Local Memor y- A ware K ernel Perforation
Daniel Maier
daniel.maier@tu- berlin.de
T e chnische Universität Berlin
Germany
Biagio Cosenza
cosenza@tu- berlin.de
T e chnische Universität Berlin
Germany
Ben Juurlink
b .juurlink@tu- berlin.de
T e chnische Universität Berlin
Germany
Abstract
Many applications provide inher ent resilience to some amount
of error and can potentially trade accuracy for performance
by using approximate computing. Applications running on
GP Us often use local memor y to minimize the number of
global memory accesses and to sp eed up execution. Lo cal
memory can also b e very useful to improve the way appr oxi-
mate computation is performed, e.g., by improving the qual-
ity of approximation with data r econstruction techniques.
This paper introduces local memor y-aware perforation te ch-
niques specifically designe d for the acceleration and approx-
imation of GP U kernels. W e propose a local memor y-aware
kernel perforation technique that first skips the loading of
parts of the input data from global memory , and later uses
reconstruction techniques on local memor y to reach higher
accuracy while having performance similar to state-of-the-
art techniques. Experiments show that our approach is able
to accelerate the execution of a variety of applications from
1
.
6
×
to 3
×
while introducing an average error of 6%, which is
much smaller than that of other approaches. Results further
show how much the err or depends on the input data and
application scenario, the impact of local memory tuning and
different parameter configurations.
CCS Concepts • Computing methodologies →
Graph-
ics processors ;
• Software and its engineering →
Software
notations and tools ;
K eywords
approximate computing, GP U, kernel perfora-
tion
A CM Refer ence Format:
Daniel Maier, Biagio Cosenza, and Ben Juurlink. 2018. Local Memor y-
A ware K ernel Perforation. In Proceedings of 2018 IEEE/A CM Interna-
tional Symposium on Code Generation and Optimization (CGO’18) .
A CM, Ne w Y ork, N Y , USA, 10 pages. https://doi.org/10.1145/3168814
CGO’18, February 24–28, 2018, Vienna, Austria
© 2018 Copyright held by the owner/author( s). Publication rights licensed
to the Association for Computing Machinery .
This is the author’s version of the work. It is posted here for your personal
use. Not for redistribution. The definitive V ersion of Record was published in
Proceedings of 2018 IEEE/ACM International Symposium on Code Generation
and Optimization (CGO’18) , https://doi.org/10.1145/3168814 .
1 Introduction
A pproximate Computing ( A C) exploits the gap between the
accuracy provided by a system and the accuracy required
by an application. Many applications are resistant to some
amount of error and earlier w orks in this field have pro ven
that there is potential for significant improv ements in terms
of execution time or energy consumption if a small amount
of error can be accepted [
3
]. The rationale behind A C is
that an application can provide acceptable output quality
even though the system e xecuting the application was in-
exact in some way . This property is shared by applications
from many domains, including signal processing, machine
learning, audio and video processing [
20
,
21
]. Research of
approximate techniques has also been conducte d from many
different perspectives: related work ranges from softwar e-
based approaches [
10
,
15
,
19
] and programming language
support [
7
,
12
,
17
,
24
] to compiler-based approaches [
16
,
19
]
and hardware-based techniques [ 18 ].
Applications suited for acceleration by A C provide inher-
ent application resilience [
3
], i.e., the y can produce acceptable
results despite some of their underlying computations being
incorrect or approximate . For example, in a photo, pixels
that are in an adjacent location potentially have similar val-
ues. This property is well-known and exploited by many
applications, e.g., by image/video and data compression. Ap-
plications in the context of digital audio signals created by
sampling continuous analog signals already introduce some
amount of error because of quantization noise. This and sim-
ilar contexts, therefor e, ar e already required to deal with
some amount of error .
As one of the goals of A C is to improve performance, se v-
eral implementations have combined the high-throughput of
massively parallel architectures such as GP Us with approxi-
mate computing techniques [
5
,
8
,
15
,
16
,
23
]. The GP U is a
very interesting architecture for the application of such tech-
niques. GP U programming models expose different types of
memory , which are explicitly selected by the programmer .
On the one hand, there is a large amount of global memory
1
.
When accessing global memory , a fairly high latency has
to be accepte d. While this latency can be hidden partly by
scheduling a different batch of threads, in fact many applica-
tions are memory b ound. On the other hand, there is a small
amount of local memor y , which is accessed with very low
1
Throughout this paper we use OpenCL terminology . In CUDA terminology
work groups are also known as thread blocks , and local memory is also known
as shared memory .

CGO’18, February 24–28, 2018, Vienna, A ustria Daniel Maier, Biagio Cosenza, and Ben Juurlink
latency . Additionally , local memor y is shared by all threads
in a work group .
In previous w ork, GP U applications have been accelerated
by perforating the execution of loops [
15
,
19
]. Howe ver , these
works are limited in two aspects. First, the acceptable error
is chosen to be 10% on average . That is unacceptable for
many applications that cannot tolerate such high amount of
error . Therefore, the potential impact of such techniques is
limited. Second, most of the use d benchmark applications
do not use local memor y . Among the few applications that
actually do use local memor y , there is no exploitation of local
memory for the approximation technique. This means that
existing appro ximation te chniques do not take the model of
the specificity of the GP U memory mo del into account.
This paper presents a novel appr oach to p erform loop
perforation on GP Us, namely kernel p erforation , which is
aware of the GP U memory hierarchy and makes use of the
fast local memor y in order to achiev e higher accuracy with
the same performance. The central idea of our approach is to
save memory accesses by approximating reads fr om the input
buffers of the kernel. Being aware of the GP U memory mo del,
our approach focuses on (1) loading only few data points
from (the slow er) global memory while (2) using the (faster)
local memor y to cache adjacent data points and (3) improv e
accuracy with a local data reconstruction technique. Local
reconstruction techniques attempt to reconstruct the input
value out of a sparse set of globally-fetched data p oints, thus
exploiting inherent application r esilience of the applications.
The approach also extends e xisting work on perforation also
with novel memory patterns and an analysis of the accuracy
on six applications with different input data.
The contributions of this paper are:
•
a novel local memory-aware approximation appr oach
for OpenCL kernels based on lo op perforation ( ker-
nel perforation ) that approximates the input data of
GP U applications by reducing the amount of data
loaded from global memory and reconstructing a high-
accuracy approximation with a local reconstruction
technique;
•
a set of local reconstruction techniques that work on
local memor y and efficiently combine the sparse data
fetched by global perforation schemes while consis-
tently improving the accuracy of the appr oximation;
•
experimental results on six benchmarks with different
input data and different combination of parameters
(perforation schemes, reconstruction techniques, lo cal
work-group size), wher e we show spee dups of 1
.
6
×
to
3
×
while maintaining a moderate amount of error on
an AMD FirePro W5100 GP U.
2 Related W ork
Approximate computing has become a hot topic with appli-
cations in many different fields and differ ent approaches [
4
,
6
,
8
,
11
,
24
]. A thorough ov er view can be found in the sur vey
paper of Mittal [ 14 ].
Lipasti et al. [
9
] presented a hardware-based approach
called Load V alue Prediction, which skips the execution stall
due to a cache miss by predicting the value based on local-
ity . Howe ver , if the error of a predicted value is too large a
rollback is necessary . Load V alue Approximation [
11
] over-
comes this limitation by not verifying the predicted values,
thus not involving the burden of r ollbacks.
Y azdanbakhsh et al. [
25
] presented a similar approach for
GP Us that focuses on memor y bandwidth, instead of the sole
latency . A fraction of cache misses is appro ximated without
any checking for the quality of the predictions. The predictor
utilizes value similarity across thr eads. The programmer
must specify which loads can b e approximated and which
are critical. The fraction to be approximated is used as a knob
to control the appro ximation error .
Several r elate d works use software-based approaches for
leveraging application’s r esilience to some amount of error .
An analysis of inherent application r esilience has be en con-
ducted by Chippa et al. [
3
]. They presented a framew ork for
A pplication Resilience Characterization (ARC) that partitions
an application into resilient and sensitive parts, and pr op osed
approximation models to analyze the resilient parts.
Loop perforation has b een presented by Sidiroglou et
al. [
19
]. While many applications are designed to trade-off
accuracy for performance using domain-spe cific techniques,
loop perforation is a generalize d approximation technique
that can be applied in a variety of contexts. Once critical
(tunable) loops are identified, the number of iterations of
such loops can b e decreased. By limiting the amount of error
of the final result to 10% at maximum, a typical spee dup of
2 × can be achieved.
Li et al. [
8
] introduced a GP U-sp ecific approach based on
the Special Function Unit (SF U) commonly available in, e.g.,
N VIDIA GP Us, which provides acceleration for transcen-
dental functions. A tunable approximation is achie ved by
dividing the work into warps that e xecute the accurate func-
tions and warps that execute the SF U-based approximate
functions.
Samadi et al. [
16
] presented Sage , a frame work consisting
of a compilation step in which a kernel is optimized using
approximation techniques and a runtime system that en-
sures that the target output quality criteria are met. The y
employed three GP U-specific optimization techniques: dis-
carding of atomic operations, data packing/compression, and
thread fusion. Even though Sa ge takes into account GP U-
specific limitations, it do es not exploit the compute unit’s
local memor y to benefit from the low latency and shared
memory .
P arapro x [
15
] is a framework for transpar ent and au-
tomated approximation of data-parallel applications. Input
to the framework is an OpenCL or CUD A kernel, which

Local Memor y- A ware Kernel Perforation CGO’18, February 24–28, 2018, Vienna, A ustria
is parametrized by applying different appro ximation tech-
niques, depending on the detecte d data-parallel pattern. A
runtime helper is used to cho ose those kernel parameters
that meet the spe cified output quality . For an error budget
of 10% they reported an average performance gain of 2 . 7 × .
Mitra et al. [
13
] recognized that there are differ ent phases
in many applications, each with very different sensitivity
to approximation. The y presented a framew ork that detects
these phases in applications and searches for specific approx-
imation levels for each of the phases. For an err or budget of
5% they report a speedup of 16%. By allowing for an error
budget of 20% the spee dup increases to 72% on average .
Lou et al. [
10
] presented image perforation, a technique
specifically designe d for accelerating image pipelines. By
transforming loops so that they skip certain samples that
are particular expensive to calculate speedups of 2
×
up to
10
×
were r ep orted. Subsequent pip eline stages rely on the
presence of these samples, and they can be reconstructed
using different methods (nearest-neighbor , Gaussian and
multi-linear interpolation). The pipeline can b e modifie d
by a storage optimization that replaces accesses to skipped
samples with on-demand reconstruction code.
3 Over view
This paper introduces a novel appro ximation te chnique that
is specifically designe d to approximate general-purpose GP U
kernels. The proposed approach extends state-of-art appr ox-
imation techniques such as the row-/column-based schemes
used in P arapro x [
15
] by exploiting the GP U’s fast local
memory to deliver more accurate solutions.
Input
buffer
Kernel
execution
Output
buffer
(a) A ccurate GP U application
Input
buffer
Data per-
foration
Data
recon-
struction
Kernel
execution
Output
buffer
(b) Our approach
Figure 1.
Accurate GP U application and lo cal memory-
aware kernel perforation approach.
In typical GP U applications, as depicted in Figure 1a , a
GP U kernel first fetches data from the input buffer in global
memory , then it p erforms its computations, and finally it
writes the result to the output buffer in global memory . The
penalty for accessing the global memor y is in general very
high, although it can be hidden to some extent by the mas-
sively parallel ar chitecture of the GP U and its scheduler . A
way to improv e the p erformance of GP U kernels is to make
use of fast local memor y , whose access latency is significantly
smaller than the one for global memory .
The rationale behind our approach is that fast local mem-
ory can also b e exploited for more accurate appro ximation.
Figure 1b shows ho w our local memor y-aware kernel perfo-
ration approach extends the original application with tw o ad-
ditional steps: a data perforation phase that fetches a part of
the input data; a data reconstruction phase that reconstructs
the missing data elements and works on local memory .
In Section 4 we describe how a kernel is perforate d and
what part of the input data is selecte d to be fetched from
memory . The successive step implementing the reconstruc-
tion phase is describe d in Section 5 . Experimental evaluation
and discussion are presented in Section 6 . Finally , we con-
clude our work in Section 7 .
4 K ernel Perforation
Sidiroglou et al. [
19
] introduced loop p erforation , an approxi-
mation technique that improves the performance of a loop
execution by skipping some iterations. Loop perforation has
be en originally applied to sequential code and can b e eas-
ily parametrized through tunable loops in order to trade
accuracy for performance.
In this work, we apply perforation to parallel Op enCL
kernels: a loop is a kernel whose iterations correspond to
OpenCL work-items. Therefor e, we apply kernel perforation
to a parallel kernel instead of a sequential loop.
4.1 From Loop to K ernel Perforation
When applying perforation to a kernel, there are differ ent
aspects to consider . While lo op perforation works at loop
iteration level, our perforation appr oach focuses on the data
( e.g., buffers) used by a kernel because memor y accesses are
an important component of GP U performance.
In particular , we distinguish between input approximation
and output approximation : input appr oximation is the pr ocess
of approximating data on the input of a GP U kernel while
output approximation does the opposite, i.e ., approximating
data on the output of a GP U kernel.
T o illustrate this concept, consider the following accurate
program ( we use loops for simplicity , but te chniques apply
to kernels similarly ):
for ( i = 0 ; i < n ; i + + ) {
o u t p u t [ i ] = c a l c ( i n p u t [ i ] ) ;
}
For every input element the function
calc
is called. Its result
is written to the output element. In this example , the function
calc
requires the
i
-th input data element and the loop is
executed n times.
By applying loop perforation, e.g., every three iterations,
we are actually appr oximating the output array , which con-
tains the results of the computations. Therefor e, we are per-
forming an output perforation of the loop.
for ( i = 1 ; i < n ; i + = 3 ) {
o u t p u t [ i ] = c a l c ( i n p u t [ i ] ) ;

CGO’18, February 24–28, 2018, Vienna, A ustria Daniel Maier, Biagio Cosenza, and Ben Juurlink
o u t p u t [ i + 1 ] = o u t p u t [ i ] ;
o u t p u t [ i + 2 ] = o u t p u t [ i ] ;
}
The output is calculated for the
i
-th element, while it is ap-
proximated for
output[i+1]
and
output[i+2]
. The loop is
executed
n
3
times. Approaches such as P arapro x are output
approximation, because the output of the kernel execution
is approximated.
While output approximation grants high performance
improv ements, it has two limitations: It usually introduces a
very high error , and it do es not take into account the possible
memory reuse of the input data. A way to overcome this
problem is to implement the data perforation in the input
data of the loop, e.g.:
for ( i = 0 ; i < n ; i + = 3 ) {
x 0 = i n p u t [ i ] ; / / d a t a p e r f o r a t i o n
x 2 = i n p u t [ i + 2 ] ;
x 1 = ( x 0 + x 2 ) / 2 ; / / d a t a r e c o n s t r u c t i o n
o u t p u t [ i ] = c a l c ( x 0 ) ;
o u t p u t [ i + 1 ] = c a l c ( x 1 ) ;
o u t p u t [ i + 2 ] = c a l c ( x 2 ) ;
}
In this example , first
x0
and
x2
are loaded from the input
array . Then
x1
is approximated by calculating the linear
interpolation b etween
x0
and
x2
. Finally , program e xecution
continues and the result for the input data at position
i , i +
1 , i + 2 is calculated analogous to the accurate program.
The approximation schemes pr esented in this paper p er-
form input approximations, wher e the input buffers ar e ap-
proximated before they serve as an input to the OpenCL
kernel computation.
While this is a simple one-dimensional application of
the approach, it can be easily extended to two- and three-
dimensional kernels, where perforation is performed, e.g.,
at row or column le vel. T wo-dimensional appro ximation
schemes are further described in Section 4.3 .
4.2 Local Input Perforation
The approach of input appro ximation is motivated by two
observations: Many applications are inherently resilient to
the input as well as the output and, ther efore, the y can toler-
ate small errors. Memory accesses on GP Us have a very long
latency , and approximation of the input may take advantage
of low-latency local memory to improve the appr oximation.
Most real-life data contain r edundancy , for example there
is a spatial locality in digital images. Additionally , this data
often contains noise, ( e.g., quantization noise), and hence
is inaccurate. Such data is the input to many applications.
Input approximation w orks by skipping the loading of some
of the input data. If the input data is two-dimensional, e.g.,
an image, a possible input perforation scheme may skip
every other row . Figure 2 shows an example of r ow-based
approximation. The err or introduced by data perforation
(a) original (b) perforated (c) approximated
Figure 2. Original, perforated and approximated data [ 22 ].
is visible as black lines in Figure 2b ; Figure 2c sho ws the
input data reconstruction. In general, input approximation
can be a suitable acceleration te chnique for any application
that processes data with redundancy and is resilient to some
amount of error in its input data. This is an advantage ov er
output approximation techniques that requir e spatial locality
in the output data. Although it has be en shown that output
approximation can be used for many applications, this is a
conceptual limitation.
The usage of local memor y to prefetch data fr om global
memory is a well-known technique to accelerate GP U ker-
nels. Applications’ execution time usually benefits from the
usage of local memor y if there is significant r euse of data
across threads. Data reuse means the data loaded by a thread
is also used by other threads which in turn also load data.
In the OpenCL programming model, this is usually im-
plemented using the local memor y , which is shared among
all threads in a work gr oup. On GP Us, the latency of lo cal
memory is far lower compared to the latency of accessing
the global memory , but its size is rather limite d. Therefore ,
we use the local memory to implement the steps (Ia) data
perforation and (Ib) data reconstruction shown in Figure 1b .
4.3 Perforation Schemes
P arapro x first presented an implementation of kernel perfo-
ration [
15
] by detecting different data-parallel access schemes
and generating new kernels that use scheme-specific ap-
proximation techniques. For instance , when a stencil ac-
cess scheme is detected, three approximated kernel versions
are generated, each implementing a different appr oxima-
tion scheme. The scheme determines which elements ar e
computed and which elements are to be approximated, and
the approximation of elements is accomplished by copying
adjacent ( calculated) values. Figure 3 shows a visual r epre-
sentation of the schemes on a 2D kernel. Colored elements
(a) Ro w (b) Column (c) Center
Figure 3. Appro ximation schemes use d in P araprox .

Local Memor y- A ware Kernel Perforation CGO’18, February 24–28, 2018, Vienna, A ustria
are calculated and white elements are copied from adjacent
calculated values. Scheme (a) calculates a ro w of results and
uses this row to appr oximate the adjacent ro ws on the top
and on the bottom of this row . Scheme ( b) procee ds analo-
gous but with columns instead. The most aggressiv e scheme
is ( c), which only calculates the value in the center and ap-
proximates all adjacent values.
In P araprox , appr oximation is accomplished by cop ying
the calculated result to adjacent result values. A s described
later in Section 5 , we introduce more accurate ways to recon-
struct these values with different r econstruction te chniques.
The reported spe edup for these schemes ranges from more
than 1
.
7
×
for
ConvolutionSeparable
to more than 3
×
for
Gaussian
and
Mean
. The spee dup can be mainly attribute d
to a reduced number of global memor y accesses, as the ap-
plications contain only a few multiplications and additions
for each calculated output element. Howev er , in the case
of the Gaussian filter , a 3
×
3 filter kernel leads to 9 data
elements read. If w e assume that scheme ( c) is applied, every
third data element in x and y dir ection is calculated and the
remaining data elements are appr oximated. Nonetheless, ev-
ery data element is loaded once from global memory , as the
calculation of the not approximated data elements depend on
them (assuming a 3
×
3 -sized filter kernel). Finally , P araprox
assumes a very high maximum error of 10%.
4.4 GP U Perforation Schemes and Halos
The implementation of an input perforation scheme deter-
mines which data is loaded from global memory to lo cal
memory . Three imp ortant aspects are considered. First, the
scheme needs to match with the memor y architecture in
a way that preferably no data that is loaded from memory
is discarded by the scheme. For GP Us the memory access
granularity depends on different factors but in general is at
least 32 bytes. Second, the scheme also nee ds to match the
applications input data structure . For example , if the input
data contains a line-shaped structure, skipping lines while
loading the data increases the error much mor e than skip-
ping, e.g., columns. And thir d, the scheme needs to take into
account the organization of threads in work gr oups.
W e present two types of schemes that ov ercome such
limitations. Both of them assume that the input data contains
spatial locality , which means that adjacent data elements
have a high similarity . This is often the case for vide o or
image data. The approach is not limited to image processing
applications. Furthermore , the redundancy does not nee d to
come from spatial locality . Any input data that contains a
known redundancy structure can be used as a template for
designing a perforation scheme.
From a statistical point of view , a scheme with randomly
selected data elements to b e approximated would be the
best choice, be cause then the error due to appr oximation
is equally distributed over the input data. Furthermore , a
random scheme is less likely to hide structures in the input
data. Howe ver , such a random scheme would interfere with
the way memory is accesse d on a GP U , where whole lines
of memory are fetched in one transaction.
Row A pproximation Scheme
Fetching one data element
from memory also induces the fetching of data elements
in the same row . As global memory accesses are affected
by a relatively long latency , it is clear that any data that is
actually fetched from global memory should also b e used to
improv e the approximation. Our row appr oximation scheme
(Figure 4 ) skips the loading of ro ws in a work group tile and,
therefore , adjacent elements in a row are always used. As
in adjacent work groups the same appr oximation scheme is
applied, the schemes match each other .
(a) Ro w scheme 1 (b) Row scheme 2
Figure 4. Row appr oximation scheme.
Stencil approximation scheme
Figure 5 shows a stencil
approximation scheme for a tile size of 6
×
6 and a stencil
kernel size of 3
×
3 . T o compute the accurate result, an extra
row on top and on the bottom as well as an e xtra column
left and right need to b e fetched additionally . This approxi-
mation scheme only fetches the block in the center (drawn
in blue) and approximates the e xtra data elements based on
their neighbors. The data elements on the b oundaries have
influence on a smaller number of the stencil calculations
than data elements, that are not on the border . This property
is leveraged by the appro ximation scheme.
Figure 5. Stencil appro ximation scheme.
5 Reconstruction
By skipping elements in the input data, an error is intro-
duced. The purpose of the reconstruction step is to minimize
this error . Such reconstructions may use very different tech-
niques, depending on the type of application and the input
that is targeted. The simplest approach is to appro ximate
the missing elements by copying the values of adjacent data

CGO’18, February 24–28, 2018, Vienna, A ustria Daniel Maier, Biagio Cosenza, and Ben Juurlink
elements. Depending on the data access scheme, that deter-
mines which elements from the input data is needed for the
calculation of one element in the output data, this approach
can be quite effe ctive .
In general, the usage of local memor y can accelerate the
execution of applications, if there is data reuse acr oss differ-
ent threads. By using local memory , which is shared among
all threads in a w ork group, data r euse can b e accomplished.
This approach is well-kno wn and can be considered a stan-
dard optimization technique for GP U applications.
Howe ver , if output approximation is applied to an ap-
plication that already uses local memory and there is data
reuse , the advantage of the approximation is v er y small, as
the whole input data needs to b e loaded ( be cause of data
reuse) and there is global memory access that can be saved
by approximating it.
An example of this situation can be taken from P arapro x :
Consider a filter kernel size of 3
×
3 and the proposed stencil
approximation techniques that skip the calculation of either
every other row , every other column, or both. The last option
calculates the result for only 1 out of 9 elements in the output
data, as Figure 3 shows. Ho wev er , all data elements in the
input data are accessed at least once. If we assume that local
memory is use d to prefetch the data plus the surr ounding
elements, the number of approximated memory accesses is
zero and hence the speedup, that was due to approximated
global memory accesses, de clines.
5.1 Reconstruction T echniques
After loading the incomplete data to the local memor y (data
perforation), the missing data nee ds to be reconstructe d.
Ideally , a perfect reconstruction of the missing data is de-
sired. Howe ver , as there is information missing, a perfect
reconstruction is not possible. Therefor e, w e aim to mini-
mize the error . For the reconstruction different options are
possible. In this work w e compare two differ ent types of
data reconstruction techniques on the sparsely fetched data:
nearest-neighbor interpolation and linear interpolation.
The data reconstruction method depends on the approxi-
mation scheme that was used and not all combinations are
possible.
Nearest-neighbor Interpolation
A straight-forward ap-
proach for the completion of the perforated data is nearest-
neighbor interp olation. Data elements that wer e not loaded
are appro ximate d by picking the nearest value that was
loaded as a replacement value.
Linear Interpolation
Another well-known technique is
linear interpolation. For this method, it is ne cessary that the
element to be approximated has adjacent elements on both
sides. This requirement is not always true , see for example
the edges of the stencil scheme in Figure 5 . In this case we
employ nearest-neighbor interpolation.
6 Experimental Evaluation
T o evaluate our approach, we have r eproduce d the approx-
imation schemes used by P arapro x (as described in Se c-
tion 4.3 ) and compared them with our approach in terms of
error as well as speedup. Furthermore , we extended P ara -
pro x ’s schemes with a more aggressive perforation scheme
that approximates 4 instead of 2 rows or columns. W e apply
their approach to a variety of benchmarks. Howe ver , we are
not able to reproduce the numbers that were reported in
the original work. This can be explained by the usage of dif-
ferent hardwar e and different benchmark implementations.
Moreover , some benchmarks are more sensitive to differ ent
input data, as we show in Figur e 6 .
Our results wer e conducte d on an AMD FirePro W5100
GP U with 3.5 GB memory using OpenCL driver AMD- APP
version 17.10-414273 supporting OpenCL version 1.2.
T able 1.
Details of the applications that have been used in
the evaluation.
Application Domain Error Metric
Ga ussian Image processing Mean relative err or
Median Medical imaging Mean relative err or
Hotspo t Physics simulation Mean relative err or
Inversion Image processing Mean r elative error
Sobel Image processing Mean err or
6.1 Benchmarks
W e manually applied our approach to six benchmarks. An
overview can be found in T able 1 . For all except one bench-
mark we use the mean r elative error (MRE) as metric. The
MRE is determined by calculating the difference of result
and the value and then dividing by the true value:
x t r u e − x t e s t
x t r u e
.
Howe ver , when
x t r u e
is zero or close to zer o the MRE is ei-
ther very high or undefined. The Sobel 3/ Sobel 5 applications
are particularly prone to such situations. Ther efore, we opt
to use the mean error as metric for these two applications
instead, which does not suffer from this limitation. Input and
output are grayscale images sized 1024
×
1024 for Ga ussian ,
Inversion , Median and Sobel 3/ Sobel 5. For Hotspo t we
used the 1024 sized input data sets provided by Rodinia [ 2 ].
The Ga ussian filter is a well-known low-pass filter . Low-
pass filtering has many applications, e.g., in electronics and
signal processing. A reduction in noise and detail is an impor-
tant preprocessing step in image processing, e .g., for building
edge detection applications, as they are particularly sensitive
to noise. The Ga ussian has data-reuse across thr eads and
therefore benefits from the use of local memory in general.
The filter kernel size is 3 × 3 .
The Inversion filter is an application that computes the
digital negative of an image. W e use this artificial b enchmark
to assess the performance of applications with 1
×
1 filter

Local Memor y- A ware Kernel Perforation CGO’18, February 24–28, 2018, Vienna, A ustria
kernels. As ther e is no data reuse across thr eads, such appli-
cations usually do not benefit from the use of local memor y .
The Median filter is a nonlinear spatial filter with applica-
tions in medical imaging and image processing. It is particu-
larly effective in reducing salt-and-pepper noise , which are
sudden and sharp signal disturbances. By applying the filter
to a signal, each sample is replaced by the median of the
samples in the neighborho od. T o calculate the result of the
filter , first all values of the filter mask need to b e sorted by
their value. The median value is selected and use d as result.
Our implementation is using local memor y for prefetching
of data elements from global memory . Additionally , we are
using private memory to load all samples in the current filter
kernel. Then we follow appr oach of Blum et al. [
1
] to deter-
mine the median of medians . Therefore, our implementation
is already highly optimized. The filter kernel size is 3 × 3 .
Hotspo t is a thermal simulation tool and part of the Ro-
dinia benchmark suite [
2
]. The application consists of a 2D
transient thermal simulation kernel that iteratively solv es
differential equations. Input to the hotspot application are
two square matrices: The first matrix r epresents pow er data
and the seconds represents temperature data. Output is a
matrix of the same size that contains the temperature.
The Sobel operator is used in image processing and com-
puter vision to build edge-detection applications. It computes
an approximation of the gradient that emphasizes on the
edges in an image. The calculation is done in a horizontal
and a vertical convolution step . W e use the Sob el operator in
two applications. Sobel 3 is using a 3
×
3 filter kernel mask
and Sobel 5 is using a 5 × 5 filter kernel mask.
6.2 Input Data Sensitivity
The results of our work sho w that the amount of error that is
introduced by the approximation depends on the input to the
applications. In contrast, the spee dup only depends on the
0 . 00
0 . 05
0 . 10
0 . 15
0 . 20
0 . 25
0 . 30
0 . 35
Error
gaussian
inversion
median
hotspot
sobel3
sobel5
0 . 0
0 . 5
1 . 0
1 . 5
2 . 0
2 . 5
3 . 0
3 . 5
Spee dup

Figure 6.
Error distribution on differ ent input data. On the
bottom is the sp eedup of our approach compared to the
state-of-the-art baseline implementation depicted.
selected approximation scheme. W e apply our technique to
six benchmarks to show the sensitivity to input data. Ga uss-
ian , Inversion , Median and Sobel 3/ Sobel 5 are executed
on a set of 100 input data sets taken from the USC-SIPI Im-
age Database [
22
] and consisting of a subset of the misc and
pattern catalogue. For each of the applications w e selected
one of the Pareto-optimal configurations. For Ho tspot and
Inversion row scheme 1 was used. For the other applica-
tions stencil scheme was use d. Figure 6 shows the r esults.
The upper part of the figure shows the distribution of the
error for the applications. The av erage error is almost always
less than 5%. Only Sobel 5 shows a higher average . For all
image-based applications, there are some outliers that have
an error of up to 20%, e xcept Sobel 3 and Sobel 5 which have
a higher error .
The Ga ussian application is spe edup by 2
.
2
×
by our ap-
proach. The median error is less than 4% and the variance
of the error is small, ev en considering that there are some
outliers in the error distribution of up to 17%.
The Inversion application has a spee dup of 1
.
59
×
. The
median error is about 5%. The variance of the error is larger
and there are outliers of up to 20% err or .
A spee dup of 1
.
62
×
is shown for the Median application.
This is particularly interesting considering the application is
working also with private memory in the implementation of
median of medians as explained in Se ction 6.1 . The reported
spee dup is therefore on top of an alr eady optimized applica-
tion. The median error is about 5%. The variance of the error
is about the same as for the Inversion application.
W e observe a spe edup of 1
.
98
×
for the Hotspo t applica-
tion. As w e rely on the input data pr ovided by the application
we have 8 differ ent input data sets, that differ in their size.
The input data is generated using a tool shipp ed with the
Hotspo t application. The results show that in general e xe-
cuting the application with a perforated data set introduces
only a very small error in the result. Furthermore , compared
with other applications the variance of the error is very small.
A spee dup of 1
.
79
×
can be se en for the Sobel 3 application.
While the median error is 5% the variance of the error is
larger than for the previous applications. For about 75% of
the measurements, the error is less than 15%.
(a) 0.12% (b) 5.05% (c) 19.32%
Figure 7. Input data and corresponding err or [ 22 ].

CGO’18, February 24–28, 2018, Vienna, A ustria Daniel Maier, Biagio Cosenza, and Ben Juurlink
0 . 00 0 . 05 0 . 10 0 . 15 0 . 20
Runtime
.00
.02
.04
.06
.08
.10
Mean Relative Error
Rows1:NN
Rows2:NN
Rows1:LI
Stencil1:NN

(a) Ga ussian
0 . 00 0 . 05 0 . 10 0 . 15 0 . 20
Runtime
.00
.02
.04
.06
.08
.10
Mean Relative Error
Rows1:NN
Rows2:NN
Rows1:LI

(b) Inversion
0 . 00 0 . 05 0 . 10 0 . 15 0 . 20
Runtime
.00
.02
.04
.06
.08
.10
Mean Relative Error
Rows1:NN
Rows2:NN
Rows1:LI
Stencil1:NN

(c) Median
Figure 8. Perforation schemes with different parameters.
The highest spee dup in our study is 3
.
05
×
for the Sobel 5
application. The higher spee dup in comparison to Sobel 3
can be partly attributed to the larger filter kernel size and
therefore to mor e data reuse across thr eads. While the me-
dian error is 15% and ther efore significantly higher than for
Sobel 3, the distribution of the error is more dense and 75%
of the measurements have an err or smaller than 20%.
These results show that the amount of err or introduced by
our approach can differ by or ders of magnitude depending
on the input. T o illustrate this further we show exemplary
input and corresponding error for the Median application
in Figure 7 . Input data that contains larger ar eas of the same
color can be approximated with a very small error of only
0.12% (Figure 7a ). Countryside photographs (Figure 7b ) pro-
duce an error of 5.05% that is about the median of our test
input data set. Pattern-images (Figure 7c ) contain a lot of
high frequency and therefore ar e prone to perforation. They
yield a larger error , in this case 19.32%.
Applications that execute filter kernels with no or small
halo areas ( Ga ussian , Inversion , Median ) and therefore
also a small area of prefetching, hav e a smaller variance in
the error . These applications also have center-weighted filter
kernels. Compared to that, Sobel 3 has as larger variance but
still a comparable low median error of about 5%. Sobel 5 has
a median error of 13% and an even higher variance .
6.3 Parametrization
Perforation Schemes
In Figure 8 , we compar e different
perforation approaches. W e conducted our study for the ap-
plications Ga ussian , Inversion and Median . On the x-axis
is the runtime of the applications in 1
/
100 seconds depicted
while the y-axis shows the mean relativ e error .
W e compare four configurations:
Rows1:NN
is perfora-
tion of every other row and r econstruction using nearest-
neighbor interp olation.
Rows2:NN
is perforation of 3 out
of 4 rows and r e construction using nearest-neighbor inter-
polation.
Rows1:LI
is perforation of every other row and
reconstruction using linear interpolation.
Stencil1
is per-
foration of the boundaries of a work group tile and uses
nearest-neighbor interpolation for reconstruction.
As e xpecte d, the error is higher if the appro ximation
scheme is more aggressiv e, e.g., if more input data is perfo-
rated. The error for
Stencil1
is very low and always less
than 1%, as this perforation scheme is approximating only
a small amount of data. The error of
Rows1:NN
is about
half of the error of
Rows2:NN
. Howe ver , the runtime is for
all applications the same. This might be attributed to the
specific implementation or the memor y architecture. The
error for
Rows1:LI
is smaller than for
Rows1:NN
( Ga ussian :
-45%, Inversion : -21%, Median : -34%). Howe ver , the error
of
Rows1:NN
is already small and less than 4% for all thr ee
applications. The error of
Stencil1
is less than 1%. This is
due to the small amount of data that is approximated.
The runtime of
Rows1:NN
,
Rows2:NN
and
Rows1:LI
is sim-
ilar for the Ga ussian and Inversion application. Howev er ,
it is different for the Median application, which is explained
by the use of private memory .
Local W ork Group Size
W e compare performance with
local work group size in Figur e 9 . W e use nearest-neighbor
interpolation for all applications. The baseline applications
use local memor y for Ga ussian and Median . Median is
also using private memory as describ ed in Section 6.1 . The
accurate Inversion application does not use local memor y
as a prefetching step would incr ease runtime.
T wo properties are remarkable: First, all configurations
have a larger or equal x than y component. This is due to
the better alignment of these configurations to the memor y
interface. Second, the optimal work gr oup configuration
for an application is different for the accurate
Baseline
and for the approximated versions. Ther efore , work group
configuration needs to match the approximation scheme.
6.4 Pareto Optimality
W e compare our appr oach with P araprox ’ state-of-the-art
solution [
15
] in Figure 10 . The state-of-the-art solution is
plotted using a
•
marker and our approach is plotted us-
ing a
×
marker . The Pareto-optimal solutions ar e connected
using a gray dashed line. Center , rows, and cols ar e three
output approximation schemes, see Figure 3 . The numbers

Local Memor y- A ware Kernel Perforation CGO’18, February 24–28, 2018, Vienna, A ustria
2,128
4,64
8,8
8,16
8,32
16,8
16,16
32,8
64,4
128,2
W ork gr oup size
0 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0
1 . 2
Runtime
Stencil1
Rows1
Baseline

(a) Ga ussian
2,128
4,64
8,8
8,16
8,32
16,8
16,16
32,8
64,4
128,2
W ork gr oup size
0 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0
1 . 2
Runtime
Rows1
Baseline

(b) Inversion
2,128
4,64
8,8
8,16
8,32
16,8
16,16
32,8
64,4
128,2
W ork gr oup size
0 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0
1 . 2
Runtime
Stencil1
Rows1
Baseline

(c) Median
Figure 9. Local work gr oup size tuning.
0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 2 . 0 2 . 2
Speedup
.00
.02
.04
.06
.08
.10
Mean Relative Error
1
2
1
2
1
2
Accurate
Center
Rows
Cols
Stencil1
Rows1

(a) Ga ussian
0 . 7 0 . 8 0 . 9 1 . 0 1 . 1 1 . 2 1 . 3
Speedup
.00
.02
.04
.06
.08
.10
Mean Relative Error
1
2
1
2
1
2
Accurate
Center
Rows
Cols
Rows1

(b) Inversion
1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 2 . 0 2 . 2 2 . 4
Speedup
.00
.02
.04
.06
.08
.10
Mean Relative Error
1
2
1
2
1
2
Accurate
Center
Rows
Cols
Rows1
Stencil1

(c) Median
Figure 10. Pareto-optimal solutions of the pr op osed and P arapro x ’ state-of-the-art solutions.
next to the points indicate the perforation scheme: (1) ap-
proximate 2 r ows/columns; (2) appro ximate 4 rows/columns.
Our approach is compared using two perforations schemes.
Stencil1
is approximating the w ork group b oundaries, and
Rows1
is approximating e very second row , see Figure 4 and
Figure 5 . The speedup of all applications is calculate d with
respect to the baseline implementation from P arapro x .
Figure 10a shows the Ga ussian application. The Pareto-
optimal solutions are
Stencil1
and
Rows1
. The error for
Stencil1
is with 0.45% very low and the spee dup is 2
.
1
×
.
The error for
Rows1
is 2.9%. The increased error is e xplained
by the larger amount of approximated input data. This also
explains the higher speedup of 2
.
2
×
. The second highest
spee dup is 2
.
08
×
by the state-of-the-art approach
Rows
that
approximates 2 out of 3 rows. This is also the e xplanation of
the much higher error of 7.5%.
The Inversion application is shown in Figure 10b . Par eto-
optimal solutions are
Rows
and
Rows1
.
Stencil1
cannot be
used as the application has a filter kernel size of 1
×
1 .
Cols
be-
comes slower , which is explained by the improper alignment
of column-shaped p erforation and memory data layout.
The results of the Median application are sho w n in Fig-
ure 10c . Pareto-optimal configurations ar e
Stencil1
,
Rows1
,
Rows
and
Cols
. Spee dup and error of
Stencil1
and
Rows1
are 1
.
29
×
and 0.5%; 1
.
36
×
and 3.3%. As the baseline imple-
ments the median of medians and therefore uses private mem-
ory which is faster than lo cal memory .
In general, the error of our appr oach is improv ed signif-
icantly compared with P araprox while w e reach a similar
spee dup. Our appr oach is not limited to applications that
generally benefit from the use of local memor y as shown by
the results of the Inversion application.
7 Conclusion
W e introduce local memor y-aware kernel perforation, a
novel technique for the acceleration of GP U kernels using
approximate computing. Our appr oach first skips loading
part of data from global memory and later uses local mem-
ory enable d reconstruction methods. W e present a general
perforation scheme that skips loading rows of input data.
Additionally , we present a stencil perforation scheme that
skips loading the data elements close to the borders of the
work group tiles.
The experimental evaluation shows that our appr oach is
able to accelerate a variety of application from 1
.
6
×
to 3
×
while maintaining an average error of 6%. Our r esults show
that the amount and distribution of the error depends on
the input data. W e wer e able to significantly lower the err or

CGO’18, February 24–28, 2018, Vienna, A ustria Daniel Maier, Biagio Cosenza, and Ben Juurlink
while keeping similar performance than the state-of-the-art
approach P arapro x .
In a parameter exploration study , we show that, depending
on the employed perforation approach and r econstruction
technique, the error can be tuned from 0.5% to 7% depending
on the input data. W e show that the optimal local work
group size for the baseline kernel and appr oximate kernels
are different. Therefore , a system optimize d for the baseline
kernel will not perform optimal for approximate kernels.
W e investigate the Par eto-optimality of our approach. Our
experiments show that our approach can impr ove the error
and the spee dup significantly with respect to state of the art.
In a following work we will implement our currently man-
ual approach in a fully automatic compiler-based framew ork.
As w e have shown, that the technique gives pr omising re-
sults for a set of general-purpose kernels, a librar y can au-
tomatically apply and tune the technique to approximable
kernels and memory regions and accelerate a large set of
applications.
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