Coolen
Coupled dynamics in the XY spin glass
G Jongen, J Anemüller, D Bollé et al.
Statistical mechanics of learning in the presence of outliers
Rainer Dietrich and Manfred Opper

J. St at. Mec h. (20 1 5) P090 1 6
Learning of couplings for random
asymmetric kinetic Ising models
revisited: random correlation matrices
and learning curves
Ludovica Bachschmid-Romano and Manfred Opper
Department of Artificial Intelligence, Technische Universit ä t Berlin,
Marchstra ß e 23, Berlin 10587, Germany
E-mail: [email protected] and
manfred.opper@tu- berlin.de
Received 13 March 2015
Accepted for publication 21 July 2015
Published 16 September 2015
Online at stacks.iop.org/JSTAT/2015/P09016
doi:10.1088/1742-5468/2015/09/P09016
Abstract. We study analytically the performance of a recently proposed
algorithm for learning the couplings of a random asymmetric kinetic Ising
model from finite length trajectories of the spin dynamics. Our analysis shows
the importance of the nontrivial equal time correlations between spins induced
by the dynamics for the speed of learning. These correlations become more
important as the spin ’ s stochasticity is decreased. We also analyse the deviation
of the estimation error
Keywords: network reconstruction, learning theory, statistical inference,
kinetic Ising model

L Bachschmid-Romano and M Opper
Learning of couplings for random asymmetric kinetic Ising models re visited
Printed in the UK
P09016
JSMTC6
© 2015 IOP Publishing Ltd and SISSA Medialab srl
2015
15
J. Stat. Mech.
JST A T
1742-5468
10.1088/1742-5468/2015/09/P09016
PaPers
9
Journal of Statistical Mechanics: Theory and Experiment

© 2 0 1 5 IOP Publ ishin g Ltd a nd sI ssa Me di ala b srl
ournal of Statistical Mech anics:

J

Theory and Experiment

IOP

174 2 - 5 4 6 8/ 1 5/ P 0 9 0 1 6 +1 9 $ 3 3 . 0 0

Learning of couplings for random asymmetric kinetic Ising models revisited
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doi: 10.1088/1742-5468/2015/09/P09016
J. St at. Mec h. (20 1 5) P090 1 6
1. Introduction
Recently, the learning of synaptic couplings for a recurrent neural network modelled by
a kinetic Ising model with random couplings has attracted attention in the statistical
physics community, see e.g. [ 1 – 10 ]. The model is defined by a system of N Ising spins

σ
i

connected through couplings J ij . We assume throughout the paper that the interactions
are non – symmetric, i.e. we have

≠ JJ

ij ji and J ii = 0. The system evolves in discrete time
according to a synchronous parallel dynamics, where spins at time t + 1 are updated
independently with transition probability (specialised on the case of no external fields)
∣ σσ βσ
{− }= ∑ −

βσ σ

=
∑

−
P

tt Jt
(( )( 1) ) e
2c osh(

(1

))

.

ij j
N
tJ t
j
ij j
1
() (1 )
i
j
ij j
(1)
We are interested in learning the spin couplings J ij , assuming that a complete trajec-
tory

σ σ {} ={ }

=… =…
t ()
T i iN

tT

0: 1, ,, 1, , of length T for all spins is observed. A well known
solution to this problem is given by the method of maximum likelihood, which leads
Contents
1. Introduction 2
2. Estimators 3
3. Learning curves from the replica approach 5
4. Statistics of correlation matrices 8
5. Results 11
6. Outlook 14
Acknowledgments 14
Appendix A. Details of the replica calculation of the free energy 15
Appendix B. Derivation of the generating function 16
Appendix C. Independence of the
J
and
B
(
t
) matrices: an example 16
Appendix D. Pad è Approximant 17
Appendix E. Details on the statistics of the correlation matrix 17
Appendix F. Asymptotic order parameters for ML estimator 18
References 18

Learning of couplings for random asymmetric kinetic Ising models revisited
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to a set of coupled nonlinear equations which have to be solved by iteration. A com-
putationally much simpler and elegant solution valid for large networks with random
couplings which avoids an iterative solution was recently presented in [ 1 ]. This solu-
tion is based on an exact mean field (EMF) expression for spin correlations which can
be explicitly solved for the couplings. The EMF estimator replaces exact correlations
by empirical correlations which can e.g. be computed from a single spin trajectory.
Simulations have shown good agreement between true and estimated couplings [ 1 ].
Of course, if there is only a limited number of observations available there will be
a nonzero estimation error for the EMF method. One may then ask how much one
has to pay for the numerical efficiency of the algorithm in terms of a loss in statistical
efficiency. Hence, we would like to investigate at what rate the error decreases with
growing length of trajectories and if the decrease is slower than that of a statistically
efficient estimator such as the maximum likelihood estimator which has an optimal
asymptotic rate [ 11 ]. Using the replica method we will compute the estimation error of
the EMF method in the thermodynamic limit

→∞ N

assuming that the data are gener-
ated from a kinetic Ising model with true couplings drawn at random from a Gaussian
distribution. The analysis of the statistical properties is significantly simplified by the
fact that kinetic Ising models with non – symmetric random couplings have spin correla-
tions which decay after a single time step (see for example [ 12 ]) and computations of
learning curves resemble those for temporally independent data. A nontrivial aspect
however is the occurrence of equal time spin correlations of the spin dynamics. We
compute an exact result for the statistics of the random correlation matrix. From this it
is possible to obtain an explicit expression for the learning curve for the EMF algorithm
and the asymptotics of the ML estimator.
2. Estimators
The EMF estimator [ 1 ] is based on a linear relation between the time-delayed and the
equal time correlator matrices,

δσ δσ δσ δσ == + Ct

tD

tt

() ()

,(

1) ()

,

ij ij ij

ij
⟨⟩ ⟨⟩

(2)
for the spin fluctuations

 δ σσ −

tt

mt

() ()

()

jj j , where m j ( t ) denotes the local magnetisa-
tion at time t and the brackets

… ⟨ ⟩

denote expectation with respect to the spin dynam-
ics ( 1 ). Here we assume stationarity for which the matrices are time independent. If the
couplings J ij are assumed to be mutually independent Gaussian random variables, with
zero meand and variance 1/ N , the following mean field relation is found to be exact in
the thermodynamic limit

→∞ N

:

∑

= Da JC

,

ij i

k

ik kj
(3)
where
D
∫ ∑
ββ = 

 − 

 +∆


 

 ∆= −
()

a

xH

xJ

m 1t an h, (1

)
ii

i
j
ij j
2e xt

22

(4)

Learning of couplings for random asymmetric kinetic Ising models revisited
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and

D x

is the normal Gaussian measure. Throughout the paper we will specialise to
the case of zero external field and vanishing initial magnetisations. In this case we have
m i ( t ) = 0,

= H 0

ex t ,

∆ = 1
i

and
D
∫

ββ

== −

a

ax x [1 tanh

() ]

i 2
(5)
is independent of time. For the estimator the exact correlation matrices

C

and

D

are
approximated by empirical averages using a long trajectory of spins (assuming zero
magnetisations):

∑∑

σσ

σσ

→= →= +

==
CC

T tt DD T

tt

ˆ 1 () () , ˆ 1 (1 )(

).

ij ij

t

T
ij ij ij

t

T

ij
11

(6)
One can then obtain the couplings by inverting ( 3 ) as follows:
∑
= −

J

a DC
1 ˆ ˆ

.

ij

k

ik kj
1
(7)
It is easy to see that the EMF estimator can be rephrased as the minimiser of the
following cost function
∑∑

σσ

=







−−








=
ta Jt E 1
2 ()

(1

)
t
T
i
j
ij j
MF
i
1
2
(8)
with respect to the couplings

{}

=
J ij j
N
1 . Note that the estimation of the ingoing couplings

{}

=
J ij j
N
1 for each spin i can be treated separately for the coupling distribution we are con-
sidering. The EMF estimator is based on simple explicit computation (inversion of the
correlation matrix in ( 7 ), which is possible if the parameter

α = TN

/ is grater than 1)
which makes the method fast. Other estimators such as the well known maximum
likelihood method (ML) have to resort to numerical optimisations using iterative algo-
rithms which could become computationally involved for large system sizes N and a
large number of data T . The ML estimator maximises the probability of spin histories

σ {} T 0:

given by
∏∏
σ σσ σ {} ={ −}

==

=
J

P

Pt

tP

() (( )( 1) )(

(0)),

T

i

N

t

T
ij j
N
0:

11

1
∣∣
(9)
where

σ P

(( 0)

)

is the initial probability of spins. Since this probability factorises in the
spins i and J ij are assumed independent, the ML estimator for all couplings {}
=
J ij j
N

1

pointing into spin i minimises the cost function
∑∑ ∑
βσ

σβ

σ =









−− +





 −




















=
tJ

tJ

t E () (1 )l n2 cosh

(1 ).

t
T
i
j
ij j
j
ij j
ML
i
1
(10)
While minimizing the cost function ( 8 ) just requires the computation of the empiri-
cal averages

C

ˆ and

D

ˆ , in order to minimize ( 10 ) with respect to J ij one needs to com-
pute the quantity

σβ σ
∑∑
tJ

t () tanh

(( ))

t j j ij j that explicitely depends on the current

Learning of couplings for random asymmetric kinetic Ising models revisited
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value of J ij and has to be recomputed at each step of the algorithm, adding a

⋅ NT

step
opera tion to the calculation. We observe that in order to avoid second order methods in
the solution we need a fine tuning of the step size which makes the algorithm fairly slow
for large N . Although it is more computationally expensive, the ML estimator has the
important property that it is asymptotically (i.e. for

→∞ T

) efficient . This means that
the asymptotic convergence of the mean squared estimation error to zero (assuming the
model is correct) happens at a rate which is minimal for any (asymptotically) unbiased
estimator [ 11 ]. In the following we will compute the error of the EMF algorithm in
the thermodynamic limit

→∞ NT

, , keeping α fixed and compare with the asymptotic

α →∞

optimal error rate of the ML estimator.
3. Learning curves from the replica approach
In this section we will introduce the replica method for computing the EMF prediction
error as a function of the scaled number of observed data. We will work in a teacher –
student scenario [ 13 , 14 ], where the data are assumed to be generated at random from
the dynamics of a teacher network with random couplings

J *

ij . We will use the scaling
=

JW

N

**

/
ij ij and assume that the

W

*
ij are independent Gaussian random variables
with N ∼

W

* (0 ,1

).

ij We can treat the estimation of the ingoing couplings ≡{ } =
W W
* *
ij j
N

1

for each spin i separately. For the sake of simplicity, in the following we will drop the
index i and define

 WW
ji

j . The average square prediction error for any estimator of
the couplings given by

W

is defined as
*
 ε ρ =− =− + WW

N

Q
1 12

,

2
(11)
where we defined



ρ =⋅ =

−−

WW W

NQ

N
*

,.
11

2
(12)
The bar denotes an average over the spin trajectories

σ {} T 0:

generated with cou-
plings

W *

and over the teacher couplings. We will now analyse the performance of
algorithms which minimise a cost function of the type
E

∑∑
σσ ==

−
=
Et hh
N Wt (( ), ), 1

(1 ),

t
T
tt
j
jj
1
such as ( 8 ) and ( 10 ), on a random finite set of spin trajectories of size T . One can
compute average properties such as the order parameters ρ and Q by introducing an
auxiliary probability density of couplings,
= ν −
W q

Z

() 1

e,

W E

()

(13)
with a formal inverse ‘ temperature ’ parameter ν and the partition function

Learning of couplings for random asymmetric kinetic Ising models revisited
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∫
σ = ν −
W Z () de .
W E

()

(14)
For any ν , we can compute disorder averages of ‘ thermal averages ’ of variables such
as ρ and Q from the quenched average of the free energy per coupling, defined by

σσ

νν =− =−

∂
∂

−− −
→
−
FN Z

n NZ

lo g( ) lim lo g(

).

n
n 11 1
0
1
(15)
By taking finally the limit

ν →∞

(zero ‘ temperature ’ ), the probability density ( 13 )
concentrates at the minimum of W

E ()

and we can extract the desired order parameters.
To compute the average, we will make the following assumptions. While the spins

σ

t

()

i are
still treated as binary random variables, in computing expectations over

σ

t

()

j for

≠ ji

we
assume a central limit theorem to be valid for the fields h t as sums of a large number of
weakly dependent random variables. Hence, we consider only the second order statistics of
these variables and treat them as Gaussian random variables. For equal times the corre -
sponding Gaussian density would be N

σ {} =

≠ C p t (( )) (0

,)

j ji , where the stationary covari -
ance matrix

C

is a random matrix which itself depends on the random matrix of teacher
couplings

W *

of the entire network. For different times

≠ ′
t t

, dependencies between
spins

σ

t

()

j and

σ ′

t

()

k are neglected. This is in accordance with our previous assumptions
for

−>
′

tt

1 ∣ ∣

, but we need an extra argument to justify neglecting D jk giving the correla -
tions at times t and t + 1. In principle,

D

might enter the computation of order param -
eters as well. Equation ( 3 ) shows a relation between the

D

and

C

matrices involving the
teacher couplings linearly. The arguments presented later in section 4 indicate that for the
asymptotic random matrix calculations involving similar relations we can treat teacher
couplings and random matrices

C

as asymptotically independent. Hence, we argue that in
an expectation over teacher couplings the contributions due to

D

vanish. We will see later
that the statistical properties of the matrix

C

will enter the final result of the learning
curve through the self averaging moment 
−
−
C

C

Tr

N

1 1 1 . We will then show in section 4
how this and other moments can be computed. Thus we will include the average over the
teacher couplings

W

*
kj for

≠ ki

in the statistics of

C

, but we need to perform the average
over the teacher couplings ≡

WW
**
ji

j pointing to spin i explicitly. Finally, the dependencies
between random correlation matrices

C

at different times are also neglected for

→∞ N

.
This results in an effective statistical weight over spin histories given by
 ∫ ∏
σ
σ
σ























  ∑ − 

 
{}





















βσ σ
β
−⋅
=
∑ −
≠
W

P

Wt
pt () de e
2c os h( 1)
((

)) ,

WW
t
T tW t
N j
j j
j ji
1
2
1
() (1 )
i N j
j j
1
*
*
**
*
(16)
where the Gaussian measure accounts for our prior knowledge on the teacher couplings
distribution. Hence, for large N , we are effectively dealing with the statistical mechan-
ics of a learning problem for a binary classifier neural network (aka logistic regres sion),
where the ‘ input ’ data

σ −

t

(1 )

j are used to predict the ‘ outputs ’

σ

t

()

i ; the input variables
are independent for different t , but have nontrivial ‘ spatial ’ correlations given by the
matrix

C

. The calculation of the free energy follows the steps of replica calculations for

Learning of couplings for random asymmetric kinetic Ising models revisited
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perceptron learning problems [ 13 – 15 ]. Averages over

σ

t

()

j factorize over time and can
be expressed through Gaussian fields h a for each replicated coupling variable W a , and
fields

*

σ = ∑ −

uW

t

(1 )
N

j j j
1 for the teacher. Under the replica symmetry assumption,
which is plausible to be correct for convex cost functions, the covariances are expressed
by order parameters
∑

==

u N WC W
1

**

1,
ij
i ij j
2
(17)

∑
= hu N WC

WR

1 * ,
a
ij
i
a ij j
(18)

∑
= h N WC

Wq

1 ,
a
ij
i
a ij j
a 2 0
(19)

∑

=≠

hh N WC Wq

ab

1
ab
ij
i
a ij j
b
(20)
and the free energy ( 15 ) is computed as (appendix A ):
DD
D E
}
∫
∫
∑
ν
α
β
=−









−
− −− −
+ 
  




 −+ 



 
  
σ
βσ
νσ




 −+ 




−− +
F qR
qq qq N C
ty

ty

z
Extr 11
2
1
2 log( ) 1
2 Tr lo g
e
2c os h1
lo ge .
qR q
ty
R
q
R
q
qq zq y
,, 0 2
0
0
1
(, )
R
q R
q
0
0
0
2
2
00

(21)
The limit

ν →∞

will occur with

→ q q

0 , since the different solutions

W

have to
converge to the same minimum. In this limit, keeping the quantity

 ν − x qq ()

0 finite,
we finally get
DD
E
∫
∑ α
β
σ
=−















− + 
  




 −+ 



 
  

   −− + 
  

















σ
βσ 



 −+ 




F qR
x ty

ty

z xz qy
Extr 2
e
2c os h1
2 (, ).
qR xz
ty
R
q
R
q
,, ,
2 1
2
0
R
q R
q
0
0
2
2

(22)
Remarkably, the explicit dependence of F on the correlation matrix (last term in the
first line of equation ( 21 )) drops when taking the limit

ν →∞

. Hence, the result we get
for F and for the order parameters extremizing F is the same that we would get if the

Learning of couplings for random asymmetric kinetic Ising models revisited
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spins over which we are computing the expectations were independent and the matrix C
was not included in the calculation. Still, the correlation matrix affects the error through
the parameters ρ and Q defined in ( 11 ), which are found to be (appendix A )

ρ =

R ,
(23)
=+ − −
C QR qR

N

()
1 Tr ,
22 1
(24)
where R and q are the order parametrs extremizing the free energy ( 22 ). Inserting the
above equations in ( 11 ) we find the following result for the error:
ε =− ++ −





 −







−
C Rq qR

N

12 ()
1 Tr

1.
21

(25)
The last term represents the effect of the correlations of the data on the error and
vanishes when

C

equals the unit matrix. This term can be shown to be positive and
leads to an increase in error. In section 5 we will give explicit results for the error of
the EMF algorithm.
4. Statistics of correlation matrices
In this section we show how one can compute the stationary value of the negative inte-
ger moment of the spin correlations
≡
−

→∞ →∞

−
C

C N

t lim lim

1

Tr

() ,
tN

1 1
(26)
necessary for the estimation error ( 25 ). Here the bar denotes expectation with respect
to independent random Gaussian couplings with zero mean and variance 1/ N . Our
analysis begins with the time evolution for the correlation matrix C t

()

assuming zero
magnetisations m j ( t ) = 0. Following [ 1 ], we can assume that in the limit of large N
the random variables g i and g j , where

σ = ∑
gJ

t

()
i k

ik k , are zero mean Gaussian random
variables with

=

∑
gg JC

tJ

()
ij kl ik kl lj

⟨ ⟩

and = g

1
i

2

⟨ ⟩

. An expansion with respect to weak
correlations similar to equations (15) – (16) in [ 1 ] yields the time evolution


γ += +

CI JC J tt

at

(1 )(

)( ),

2
(27)
where

I

is the unit matrix,

= CI

(0 ) and

J

is the

× NN

coupling matrix. The self-
averaging quantity γ must be determined such that C ii ( t ) = 1 yielding the condition
that 
γ =− JC J ta t () 1T r( )
2 . Defining =
γ

B

C

tt

()

()

t
1
() and assuming

γγ +≈ tt

(1

)( )

one finds the simplified iteration



∑
+=

+=
=

B IJ BJ

BJ

J ta

tt

a (1 )( ), havin gt he solution ()

() .
k
t

kk k 2

0

2


(28)

Learning of couplings for random asymmetric kinetic Ising models revisited
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Note that in the limit of small β (small a ) one could choose to truncate the sum
in ( 28 ) to the first order in a (corresponding to k = 0) and thus approximatig

B

by
the unit matrix, or to keep the first two orders in a (up to k = 1) and thus getting
the sum of the unit matrix and a Wishart matrix. From the above equations we get
 γ γ
→ =
∞

+

t lim ()
t

a

1

1

2 . We can use ( 28 ) to derive an iteration for the generating func-
tion of integer moments. In the thermodynamic limit the calculation simplifies remark-
ably. Consider e.g. the computation of → +
∞ B t l im Tr

(1

)
N

N

k
1 for some integer k . One
would have to deal with terms of the form

 
JB JJ BJ JB J

N

tt t
1 Tr (( )(

)( )) .

(29)
Writing B t

()

as the sum in ( 28 ) one is left with a sum of averages involving only the

J

and


J

matrices. Given the Gaussian form of the

J

random matrix, Wick ’ s theorem
applies and the expectation in ( 29 ) can be computed using diagrammatic techniques.
As is well known [ 16 ], for

→∞ N

only the planar diagrams, i.e. the ones for which lines
are not crossing, will contribute to the limit. Besides, note that in the evaluation of ( 29 )
the terms containig J ... J and J ... J pairings will vanish because of the asymmetry
of the

J

matrix. It is easy to see (an example is given in appendix C ) that this implies
that also pairings of the kind B ( t ) ... J and B ( t ) .. . J are forbidden, where B ( t ) .. . J
is a shortcut to indicate the pairing between

J

and any of the

J

s contained in

B

t

()

.
Hence, in computing moments by iteration over time, we can formally treat

B

t

()

as
independent from

J

k . We will not pursue the diagrammatic approach further but use
this independence directly in the selfconsistent computation of the generating function
S ( x ) of the asymptotic integer moments. This is given by
∑
== −
∞

=
∞

→

Sx

Sx

xB

() lim () ()

,

t t

k

k k

0

where
 +

∞

−
→
B

Sx N

Ix t () lim 1 Tr

((

))

,

t

N

1

(30)
=
→∞ →∞
B

B N

t lim lim 1 Tr

()

,
k tN
k
(31)
Finally, from S ( x ) we can also deduce ( 26 )
γ
=
−
→∞

Cx

Sx

1

lim

() .

x
1
(32)
We use an expression for S t ( x ) based on the Gaussian ensemble of auxiliary
N -dimensional vectors

y

. This is defined by the partition function

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
 
∫
∫
∏
∏
=





 −+ +






= 


 −+ − 



+ yB y
yy yJ BJ y
Zx yI xt
yx
ax t
() de xp 1
2 (( 1))
de xp 1
2 (1 ) 2 () ,
t
i
i
i
i
1
2

(33)
from which the generating function is obtained as

=
+

→∞

+
yy

Sx N

() lim

1 ,

t

N

t
1 1
⟨⟩
(34)
where the brackets denote expectation wrt to ( 33 ). We compute the average over ran-
dom matrices

J

, using the fact that we can neglect the dependency between the ran-
dom matrices

J

and

B

t

()

in the partition function ( 33 ). An annealed average of ( 33 )
and the limit

→∞ t

(appendix B ) yields the self consistent equation
=

+
Sx x

Sa xS x () 1

1 (( )).

2
(35)
The explicit computation of moments is facilitated by introducing an auxiliary
function ϕ , its power series expansion (whose coefficients are denoted by M k ) and its
inverse by
∑
φ = −



 −




 =−
=

∞

x ax
ax
S x
ax

xx

M () (1

),

k
kk k
2
22 0
(36)
φ =






 +








ay Sy ay
y
() 1 .
2 2
(37)
From ( 30 ), ( 36 ) and taking the limit

→∞ y

in ( 37 ), we obtain
∑
γ φ γ
== −
−

=
∞
C

a

aa

M
1 () 1 () .

k

k k 1 2
2

0

2
(38)
We will next see how to obtain closed form expressions for the B k and M k recur-
sively. Let us first show that for known values of

… B

B ,,
n 1 , we can compute M n . From
( 35 ) and ( 36 ) we get the expression

φ φ =

xx

Sx

() ((

)).

(39)
Applying Lagrange ’ s inversion formula [ 17 ] to ( 39 ) one can express the coefficients
of the power series expansion of

φ

x

()

in terms of those of S :
∑
φφ

φφ

= −
+ {} = −
+


















 − 


















+
=
∞
M n S n B
(1 )
1 [] (( )) (1 )
1 [] (1

),

n
n
nn
n
n
k
kk k
n
1
0
(40)

Learning of couplings for random asymmetric kinetic Ising models revisited
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where

φ []
n

denotes the coefficient of

φ n

in a power series expansion of the mathematical
expression in the brackets

{…}

. Finally, we insert in ( 40 ) the expansion of S ( 30 ). One
can see that the coefficients are of the form

=+ …

−
MB fB B (,

,) ,

nn

n

n

11

(41)
where the functions f
n can be computed in closed form for any n with a computer
algebra programme such as Mathematica. To obtain a relation for B n , we expand both
sides of ( 36 ) into powers of y . Using elementary properties of binomial coefficients and
comparing coefficients of y n yields the second explicit relation

∑∑

= 







 =+ 









==
−
B

a n
l Ma Ma
n
l M

.

n

l

n
l l n n

l

n
l l

0

22

0

1
2
(42)
Hence, inserting ( 41 ) into ( 42 ), we obtain
∑
= −







 …+ 
















−
=
−

B

a af BB a n
l M
1
1 (, ,)

.

n n
n n n
l
n
l l
2
2 11
0
1
2
(43)
Unfortunately, the series ( 38 ) turns out to be an asymptotic one. Coefficients M n
diverge for

→∞ n

and one has to use a regularisation method such as the Borel summa-
tion or the Pad è approximation in order to extract a useful result out of a finite num-
ber of coefficients. We have resorted to the latter method (appendix D ). Our results
obtained in this way are in excellent agreement with simulations of the kinetic Ising
model for N = 200 and T = 1000. Figure 1 shows that for small values of a , i.e. small
β , the matrix

≈ CI

. For increasing β also

−

C

1

increases but remains finite. Note, that
for

β →∞

, the parameter a converges to the value π =

a

2/ .
5. Results
In the case of the EMF estimator ( 8 ) the free energy ( 22 ) becomes:
DD
∫
∑
α β
σ
=− 






− +



 −− −+ +− 












σ
βσ
(( ))
F qR
x uv u
z ax zR uq Rv
Extr 2
e
2c osh( )
2
1
2 .
qR xz
u
,, ,
2
2
0 2 2
0
0

(44)
Integration by part shows that D

∫

β = uu

ua

tanh( ) , thus the above equation
reduces to
α
=













− − + +−













F qR
xa x aq aR Extr 21 (1

2) ,

Rq x ,,
2
2

22

(45)
and the extremum conditions yield the following equations for the order parameters:

Learning of couplings for random asymmetric kinetic Ising models revisited
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= R 1

(46)
α
α
=

−+

−
q a
a
(2

)1
(1

)
2
2
(47)
α
=
−
x a
1

(1

) .
2
(48)
Inserting the above equations in ( 25 ) the error is computed as follows:
ε

α

=

−

−
−
a

aN

C
1

1

11
Tr

.

EM F
2
2
1
(49)
We defer a detailed analysis of the finite α performance of the ML estimator to a
future publication. Here we are interested in the leading behaviour of the decay of the
prediction error as

α →∞

. It is well known that ML estimators are asymptotically effi-
cient, i.e. the errors decay at an optimal speed. Hence, our asymptotic result should be
a yardstick that allows for a comparison of algorithms. The calculation in appendix F
shows that for large values of the α parameter this optimal error decays as


ε

βα
−
aN C
11
Tr

.

opt 1
(50)
Hence, for

α →∞

, we have

ε

ε
β
=
−
α →∞

a

a
lim (1 )

.

opt
EM F 2
(51)
For small β , i.e. large stochasticity of the spins, we have

 β a

and both algorithms
decay at the same rate. This can still be seen in figure 2 for

β = 1

, where the EMF
Figure 1. The analytic result(black line) for =
−
−

C N

C
1 Tr
1 1 is compared with the
values obtained from simulation (blue line) for N = 200 and T = 1000. Results are
averaged over 50 istances of the network and error bars are negligible.
0.1 0.2 0.3 0.4 0.5 0.6
a 2
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Tr C -1 /N

Learning of couplings for random asymmetric kinetic Ising models revisited
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algorithms performs close to optimal. For larger β , the spins behave more deterministi-
cally and as shown in figure 3 the EMF algorithm deviates significantly from optimal-
ity. We have also included data points from a simulation of a penalised ML estimator,
where we have minimised the cost function

+ E

WW

ML

2

numerically by a gradient
Figure 2. Mean squared error of the couplings inferred with the EMF method
(red dots) for a system of size N = 200 with

β = 1

. Results are averaged over 25
istances of the network. Error bars are negligible. The red line corresponds to the
replica result for the EMF prediction error, the blue line to the replica result for
the asymptotic optimal prediction error.

11

00
α
1

ε
β=1

Figure 3. Mean squared error of the couplings inferred with the EMF method
(red dots) for a system of size N = 200 with

β = 5

. Results are averaged over 25
instances of the network. The red line corresponds to the replica result for the EMF
prediction error, the blue line to the replica result for the optimal prediction error.
The blue dots are results from simulations of a penalised ML algorithm. Error bars
are negligible. For large values of α , the EMF method displays finite-size effects
(see the red dot at

α = 50

), which are stronger for larger β . The green dot takes
into accout finite-size corrections, and it is obtained as explained in figure 4 .
1 100
α
0.0001
0.01
1

ε

β=5

Learning of couplings for random asymmetric kinetic Ising models revisited
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descent algorithm. Note that the penalty term we chose is equivalent to the prior
and we are thus maximizing the log-posteror. One can see that this type of algorithm
achieves asymptotic optimality. Finally, with increasing β the ratio ( 51 ) decays to zero.
While the decay rate of the EMF algorithm converges to a nonzero value (note that
for

β →∞

, we have π →

a

2/ ), the optimal asymptotic error rate converges to zero
indicating a transition to a faster decay than

α 1/

in the limit. It is also interesting to
note that for larger β simulations of the EMF algorithms show strong finite size effects
in N and the error reaches a plateau for increasing α . Hence, we had to apply a finite
scaling for the last simulation point in figure 3 .
6. Outlook
It will be interesting to develop and study algorithms which include prior knowledge
about the couplings to be learnt. This could be done within a Bayesian approach where
a prior probability density over couplings is specified. In this way one may e.g. introduce
sparsity. Using a similar replica approach, one could compare the performance of differ -
ent algorithms to that of the Bayes estimator, which is optimal on average over teacher
networks drawn at random from the prior. A nontrivial question is that of an algorithmic
realisation of the Bayes predictor. We expect that cavity approaches (TAP equations)
could be applied to get a tractable approximation which becomes exact in the thermody -
namic limit. We also expect that one should include explicit knowledge of the statistics
of the spin correlations into such an approach in order to get optimal performance.
Acknowledgments
This work is supported by the Marie Curie Training Network NETADIS (FP7, grant
290038).
Figure 4. EMF prediction error for fixed

α = 50

and

β =

5 as a function of N .
Fitting a power law to the data we find the asymptotic value valid for large N ,
which corresponds to the green dot in figure 3 .
0.015
0.0 2
0.025
0.0 3
0.035
0.0 4
0.045
500 1000 1500 2000 2500 3000 3500 4000

ε

N

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Appendix A. Details of the replica calculation of the free energy
After some standard manipulations [ 13 – 15 ], the quenched free energy ( 15 ) is
computed as
DD
D E
∫
∫
∑
ν α
β
=−















+ 
  




 −+ 



 
  

















σ
βσ
νσ




 −+ 




−− +
FG Rq qt y

ty

z
Extr 1 (, ,) e
2c os h1
lo ge ,
qR q
ty
R
q
R
q
qq zq y
,, 0
1
(, )
R
q R
q
0
0
0
2
2
00

(A.1)
where G ( R , q , q 0 ) is the weight of the coupling vectors

W

which are constrained by the
order parameters:
=

∂
∂

→

GR

qq

nN

Z (, ,) lim 1 ln

,

n
0 0 coup
(A.2)
with
∫ ∏∏ ∑
∏∑ ∏∑
δ

δδ

= 




 − 











 − 











 − 






−⋅
<
WW

ZW

CW Nq
WC WN RW CW Nq
d * de *
.
WW
a
a
a ij
i
a ij j
a ij
i
a ij j
a
ab ij
i
a ij j
b
coup
1
2 ** 0

(A.3)
We can decouple the integrals over different spins by diagonalising


=Λ CU U

and
transforming to new variables

 → U WW
aa

,

 → U WW
**

which we give just the same
name:
∫ ∏∏ ∑
∏∑ ∏∑
δ

δδ

= 




 Λ−













 Λ−











 Λ−






−⋅
<
WW

ZW WN

q
WW NR WW Nq
d * de *
.
WW
a
a
a i
i
a i i
a i
i
a i i
a
ab i
i
a i i
b
coup
1
2 ** 0

(A.4)
The integration over the couplings and the auxiliary parameters gives rise to the
following equation for G :
=

−

−
−− −

GR

qq qR
qq qq N C (, ,) 1
2
1
2 log( ) 1
2 Tr lo

g.

0 0 2
0
0
(A.5)
In order to compute the parameters ρ and Q from the free energy F , we introduce
the auxiliary variables

ηη {}

,

12

in the partition function

Z

coup ( A.4 ) as follows:

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∫

∏∏

∏∏
= ∑
∑∑

ηη

−⋅





 Λ−












 Λ+ − 






<





 Λ+ − 






WW Zd qd Rd q d * d ˆ ˆ ˆ ee
ee .
WW
a
a
a
qW

WN

q
a
RW WN R
ab
qW WN q
coup 0
1
2 ** i ˆ *
i ˆ () i ˆ ()
i
i
a i i
i
i
a i i
a
i
i
a i i
b

00
12

(A.6)
By derivatives with respect to

ηη {}

,

12

and taking the limit

η η →→

0, 0

12

one recov-
ers ( 24 ).
Appendix B. Derivation of the generating function
For a Gaussian model without external field we have

=

y

0
i
⟨ ⟩

, hence = ∑ =

q

y 0

N

i i
1 2
⟨⟩
and there is no need to introduce replicas, (absence of spin – glass ordering) and we can
restrict ourselves to an annealed average. Decoupling the quadratic form in the expo-
nent of ( 33 ) using correlated Gaussian random vectors with covariance

 =

zz B t

()
c
⟨ ⟩

,
we get





∫
∫
∫
∫
∏
= 
   −+ 
  




 − 





∝ 

   −+


  




 −
∝ 
   −+

   +
= 



  −+ −+ 



 
+
∞ +
∞ + −
∞ +
∣∣
yy zz yy
zz
B
B
Zx yx ax
N
ss N xs ax
N s
ss N xs Ia xs t
ss N xs Ia xs t
() de xp 1
2 (1 )e xp 2 ()

()

de xp 2 (1 )e xp 2 ()
de xp 2 (1 )( )
de xp 2 (1 ) 1
2 Tr ln (( )) ,
t
i
i
z
N
z
N
N
1
2
0
1
2
2
0
1
2 2 1/2
0
1
2 2

(B.1)
where in the second line we have introduced polar coordinates 
=

yy s N

1 . We
compute the final integral for

→∞ N

by Laplace ’ s method, and use the fact that
from ( 34 ) the maximiser of the integral gives 

==

+
yy

sS

x

()
N

t
1 1
⟨⟩ . Finally from

−

+= + B Ia xs

tZ

ax s Tr ln (( )) cons tl n(

)

t
1

2
22

we get the recursion
=

+
++
Sx x

Sa xS x () 1

1 (( )).
tt

t 1 2 1
(B.2)
Taking the limit

→∞ t

yields ( 35 ).
Appendix C. Independence of the
J
and
B
(
t
) matrices: an example
To better illustrate the independence of the

J

and

B

t

()

matrices, let us give an exam-
ple and consider the evaluation of one of the terms needed for the computation of
→ +
∞ B t l im Tr

(1

)
N

N

k
1 (see ( 29 )):

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

JB JJ BJ

N tt

1 Tr (( )(

)) .

(C.1)
The only sets of contractions giving nonzero contribution in the large N limit are
the following two:
1
N Tr ( J B ( t ) J J B ( t ) J )= 1
N Tr B ( t ) 2 ,
1
N Tr ( JB ( t ) J JB ( t ) J )= 1
N Tr ( B ( t ) 2 . )
(C.2)
The contractions involving the pairing of a

J

with a

B

t

()

vanish, since they involve
either J ... J ( J ... J ) pairings or crossing lines (resulting in non planar diagrams),
as shown in the two examples below:
1
N Tr ( JB ( t ) J JB ( t ) J )=0 , 1
N Tr ( JB ( t ) J JB ( t ) J

)=0

.
(C.3)
Appendix D. Pad è Approximant
The so called Pad è approximant [ 18 ], is a rational function (of a specified order) whose
power series expansion agrees with a given power series to the highest possible order.
Given a rational function of the form

∑∑

≡







 +









==

Rx ax bx ()

1,

k
M
k k
k
N
k k

01

(D.1)
then R is said to be the Pad è approximant to the series
∑
=

=
∞

f

xc

x ()

k

k k

0

(D.2)
if the following set equations is satisfied:
=

Rf

(0

)(

0)
(D.3)

== …+

==
x Rx x fx

kM

N
d
d () d
d () 1,

,,

k
k x
k
k x 00
(D.4)
which gives M + N + 1 equations for the unknowns

… a a ,,

M 0 and

… bb

,,
N 0 .
Appendix E. Details on the statistics of the correlation matrix
The iterative methods explained is section 4 allows us to calculate the moments B k and
M k , defined respectively in ( 31 ) and ( 36 ), for any given k . As an example, in the follow-
ing we will enumerate the first three moments.

Learning of couplings for random asymmetric kinetic Ising models revisited
18
doi: 10.1088/1742-5468/2015/09/P09016
J. St at. Mec h. (20 1 5) P090 1 6
=−

−
Ba

(1 )
1

21

(E.1)
=− −

−−

Ba a (1 )(

1)

2 41

22

(E.2)
=+ −− −
−− −
Ba aa a (1 2 )(1 )( 1) (1 )
3 46 14

12

3
(E.3)
=−
−

Ma

(1 )
1

21

(E.4)
=−

−−
−−

Ma

aa

(2 )(1 )(

1)

2 42

24

1
(E.5)
=+ −+ −− ++

−−

Ma aa aa

aa

(5 4 )(1 )( 1) (1 ).
3 46 10 24 41

24

(E.6)
Appendix F. Asymptotic order parameters for ML estimator
The free energy for the ML estimator is given by
DD
∫
∑ α
β
βσ β
=−















− + 
  




 −+ 



 
  
−+ +− +

















σ
βσ 



 −+ 




F qR
x ty

ty

z xz qy xz qy
Extr 2
e
2c os h1
[ 2 () lo g2c osh[

()

]] .
qR xz
ty
R
q
R
q
,, ,
2 1
2
R
q R
q
2
2

(F.1)
It is possible to show that for

α →∞

one can assume that

−→ q

R

0

2 ,

→ x 0

and

→ q 1

. Expanding the α dependent part of ( F.1 ) for small

x

, solving for z and finally
taking the limit →

q

R 2 , we obtain
 D
∫
α β β −













− + 



 ++ 

















F qR
x
ax Rb

yq

y Extr 22 lo g2c osh(

).

qR x ,,
2
(F.2)
This yields the following asymptotic scaling of order parameters:
 

αα

− Rx

b

qR

b

1, 1 , 1

.

2
(F.3)
Inserting the above expressions in the definition ( 25 ) one obtains ( 50 ).
References
[1] M é zard M and Sakellariou J 2011 Exact mean-field inference in asymmetric kinetic Ising systems J. Stat.
Mech. L07001
[2] Roudi Y and Hertz J 2011 Mean field theory for nonequilibrium network reconstruction Phys. Rev. Lett.
106 048702

Learning of couplings for random asymmetric kinetic Ising models revisited
19
doi: 10.1088/1742-5468/2015/09/P09016
J. St at. Mec h. (20 1 5) P090 1 6
[3] Roudi Y and Hertz J 2011 Dynamical TAP equations for non-equilibrium Ising spin glasses J. Stat. Mech.
P03031
[4] Aurell E and Mahmoudi H 2012 Dynamic mean-field and cavity methods for diluted Ising systems Phys.
Rev. E 85 031119
[5] Huang H and Kabashima Y 2013 Dynamics of asymmetric kinetic Ising systems revisited arXiv: 13105003
[6] Dunn B and Roudi Y 2013 Learning and inference in a nonequilibrium Ising model with hidden nodes Phys.
Rev. E 87 022127
[7] Tyrcha J and Hertz J 2014 Network inference with hidden units Math. Biosci. Eng. 11 149
[8] Bachschmid-Romano L and Opper M 2014 Inferring hidden states in a random kinetic Ising model: replica
analysis J. Stat. Mech. P06013
[9] Mahmoudi H and Saad D 2014 Generalized mean field approximation for parallel dynamics of the Ising
model J. Stat. Mech. P07001
[10] Battistin C, Hertz J, Tyrcha J and Roudi Y 2015 Belief propagation and replicas for inference and learning
in a kinetic Ising model with hidden spins J. Stat. Mech. P05021
[11] Schervish M J 1995 Theory of Statistics ( Springer Series in Statistics ) (New York: Springer)
[12] Eissfeller H and Opper M 1994 Mean-field Monte Carlo approach to the Sherrington-Kirkpatrick model with
asymmetric couplings Phys. Rev. E 50 709 – 20
[13] Engel A and Van den Broeck C 2001 Statistical Mechanics of Learning (Cambridge: Cambridge University
Press)
[14] Opper M and Kinzel W 1996 Statistical mechanics of generalization Models of Neural Networks III
ed E Domany et al (New York: Springer)
[15] Nishimori H 2001 Statistical Physics of Spin Glasses and Information Processing: an Introduction (Oxford:
Oxford University Press)
[16] Hooft G 1974 A planar diagram theory for strong interactions Nucl. Phys. B 72 461 – 73
[17] Wilf H S 2006 Generatingfunctionology (Wellesley, MA: A K Peters)
[18] Press W, Teukolsky S, Vetterling W and Flannery B 2007 Numerical Recipes (Cambridge: Cambridge
University Press)

4 Learning in kinetic Ising mo dels
ERRA T A CORRIGE: Equation (22) should b e replaced b y
F = − Extr q ,R,x 


q − R 2
2 x + α X
σ 0 Z D t D y e β σ 0 ( q 1 − R 2
q t + R
√ q y )
2 cosh[ β ( q 1 − R 2
q t + R
√ q y )]
Max z  − z 2
2 + E ( σ 0 , √ xz + √ q y )  .
4.3 F urther results
4.3.1 The optimal linea r estimato r
W e discuss the p erformance of another estimator, obtained b y minimizing a
cost function of the same form as the one of the linear mean-field estimator:
E i = 1
2
T
X
t =1 σ i ( t ) − a X
j
J ij σ j ( t − 1) ! 2
, (4.2)
where no w a is a free parameter. This allo ws us to deriv e the optimal linear
estimator.
The replica calculation explained in P ap er 2 is used to find the estimation
error. The order parameters of the mo del are found from equations (P ap er 2,
43). Ho w ev er, for the mean field estimator, equations (P ap er 2, 44-46 ) follo w
from the explicit definition of the parameter a (P ap er 2, 4). If we no w consider
a as a free parameter, the saddle p oin t equations b ecome:
R = R D x x tanh( β x )
a , (4.3)
q =  R D x x tanh( β x )  2 ( α − 2) + 1
( α − 1) a 2 , (4.4)
x = 1
2( α − 1) a 2 (4.5)
and the error (P ap er 2, 25) is
ε =1 − 2 R D x x tanh( β x )
a (4.6)
+ ( α − 1 − 1
N T r C − 1 )  R D x x tanh( β x )  2 + 1
N T r C − 1
( α − 1) a 2 . (4.7)
92

4.3 F urther results
The v alue of a whic h minimizes (4.7) is
a opt = Z D x x tanh( β x ) +
1
N T r C − 1 h 1 −  R D x x tanh( β x )  2 i
( α − 1) R D x x tanh( β x ) , (4.8)
with the corresp onding minimal error
ε opt =
1
N T r C − 1 h 1 −  R D x x tanh( β x )  2 i
( α − 1 − 1
N T r C − 1 ) R D x x tanh( β x ) + 1
N T r C − 1 . (4.9)
The error sho ws no div ergence for α = 1 (Figure 4.1) and reac hes the same
asymptotic v alue as the mean field estimator. Since a opt is indep enden t on
the couplings and can b e directly estimated from data, the asso ciated linear
estimator
J ij = 1
a opt X
k
ˆ
D ik ˆ
C − 1
k j (4.10)
only relies on one matrix in v ersion and it is v ery fast to compute. Also,
with resp ect to the linear mean field estimator, the optimal linear estimator
sho ws w eak er finite size effects, pro viding a faster and b etter algorithm. Still,
comparison with the asymptotic error of maxim um lik eliho o d sho ws that linear
estimators are sub optimal for large v alues of β .
4.3.2 Ba y esian inference
In a Ba y esian setting, if the correct prior kno wledge on the distribution of the
parameters is in tro duced, one can design an algorithm that is asymptotically
optimal: it is the Ba y es optimal estimator giv en b y the p osterior exp ectation of
the parameters. Ho w ev er, p osterior a v erages require high dimensional in tegrals
to b e computed exactly . Here, w e prop ose an analytic appro ximation to the
p osterior exp ectations based on ca vit y argumen ts.
Let us recall the lik eliho o d of a spin sequence
σ = { σ (0) . . . σ ( T ) } for giv en couplings W :
P ( σ | W ) =
T
Y
t =1
N
Y
i =1
e β
√ N σ i ( t ) P j W ij σ j ( t − 1)
2 cosh β
√ N P j W ij σ j ( t − 1) P ( σ 0 ) , (4.11)
where P ( σ 0 ) is the initial distribution of spins. As prior distribution o v er the
couplings, w e consider a univ ariate Gaussian distribution:
W ij ∼ N (0 , 1) (4.12)
93

4 Learning in kinetic Ising mo dels
1 100
α
0.01
1
ε
β=5

Figure 4.1: Mean squared error of the couplings inferred with differen t algo-
rithms as a function of α . Ligh t blue dots correp ond to the linear
optimal algorithm, red dots corresp ond to the mean field algorithm
of P ap er 2, green dots to p enalized maxim um lik eliho o d. W e con-
sider a system of N = 200 spins with β = 5. Results are a v eraged
o v er 25 istances of the net w ork. Con tin uous line refer to the a v-
erage error from the replica calculation, ligh t blue for the linear
optimal and red for the mean field algorithm. The green dotted
line sho ws the asymptotic error for the maxim um lik eliho o d algo-
rithm.
indep enden tly for an y i = 1 , . . . , N and j = 1 , . . . , N . The p osterior distribu-
tion
P ( W | σ ) ∝ P ( σ | W ) P ( W ) (4.13)
represen ts the information ab out lik ely couplings, when a spin tra jectory σ
is observ ed. The Bayes optimal prediction for the couplings is giv en b y the
p osterior mean h W i , where the brac k ets denote an exp ectation o v er the p oste-
rior. Since the couplings are non-symmetric, coupling v ectors W ( i ) at differen t
neurons i are neither in teracting in the lik eliho o d (4.11) nor in the prior (4.12).
Hence, inference for differen t coupling v ectors can b e done indep enden tly for
eac h W ( i ) .
W e write the p osterior distribution of the coupling v ector W ( i ) as the pro duct
of factors:
p ( W ( i ) | σ ) = 1
p ( σ )
T
Y
t =0
f t ( W ( i ) ) , (4.14)
94

4.3 F urther results
where, for t = 0, the factor f 0 ( W ( i ) ) coincides to the prior
f 0 ( W ( i ) ) = 1
√ 2 π e − 1
2 P j ( W ( i )
j ) 2 , (4.15)
while the other factors corresp ond to the lik eliho o d:
f t ( W ( i ) ) = e β σ i ( t ) 1
√ N P j W ( i )
j σ j ( t − 1)
2 cosh β
√ N P j W ( i )
j σ j ( t − 1) for t = 1 ...T . (4.16)
The normaliser is giv en b y the partition function
Z ( σ ) = Z dW ( i )
T
Y
t =0
f t ( W ( i ) ) . (4.17)
T o lighten notation, in the follo wing w e will drop the sup erscript i of the and
consider the inference problem for a singular spin v ector.
4.3.3 App ro ximating the p osterio r b y cavit y a rguments
W e are in terested in computing an appro ximation to the p osterior statistics
of W j . Since the prior is Gaussian, the follo wing exact represen tation for the
first t w o momen ts is deriv ed using in tegration b y parts:
h W j i = β
T
X
t =1
σ j ( t − 1)
√ N { σ ( t ) − h tanh( β h t ) i} (4.18)
1
N X
j  h W 2
j i−h W j i 2  = 1 − β 2
N
T
X
t =1 {h ( h t − h h t i ) tanh( β h t ) i} . (4.19)
where the brac k ets denote exp ectation with resp ect to the p osterior distribu-
tion (4.14) of the field h t :
h t = 1
√ N X
j
W j σ j ( t − 1) . (4.20)
W e will no w deriv e an appro ximation to the distribution p ( h t ) of h t using a
ca vit y argumen t. W e first write
p ( h t ) ∝ f t ( h t ) p \ t ( h t ) , (4.21)
where p \ t ( h t ) is the ’ca vit y distribution’ of h t , i.e the distribution o v er a system
where the term f t w as left out of the p osterior. Using standard argumen ts (see
95

4 Learning in kinetic Ising mo dels
section 3.A.1) based on cen tral limit theorem, in the large N limit, p \ t ( h t ) is
assumed to b e a Gaussian:
p \ t ( h t ) = 1
√ 2 π λ \ t e − 1
2 λ \ t ( h t − γ \ t ) 2 . (4.22)
T o close the system of equations, w e express the mean γ \ t and the v ariances
λ \ t in terms of the ’full’ exp ectations h W j i and 1
N P j  h W 2
j i−h W j i 2  . This
yields the follo wing relations:
λ \ t = λ = 1
N X
j  h W 2
j i−h W j i 2  , (4.23)
1
√ N X
j h W j i σ j ( t − 1) = h h t i = h h t f t ( h t ) i \ t
h f t ( h t ) i \ t
= γ \ t + β λ \ t { σ ( t ) − h tanh( β h t ) i} . (4.24)
In the first equation, w e ha v e neglected correlations b et w een couplings and
assumed that total ca vit y v ariance and ’full’ v ariance of couplings are equal in
the thermo dynamic limit. T o deriv e the second equation, w e used the Gaussian
form of the ca vit y distribution and an in tegration b y parts.
4.3.4 A simple exp ectation p ropagation algo rithm
It is not a priori clear ho w the sets of coupled nonlinear equations (4.18), (4.19),
(4.23) and (4.24) can b e solv ed in an efficien t w a y to get explicit predictions.
W e hav e resorted to the so–called Exp e ctation Pr op agation (EP) algorithm,
an appro ximate inference tec hniques widely used in mac hine learning [O W00,
Min01]. W e will state the algorithm first and then sho w that its fixed p oin ts
agree with the solution of the ca vit y equations (4.23) and (4.24).
The algorithm is based on an auxiliary Gaussian appro ximation q ( W ) to the
p osterior, whic h is used for b o ok–k eeping of the first and second order momen ts
of the W j and their resp ectiv e ca vit y statistics. This pseudo–p osterior q ( W )
can b e written as a pro duct of factors:
q ( W ) = 1
˜
Z
T
Y
t =0
˜
f t ( W ) , (4.25)
where ˜
Z is a normalizing term, ˜
f 0 ( W ) = f 0 ( W ) and for t = 1 , . . . , T
˜
f t ( W ) =  2 π ˜
λ ( t )  − N/ 2 exp " − 1
2 ˜
λ ( t ) X
j
( W j − ˜ µ j ( t )) 2 # . (4.26)
96

4.3 F urther results
W e are approximating eac h factor f t ( W ) of the true p osterior (4.14) with
one Gaussian factor ˜
f t ( W ). In this v ersion of the algorithm, whic h w e call
’Naiv e EP’, w e are making the simplifying assumption that the co v ariance
matrix of (4.26) is diagonal. W e will later discuss the case where the full
co v ariance matrix is considered. In order to determine the set of parameters
of the appro ximate p osterior, the factors ˜
f t ( W ) of the o v erall appro ximation
are optimized sequen tially . Supp ose w e wish to refine the factor ˜
f t ( W ). W e
first remo v e it from the curren t appro ximation of the p osterior to get the
unnormalized ’ca vit y’ distribution, whic h is also Gaussian:
q \ t ( W ) = q ( W )
˜
f t ( W ) . (4.27)
A new appro ximate p osterior q new ( W ) is then computed b y in tro ducing the
follo wing distribution, 1
Z t
f t ( W ) q \ t ( W ) , (4.28)
whic h corresp onds to the old q ( W ) where one Gaussian factor ˜
f t ( W ) has b een
replaced b y one factor f t ( W ) of the true p osterior, and Z t is normalizing
the distribution to 1. In particular, the appro ximate p osterior is up dated b y
minimizing the Kullbac k-Leib er div ergence
K L  1
Z t
f t ( W ) q \ t ( W )     q new ( W )  . (4.29)
Since the appro ximating distribution q new ( W ) is Gaussian, it is easy to pro v e
[Bis06] that minimizing (4.29) is equiv alent to matc hing the exp ected sufficien t
statistics of q new ( W ) to the corresp onding momen ts of (4.28). Finally the
revised form of the factor ˜
f t ( W ) is obtained as
˜
f t ( W ) = Z t
q new ( W )
q \ t ( W ) . (4.30)
The algorithm in v olv es the computation of three Gaussian distributions, (4.27)
(4.28) and q ( W ) b y sequen tially up dating their sufficien t statistics. W e denote
the mean of the appro ximate p osterior q ( W ) b y µ and its co v ariance matrix
b y Λ = I λ . A summary of the algorithm is describ ed as follo ws.
• Set ˜
f 0 equal to the prior
• Initialize all the factors ˜
f t to 1 for t = 1 ...T
• Iterate un til con v ergence:
97

4 Learning in kinetic Ising mo dels
1. F or t=1...T
a) Up date the momen ts of the ca vit y distribution (4.27):
µ \ t
j = ˜
λ ( t ) µ j − λ ˜ µ j ( t )
˜
λ ( t ) − λ
Λ \ t = I λ \ t , λ \ t = ˜
λ ( t ) λ
˜
λ ( t ) − λ .
(4.31)
b) Matc h the first and second momen ts { µ j , λ j } of the appro xi-
mate p osterior with the ones of the distribution (4.28). The
latter momen ts can b e computed as deriv ativ es of the generat-
ing function:
Z t ( ψ ) = Z dW q \ t ( W ) f t ( W ) e P j W j ψ j , (4.32)
in the limit ψ → 0. F or the first momen t w e obtain the follo wing
condition (see 4.E for details):
µ j .
= µ \ t
j + β λ \ t
√ N σ j ( t − 1)  σ ( t ) − h tanh( β h \ t ) f t ( h t ) i \ t
h f t ( h t ) i \ t  ,
(4.33)
where the a v erage is o v er a gaussian field with v ariance λ \ t and
mean γ \ t ,
γ \ t = 1
√ N X
j
σ j ( t − 1) µ \ t
j . (4.34)
The second momen ts λ j turn out to b e indep enden t of j , due
to the prop ert y σ 2
j = 1 of Ising spins. One gets (4.E)
λ .
= λ \ t + β 2 ( λ \ t ) 2
N  2 h tanh( β h \ t )[tanh( β h \ t ) − σ ( t )] f t ( h t ) i \ t
h f t ( h t ) i \ t
−  σ ( t ) − h tanh( β h \ t ) f t ( h t ) i \ t
h f t ( h t ) i \ t  2 ) . (4.35)
c) Ev aluate and store the new factors ˜
f t ( W ) using (4.30). Its
momen ts are:
˜ µ j = λ \ t µ j − λµ \ t
j
λ \ t − λ ,
˜
λ = λ \ t λ
λ \ t − λ .
(4.36)
98

4.3 F urther results
F rom (4.25) we see that, after con v ergence, w e can compute the the momen ts
of the p osterior distribution as:
µ j = λ
T
X
t =1
˜ µ j ( t )
˜
λ ( t ) , (4.37)
λ = 1 +
T
X
t =1
1
˜
λ ( t ) ! − 1
. (4.38)
In 4.F w e sho w that those fixed p oin t equations are equiv alen t to the exp ected
momen ts (4.23) and (4.24) of the couplings obtained from ca vit y argumen ts.
4.3.5 Average case: a replica analysis
The a v erage prediction error for the Ba y es optimal estimator can b e computed
in a studen t-teac her setting with a replica analysis, analogously to the analysis
of P ap er 2. W e no w w ork in a Ba y esian framew ork, where the studen t has
prior kno wledge ab out the teac her. The distribution of the studen t couplings
is giv en b y the p osterior distribution (4.14) and the partition function Z ( σ )
is the normalizer (4.17) of the p osterior distribution:
Z ( σ ) = Z d W 1
√ 2 π e − 1
2 P j ( W j ) 2 Y
t
e β σ i ( t ) 1
√ N P j W j σ j ( t − 1)
2 cosh β
√ N P j W j σ j ( t − 1) (4.39)
Since (4.14) also represen ts the p osterior distribution corresp onding to a prior
distribution of random teac hers, Z ( σ ) will b e prop ortional to the total prob-
abilit y P ( σ ):
P ( σ ) = Z ( σ )
C , (4.40)
where C = P σ Z ( σ ) is the normalization factor. Hence, the teac her and the
studen t net w ork en ter the calculation in a completely symmetric w a y: the
a v erage studen t-teac her o v erlap equals the a v erage studen t self-ov erlap, and
the error is
ε = 1
N ( W ∗ − h W i ) 2 = 1 − h W a · W b i , (4.41)
where h . . . i denotes a v eraging with resp ect to the distribution of couplings.
The error can b e computed from the free energy using the replica tric k as
follo ws:
F = − N − 1 X
σ
P ( σ ) log Z ( σ ) = − lim
n → 1
∂
∂ n N − 1 log X
σ
Z n ( σ ) . (4.42)
99

4 Learning in kinetic Ising mo dels
In order to compute the a v erage o v er the spin tra jectories, w e consider the
same appro ximation describ ed in the pap er, that w e argue to b e correct in
the limit N → ∞ . F or the cen tral spin σ 0 , the v ariables σ 0 ( t ) are binary
at all times t . The other spins { σ j ( t − 1) } j 6 =0 en ter the partition function
through the fields P j 6 =0 W a
j σ j ( t − 1). Since the system is w eakly coupled, the
cen tral limit theorem tells us that suc h fields are Gaussian distributed and
w e treat the spins { σ j ( t − 1) } j 6 =0 themselv es as Gaussian random v ariables:
p ( { σ j ( t ) } j 6 =0 ) = N (0 , C ). The stationary co v ariance matrix C tak es in to
accoun t equal time spatial correlations among the spins { σ j ( t − 1) } for j 6 = 0,
while dep endences at differen t time steps are neglected. The partition function
is:
X
σ ( t )
Z n ( σ ) = Z n
Y
a =1
d W a 1
√ 2 π e − 1
2 P a W a . W a


 X
σ 0 Y
j 6 =0
dσ j
1
p | 2 π C | e − 1
2 P i,j 6 =0 σ i C − 1
ij σ j
n
Y
a =1
e β σ 0 1
√ N P j 6 =0 W a
j σ j
2 cosh β
√ N P j 6 =0 W a
j σ j 


T
.
(4.43)
W e ha v e remapp ed the kinetic Ising mo del in a num b er T of logistic regression
mo dels, whose inputs are not indep enden t but correlated through the matrix
C . The stationary v alue of the matrix C dep ends on the (teac her) couplings
and enco des for non-trivial equal time correlations among spins. W e computed
its statistics in P ap er 2. Av erages o v er the spins σ j can b e expressed through
Gaussian fields
h a .
= 1
√ N X
j
W a
j σ j (4.44)
whose co v ariances in the limit N → ∞ will b ecome self a v eraging order pa-
rameters. Under the assumption of replica symmetry , correct for con v ex cost
functions, w e ha v e:
 h 2
a  = 1
N X
ij
W a
i C ij W a
j = 1 , (4.45)
h h a h b i = 1
N X
ij
W a
i C ij W b
j
.
= q a 6 = b. (4.46)
The first equalit y follo ws from the fact that the matrix C represen ts the cor-
relation b et w een all the spins but σ 0 ; hence, it is indep enden t of W 0 j . The
100

4.3 F urther results
calculation, detailed in 4.H, yields:
F = − Extr q , ˆ q
1
2  ˆ q ( q − 1) − 1
N T r(1 − ˆ q C )
+ α X
σ Z D y A ( σ , y , q ) log A ( σ, y , q ) ) ,
(4.47)
where
A ( σ , y , q ) = Z D x e β σ ( x √ 1 − q + y √ q )
2 cosh[ β ( x √ 1 − q + y √ q )] . (4.48)
Here, D x = ( dx/ √ 2 π ) e − x 2 / 2 , D y = ( dy / √ 2 π ) e − y 2 / 2 and the parameter α =
T / N represen ts the rescaled length of the tra jectories. The saddle p oin t equa-
tions for the order parameters, extremising the free energy (4.47) are:
q = 1 + 1
ˆ q  1 − 1
N T r(1 − ( ˆ q C ) − 1 ) − 1  ,
ˆ q = − α X
σ Z D y B ( σ, y , q ) / A ( σ , y , q ) ,
(4.49)
where
B ( σ , y , q ) = " β Z D x  σ − tanh( x p 1 − q + y √ q )  e β σ ( x √ 1 − q + y √ q )
2 cosh[ β ( x √ 1 − q + y √ q )] # 2
.
In order to compute the error (4.41) w e need the t ypical o v erlap b et w een
t w o studen t net w orks, that is differen t from the parameter q (4.46). It can b e
computed from (4.109), b y noticing that
1
ˆ q X
i
∂
∂ Λ i
1
N ln Z n
c = − 1
2 N X
a 6 = b X
i h W a
i · W b
i i (4.50)
F rom (4.41) and (4.50) one gets the final result for the mean square error:
ε = 1 − 2
ˆ q X
i
∂
∂ Λ i
F 0 ( q , ˆ q ) = 1
N T r( I − ˆ q C ) − 1 . (4.51)
4.3.6 Results
W e ev aluate the analytic expression of the error (4.51) from the system of equa-
tions (4.49) and from the statistics of the C matrix (P ap er 2). Figure (4.2)
compares the results with the mean square error of the couplings inferred b y
101

4 Learning in kinetic Ising mo dels
using the Naiv e Exp ectation Propagation algorithm of section 4.3.4. The data
are generated from a kinetic Ising mo del with indep enden t Gaussian couplings
with v ariance 1 / N . The Naiv e Exp ectation Propagation algorithm, where the
true p osterior distribution is appro ximated b y a Gaussian distribution with
diagonal co v ariance matrix, is in go o d agreemen t with the theoretical predic-
tions of the Ba y es estimator. F or large v alues of α , the error of Exp ectation
Propagation deviates from the replica result due to finite size effects. Figure
(4.3) sho ws that it con v erges to the replica result for large N . In particular, w e
fix α = 500 and sho w ho w the error deca y as a function of N . Fitting a shifted
p o w er lo w to the data w e obtain an asymptotic v alue ε N →∞ = 0 . 0007 ± 0 . 0004,
whic h is in go o d agreemen t with the replica v alue ε = 0 . 000746. F or small
v alues of α , Exp ectation Propagation outp erforms all other algorithms. F or
completeness, in App endix 4.G w e design a ’Complete’ Exp ectation Propaga-
tion algorithm, where the p osterior distribution is appro ximated b y a Gaussian
with full co v ariance matrix. W e tested it for v alues of α up to 10: the error
is not significan tly lo w er than the one of Naiv e EP while the required time
for con v ergence is m uc h higher. Regarding computational complexit y , eac h
iteration of b oth the Exp ectation Propagation and Maxim um Lik eliho o d al-
gorithms to estimate one coupling v ector { W j } requires a computation of the
order T N . Exp ectation Propagation, though, conv erges in m uc h few er steps
and shorter time. F or instance, for a system of N = 100 spins at α = 10, w e
needed appro ximately 284 up dates of the learning rates for maxim um lik eli-
ho o d (25 seconds) and 5 iterations o v er time for EP (2 seconds).
4.4 Conclusions
In this c hapter w e considered a kinetic Ising mo del where the couplings are
indep enden t Gaussian random v ariables with v ariance scaling as 1 / N , and
computed the error of three differen t estimators for the couplings, w orking
in a a studen t-teac her scenario. W e analysed a linear mean field estimator,
whic h can b e rephrased as the minimizer of a lo cal quadratic cost function;
an estimator based on an anlogous quadratic cost function, whic h con tains a
free parameter that is optimized to minimize the estimation error; the optimal
Ba y es estimator, where a prior distribution is in tro duced and the couplings
are estimated as their p osterior a v erages.
The replica calculation rev ealed the imp ortance of equal-time correlations
b et w een spins at differen t sites: despite b eing of the order 1 / √ N [MS11], they
significan tly affect the estimation error, esp ecially when the sto c hasticit y of the
spin dynamics is decreased. By computing an exact result for the statistics of
the random correlation matrix, w e find an explicit expression for the learning
102

4.4 Conclusions
1 100
α
0.01
1
ε
β=5

Figure 4.2: Mean squared error of the couplings inferred with differen t algo-
rithms as a function of α . Red dots corresp ond to the mean field
algorithm of P ap er, green dots to maxim um lik eliho o d, blue dots
to Naiv e EP and ligh t blue dots to linear optimal. W e consider a
system of N = 200 spins with β = 5. Results are a v eraged o v er 25
istances of the net w ork. Con tin uous line refer to the av erage error
from the replica calculation, red for the mean field algorithm (see
P ap er) and blue for the Ba y es estimator. The green dotted line
sho ws the asymptotic error for the maxim um lik eliho o d algorithm.
curv e of the three algorithms, whic h agrees v ery w ell with sim ulations.
By comparison with the asymptotic error of the maxim um lik eliho o d es-
timator, whic h has the prop ert y of asymptotic optimalit y , w e assessed the
p erformance of the considered metho ds. The error of linear estimators, suc h
as the linear mean-field one, is asymptotically close to optimal one for w eak
couplings, whereas it deviates from optimalit y for stronger couplings.
If the prior corresp onds to the true distribution of the parameters, the
Ba y es optimal estimator pro vides an asymptotically optimal estimator; the
in tractable in tegrals required to compute p osterior a v erages can b e appro xi-
mated using the ca vit y metho d of statistical ph ysics, and w e solv ed the re-
sulting set of equations b y an algorithm of the Exp ectation Propagation t yp e:
the true p osterior distribution of the couplings is appro ximated b y a Gaussian
distribution, whose mean and co v ariance are up dated iterativ ely , in suc h a w a y
that the appro ximated distribution is as close as p ossible to the true one (in
the sense of KL-div ergence). The fixed p oin t equations for the momen ts of
103

4 Learning in kinetic Ising mo dels
Figure 4.3: Mean squared error of the couplings inferred with the Naiv e EP
algorithm, plotted as a function of N for fixed α = 500 and β = 5.
W e fit a rescaled p ow er la w to the data to find an asymptotic v alue
ε N →∞ = 0 . 0007 ± 0 . 0004 and a deca y exp onen t 0 . 48 ± 0 . 007.
the p osterior distribution are equiv alen t to the ones obtained from ca vit y ar-
gumen ts. An in teresting question, that w e lea v e to future researc h, is whether
our Ba y esian estimator implemen ted via the Exp ectation Propagation algo-
rithm b ecomes exact in the limit N → ∞ (i.e., whether our approximation to
the true p osterior a v erages b ecome exact in the thermo dynamic limit).
Moreo v er, as a future direction, it w ould b e in teresting to extend our results
to other t yp es of net w orks. P articularly relev an t for practical applications are
sparse net w orks. Prior kno wledge on the couplings could b e in tro duced via a
spik e and slab distribution, widely used in mac hine learning for sparse linear
mo dels (see, e.g., [MB88, GM93, BBB + 03, BBB + 03]). Eac h w eigh t J ij of the
prior w ould b e set to zero with probabilit y 1 − π and drawn from a Gaussian
distribution with probabilit y π . This w ould allo w us to use the Exp ectation
Propagation algorithm dev elop ed in this section to infer the couplings in sparse
net w orks.
F or what concerns the analytical analysis, the basic ideas underlying our
replica formalism can b e applied to other systems. The idea of treating one
cen tral spin as the output of a p erceptron whose inputs are correlated through
a ca vit y matrix C , will inspire the analysis presen ted in the next c hapter.
104

App endix
4.A The replica metho d: from spin glasses to
neural net w o rks
The replica tric k is a long-established metho d in the analysis of disordered sys-
tems; dating bac k at least to Hardy [HLP34] as an iden tit y for computing the
a v erage of a logarithm (see also the w ork of Kac [Kac68]), it w as rein tro duced
b y Edw ards [Edw70] for a mo del of rubb er elasticit y and b ecame w ell kno wn
with its application to spin glasses [EA75, SK75]. After the seminal w ork of
Gardner [Gar87, Gar88], it has b een widely used to study learning in neural
net w orks in a statistical mec hanics framew ork [WRB93, OK96, EVdB01].
In the follo wing sections, I will first in tro duce the replica metho d in the con-
text of spin glasses; then, I will sho w ho w this statistical mec hanics formalism
is applied to the problem of learning in neural net w orks.
4.A.1 Spin glasses and the replica trick
Spin glasses are the simplest mo dels for glassy systems [P ar06]. They ha ve been
widely studied in the last 40 y ears not only to deriv e some of the main prop-
erties of glassy systems, but also b ecause they pro vided a framew ork to study
prop erties of other ph ysical systems, as fragile glasses, colloids and gran ular
materials; moreo v er, and man y ideas dev elop ed in the field w ere later applied
to com binatorial optimization problems and learning in neural net w orks.
The Hamiltonian of a spin glass with pairwise in teractions is:
H ( σ ) = − X
i,j =1 ,...,N
J ij σ i σ j − X
i =1 ,...,N
h ex
i σ i (4.52)
where σ i are Ising v ariables (i.e., σ i = ± 1) lo cated on a lattice v ertices, the
couplings J s are random v ariables lo cated on the edges of the lattice and h ex
i
are lo cal external fields.
Man y mo dels of spin glasses ha v e b een studied, according to the distribu-
tion of the couplings and the top ology of the lattice. W e will consider the
105

4 Learning in kinetic Ising mo dels
Sherrington-Kirkpatric k (SK) mo del c, in tro duced in 1975 as an exactly solv-
able mo del of a spin glass; all couplings are random v ariables with a Gaussian
or bimo dal distribution with v ariance 1 / N , and the net w ork is fully connected.
The disorder induced b y the randomness of the couplings is assumed to b e
quenc hed, whic h means that the c hanges in the J s happ en on a time scale
infinitely larger than the t ypical time scale of spin fluctuations. If the system
observ ables dep ended on J , it w ould follo w that the ph ysical prop erties of spin-
glasses are differen t for eac h differen t realization of the quanc hed disorder. In
con trast, it turns out [Ca v09] that extensiv e quan tities, suc h as the free energy ,
ha v e the prop ert y of self-a v erageness: in the thermo dynamic limit (infinite
v olume limit) they assume the same v alue for eac h realization of the couplings.
This means that analytically w e can a v erage o ver J , and the obtained result
is in agreemen t with the ph ysical v alue of the observ able.
Let us no w fo cus on the SK mo del, whose Hamiltonian is
H ( σ ) = − X
i<j
J ij σ i σ j − X
i
h ex
i σ i (4.53)
where the couplings J ij are indep enden t Gaussian random v ariables with zero
mean and v ariance J 0 / N , and h ex
i is the external field. Denoting b y f J and Z J
the free energy and the partition function of a sample with a set J of couplings:
f J = − 1
β N log Z J = − 1
β N log T r { σ } e − β H ( σ ) , (4.54)
where β is the in v erse temp erature, w e are in terested in computing the a v erage
v alue o v er the disordered distribution of the free energy ,
f = Z dJ P ( J ) f J = − 1
β N log Z J , (4.55)
where the o v erbar ( ... ) denotes the a v erage o v er the disorder distribution, whic h
is called quenche d aver age . The computation of suc h a v erage is tec hnically dif-
ficult, since it requires a v eraging the logarithm of the partition function. A
m uc h easier approac h w ould b e to consider the anne ale d aver age of the free
energy , that is the logarithm of the a v erage of the partition function. Ho w ev er,
while F is an extensiv e quan tit y , this is not the case for Z (whic h is exp onen-
tial in the system size), and therefore Z is not in general self-a v eraging. T o
o v ercome the difficult y , Edw ards and Anderson prop osed to apply the replica
tric k, whic h mak es use of the relation l og ( x ) = lim n → 0 x n − 1
n to transform the
logarithm in a p o w er la w:
− β N f = lim
n → 0
Z n − 1
n . (4.56)
106

4.A The replica metho d
One first assumes that n is in teger, and can see Z n as the partition function of
n replicas of the same system, that share the same realization of the couplings
but are non-in teracting. Then one m ust p erform the limit n → 0, and later
N → ∞ , to get the self-a v eraging v alue of the free energy (in practice, it
turns out that rev erting the order of the t w o limits is not a source of trouble).
Without going in to the details of the calculation (an example will b e giv en in
P ap er 4) w e p oin t out that the in tegral in
Z n = Z dJ P ( J )T r { σ a } exp " β X
i<j
J ij X
a
σ a
i σ a
j + β X
i
h ex
i X
a
σ a
i # , (4.57)
where a is the replica index, can b e easily p erformed; since the J s are coupled
to a quadratic term in the spins, one can use the in v erse Gaussian in tegral
(Hubbard-Stratonivic h transformation) to uncouple the spins σ a
i in the sites
and sum o v er all p ossible spin configurations. This pro cedure naturally in tro-
duces the spin o v erlap
q ab = 1
N X
i
σ a
i σ b
i , a < b, (4.58)
whic h represen ts the t ypical o v erlap b et w een t w o configurations in a giv en
state b elonging to t w o differen t replicas.
Sherrington and Kirkpatric k [SK75] considered a r eplic a symmetric (RS)
ansatz for the order parameter: the o v erlap is the same no matter what t w o
replicas are c hosen,
q ab = q (1 − δ ab ) . (4.59)
Ho w ev er, their result turned out to ha v e some unph ysical feature, suc h as a
negativ e en trop y at lo w temp eratures. It w as first though t to b e a problem
related to exc hanging the n → 0 limit with the large v olume limit N → ∞
when computing the free energy , but it later b ecame clear [D A T78, BM80] that
the problem resides in the symmetry of the replica ansatz.
The replica symmetry breaking (RSB) calculation w as presen ted in a series
of pap ers b y P arisi [P ar80a, P ar80b, P ar80b]; its solution w as ph ysically con-
sisten t and confirmed b oth b y n umerical sim ulations and b y other analytical
metho ds. Y et, it to ok o v er 20 y ears for the reluts predicted b y the RSB cal-
culation to b e rigorously pro v en b y T alagrand [T al06], using the in terp olation
metho d of Guerra [Gue03].
Our w ork will fo cus on systems for whic h the replica symmetric ansatz is
correct. Hence, w e refer the reader to the literature men tioned ab o v e and
to [MPV87] for a description of the replica symmetry breaking metho d; here,
w e just men tion a few k ey concepts.
107

4 Learning in kinetic Ising mo dels
The replica symmetry breaking (RSB) pro cedure can b e describ ed as an
iterativ e pro cess that sequen tially reparameterizes the n x n matrix with el-
emen ts q ab . The starting p oin t (0-th step) is the ansatz (4.59), where the
matrix has zero en tries on the diagonal and v alues q ab = q 0 for all non di-
agonal v alues. A t the 1-st step, the n x n matrix is div eded in n/m 1 blo c ks
of size m 1 x m 1 : the off-diagonal terms of the diagonal blo c ks tak e v alue q 1 ,
the other terms remain unc hanged. The correct solution w as found b y con-
sidering an infinite n um b er of steps of RSB. The analysis of the probabilit y
distribution of the order parameters yielded a geometrical c haracterization
of the space of solutions, that turned out to b e an ultrametric space, where
q ac ≤ min( q ab q bc ) [MPS + 84, MV85]. This is a signature of the complex free–
energy landscap e of the spin glass phase, where an infinitely large n um b er
of minima are separated b y barriers that gro w indefinitely as the system size
increases [P ar83]. The n um b er of minima (metastable states) in exp onen tially
large in the size of the system N [BM80], and so is the time sp en t b y the
system in ev ery single v alley: in the large N limit ergo dicit y is brok en [MY82].
4.A.2 Statistical mechanics of lea rning: general setup
In this section, w e in tro duce the statistical mec hanics framew ork to analyze the
theoretical p erformance of learning algorithms in the so called teac her-studen t
scenario. F or further reading, see [WRB93, OK96, EVdB01].
Let us b egin b y defining a neural net w ork as a set of no des, or neurons,
that can tak e v alues ± 1 and influence eac h other’s state through directed
connections W ij . Among the v arious arc hitectures that ha v e b een studied, w e
will fo cus on la y ered net w orks; w e refer to the the first la y er as the input, and
to the last la y er as the output. F or simplicit y , w e will further assume that the
net w ork has only one la y er of N input no des with v alues S = { S i } and one
no de in the in the output la y er, whose state is σ . The state of the neuron σ is
set to a function of the w eigh ted sum of the inputs, where the w eigh ts are the
connections W j (here W = { W i } is an N dimensional v ector):
σ ( W ; S ) = g X
j
W j S j ! , (4.60)
where g is a generic non-linear function. The term learning refers to the pro cess
of setting the w eigh ts to the v alues that make the net w ork p erform a desired
task, that is a target input-output mapping whic h will b e denoted as the rule .
W e will fo cus on sup ervise d learning, where the w eigh ts are adjusted as to
appro ximate as closely as p ossible a target function σ 0 ( S ). This is ac hiev ed
b y pro viding the net w ork with a training set, that is a set of M input/output
108

4.A The replica metho d
pairs { S ( k ) , σ ( k )
0 } M
k =1 generated b y some unkno wn mapping, and b y requiring
that the net w ork adapts its w eigh t to map eac h pair w ell 2 . W e assume that
the inputs are are generated indep enden tly at random from the input space
according to some probabilit y distribution P ( S ). The target mapping can
b e represen ted b y another net w ork with w eigh ts W ∗ , or te acher net w ork,
that kno ws the correct mapping and generated the examples. The learning
net w ork W is called the student and the prescription { S ( k ) , σ ( k )
0 } → W that,
giv en the training set, sp ecifies the studen t coupling v ector is referred to as the
le arning rule . In particular, a rule for whic h a net w ork in the studen t space
exists that realizes the target function σ 0 ( S ) is called learnable. Otherwise
the rule is called unlearnable.
In order to measure the deviation of the net w ork output σ ( W ; S ) from the
target output σ 0 ( S ), w e in tro duce an error function E ( W ; S ) whic h is zero if
teac her and studen t agree on the output to S and larger than zero otherwise.
Based on the error function, one can define an extensiv e energy , whic h scales
with the n um b er of examples; if suc h energy is defined not to dep end explicitly
on the unkno wn rule, it can b e used in a learning algorithm. A widely used
c hoice is the training energy
E ( W ) =
M
X
k =1 E ( W ; S ( k ) ) , (4.61)
and training is usually ac hiev ed b y minimizing suc h training energy , for exam-
ple via gradien t descen t.
After the studen t net w ork has learned a rule from a limited set of examples,
it can mak e predictions on no v el inputs. The abilit y of a net w ork to generalize
from a limited n um b er of examples to the whole space of inputs is measured
b y the generalization function:
ε ( W ) = Z d S P ( S ) E ( W ; S ) . (4.62)
Let us in tro duce one learning scenario that is particularly w ell suited for a
theoretical analysis and that w e will consider in c hapters 4 and 5: the case
of Gibbs learning at non-zero temp eratures. In this case training is ac hiev ed
b y minimizing a generic training energy of the form (4.61), according to a
sto c hastic dynamics go v erned b y the Langevin relaxation equation
∂ W
∂ t = −∇ W E ( W ) + η ( t ) , (4.63)
2 W e will not consider the case where the data a v ailable for training are corrupted with
noise.
109

4 Learning in kinetic Ising mo dels
where η is a white noise with v ariance
h η i ( t ) η j ( t 0 ) i = 2 T δ ij δ ( t − t 0 ) . (4.64)
A t zero temp erature the noise drops out lea ving us with a simple gradien t
descen t equation. In learning algorithms, the noise can b e useful in escaping
lo cal minima of the energy; the temp erature is slo wly decreased so that the
system settles near to the global energy minim um at T ≈ 0. The dynam-
ics (4.63) generates at long times the Gibbs probabilit y distribution on the
parameter space for a canonical ensem ble of net w orks
ρ ( W ) = 1
Z exp[ − ν E ( W )] (4.65)
where ν = 1 /T quan tifies the noise of the training pro cedure, and the normal-
ization in tegral
Z = Z d W exp[ − β E ( W )] (4.66)
measures the w eigh ted accessible v olume in the configuration space. In the
limit ν → ∞ the system settles at the global energy minim um 3 . The t ypical
b eha viour of a net w ork can b e no w computed via thermal a verages, denoted
b y h . . . i , with resp ect to the distribution (4.65). Note ho w ev er that the ab o v e
quan tities still dep end on the random c hoice of a sp ecific training set { S ( k ) } M
k =1 .
Moreo v er, w e do not w an t to consider a sp ecific realization of the teac her
net w ork, but w e assume that the teac her net w ork is dra wn at random from a
teac her rule space. Both the teac her net w ork and the data sets indep enden tly
generated from it are randomly c hosen and k ept fixed during the learning
pro cedure, and they represen t - in the language of the statistical ph ysics of
spin glasses -a quenc hed disorder. It turns out ho w ev er that the error (4.62) is
self-a v eraging in the limit of N → ∞ , whic h means that almost an y realization
of the teac her net w ork and training set will giv e the same result. W e will denote
3 This distribution, arising naturally for sto chastic algorithms, w as also in tro duced b y Levin
Tish b y Solla [L TS90] from a statistical estimation theory p ersp ectiv e, where the train-
ing pro cess in feedforw ard neural net w orks is seen as a parameter estimation problem.
The solution can b e found b y setting the parameters to the v alue that maximizes the
lik eliho o d of the training set of M indep enden t examples. Imp osing that the maximiza-
tion of the lik eliho o d b e equiv alent to the minimization of an additiv e error of the form
(4.61) for ev ery set of indep enden t training examples, the authors arrive at the Gibbs
canonical distribution on the ensem ble of all net works with the same parameter space
(i.e., net w orks with the giv en architecture). The distribution dep ends on a free p ositiv e
parameter, whic h determines the lev el of acceptable training error as w ell as the tevel of
sto c hasticit y in the training algorithm, and can b e in terpreted as an in verse temperature.
110

4.A The replica metho d
quenc hed a v erages b y o v erlines :
( . . . ) = Z M
Y
k
d S k ! d W ∗
M
Y
k
p ( S k | W ∗ ) p ( W ∗ )( . . . ) . (4.67)
The a v erage generalization error h ε ( W ) i will then dep end on the noise pa-
rameter ν in the thermal a v erage and on the n um b er of examples M in the
quenc hed a v erage, whic h w e will assume to b e prop ortional to the n um b er
of degree of freedom (i.e. indep enden t synaptic w eigh ts): M = α N , with α
finite. Giv en a distribution of the inputs and an energy function, one can
use the to ols of statistical mec hanics to calculate the quenc hed a verages and
deriv e the a v erage generalization error from deriv atives of the free energies, in
the thermo dynamic limit of N → ∞ . The calculation of the quenc hed a v erage
of the free energy p er coupling,
F = − N − 1 ν − 1 log Z , (4.68)
can b e carried out using the replica metho d. The quan tit y
lim
n → 0
1
n ln Z n
has to b e ev aluated for in teger n and then analytically con tin ued to n = 0.
The replicated partition function yields
Z n = Z n
Y
a =1
d W a ! e − N α G r [ { W a } ] (4.69)
where the replicated Hamiltonian is
G r [ { W a } ] = − ln Z d S d W ∗ p ( S | W ∗ ) p ( W ∗ ) exp[ − β
n
X
a =1 E ( W a ; S )] . (4.70)
The a v erage generalization error can then b e computed as follo ws:
h ε ( W ) i = lim
n → 0 Z n − 1 Z d W ε ( W ) exp[ − β E ( W )]
= lim
n → 0 Z n
Y
a =1
d W a ! ε ( W 1 ) e − N α G r [ { W a } ] .
(4.71)
The in tegration o v er the inputs will couple the w eigh ts of differen t replicas of
the system, whic h mak es it natural to in tro duce order parameters - represen t-
ing the o v erlap of the w eigh ts of t w o copies of the studen t net w orks and the
111

4 Learning in kinetic Ising mo dels
o v erlap b et w een the teac her and the studen t net w ork - that will con v ey the
dep endence of G r on the w eigh ts. The v alues of these order parameters are
the ones that extremize G r ; computing the saddle p oin t equations for the pa-
rameters requires making an ansatz ab out the symmetry of the parameters at
the saddle p oin t. That simplest ansatz is the replica symmetric ansatz, whose
v alidit y can b e assessed b y studying the lo cal stabilit y of the replica symmetric
saddle p oin t. In this thesis, w e will consider con v ex cost functions of the w eigh t
v ector, whic h ensures the replica symmetric ansatz to b e correct [EVdB01].
4.B Maximum lik eliho o d estimato r
Let us consider the Mark o vian dynamics for the Ising mo del that w e in tro duced
in P ap er 1, whic h is describ ed b y the transition probabilit y
p ( σ ( t + 1) | σ ( t )) =
N
Y
i
exp[ β σ i ( t + 1) h i ( t )]
2 cosh β h i ( t ) , (4.72)
where w e defined the field h i ( t ) = P j J ij σ j ( t ) + H ext
i ( t ). The log-lik eliho o d of
the system parameters is
L ( J , H ext ) = 1
T X
t X
i
[ β σ i ( t + 1) h i ( t ) − log 2 cosh β h i ( t )] . (4.73)
T o find the maxim um lik eliho o d parameters, one starts from an initial sets
of couplings and external fields, and then adjust them iterativ ely b y gradien t
ascen t; the deriv ativ es are giv en b y
∂ L
∂ H ext
i ( t ) = h σ i ( t ) i r − h tanh h i ( t ) i r ,
∂ L
∂ J ij
= 1
T X
t h σ i ( t ) σ j ( t ) i r − h tanh h i ( t ) σ j ( t ) i r ,
(4.74)
where w e assumed that N r realizations of the tra jectories can b e observ ed,
and the brac k ets h . . . i r represen t empirical a v erages o v er differen t realizations.
The deriv ativ es (4.74) can b e ev aluated in N 2 T N r computational steps, whic h
mak es the computation m uc h faster than the Lik eliho o d of the equilibrium
Ising mo del, where the normalizer of the Boltzmann distribution scales exp o-
nen tially with the system size.
112

4.C Mean field estimators
4.C Mean field estimato rs fo r the stationa ry state
Let us summarize the deriv ation of the mean field relation b et w een the coupling
matrix and the correlation matrix found in [RH11b] for a kinetic Ising mo del
with parallel dynamics.
W e start from the definition of one-step-dela y ed and equal time correlation
matrices for the spin fluctuation δ σ i ( t ) = σ i ( t ) − m i ( t ) , that can b e computed
from data:
D ij = h δ σ i ( t + 1) δ σ j ( t ) i (4.75)
C ij = h δ σ i ( t ) δ σ j ( t ) i (4.76)
where h ... i are empirical a v erages; in the stationary case, a v eraging o v er time
and rep eats w ould b e equiv alen t, so in this paragraph for an y function of time
f ( t ) observ ed o v er a tra jectory of length T , w e define
h f ( t ) i = 1
T X
t
f ( t ) .
By setting the gradien t of the lik eliho o d (4.74) to zero, one gets
h σ i ( t + 1) σ j ( t ) i = h tanh[ h i ( t )] σ j ( t ) i . (4.77)
W e now expand the effectiv e lo cal field h i ( t ) around its mean field solution.
F ormally , w e write s i = m i + δ s i , and use the naiv e mean field equation
m i = tanh[ P j J ij m j + H i ] for the magnetization. F rom (4.77), expanding
tanh[ h i ( t )] in p o w ers of δ s i w e get to the leading order
h δ σ i ( t + 1) δ σ j ( t ) i = (1 − m 2
j ) X
k
J nMF
ik h δ σ k ( t ) δ σ j ( t ) i , (4.78)
whic h can b e written as
J nMF = A nMF D C − 1 , (4.79)
where
A nM F
ij = δ ij (1 − m 2
i ) . (4.80)
The T AP in v ersion form ula is deriv ed analogously [RH11b] and also results in
the linear relation
J T AP = A T AP D C − 1 , (4.81)
113

4 Learning in kinetic Ising mo dels
where
J T AP
ij = A nM F
ij (1 − F i ) , (4.82)
and F i is the smallest ro ot of the follo wing cubic equation:
F i (1 − F i ) 2 = (1 − m 2
i ) X
j
( J nM F ) 2
ij (1 − m 2
j ) . (4.83)
4.D Mean field estimato rs fo r the transient
dynamics
In the case of out-of-equilibrium dynamics w e refer to the three mean field
theories of section 3.B; analogous relations to (4.79, 4.82) can b e found for the
the one-time-dela y ed and equal time correlation matrices, defined as
D ij ( t ) = h δ s i ( t + 1) δ s j ( t ) i
C ij ( t ) = h δ s i ( t ) δ s j ( t ) i , (4.84)
where no w the matrices dep end on time. Starting from the dynamical mean
field equation of (3.38), b y using the same expansion as in (4.C), one finds he
follo wing relation:
D ( t ) = A ( t ) J ( t ) C ( t ) (4.85)
where A ij = δ ij a i is a diagonal matrix with elemen ts
a i ( t ) =  1 − m 2
i ( t )  " 1 −  1 − m 2
i ( t )  X
j
J 2
ij  1 − m 2
j ( t − 1)  # . (4.86)
F or the mean field theory (3.41), whic h is exact for asymmetric net w orks,
the k ey observ ation is that when couplings scale as 1 / √ N , also eac h matrix
elemen t will b e of the order 1 / √ N . W e define the field fluctuation δ g i ( t ) =
P j J ij δ s j ( t − 1), and note that the join t distribution of δ g i ( t ) and δ g j ( t ) has
small co v ariance  = h δ g i ( t ) δ g j ( t ) i . By an expansion in small  , one retriev e
the relation (4.88), where no w A ij = δ ij a i ,
a i ( t ) = Z D x h 1 − tanh 2  g i ( t ) + H i ( t ) + x p ∆ i ( t ) i . (4.87)
In [MS14], the authors deriv e recursiv e equations that allo w to compute corre-
lations b et w een spins at differen t times, starting from ca vit y argumen ts. The
114

4.E Exp ectation Propagation algorithm: generating function of the momen ts
equal time and one-time-dela y ed correlation matrices are related through the
follo wing relation:
(1 − δ ) D ( t ) = A ( t ) J ( t ) C ( t ) , (4.88)
where δ is the unit matrix, so that (1 − δ ) D ( t ) con tains only non-diagonal terms
of the co v ariance matrix D , i.e. D ij i 6 = j ; its diagonal elemen ts of the form
D ii ( t ) ha v e to b e computed separately according to (3.45). Please note that
in the presen t c hapter w e are using the notation D ij ( t ) = h δ s i ( t + 1) δ s j ( t ) i , to
dra w a parallel b et w een tec hniques used for the stationary case and the ones
v alid for the transien t dynamics; in section 3.B w e w ere using C ij ( t + 1 , t ) =
h δ s i ( t + 1) δ s j ( t ) i . The elemen ts of the diagonal matrix A in (4.88) are
a i ( t ) = Z D x h 1 − tanh 2  g i ( t ) + H i ( t ) + x p V ii ( t, t ) i , (4.89)
where the definitions of g i ( t ) and V ii ( t, t ) are resp ectiv ely (3.39) and (3.43).
4.E Exp ectation Propagation algo rithm:
generating function of the moments
The momen t generating function for the distribution (4.28) is
Z t ( ψ ) = Z dW q \ t ( W ) f t ( W ) e P j W j ψ j
=  2 π λ \ t  − N / 2 Z dW e − 1
2 λ \ t P j ( W j − µ \ t
j ) 2
e β σ i ( t ) 1
√ N P j W j σ j ( t − 1)
2 cosh β
√ N P j W j σ j ( t − 1) e P j W j ψ j .
(4.90)
Enforcing the definition of h t b y delta function and in tro ducing the in tegral
represen tation of the delta, one obtains:
Z t ( ψ ) =  2 π λ \ t  − N/ 2 Z dW dh d ˆ
h e − 1
2 λ \ t P j ( W j − µ \ t
j ) 2 e β σ i ( t ) h
2 cosh( β h )
exp ( i ˆ
h " h − 1
√ N X
j
W ( i )
j σ j ( t − 1) # + X
j
W j ψ j ) .
(4.91)
115

4 Learning in kinetic Ising mo dels
The in tegration o v er dW yields:
Z t ( ψ ) = Z D φ dh d ˆ
h e β σ i ( t ) h
2 cosh( β h ) exp ( i ˆ
h " h − λ \ t
√ N X
j
σ j ( t − 1) ψ j
− 1
√ N X
j
σ j ( t − 1) µ \ t
j − φ # + λ \ t
2 X
j
ψ 2
j + X
j
ψ j µ \ t
j ) ,
(4.92)
where D x = ( dx/ √ 2 π λ \ t ) e − φ 2 / 2 λ \ t is the probabilit y densit y for a Gaussian
v ariables with zero mean and v ariance λ \ t . One reco v ers (4.33 ) from ∂ log Z ( ψ )
∂ ψ j
and (4.35) from ∂ 2 log Z ( ψ )
∂ ψ 2
j in the limit ψ 0.
4.F Fixed p oint of the Exp ectation Propagation
algo rithm
F rom (4.33), (4.36) and (4.37) we find
µ j 1
λ −
T
X
t =1
1
˜
λ ! =
T
X
t =1
β
√ N σ j ( t − 1)  σ ( t ) − h tanh( β h \ t ) f t ( h t ) i \ t
h f t ( h t ) i \ t  . (4.93)
Using (4.38), this yields equation (4.18), while (4.24) is reco vered from (4.33)
and (4.104). F rom (4.36) and (4.38) w e observ e that
λ \ t = 1 + X
τ 6 = t
1
˜
λ ( τ ) ! − 1
≈ λ for large t, (4.94)
whic h is equiv alen t to (4.23). Hence, from (4.35), we get
1
˜
λ ( t ) = ˜
λ 2 ( t )
˜
λ ( t ) λ
β 2
N  2 h tanh( β h \ t )[tanh( β h \ t ) − σ ( t )] f t ( h t ) i \ t
h f t ( h t ) i \ t
+  σ ( t ) − h tanh( β h \ t ) f t ( h t ) i \ t
h f t ( h t ) i \ t  2 )
≈ β 2
N  2 h tanh( β h \ t )[tanh( β h \ t ) − σ ( t )] f t ( h t ) i \ t
h f t ( h t ) i \ t
+  σ ( t ) − h tanh( β h \ t ) f t ( h t ) i \ t
h f t ( h t ) i \ t  2 ) ,
(4.95)
116

4.G Complete Exp ectation Propagation
where in the last equalit y w e used (4.94). Inserting the ab o v e equation in
(4.38), the follo wing expression for the p osterior v ariance is found:
λ ≈ ( 1 + β 2
N
T
X
t =1  2 h tanh( β h \ t )[ σ ( t ) − tanh( β h \ t )] f t ( h t ) i \ t
h f t ( h t ) i \ t
+  σ ( t ) − h tanh( β h \ t ) f t ( h t ) i \ t
h f t ( h t ) i \ t  2 #) − 1
.
(4.96)
This equals the com bination of equation (4.23) and (4.19), when the latter is
expressed through the ca vit y fields.
4.G Complete Exp ectation Propagation
W e now appro ximate the p osterior (4.14) b y a gaussian distribution of the
form
q ( W ) = N ( µ , C ) (4.97)
where C is the full N × N co v ariance matrix. W e assume that q ( W ) can b e
also written as a pro duct of factors,
q ( W ) = 1
˜
Z
T
Y
t =0
˜
f t ( W ) , (4.98)
where ˜
f 0 ( W ) = f 0 ( W ) and for t = 1 , . . . , T
˜
f t ( W ) =  2 π | ˜
C t |  − N/ 2 exp " − 1
2 X
ij
( W i − ˜ µ t
i )( ˜
C t ) − 1
ij ( W j − ˜ µ t
j ) # . (4.99)
As b efore, w e in tro duce the ’ca vit y’ distributions:
q \ t ( W ) = q ( W )
˜
f t ( W ) = N ( µ \ t , C \ t ) , (4.100)
where
C \ t = ( C − 1 − ( ˜
C t ) − 1 ) − 1 , µ \ t = C \ t · ( C − 1 · µ − ( ˜
C t ) − 1 · ˜
µ t ) . (4.101)
W e then up date the approximate posterior by m atc hing its first and second
momen ts with the ones of the distribution
1
Z t
f t ( W ) q \ t ( W ) .
117

4 Learning in kinetic Ising mo dels
Those momen ts can b e calculated b y deriv atives of the generating functional
Z t ( ψ ) = Z dW q \ t ( W ) f t ( W ) e P j W j ψ j
=  2 π | C \ t |  − N/ 2 Z dW e − 1
2 P ij ( W i − µ \ t
i )( C \ t ) − 1
ij ( W j − µ \ t
j )
e β σ i ( t ) 1
√ N P j W j σ j ( t − 1)
2 cosh β
√ N P j W j σ j ( t − 1) e P j W j ψ j
=  2 π | C \ t |  − N/ 2 Z D φ dg d ˆ g e β σ i ( t ) g
2 cosh β g
e − i ˆ g h g − 1
√ N µ \ t · σ ( t − 1) − 1
√ N ψ · C \ t · σ ( t − 1) − φ i e 1
2 ψ · C \ t ψ + µ \ t · C \ t
(4.102)
in the limit ψ → 0, where D φ =  N
2 π σ ( t − 1) · C \ t · σ ( t − 1)  N/ 2 e − φ 2
2
N
σ ( t − 1) · C \ t · σ ( t − 1) .
The calculation of the first momen t yields:
µ j = µ \ t
j + β
√ N X
k
C \ t
j k σ k ( t − 1)  σ ( t ) − h tanh( β h \ t ) f t ( h t ) i \ t
h f t ( h t ) i \ t  , (4.103)
where the a v erage is a o v er the gaussian field with v ariance
( σ ( t − 1) · C \ t · σ ( t − 1)) / N
and mean
γ \ t = 1
√ N X
j
σ j ( t − 1) µ \ t
j . (4.104)
F or the second moments w e get
C ij = C \ t
ij + β 2
N X
k l
C \ t
ik C \ t
lj σ k ( t − 1) σ l ( t − 1)  h tanh 2 ( β h \ t ) f t ( h t ) i \ t
h f t ( h t ) i \ t − 1  .
(4.105)
As last step of the iteration, one ev aluates and store the new factors ˜
f t ( W )
(4.99), whose first t w o momen ts satisfy the follo wing equations:
˜
C t = ( C − 1 − ( C \ t ) − 1 ) − 1 , ˜
µ t = ˜
C t · ( C − 1 · µ − ( C \ t ) − 1 · µ \ t ) . (4.106)
4.H Details of the replica calculation
It is con v enien t to split the computation of the free energy in to t w o parts. The
first one represen ts the w eigh t of the coupling v ectors W which are con strained
118

4.H Details of the replica calculation
b y the order parameters:
F 0 = − lim
n → 1
∂
∂ n N − 1 ln Z n
c , (4.107)
with
Z n
c = Z Y
a
d W a e − 1
2 P a W a · W a Y
a<b Z dq δ X
ij
W a
i C ij W b
j − N q !
= Z Y
a
d W a e − 1
2 P a W a · W a Y
a<b Z dq d ˆ q
2 π e − ˆ q
2 { P ij W a
i C ij W b
j − N q } .
(4.108)
Note, there is no need to in tro duce extra conditions on diagonal o v erlaps as
they are tak en care of b y the prior. As w e did in the P ap er, w e can decouple
the in tegrals o v er differen t spins b y diagonalising C = U Λ U > and transforming
to new v ariables U > W a → W a whic h w e giv e just the same name. Hence
Z n
c = Z Y
a
d W a e − 1
2 P a P i ( W a
i ) 2 Y
a<b Z dq d ˆ q
2 π e − ˆ q
2 { P i W a
i Λ i W b
i − N q } . (4.109)
Once the sites are decoupled w e consider the limit N → ∞ and ev aluate the
in tegral with the saddle p oin t metho d. W e get the follo wing result, where w e
write Z n
c as a function of the parameters q , ˆ q ; the ph ysical v alue of the free
energy will b e then obtained b y extremization o v er q , ˆ q :
1
N ln Z n
c ( q , ˆ q ) = 1
N X
i
ln Z Y
a
dW a e − 1
2 P a ( W a ) 2 e − ˆ q Λ i
2 P a 6 = b ( W a W b − q )
= ˆ qqn ( n − 1)
2
1
N X
i
Λ i + 1
N X
i
ln Z D z 1
p 2 π (1 − ˆ q Λ i ) e − 1
2
ˆ q Λ i z 2
1 − ˆ q Λ i ! n
= ˆ qqn ( n − 1)
2
1
N T r( C ) − n − 1
2 N T r(ln( I − ˆ q C )) − 1
2 N T r(ln[ I + ( n − 1) ˆ q C ]) .
Using 1
N T r C = 1 and (4.107) one finds:
F 0 ( q , ˆ q ) = − 1
2 ˆ q ( q − 1) + 1
2 N T r(1 − ˆ q C ) (4.110)
The second term of the free energy in v olv es a v erages ov er the Gaussian random
fields (4.44). Its computation follo ws the steps of standard calculations for the
p erceptron learning problem [EVdB01, OK96, NY96] and the result is written
in (4.47).
119

5 Lea rning Curves fo r the inverse
Ising p roblem
5.1 Intro duction
Ha ving discussed the a v erage p erformance of inference algorithms for the ki-
netic Ising mo del in the previous c hapter, w e will no w examine the corre-
sp onding problem for the equilibrium case. W e aim to compute the a v erage
error of learning the couplings from indep enden t data generated b y the equi-
librium Ising mo del. As exact inference via the maxim um lik eliho o d metho d
is computationally in tractable for large systems, a v ast amoun t of literature
has b een dev oted to design appro ximate inference algorithms (for a review,
see [NZB17b]).
As for n umerical approac hes, Mon te Carlo metho ds can b e exploited in a
maxim um lik eliho o d algorithm [FS88, BDT + 07] or b e used to directly sample
from the p osterior probabilit y distribution of the parameters [F er16].
Mean field analytical appro ximations to lik eliho o d maximization ha v e b een
deriv ed b y information geometric approac hes [T an00], ca vit y metho ds [O W01b],
w eak-coupling expansions [Ple82]; a related tec hnique is based on a p ertur-
bativ e expansion of the en trop y functional in terms of connected correla-
tions [SM09]. These approac hes are exact in the thermo dynamic limit for
densely and w eakly in teracting systems, but constitute a p o or appro ximation
when the couplings are strong.
An appro ximation that is v alid also for net w orks with strong couplings w as
dev elop ed b y [CM11, CM12]. It consists in constructing and selecting sp ecific
subsets of v ariables of increasing size, called clusters. The algorithm retains
the clusters of v ariables con tributing most to the cross-en trop y and rejects the
small con tributions. It w orks w ell when the net w orks ha v e man y short lo ops.
In the opp osite limit, i.e. for net w orks with no lo ops, the Bethe-P earls
ansatz [P ei36] of pair-wise factorised form for the spin distribution is exact; it
can b e used within a v ariational appro ximation to reconstruct the couplings;
it is exact on trees but can b e also used on net w orks that are lo cally tree-lik e
[Bet35]. A related metho d is a message passing algorithm called susceptibilit y
propagation, whic h com bines b elief propagation [P ea14] and linear resp onse
121

5 Learning Curv es for the in v erse Ising problem
theory [NB12a]. Its p erformance is in v estigated in [NB12a, MVK10].
Based on v ery differen t approac hes, t w o consisten t estimators for the cou-
plings are giv en b y minim um probabilit y flo w [SDBD09] and pseudolik eliho o d
maximisation. The latter metho d, inspired b y logistic regression, w as de-
v elop ed in the statistics comm unit y [Bes74] and recen tly b ecame v ery p opu-
lar within the ph ysics comm unit y [AE12, Bes74, ELL + 13, MDP14]. The log-
lik eliho o d function to b e maximised is replaced b y a tractable sum of lo cal log-
lik eliho o d functions (i.e., distributions of single random v ariables conditioned
on the others). With resp ect to exact maxim um lik eliho o d, the computational
complexit y is reduced from exp onen tial to p olynomial in the system size (and
in the sample size). Moreo v er, it outp erforms most of the metho ds cited ab o v e
at lo w temp eratures [AE12, NZB17b].
In this c hapter, w e pro vide a setting for a theoretical comparison of the
p erformance of some of these algorithms. W e compute the a v erage error of
learning the couplings in the teac her-studen t scenario, where the teac her net-
w ork will b e k ept fixed during the calculation. As w e sa w in the last c hapter,
the computation requires p erforming ’thermal’ a v erages o v er studen t couplings
and quenc hed a v erages o v er the spin configurations. Suc h configurations are
distributed according to the Boltzmann measure, and the in tractabilit y of the
partition function constitutes the main tec hnical difficult y . A first approac h
to this problem w as published in [KK C98], a w ork that analyses the p erfor-
mance of v arious online algorithms for learning the parameters in a spin glass
from data ab out its metastable states. The authors, inspired b y the w ork of
P almer and P ond [PP79], considered an appro ximation for the distribution of
lo cal fields that factors in the sites. They sho w ed that it is p ossible to learn
the Hamiltonian from a small set ( O ( N )) of metastable states; ho w ev er, the
reconstruction error do es not matc h w ell with the one from sim ulations due to
the crude appro ximation on the field distribution.
In P ap er 3, using ideas from the ca vity meth o ds of statistical ph ysics [MPV87],
w e dev elop a formalism that tak es in to accoun t the correlations b et ween local
fields; in this w a y , w e are able predict the error with great accuracy for the first
time. An alternativ e approac h, based on a replica calculation, can b e found
in [Ber16]. Our metho d allo ws us to simply study the p erformance of algo-
rithms based on the minimisation of a lo cal cost function, suc h as maxim um
pseudolik eliho o d and a mean-field appro ximation to maxim um lik eliho o d. 1
W e also deriv e the form of the optimal lo cal cost function that ac hiev es
minimal error. The formalism to address this problem dates to the study of
one-la y er p erceptrons, when Kinouc hi and Catic ha (1992) presen ted a mo di-
1 Another recen tly prop osed algorithm based on con v ex optimization is in teraction screening
[VMLC16]; its p erformance has b een analysed in [Ber16].
122

5.2 P ap er 3
fied v ersion of the Hebb algorithm for the one-la yer perceptron that minimises
the generalisation error. W e follo w the more recen t approac h of [A G16] (in-
tro duced for the problem of optimal regression and follo w ed in [Ber16] for the
in v erse Ising problem) and p erform a functional minimisation of the error with
resp ect to the cost function.
Our results will dep end on parameters related to the statistics of the net w ork
that generated the data; in the last part of the c hapter, w e will sho w ho w suc h
parameters can b e estimated from the data.
The explicit equations for maxim um lik eliho o d and maxim um pseudo-lik elio o d
are giv en in App endix 5.A.
5.2 P ap er 3.
Author’s con tribution : I p erformed the analytical and n umerical calcula-
tions, prepared the figures and con tributed to writing the pap er.
123

J. St at. Mec h. ( 2 01 7 ) 063406
A statistical physics approach to
learning curves for the inverse
Ising problem
Ludovica Bachschmid-Romano and Manfred Opper
Department of Artificial Intelligence, Technische Universit ä t Berlin,
Marchstra ß e 23, Berlin 10587, Germany
E-mail: [email protected] and
[email protected]
Received 24 January 2017
Accepted for publication 6 May 2017
Published 23 June 2017
Online at stacks.iop.org/JSTAT/2017/063406
https://doi.org/10.1088/1742-5468/aa727d
Abstract.   Using methods of statistical physics, we analyse the error of
learning couplings in large Ising models from independent data ( the inverse Ising
problem ) . We concentrate on learning based on local cost functions, such as the
pseudo-likelihood method for which the couplings are inferred independently
for each spin. Assuming that the data are generated from a true Ising model,
we compute the reconstruction error of the couplings using a combination of
the replica method with the cavity approach for densely connected systems.
We show that an explicit estimator based on a quadratic cost function achieves
minimal reconstruction error, but requires the length of the true coupling
vector as prior knowledge. A simple mean field estimator of the couplings
which does not need such knowledge is asymptotically optimal, i.e. when the
number of observations is much larger than the number of spins. Comparison
of the theory with numerical simulations shows excellent agreement for data
generated from two models with random couplings in the high temperature
region: a model with independent couplings ( Sherrington – Kirkpatrick model ) ,
and a model where the matrix of couplings has a Wishart distribution.
Keywords: analysis of algorithms, learning theory, statistical inference
L Bachschmid-Romano and M Opper
A statistical physics approach to learning curves for the inverse Ising problem
Printed in the UK
063406
JSMTC6
© 2017 IOP Publishing Ltd and SISSA Medialab srl
2017
J. Stat. Mech.
JSTAT
1742-5468
10.1088/1742-5468/aa727d
P APER: Interdisciplinary statistical mechanics
6
Journal of Statistical Mechanics: Theory and Experiment
© 20 1 7 IOP P ublishing Ltd and SISS A Medialab srl
ournal of Statistical Mech anics:

J

Theory and Experiment
IOP
2017
174 2 - 5 4 6 8/ 17 /0 6 3 4 0 6 +2 8 $ 3 3 . 0 0

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1. Introduction
In recent years, there has been an increasing interest in applying classical Ising mod-
els to data modelling. Applications range from modelling the dependencies of spikes
recorded from ensembles of neurons [ 1 , 2 ] to protein structure determination [ 3 ] or
gene expression analysis [ 4 ] . An important issue for such applications is the so-called
Contents
1. Introduction 2
2. Estimators for the inverse Ising model 4
3. Local learning 4
4. Teacher – student scenario and statistical physics analysis 5
5. Cavity approach I: quenched averages 7
6. Replica result 8
7. Quadratic cost functions 9
8. The optimal local cost function 11
9. Cavity approach II: TAP equations and approximate
mean field ML estimator 12
10. Reconstruction error for MF-ML estimator 14
11. Asymptotics 15
12. Numerical results 17
13. Discussion and outlook 18
Acknowledgments 21
Appendix A. Details of the replica calculation 21
Appendix B. Saddle point equations for the order parameters 22
Appendix C. Relation between order parameters 23
Appendix D. Replica result for quadratic cost functions 24
Appendix E. Asymptotics from the replica approach 25
Appendix F. Asymptotic error for pseudo-likelihood estimator 25
Appendix G. Error dependence on the system size 26
References 27

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inverse Ising problem, i.e. the statistical problem of fitting model parameters, external
fields and couplings, to a set of data. Unfortunately, the exact computation of statisti-
cally ecient estimators such as the maximum likelihood estimator is computationally
intractable for large systems. Hence, to overcome this problem researchers have sug-
gested two possible solutions: the first one tries to approximate maximum likelihood
estimators by computationally ecient procedures such as Monte Carlo sampling [ 5 ]
or mean field types of analytical computations, see e.g. [ 6 – 9 ] . A second line of research
abandons the idea of maximising the likelihood function and replaces it by other cost
functions which are easier to optimise. The most prominent example is the so-called
pseudo-likelihood method [ 10 – 14 ] . In general it is not clear which of the two methods
leads to better reconstruction of an Ising model. The quality of such estimators, e.g.
measured by the mean squared reconstruction error of network parameters, will depend
on the problem at hand.
As an alternative to analysing specific instances of problems, one may study the
typical prediction performance of algorithms assuming that the true Ising parameters
are drawn at random from a given ensemble distribution. For such random prob-
lem cases, one can apply powerful methods of statistical physics to compute ( scaled )
reconstruction errors exactly in the limit where the number of spins grows to infinity
and the number of data is increased proportionally to the number of spins. Such an
approach has been applied extensively to statistical learning in large neural networks
in the past [ 15 – 17 ]   and also to learning in an Ising spin glass with binary teacher cou -
plings [ 18 ] , where learning is performed in an online fashon. In a previous paper [ 19 ]
we have applied this method to the learning from dynamical data which are modelled
by a kinetic Ising model with random independent couplings. This problem is theor -
etical ly simpler compared to the static, ‘ equilibrium ’ Ising case discussed in the present
paper. This is because the spin statistics of the dynamical model is fairly simple in the
‘ thermodynamic ’ limit of a large network and gives rise to Gaussian distributed fields.
We will show in the following that a related approach is possible to data drawn
independently from an equilibrium Ising model when we assume that couplings are
learnt independently for each spin using local cost functions. Although the spin statis-
tics is more complicated, computations are possible, when the so-called ‘ cavity ’ method
[ 20 ] is applicable to the true teacher Ising model.
The paper is organised as follows: section 2 explains the inverse Ising problem
and maximum likelihood estimation. Section 3 introduces simpler estimators which
are derived from local cost functions. In section 4 , we review the statistical phys -
ics approach for analysing learning performances within the so-called teacher student
scenario. In section 5 we explain the cavity method for performing quenched averages
over spin configurations. Section 6 presents explicit results of our method applied to
the inverse Ising model with independent Gaussian couplings ( SK-model ) . In section 7
we study the learning performance of algorithms based on local quadratic cost func-
tions and we compute the optimal local quadratic cost function. In section 8 we show
that an optimal quadratic function provides the best local estimator for the couplings.
Section 9 introduces further applications of the cavity method which allow us to sim-
plify order parameters corresponding to the true teacher couplings. As an example,
we compute the reconstruction error for an Ising model with Wishart distributed, i.e.
weakly dependent couplings. The method is also applied to re-derive a simple mean
field approximation to the maximum likelihood estimator. Section 10 explains how the

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mean field estimator can be obtained from a local cost function and presents results for
the reconstruction errors. Section 11 discusses the asymptotics of the reconstruction
errors for large number of data and relates these results to expressions known from clas -
sical statistics. Section 12 contains comparisons of our results with those of simulations
of the estimators and section 13 presents a summary and an outlook.
2. Estimators for the inverse Ising model
Let us consider a system of N binary spin variables

σ =( σ 0 ,...,σ
N − 1 )

connected by
pairwise interactions J ij and subject to external local fields H i . The probability distribu-
tion of the spin set is given by the Boltzmann equilibrium distribution

P

( σ | J , H )= Z − 1
Ising exp



β



i<j
J ij σ i σ j + β



i
H i σ i

 ,

( 1 )
where

Z Ising

is the partition function and β is the inverse temperature. Given a set of
M independent observations

{

σ

k } M
k =1

drawn independently from ( 1 ) , the inverse Ising
problem consists of estimating the model parameters

H

and

J

from the data. A stan-
dard approach for parameter estimation is the maximum likelihood ( ML ) method,
which has the properties of consistency and asymptotic eciency [ 21 ] . Maximum like-
lihood can be formulated as the minimisation of the following cost function ( negative
log-likelihood )

E

ML ( J , H )= −

M

k =1

ln P ( σ k | J , H

)

( 2 )
with respect to the matrix of couplings

J

and the field vector

H

. As is well known, the
minimisation of ( 2 ) is equivalent to a simple set of conditions for the first and second
moments of the ensemble ( 1 ) of spins: the parameters estimated by ML lead to the
matching of the empirical ( data averaged ) magnetisations to the magnetisation given
by the model ( 1 ) . Likewise we have the matching of all empirical pair correlations of
spins with their model counterparts. Despite the simplicity of this rule, the practical
minimisation of ( 2 ) requires the computation of these spin moments for a given set of
couplings and fields which is equivalent to averaging over 2 N spin configurations, which
is intractable for larger N . An approximation of such averages by Monte Carlo sampling
is possible but requires sucient time for equilibration. Alternatively, dierent approx-
imation techniques have been developed to provide a good estimate of the parameters
at a smaller computational cost, see e.g. [ 8 , 9 , 11 , 12 , 22 – 25 ] .
3. Local learning
If we neglect the symmetry of coupling matrix, i.e. the equality

J ij = J ji

, we can develop estimators which learn the ‘ ingoing ’ coupling vectors J ij for

j =0 ,...,i − 1 ,i +1 ,...,N − 1

for each spin

σ i

independently. It turns out that the

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corresponding ( local ) algorithms can often be performed in a much more ecient
way compared to the ML method.
In the following we will concentrate on the estimation of the couplings only and set
the external fields H i to zero. We will specialise on the couplings for spin
σ 0

and assum-
ing that the typical couplings J ij are variables with magnitude scaling like
1

/
√

N for
large N . We define a vector of rescaled couplings ( weights ) as
W

=( W
1

,...,W
N − 1

)
.

=
√

N ( J
01

,...,J
0 N − 1

)
.

( 3 )
We will assume that an estimator for
W

is defined by the minimisation of a cost
function
E ( W )=
M

k =1

E ( W ; σ k
)

( 4 )
which is additive in the observed data. An important and widely used case is the
pseudo-likelihood approach, where the cost function
E ( W ; σ )= − ln P ( σ

0
| σ

\ 0
, W )

= − βσ 0


j


=0
W j σ j
√ N + ln  2 cosh β


j


=0
W j σ j
√ N


( 5 )
is given by the negative log-probability of spin
σ 0

conditioned on all other spins
σ \ 0

.
In contrast to the ML approach, the gradient of this function can be computed in an
ecient way.
4. Teacher – student scenario and statistical physics analysis
We assume in the following that data are generated independently at random from
a ‘ teacher ’ network with coupling matrix
J ∗
ij

. A local learning algorithm based on
the minimisation of ( 4 ) produces ‘ student ’ network couplings
W

as estimators for
the teacher network couplings
W ∗ = √ N ( J ∗
01 ,...,J ∗
0 N − 1 )

. To measure the quality of a
given local learning algorithm, we will compute the average square reconstruction error
given by
ε = N
− 1

( W
∗
−

W ) 2 = Y
−

2 ρ +
Q,

( 6 )
where we define order parameters
Y = N
− 1

( W
∗

) 2 Q = N
− 1

( W ) 2 ρ = N
− 1

W
∗
·

W
,

( 7 )
representing, respectively, the squared lengths of the teacher and student coupling vec-
tors and the overlap between teacher and a student coupling vectors. Here the overline
defines an expectation over the ensemble of
M = αN

training data drawn at random
from an Ising model with teacher couplings
J ∗

, i.e.
( ... )=

σ

1
,...,σ

M
M


k =1
P ( σ k | J ∗ )( ... )
.

( 8 )

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Since there is often no explicit analytical solution to the minimisers

W

of ( 4 ) , we will
resort to a statistical physics approach which has been successfully applied to the analy -
sis of a great variety of problems related to learning in neural networks [ 15 – 17 ] . In this
approach one defines a statistical ensemble of student weights by a Gibbs distribution [ 26 ]
p ( W )=

1
Z

exp[

−

νE ( W )]

,

( 9 )
with the partition function

Z

=



d W exp[

−

νE ( W )]

,

( 10 )
where

1 /ν

represents an eective temperature which controls the fluctuations of the
‘ training energy ’

E ( W )

. Using techniques from statistical physics of disordered sys-
tems one computes order parameters at nonzero temperature and performs the limit

ν →∞

at the end of the calculation. The ‘ thermal average ’

 W 

with respect to the
distribution ( 9 ) converges to the minimiser of the cost function

E ( W )

. Order param-
eter s can be extracted from the quenched average of the free energy F corresponding to
( 8 ) using the replica method:
F =

−

N − 1 ν − 1 ln Z =

−

lim

n → 0

N − 1 ν − 1

∂
∂n

ln Z n

,

( 11 )
where the average replicated partition function for integer n is given by
Z n =



n


a =1

d W a

 
σ

P ( σ | J ∗ ) exp[ − ν
n


a =1

E ( W a ; σ )]

 αN
.

( 12 )
To allow for an analytical treatment, we assume that the local cost function

E ( W a ; σ )

depends on the spins and couplings only via

σ 0

and the local field

h .

=

1
√ N 

j



=0 W j σ

j

in the following way:

E ( W ; σ ) = Φ( σ 0 h ) .

( 13 )
Obviously, the pseudo-likelihood cost function ( 5 ) belongs to this class of functions.
The goal of the following section is to perform the expectation ( 12 ) . The resulting
expression depends on a set of order parameters and can for integer n be evaluated by
standard saddle-point methods in the limit

N →∞

. Performing an analytical continu-
ation for

n → 0

yields both the free energy and the self-averaging values of these order
parameters. While in most previous applications [ 15 – 17 ] of this programme to learn-
ing in neural networks, the quenched average over data in ( 12 ) is straightforward, the
required average over Ising spin configurations drawn from the distribution ( 1 ) cannot
be performed ( for arbitrary N ) in closed form. One might attempt a solution to this
problem by introducing a second set of replicas which would deal with the partition
function

Z − 1
Ising

in the denominator of ( 1 ) . We expect that such an approach can be car-
ried out for random teacher couplings but may lead to complicated expressions which
have to be carefully evaluated for

N →∞

. In the next Section we will use a simpler
approach using ideas of the cavity method [ 20 ] which allows, under certain assump-
tions on the teacher coupling matrix

J ∗

, the explicit computation of the quenched
average for

N →∞

.

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5. Cavity approach I: quenched averages
In order to perform the quenched averages in ( 12 ) , we will combine the replica approach
with ideas of the so-called cavity method. In doing so we write the Gibbs distribution
( 1 ) corresponding to the teacher couplings in the form
P

( σ | J ∗ ) ∝ exp


βσ 0

j  =0

J ∗
0 j σ j


P c av ( σ \ σ 0 )
,

( 14 )
where
P c av

denotes the distribution of the remaining spins in a system where the spin
σ 0

was removed , creating a cavity at this site, which gives the method its name. The
replicated partition function depends only on the fields
h

a
.

=
1
√ N  j  =0

W a
j σ j where
a ∈ {∗ , 1 ,...,n }

. The cavity assumption for the statistics of such fields in densely con-
nected systems can be summarised as follows: in performing expectations over
P c av

,
we can assume that dependencies between spins are so weak that random variables h a
become jointly Gaussian distributed in the limit
N →∞

. Hence, the joint distribution
of spin
σ 0

and the fields can be expressed as
P

( σ 0 ,h
∗ ,h
1 ,...h
n )=
1
Z 0

e βσ 0 h ∗ p c av ( h ∗ ,h
1 ,...h
n
)

( 15 )
with the normalisation
Z

0 =2


cosh( βh ∗ ) p cav ( h ∗ )d h ∗
.

( 16 )
Assuming that in absence of external fields we have vanishing magnetisations ( para-
magnetic phase ) , the distribution
p c av ( h ∗ ,h
1 ,...h
n )

is a multivariate Gaussian density
with zero mean and covariance


h a h b  =
1

N

i,j  =0

W a
i C \ 0
ij W b
j
.

( 17 )
The matrix
C \ 0

is the correlation matrix of the reduced spin system ( without
σ 0

) , which
does not depend on the couplings
W ∗

. We have C \ 0
ii =1

and assume that typically
C \ 0
ij = O ( 1
√ N )

for
i  = j

and large N . However, this scaling does not mean that we can
neglect the non-diagonal matrix elements. We will later see that they give nontrivial
contributions to the final reconstruction error. Within this framework, the quenched
average in ( 12 ) is rewritten in terms of integrals over the random variables h a as follows:


σ
P ( σ | J ∗ ) exp[ − ν
n


a =1
E ( W a ; σ )]
=

σ 0 

d h ∗
n

a =1

d h a
1
Z 0
exp [ βσ 0 h ∗ ] exp  − ν
n

a =1

Φ( σ 0 h a )  p c av ( h ∗ ,h
1 ,...h
n )
.

( 18 )
This result can be expressed by the covariances ( 17 ) which in the limit
N →∞

will
become self averaging order parameters which will be computed by the replica method
( appendix A ) . Under the assumption of replica symmetry ( which is expected to be

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correct for convex cost functions, which holds e.g. in the case of pseudo-likelihood ) ,
these new order parameters and their physical meaning are denoted as:
V .
=

1

N



i,j  =0
W ∗
i C \ 0
ij W ∗
j

R

.
=
1
N 
i,j  =0
W ∗
i C \ 0
ij  W j  w =
1
N 
i,j  =0
W ∗
i C \ 0
ij W a
j a  = ∗ ,

q

0
.
=
1
N 
i,j  =0
 W i W j  w C \ 0
ij =
1
N 
i,j  =0
W a
i C \ 0
ij W a
j a  = ∗ ,

q

.
=
1
N



i,j



=0
 W i  w C \ 0
ij  W j  w =
1
N



i,j



=0
W a
i C \ 0
ij W b
j a  = b  = ∗

,

( 19 )
where the brackets



...

 w

denote averages with respect to the distribution of
couplings ( 9 ) .
6. Replica result
Using a replica symmetric ansatz, the computations follow the approach summarised in
appendix A . In the zero temperature limit

ν →∞

the fluctuations of student couplings
vanish and we obtain the convergence of the order parameters

q 0 → q

with the limiting
‘ susceptibility ’

x .

= lim
ν →∞ ( q 0 − q ) ν = lim
ν →∞

ν

N


i,j  =0


 W i W j  w − W i  w  W j  w



C \

0

ij
remaining finite and nonzero. As a main result, we find that the auxiliary order param-
eters ( 19 ) are obtained by extremizing the limiting free energy function
F =

−

extr
q ,R,x



1

2

q − R

2

/V

x

+ α



d vG
β R,q ( v ) max
y

 −

( y − v )

2
2 x
−

Φ( y )

 ,

( 20 )
where G

µ,ω

( v

)

denotes a scalar Gaussian density with mean μ and standard deviation
ω . Remarkably, this free energy does ( for any fixed cost function

Φ

) only depend on the
teacher couplings

J ∗

via the order parameter V , defined in equation ( 19 ) . To compute
the prediction error, however, we need the ‘ original ’ order parameters ( 7 ) . These can be
expressed by the auxiliary ones q , R and x . This relation can be derived from the free
energy ( appendix C ) in a standard way by adding corresponding external fields to the
‘ Hamiltonian ’ in the Gibbs free energy ( 9 ) . This relation brings back further statistics
related to the teacher couplings

J ∗

via
ρ =

RY

V ,
Q =( q

−

R 2

V

) 1

N

Tr C − 1 + R 2 Y

V

2

,

( 21 )
with the corresponding reconstruction error

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ε =( q
−

R
2
V

) 1
N

Tr C − 1 + Y (1
−

R
V

) 2
.

( 22 )
In deriving these results, we have also assumed that for
N →∞

, 1
N

Tr ( C \ 0 ) − 1
→

1
N

Tr ( C ) − 1 .
Note that the prediction error is larger than the one we would get if we had neglected
the o-diagonal elements of the correlation matrix
C \ 0

. The error ( 22 ) depends on the
teacher couplings
J ∗

through the parameter Y and the parameter V ( the cavity variance
of the teacher field ) and through the trace of the inverse correlation matrix
C

corresponding to the teacher ’ s spin distribution. We will show later that the latter
quantity can be expressed by the former using a second application of the cavity
method. In the next section, we will see that the parameter V can be estimated from
the data.
We will illustrate the result ( 22 ) for the case of random teacher couplings
J ∗
ij

drawn
independently for i < j from a Gaussian density of variance 1. This corresponds to the
celebrated Sherrington – Kirkpatrick ( SK ) model [ 27 ] . For
β< 1

, i.e. outside of the spin-
glass phase, our simple form of the cavity arguments are known to be correct [ 20 ] and
one finds the values
V = Y =1 ,

lim
N →∞

1
N

Tr ( C ) − 1 = 1+ β 2
,

( 23 )
for zero magnetisations m i = 0 in the literature [ 28 ] . A comparison of the theory ( 22 )
with numerical simulations is shown in section 12 .
7. Quadratic cost functions
Among the simplest functions satisfying the property ( 13 ) , we consider quadratic cost
functions of the form
E

η ( W )=
1

2

i  =0 ,j  =0

W i ˆ
C ij W j − η
√

N

j  =0

ˆ
C 0 j W j
,

( 24 )
where the empirical correlation matrix is defined as
ˆ
C ij
.
= 1
M
M

k =1

σ k
i σ k
j
.

( 25 )
These allow for an explicit computation of the estimator in terms of a matrix inversion.
The estimator minimizing ( 24 ) is given by
W

η
i = η
√

N

j  =0

( ˆ
C − 1
− 0 ) ji ˆ
C 0 j i 
=0 ,

( 26 )
where the matrix
ˆ
C − 0

is the submatrix of
ˆ
C

where the 0th column and 0th row are
deleted ( not to be confused with the cavity matrix
C \ 0

) and η is a free parameter. The
estimation error can be computed from the free energy ( 20 ) by setting

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Φ(

h )= h

2
2
− η h,

( 27 )
and gives ( see appendix D )
ε =



βη
1+ β 2 V
− 1

 2

Y + η 2
( α

−

1)(1 + β 2 V )
1
N Tr C − 1

.

( 28 )
The optimal choice for the quadratic cost function ( 24 ) is found by fixing the parameter
η to the value that minimizes the error ( 28 ) , namely

η

opt = ( α − 1)(1 + β

2

V ) βY
( α

−

1) β 2 Y +( 1+ β 2 V ) 1

N

Tr C − 1

,

( 29 )
with the corresponding minimal error
ε opt = (1 + β 2 V ) Y 1
N Tr C − 1
( α

−

1) β 2 Y +( 1+ β 2 V ) 1

N

Tr C − 1

.

( 30 )
In general, the computation of the optimal parameter

η opt

requires the knowledge of
the three parameters Y , V and 1

N

Tr C − 1 which characterise the statistical ensemble to
which he unknown teacher matrix

J ∗

belongs. However, ( 29 ) simplifies as

α →∞

and
we get
lim

α →∞

η opt = 1+ β

2

V

β .

( 31 )
We will now show that the remaining parameter V can be estimated from the observed
data. We use the fact that at its minimum, the cost function ( 24 ) equals
E η ( W η )=

−
N
2

η 2 ∆

,

( 32 )
where we have used ( 26 ) and defined

∆= 

i



=0 ,j



=0

ˆ

C 0 i (

ˆ

C − 1
− 0 ) ij

ˆ

C 0 j

,

( 33 )
which only depends on the spin data. On the other hand in the situations where our
statistical physics formalism applies, the minimal training energy ( 32 ) will be self-aver-
aging in the thermodynamic limit

N →∞

and can be computed as the zero temper ature
limit of the free energy , i.e. the free energy function ( 20 ) evaluated at the stationary
values of the order parameters. The calculation in ( appendix D ) yields

∆=

1+ αβ

2

V
α (1 + β 2 V )

.

( 34 )
This shows, that the unknown parameter V and the asymptotically optimal parameter
η can be directly estimated from the observed spin correlations.
In the next section, we will show that the optimal quadratic cost function yields in
fact the total optimum of the reconstruction error with respect to free variations of the
cost function

Φ

.

a statistical physics approach to learning curves for the inverse Ising problem
11
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J. St at. Mec h. ( 2 01 7 ) 063406
8. The optimal local cost function
In this section, we will derive the form of the optimal local cost function
Φ

within the
cavity/replica approach and show that it is quadratic. Hence, the results of the previ-
ous section can be applied, where the optimal quadratic cost function was already com-
puted. We will give a derivation of this fact for the case of finite inverse ‘ temperature ’
ν , assuming that the argument can be continued to
ν →∞

.
The optimisation of cost functions for learning problems within the replica approach
goes back to the work of Kinouchi and Caticha [ 29 ] . We will follow the framework of [ 30 ]
( see also [ 31 ]) . Our goal is to minimise an error measure for a learning problem which is
of the form
ε ( R, q , q 0 )

such as ( 22 ) . It depends on order parameters which are computed
by setting the derivatives of a free energy function
F Φ ( R, q , q 0 )

( such as A.10 ) equal to
zero. The main idea is to take these conditions into account within a Lagrange function
ε

( R, q , q 0 )+

S ∈ R,q ,q 0

λ S
∂

∂S F Φ ( R, q , q 0 )
,

( 35 )
where the
λ S

are the corresponding Lagrange multipliers. The optimal function
Φ

is
obtained from the variation
δ
δ

Φ

S ∈ R,q ,q 0

λ S
∂

∂S F Φ ( R, q , q 0
)=0 .

( 36 )
For our problem, we can write ( see ( A.2 ) and ( A.10 ))
F Φ ( R, q , q 0 )= F 0 ( R, q , q 0 )
−
α
ν 

G β R,q ( v ) ln Ψ q 0 − q ( v )d
v,

( 37 )
where
F 0 ( R, q , q 0 )

is independent of
Φ

and
G µ,ω

( v
)

denotes a scalar Gaussian density
with mean μ and standard deviation ω . The free energy depends on
Φ

through the
function
Ψ

q 0 − q ( v ) .
=


G v ,q 0 − q ( y )e
− ν Φ( y ) d
y.

( 38 )
We will first derive a condition on the form of the optimal function
Ψ

from the variation
δ

δ Ψ

S ∈ R,q ,q 0

λ S
∂

∂S


G β R,q ( v )Ψ q 0 − q ( v )d v
=0 .

( 39 )
From this, we will recover the form of the optimal
Φ

. To obtain the derivatives with
respect to the order parameters we use the following rules for expectations over Gaussian
measures, which can be easily derived using integration by parts
∂
∂µ 

G µ,ω ( v ) f ( v )d v =


G µ,ω ( v ) ∂ v f ( v )d
v,

( 40 )
∂
∂ω 

G µ,ω ( v ) f ( v )d v = 1
2

∂
2
∂µ

2


G µ,ω ( v ) f ( v
)

( 41 )

a statistical physics approach to learning curves for the inverse Ising problem
12
https: / / doi.org/10.1 0 8 8 /1 7 4 2 - 5 4 6 8 / a a 7 2 7 d
J. St at. Mec h. ( 2 01 7 ) 063406

= 1
2 

G µ,ω ( v ) ∂ 2
v f ( v )d

v.

( 42 )
Hence, the derivatives required for ( 39 ) are

d
d R 

G β R,q ( v ) ln Ψ q 0 − q ( v )d v = β



G β R,q ( v ) ∂ v ln Ψ q 0 − q ( v )d

v,

( 43 )
d
d q 0



G β R,q ( v ) ln Ψ q 0 − q ( v )= 1
2



G β R,q ( v ) ∂

2

v Ψ q 0 − q ( v )
Ψ q 0 − q ( v ) d

v

= 1

2 

G β R,q ( v )



∂ 2
v ln Ψ q 0 − q ( v )+( ∂ v ln Ψ q 0 − q ( v )) 2



d

v,

( 44 )
d
d q



G β R,q ( v ) ln Ψ q 0 − q ( v )d v = β

2

2



G β R,q ( v ) ∂ 2
v ln Ψ q 0 − q ( v )d

v
−

d

d q 0 

G β R,q ( v ) ln Ψ q 0 − q ( v )d

v.

( 45 )
An application of standard variational calculus to a linear combination of these order
parameter derivatives shows that

∂ v ln Ψ q 0 − q ( v )= c 1 + c 2 ∂ v ln G β R,q ( v ) ,

( 46 )
where c 1,2 are independent of v . Since the logarithm of the Gaussian density

ln

G β R,q ( v

)

is a quadratic function in v , we conclude that also

ln Ψ

q

0 −

q ( v

)

is a quadratic expression
in the variable v , making

Ψ

q

0 −

q ( v

)

a ( non-normalised ) Gaussian density.
To conclude our argument on the optimal form of

Φ

, we use relation ( 38 ) . This
shows that the Gaussian density Ψ q

0 −

q ( v

)

is the convolution of a ( non-normalised )
Gibbs density

e
− ν Φ( y )

of a random variable y with the density

G

v ,q

0 −

q ( y )= G y ,q

0 −

q ( v

)

of a Gaussian random variable v . As a convolution corresponds to the addition two
random variables, we know that v + y is also a Gaussian random variable. Since v is
Gaussian, then

e
− ν Φ( y )

is also a Gaussian density and

Φ( y )

is quadratic in y . We have
a already computed the best quadratic cost function in the previous Section, and we
conclude that the estimator ( 26 ) with ( 29 ) is the best local estimator of the couplings.
9. Cavity approach II: TAP equations and approximate mean field ML estimator
So far we have ignored the symmetry of the coupling matrix by restricting ourselves
to estimators derived from local cost functions. In this Section, we will discuss a well
known approximation [ 32 ] of the ( symmetric ) maximum likelihood estimator which is
based on mean field theory. We will re-derive this estimator using the more advanced
( adaptive ) TAP mean field theory, because its results for the spin correlation matrix
will also be needed in the following. We will later compute its reconstruction error in
section 10 . Our starting point is a generalisation of the well known TAP mean field
approach developed for the SK model. Using the cavity approach [ 32 ] one derives the
following ‘ adaptive ’ TAP equations for the magnetisations

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