Learning to Defer: A Survey

Learning to Defer: A Survey Joshua Strong 1*, Emma Sun 1, Harry Rogers 1, Helen Higham 2, and J. Alison Noble 1 1Department of Engineering Science, University of Oxford, UK , 2Nuffield Department of Clinical Neurosciences, University of Oxford, UK *Correspondence: [email protected] Preprint — December 6, 2025 Abstract Learning to defer (L2D) enables AI systems to choose between autonomous prediction and deferral to experts. This survey consolidates the fast-growing literature through a four-branch taxonomy: methodological frameworks; optimization and theory; task generalizations; and real-world adaptations. We outline contrasts between score-based and predictor–rejector formulations; one-stage, two-stage, and post-hoc training; unify surrogate losses with theoretical guarantees; and synthesize extensions to regression, multitask prediction, top𝑘 committees, sequential settings, and causal pipelines. Practical considerations include limited annotations, dynamic expert pools, workload/budget control, fairness, interpretability, robustness, and uncertainty handling, concluding with open challenges for reliable human–AI decision systems. Keywords: learning to defer, selective prediction, rejection learning, human–AI collaboration, uncertainty, decision referral, human-in-the-loop 1 Contents 1 Introduction 3 1.1 Contributions and Organization of This Survey .................... 3 2 Related Fields 5 2.1 Direct Predecessors of Learning to Defer ........................ 5 2.2 Alternative Paradigms of Human-AI Collaboration .................. 7 2.3 Related Algorithmic Frameworks and Architectures ................. 7 3 The Learning to Defer Problem Formulation and Introduction 8 3.1 Notation and Setup .................................... 9 3.2 Architectural Formulations (Score-Based vs. Predictor-Rejector) .......... 9 3.3 The Deferral Target Loss and Optimization Objective ................ 10 3.4 The Bayes-Optimal Strategy and the Need for Surrogates .............. 11 4 Methodological Frameworks in L2D 13 4.1 One-Stage (Joint) Learning ................................ 13 4.2 The Two-Stage Setup: Deferral for Fixed Predictors .................. 14 4.3 Post-Hoc Fine-Tuning and Calibration ......................... 15 4.4 The Trade-Off Between Frameworks .......................... 15 5 Optimization & Theoretical Foundations (Surrogate Losses & Guarantees) 16 5.1 Background: Theoretical Guarantees for Surrogate Loss Functions ......... 16 5.2 Optimization in the One-Stage Setup .......................... 18 5.2.1 Single-Expert Settings: Consistency, Calibration, Underfitting, and Realizability ....................................... 18 5.2.2 The Multi-Expert Problem: Generalization and Stronger Guarantees . . . 21 5.3 Optimization in the Two-Stage Setup .......................... 22 5.4 An Alternative Formulation: Dependent Bayes Optimality ............. 24 5.5 Key Considerations for Implementation ........................ 24 6 L2D Task Generalizations 24 6.1 Regression with Deferral ................................. 25 6.2 Multi-Task Learning ................................... 26 6.3 Top-𝑘Classification and Deferral ............................ 28 6.4 Sequential Learning to Defer .............................. 29 6.5 Causality and Learning to Defer ............................ 30 7 Real-World Adaptations for L2D 31 7.1 Learning with Limited Expert Annotations ...................... 31 7.2 Handling Dynamic Expert Pools ............................ 32 7.3 Integrating Policy Constraints into Deferral Frameworks .............. 33 7.4 Enhancing L2D Safety .................................. 33 7.5 Controlling Workload Distribution to Experts ..................... 35 8 Future Directions, Open Challenges and Concluding Remarks 37 2 Figure 1: L2D (inference) framework. An input 𝑥 is passed through a rejector and predictor. Depending on the output of the rejector, the system decides to either defer to the expert, or to autonomously predict using the predictor’s output. The Figure depicts the predictor-rejector formulation, as described in §3.2. Notation in parentheses is introduced in §3.1. 1 Introduction Artificial intelligence (AI) is increasingly being deployed in high-stakes environments where the cost of an error is significant: an error in a healthcare setting could lead to incorrect treatment and patient harm, an error in finance could mean failing to stop a major fraudulent act, and a failure in an autonomous car could cause a catastrophic accident. To mitigate these risks, Learning to Defer (L2D) is emerging as a prominent area of research within Human-AI Collaboration (HAIC). L2D equips an AI model with the ability to either make an autonomous prediction or to defer its decision to an external human expert. In this paper, we use “expert” to mean a skilled human; experts vary in their decisions and may be biased or fallible. This deferral acts as a fail-safe, redirecting challenging or uncertain cases to an expert who can provide more reliable or nuanced judgment. Deferral can provide additional benefits beyond improved task performance, such as shared accountability in decision-making, increased trust [ 66 ] and hence adoption, and a more efficient and cost-effective use of human time and expertise. Overall, the central objective of L2D is to optimize the overall system performance of this human-AI collaboration, by finding the optimal decision between deferring or autonomously predicting for each case. The L2D problem can be explained as a system with three principal components: a task predictor model, a rejector model, and a human expert (Figure 1). In this framework, for any given input, the predictor generates a potential prediction. The rejector model then decides whether to accept this prediction or to defer to the expert. This decision is governed by a cost structure, which typically includes a penalty for incorrect predictions and a specified cost for querying the expert. The fundamental task in L2D is therefore to learn prediction and deferral policies that jointly minimize the total expected cost of this hybrid Human-AI system. 1.1 Contributions and Organization of This Survey While L2D is a rapidly expanding field, there is a distinct lack of a comprehensive survey. This survey fills that gap. The most relevant existing work is the 2022 perspective paper by Leitão et al . [36] . This work provided a good introduction of the field in its infancy and identified several critical open challenges, such as data requirements, expert capacity management, and model brittleness. However, the L2D field has advanced considerably since this work. Since the original paper was based on seven L2D publications, there have been over forty new ones that have not only addressed many of the initial challenges but have also introduced new research 3 Learning to Defer (L2D) §4: Methodological Frameworks One-Stage (Joint Learning) Two-Stage (Fixed Predictor) Post-Hoc Fine-Tuning §5: Optimization & Theoretical Foundations Surrogates & Guarantees Bayes-Consistency Realizable ℋ-Consistency ℋ-Consistency BoundsSingle-Expert Surrogates Multi-Expert Surrogates Alternative Formulations §6: Task Generalizations Regression with Deferral Multi-Task Learning Top-𝑘&𝑘(𝑥)Committees Sequential Deferral Causality §7: Real-World Adaptations Data & Expert Dynamics Limited Expert Annotations Dynamic Expert Pools Operations & Safety Workload & Budgets Policy Constraints Robustness & Uncertainty Interpretability Figure 2: A taxonomy of learning to defer (L2D), with the literature organized into methodological frameworks, optimization and theory, real-world adaptations, and task generalizations. directions. These include significant advancements in the theoretical understanding of surrogate losses [ 47 , 49 , 56 ], the introduction of new methodological frameworks (e.g., two-stage learning) [ 43 , 50 , 52 ], the generalization of L2D to complex tasks (e.g., regression, multi-task learning) [ 48 , 53 ], and sophisticated frameworks for handling real-world constraints (e.g., dynamic expert pools, fairness) [64,67,68]. The rapid development of these new theories and methods has led to a fragmented body of literature, making it difficult for researchers to gain a holistic view of the field. This survey provides a structured, comprehensive analysis and the first categorization of the L2D field, addressing this gap. The primary contributions of this survey are: • A four-branch taxonomy of the L2D field: We introduce a taxonomy that organizes L2D research into four core themes: Methodological Frameworks,Optimization & Theoretical Foundations,Task Generalizations, and Real-World Adaptations (Figure 2). • Systematic review of methodological frameworks: We contrast one-stage (joint), two-stage (fixed predictor), and post-hoc fine-tuning frameworks, and analyze their trade-offs in 4 performance, practicality, and robustness (§4). • Comprehensive synthesis of optimization and theory: We tracethe developmentofsurrogate objectives across one-stage and two-stage settings and clarify their theoretical guarantees, alongside calibration and cost-sensitivity issues; we also discuss the alternative formulation of dependent Bayes optimality (§5, §5.4). • Task generalizations: We cover extensions beyond the canonical classification setting to regression, multi-task learning, and top-𝑘deferral (§6). • Analysis of practical adaptations: We synthesize approaches for limited expert data, dynamic expert pools, workload/budget control, and responsible AI considerations such as fairness, interpretability, adversarial robustness, and uncertainty-aware abstention (§7). • Delineation of open challenges and a research agenda: We identify gaps and outline promising directions for future work (§8). 2 Related Fields Figure 3: Differences between rejection learning, selective classification and L2D. Notation in parentheses is introduced in §3.1. 2.1 Direct Predecessors of Learning to Defer The L2D field builds upon a well-established history of research into models that have the ability to abstain from making a prediction. This family of methods acknowledges that it can be beneficial for a model to recognize its own uncertainty and reject an instance rather than risk an erroneous low-confidence prediction. The primary differentiator for L2D lies in its reconceptualization of what happens after a model abstains and, crucially, how that action is valued within the system. We discuss two major fields in this domain: Rejection Learning and Selective Classification. Rejection Learning Learning with a reject option, or rejection learning, is the most direct ancestor of L2D. First formalized by Chow [13 , 14] and further developed by Cortes et al . [16] , this concept introduces an additional option for a classifier, alongside the standard class predictions: the reject option. 5 The AI model can choose to abstain from making a prediction if its confidence in its own output is below a certain threshold. This rejection is generally prompted by one of two conditions: ambiguity rejection, for instances that are confusing or near a decision boundary, or novelty rejection, for instances that are far away from the training data [ 29 , 75 ]. The key assumption in the standard rejection learning framework is that the act of rejection incurs a predefined, constant cost. The model is therefore trained to optimize a trade-off between the classification error on the instances it accepts and the fixed penalty it pays for the instances it rejects. The goal is to improve the model performance on the non-rejected subset of data. A key limitation of this method is that it is inherently non-adaptive to any downstream human expert’s performance, as it only considers the model uncertainty, in isolation from external factors. This leaves it open to a significant limitation: what if the expert performs poorly on deferred cases, or outperforms AI on those retained by the model? L2D can be viewed as an adaptive generalization of the learning with rejection framework. This generalization is achieved by replacing the fixed cost with a variable, instance-dependent cost, which can represent the performance of the downstream expert on that specific instance (Figure 3). The seminal work from Madras et al . [40] introduces adaptive rejection learning, termed learning to defer, where the decision to defer is based not only on the model confidence but also on the expected expert competence. The model must learn not only about its own weaknesses but also the specific strengths and weaknesses of the expert it is collaborating with, in relation to itself. This original framework conceptualizes the system as a mixture of Bernoullis such that it uses a learned switch that, for each case, chooses whether the final decision comes from the model or from the human expert. This arrangement functions similarly to a mixture-of-experts (MoE) model [ 30 ], but with the distinction that the learning algorithm refines only the automated model and its deferral strategy, while the characteristics of the external expert are considered fixed. Conceptually, L2D serves as a sophisticated form of ambiguity rejection, as it directly learns which uncertain instances are best handled by the model versus the expert. Selective Classification A closely related approach is selective classification [ 20 ]. As with rejection learning, it allows a model to abstain from predicting. However, its objective is different; instead of rejection carrying a cost, the aim is to maximize accuracy without violating a predefined constraint on the rejection rate or coverage. For example, the goal might be to maximize accuracy while ensuring the model predicts on at least 80% of the data. The focus is on controlling the volume of abstention to guarantee a certain level of performance on the selected examples [23,24]. In comparison, the optimal deferral rate in L2D is not predefined but emerges from systemlevel cost optimization (explained further in §3.3). The model defers when it is economically rational to do so for the overall system. While this principle is central to L2D, a key insight from the field’s evolution is that this unconstrained optimization can lead to an impractically high deferral rate, overwhelming the human expert [ 1 ]. Recognizing this, more recent L2D frameworks introduce explicit mechanisms for workload balancing and budget control, a topic we explore further in Section 7.5. Furthermore, L2D explicitly models the downstream expert, whose varying performance is central to the problem, a component absent from the standard selective classification setup. 6 2.2 Alternative Paradigms of Human-AI Collaboration Beyond methods that simply abstain, L2D can be further contextualized by comparison to other established frameworks for HAIC such as Human-in-the-loop (HITL). While these fields also involve collaboration between automated systems and humans, they differ fundamentally in whether the human or AI has decision-making authority, the workflow and the purpose of the interaction. Human-in-the-Loop Machine Learning L2D can be situated within the broader field of HITL, which encompasses various techniques where humans and models interact. These interactions are often categorized by who, or what, is in control of the process: the model, the human expert, or a shared partnership [ 55 ]. L2D presents a specific configuration of this relationship, which becomes clear when contrasted with other common HITL approaches such as Active Learning and AI-assisted decision-making. In Active Learning, the model is in control of the decisions made. Active learning is a training procedure designed to minimize the cost of data labeling by having a model select the most informative examples for a human to label [ 37 ]. In this setup, the model queries a human for ground truth labels on instances where it is most uncertain. The goal is to achieve higher accuracy with fewer labeled examples compared to random sampling. The human’s input is used to improve the model itself during the training phase. The fundamental difference from L2D lies in the phase and purpose of the human interaction. Active learning is a training-time procedure where the human’s role is to provide ground truth to improve the model. In contrast, L2D is an inference-time procedure. The L2D model defers to the expert during deployment, and the expert’s decision becomes the final output for that instance. Anothercommon HITLconfigurationinvolvesanAIsystem that provides aninitial prediction or recommendation, which a human expert then reviews. The expert holds the final authority and can choose to accept, reject, or modify AI output. This model is prevalent in current medical diagnosis, where an AI might flag potential issues on a medical scan for a radiologist to confirm. The human is responsible for oversight and final accountability [ 5 , 6 ]. Mozannar and Sontag [57] explicitly refer to L2D as the “reverse setting” compared to existing AI-assisted decision-making, where the human expert has the final say and acts as a check on the model. In L2D, the model has the initial agency to decide whether to handle a case itself or to defer it to the human. The decision flow is initiated by the model’s self-assessment of its capabilities relative to the expert, rather than the expert assessment of the model output. 2.3 Related Algorithmic Frameworks and Architectures The implementation of L2D relies on adapting existing algorithmic frameworks to its unique problem setting. By examining these underlying structures, we can better understand both the mechanics of L2D and the novel constraints it imposes on these familiar architectures. Cascading Models Cascading classifiers are multi-stage architectures designed to improve computational efficiency and, in some cases, accuracy. A typical cascade consists of a sequence of classifiers, starting with a simple, fast model and progressing to more complex, computationally expensive ones (Figure 4(i)). The initial models filter out “easy” or obvious negative examples, allowing the more powerful models to focus only on the more difficult or ambiguous instances that remain [ 18 ]. 7 Figure 4: (i) Cascading Models framework, where increasingly complex models output a prediction if a confidence threshold is met, and otherwise pass the problem to the next model in the cascade, and (ii) Mixture-of-Experts framework, where a gating function (analogous to the rejector) combines the output of multiple expert predictions on the same input. This structure is conceptually similar to L2D, where the classifier acts as the first stage and the expert as the second. While structurally analogous, the relationship has been recently formalized, with research demonstrating that classical model cascades arise as a restricted special case of generalized L2D frameworks [ 52 ]. The key distinctions lie in flexibility and the scope of the objective. Cascading models are typically designed with a fixed sequence of models to optimize for computational efficiency, stopping at the first model that meets a confidence threshold. L2D, by contrast, learns a dynamic, input-dependent policy to route queries to the most cost-effective agent (or even a committee of agents), optimizing for overall system performance, which can include metrics such as accuracy and fairness [ 40 ]. Furthermore, the final stage in a cascade is another trainable model, whereas in L2D, it is often a human expert, whose performance characteristics are taken to be inherent, unchangeable, and often non-uniform. Mixture-of-Experts (MoE) The Mixture-of-Experts (MoE) model [ 31 ] is a commonly utilized ensemble technique. It consists of several expert networks, each specializing in a different part of the input space, and a gating network that learns to assign weights to each expert’s prediction for a given input. The final output is a weighted combination of the expert outputs (Figure 4(ii)). The single-expert L2D framework can be viewed as a specialized form of a two-expert MoE system, where the gating function corresponds to the rejector, and the two “experts” are the AI classifier and the human. The important distinction is that in L2D, one of the experts (the human) is a fixed, external, and non-differentiable component. Unlike in a standard MoE, where all experts and the gating network are trained jointly via backpropagation, the L2D model has no control over the human expert’s parameters or behavior. The optimization is constrained: the system can only learn the parameters of one expert (the classifier) and the gating function (the rejector) to best complement the fixed, pre-existing human expert. This makes the problem fundamentally different from standard end-to-end MoE training. Mozannar and Sontag [57] further critique the direct application of a MoE loss from Madras et al . [40] , proving it is not classification-consistent and can lead to suboptimal behavior where the model learns never to defer. 3 The Learning to Defer Problem Formulation and Introduction This section gives a compact formalization of Learning to Defer (L2D) and orients the reader to the four-part taxonomy developed in this survey (Figure 2). At a high level (Figure 1), an L2D system decides, per input, whether to act with an automated predictor or defer to an external 8 expert (human or stronger model) so as to minimize a system-wide cost. 3.1 Notation and Setup Table 1: Summary of mathematical notation for Learning to Defer (L2D) in this survey paper. Symbol Meaning Symbol Meaning 𝒳Input space 𝒴Output label space (𝑋, 𝑌)Input/label random variables (𝑥, 𝑦)A sampled data point from 𝒟 𝒟Joint distribution over 𝒳 × 𝒴 𝑐(𝑥, 𝑦)Cost of deferral ⊥Deferral action 𝒴⊥Extended label space ℎPredictor function ℋPredictor hypothesis class 𝑟Rejector function ℛRejector hypothesis class 𝑠Scoring function ℰExpert function 𝑚Expert prediction ℳExpert prediction space 𝐾Number of classes 𝐽Number of experts The Learning to Defer (L2D) framework extends standard supervised learning by enabling a model to either autonomously predict or defer the decision to an expert. This expert could be a human, or a more powerful AI model. This framework is built upon three core components, which we collectively refer to as the system: an autonomous predictor ( ℎ ), a fixed expert ( ℰ ), and a deferral mechanism ( 𝑟 or 𝑠 , cf. §3.2) that routes inputs to either the predictor or the expert to minimize a system-wide loss. Here, we introduce the L2D problem formulation for the standard task of multi-class classification. Let 𝒳 be the input space and 𝒴={1, . . . , 𝐾} be the output label space. We assume there is a joint probability distribution 𝒟 over the space 𝒳 × 𝒴 . We denote the random variables for the input and true label as (𝑋, 𝑌) , and a single data point (𝑥, 𝑦) is a realization sampled from 𝒟 . The expert ℰ is represented by a fixed, deterministic function ℰ:𝒳 → ℳ , where ℳ is the expert prediction space (typically ℳ ≡ 𝒴 ). For any input 𝑥 , the expert’s prediction is 𝑚=ℰ(𝑥) . For training, we assume access to triplets (𝑥, 𝑦, 𝑚) , where 𝑚 is generated by the expert. The core mathematical notation is summarized in Table 1. 3.2 Architectural Formulations (Score-Based vs. Predictor-Rejector) A central design choice in L2D lies in the architecture of the deferral mechanism: a unified, score-based approach that integrates prediction and deferral into a single model, and a modular, predictor-rejector approach that decouples them into separate models. Understanding the trade-offs between these two formulations is useful not only for deeply understanding L2D, but also for appreciating why frameworks are designed in particular ways and for developing new tasks. We further detail these formulations and their optimization implications in §4: Methodological Frameworks in L2D. The Score-Based Formulation The Score-Based (SB) formulation (Figure 5(i)), introduced in [ 57 ], frames the deferral problem as a single, unified classification task. This is achieved by extending the label space to 𝒴⊥=𝒴 ∪ {⊥} , where the new label ⊥ represents the action of deferring to the expert. A single, multi-output scoring function, 𝑠:𝒳 × 𝒴⊥→R , is then learned to assign a score to each possible class and to the deferral action. The system’s decision is given by a single argmax : 9 Table 2: Trade-offs among L2D training frameworks. Criterion One-Stage (Joint) Two-Stage (Fixed Predictor) Post-Hoc Fine-Tuning Performance potential Highest ceiling via full co-adaptation and specialization. Bounded by the fixed pre-trained predictor; learns optimal routing only. Intermediate: some co-adaptation without training from scratch. Computational cost (training) Highest (end-to-end training). Lowest for deferral (train a lightweight rejector only). Moderate (fine-tune predictor + train deferral). Robustness to expert changes Lower; specialization can be brittle if expert identity/availability or constraints change. Higher; general-purpose predictor remains intact. Intermediate; partial specialization can reduce robustness. Specialization behavior Strong specialization on non-deferred regions. Minimal specialization (predictor is frozen). Some specialization due to co-adaptation during fine-tuning. Model accessibility Requires full “white-box” access to predictor. Works with black-box/API predictors. Requires write access to predictor weights. Modularity Low; predictor and deferral tightly coupled. High; plug-and-play rejector around a fixed predictor. Moderate; coupled through fine-tuning. 5 Optimization & Theoretical Foundations (Surrogate Losses & Guarantees) The choice of methodological framework (§4) dictates the optimization strategy. A central challenge across all frameworks is the design of tractable surrogate losses suitable for the task that provide strong theoretical guarantees. This section synthesizes the history of surrogate loss design for the one-stage setting (§5.2). Following this, we discuss optimization the more recently proposed two-stage setting (§5.3). We begin with background on the related theoretical guarantees for surrogate loss functions required to comprehend this section (§5.1). Notation. Symbols follow §3.1 (Table 1). Target deferral losses are the score-based (SB) and predictor-rejector (PR) objectives in Eqs. 1–2. 5.1 Background: Theoretical Guarantees for Surrogate Loss Functions Having introduced the problem formulations (§3) and methodological frameworks (§4) of L2D, we now turn to the question: what guarantees do we have that minimizing a surrogate loss will lead to a system that is near-optimal for the original deferral task? In this subsection, we describe a hierarchy of theoretical guarantees proposed to evaluate such surrogate losses. These guarantees apply generally to both the PR and SB formulations. We review the three principal guarantees of Bayes-consistency,realizable ℋ-consistency, and ℋ-consistency bounds. To formalize these guarantees, we define the expected risk. For a distribution 𝒟 , a loss function 𝐿 , and a hypothesis 𝑓 from a class ℱ (where 𝑓 represents the learnable components, e.g., (ℎ, 𝑟)or 𝑠), the expected risk is ℰ𝐿(𝑓)=E(𝑥,𝑦)∼𝒟[𝐿(𝑓 , 𝑥, 𝑦)]. The optimal risk within the class is the best-in-class risk, ℰ∗ 𝐿(ℱ ) =inf 𝑓∈ℱ ℰ𝐿(𝑓). The core objective in surrogate analysis is to ensure that minimizing the estimation error of the surrogate, ℰℒsurr(𝑓)−ℰ∗ ℒsurr(ℱ ) , also minimizes the estimation error of the true deferral loss, ℰℓdef(𝑓)−ℰ∗ ℓdef(ℱ ). 16 Bayes-Consistency Bayes-consistency is the most fundamental requirement for a surrogate loss. It addresses an asymptotic question: if we could minimize the surrogate loss perfectly over the space of all possible measurable functions (ℱall), would we recover the Bayes-optimal strategy for the true deferral loss? Definition 1 (Bayes-Consistency [ 74 ]).A surrogate loss ℒsurr is Bayes-consistent with respect to the target deferral loss ℓdef if, for any sequence of hypotheses {𝑓𝑛}𝑛∈N⊂ ℱall , the convergence of the surrogate excess risk to zero implies the convergence of the deferral excess risk to zero: hℰℒsurr(𝑓𝑛)−ℰ∗ ℒsurr(ℱall)i𝑛→∞ −−−−→ 0=⇒hℰℓdef(𝑓𝑛)−ℰ∗ ℓdef(ℱall)i𝑛→∞ −−−−→ 0. Bayes consistency is a population-level guarantee over an unrestricted function class (effectively ignoring model capacity and optimization error). Hence the need for ℋ -dependent guarantees that tie surrogate regret to 0–1 regret for a fixed class. Realizable ℋ-Consistency To address the practical limitations of Bayes-consistency, realizable ℋ -consistency provides a stronger guarantee for the important realizable case. This setting assumes there exists a “perfect” hypothesis 𝑓∗∈ ℱ that achieves zero deferral loss, i.e., ℰℓdef(𝑓∗)=0. Definition 2 (Realizable ℋ -Consistency [ 39 ]).A surrogate loss ℒsurr is realizable ℋ -consistent with respect to ℓdef if, for any realizable distribution, any hypothesis 𝑓∈argmin𝑓′∈ℱ ℰℒsurr(𝑓′) also achieves zero deferral loss, i.e., ℰℓdef(𝑓)=0. Thispropertyishighlydesirableasit guaranteesoptimality innoise-freesettings. Theimportance of this guarantee was highlighted when it was shown that some prominent Bayes-consistent L2D surrogates do not satisfy realizable ℋ -consistency [ 56 ]. This result has motivated the design of new loss functions that are provably both Bayes-consistent and realizable ℋ-consistent [47]. ℋ-Consistency Bounds While realizable ℋ -consistency provides a crucial guarantee for the noise-free case, ℋ - consistencyboundsofferthemostpowerfulandgeneralframework. Theyarenon-asymptotic—i.e., they give a quantitative inequality at the population level without taking 𝑛→∞ —and they tie surrogate performance directly to target deferral loss for a specific (possibly restricted) hypothesis class ℱand for any data distribution. Definition 3 ( ℋ -Consistency Bound [ 3 , 4 ]).A surrogate loss ℒsurr admits an ℋ -consistency bound with respect to ℓdef if there exists a non-decreasing concave function Γ:R+→R+ with Γ(0)=0such that, for all 𝑓∈ ℱ and all distributions 𝒟, ℰℓdef(𝑓)−ℰ∗ ℓdef(ℱ ) + 𝑀ℓdef(ℱ ) ≤ Γℰℒsurr(𝑓)−ℰ∗ ℒsurr(ℱ ) + 𝑀ℒsurr(ℱ ). An ℋ -consistency bound implies Bayes-consistency (taking ℱ to be unrestricted and noting the gaps vanish). The function Γ acts as a transfer (or calibration) function: small surrogate excess risk forces small target excess risk (up to the additive gaps). Note that, a key feature of this modern formulation is the minimizability gap, 𝑀𝐿(ℱ ) =ℰ∗ 𝐿(ℱ ) − E𝑥inf 𝑓∈ℱ E𝑦|𝑥𝐿(𝑓 , 𝑥, 𝑦), 17 which measures how far the best-in-class global risk is from the expected per𝑥 best-in-class conditional risk. It vanishes when ℱ contains a conditional risk minimizer almost surely, and it isolates limitations intrinsic to the chosen class ℱ (distinct from the usual approximation error that compares ℱ to an unrestricted class). Accounting for 𝑀𝐿(ℱ ) yields bounds that are both sharper and more faithful to practical modeling constraints. Recent work derives L2D surrogates that admit strong ℋ -consistency bounds, providing theoretical justification for their use in practice [46]. The Interplay Between Consistency Guarantees in L2D At first glance, these guarantees appear to form a strict hierarchy of strength: ℋ -consistency bounds imply Bayes-consistency, and one might assume they also imply realizable ℋ -consistency. In standard classification, this is often the case. However, a key theoretical insight in the L2D literature is that this relationship breaks down. As shown by Mao et al . [47] , the minimizability gap for some L2D surrogates does not necessarily vanish, even in the realizable case. This seemingly technical detail has a profound implication: an ℋ -consistency bound no longer guarantees realizable ℋ-consistency in the L2D setting. Consequently, because 𝑀𝐿(ℱ ) can remain positive even when ℱ is realizable, the two guarantees decouple. The independence manifests in both directions: • A surrogate may have strong ℋ -consistency bounds yet fail in the simple realizable case. For instance, the SB L2D cross-entropy surrogate ( 𝐿𝐶𝐸 ) [ 57 ] has such bounds but is not realizable ℋ-consistent [44,56]. • Conversely, a surrogate may be realizable ℋ -consistent yet lack the stronger non-asymptotic guarantees of an ℋ-consistency bound for general, non-realizable distributions. This distinction is not just academic; it clarifies why the search for a single surrogate loss that satisfies all three properties became a central goal of modern L2D research, a goal first achieved by the unified frameworks of Mao et al. [47]. 5.2 Optimization in the One-Stage Setup In this subsection, the historical developments of surrogate loss functions in the one-stage setup for singleand multi-expert settings are detailed chronologically. Findings are summarized in Table 3. Considerations for selecting surrogate loss following the findings of this subsection are outlined in §5.5. 5.2.1 Single-Expert Settings: Consistency, Calibration, Underfitting, and Realizability The research on single-expert L2D surrogates began with the initial goal of establishing Bayesconsistency (Def. 1) [ 57 ], then moved to address a series of practical and theoretical challenges that arose (calibration, underfitting, and realizability) [ 8 , 38 , 70 ], and culminated in a unified framework satisfying three key theoretical guarantees [ 47 ]. We categorize this progression in three distinct generations: (1) Foundational Bayes-Consistent Surrogates and Their Practical Limitations The initial wave of research in L2D focused on establishing the initial property of Bayes-consistency [ 74 ]. The seminal cross-entropy-based surrogate, 𝐿𝐶𝐸 , proposed by Mozannar and Sontag [57] , was the first to be proven Bayes-consistent for the multi-class L2D setup. Following this, a unifying theoretical framework was proposed by Charusaie et al . [9] , who demonstrated a general method to convert any consistent multi-class surrogate loss into a consistent, cost-sensitive loss suitable 18 Table 3: A Summary of Key Developments in One-Stage L2D Surrogate Losses. The table charts the chronological progression from foundational Bayes-consistent losses to unified frameworks with the strongest theoretical guarantees. Citation Year Surrogate (Abbr.) BayesConsistent Realizable ℋ - Consist. ℋ - Consist. Bounds Key Contribution and Significance — §5.2.1 Single-Expert Settings: Consistency, Calibration, Underfitting, and Realizability — Mozannar and Sontag [57] 2020 Cross-Entropy (𝐿𝐶𝐸) Yes No (per [56]) Yes (via [44]) Seminal paper. Introduced the first Bayes-consistent surrogate loss for L2D, establishing the standard framework. Charusaie et al. [9] 2022 Cost-Sensitive Family (𝐿𝜙) Yes No Yes (via [46]) Generalized the L2D framework. Proposed a method to convert any consistent multi-class loss into a consistent L2D loss, creating a broad family of surrogates. First to provide excess risk bounds. Verma and Nalisnick [70] 2022 One-vs-All (𝐿OvA) Yes No (per [56]) Yes (via [44]) Introduced an OvA-based alternative. First to identify the calibration problem, showing the softmax surrogate leads to unbounded estimates. Cao et al . [8] 2023 Asymmetric SM (𝐿˜ 𝜓) Yes No Yes Defended the softmax approach. Showed the calibration issue was due to loss symmetry, not softmax itself. Proposed a bounded, consistent asymmetric softmax loss. Mozannar et al. [56] 2023 Realizable (𝐿RS) Yes (per [47]) Yes** Yes (per [47]) Critical turning point. Proved prior losseswere not realizable and introduced the first realizable surrogate. Raised a key open question about Bayes-consistency. Liu et al . [38] 2024 LabelSmoothing-Free (𝐿LSF) Yes No No Solved the underfitting problem caused by implicit label-smoothing when deferral costs are non-zero. Proposed a “label-smoothing-free” formulation that improves performance and robustness to cost. Mao et al . [47] 2024c Realizable L2D (𝐿RL2D) Yes Yes Yes Solved the open problem from Mozannar et al . [56] . Provided the first surrogate loss satisfying all three core consistency guarantees in the single-expert setting. — §5.2.2 The Multi-Expert Problem: Generalization and Stronger Guarantees — Verma et al . [69] 2023 Multi-Expert OvA and CE (𝐿𝐽 𝑆𝑀 &𝐿𝐽 OvA) Yes (per [46]) No Yes First to generalize L2D to multiple experts with a Bayes-consistent surrogate. Retained the same realizability gap as earlier single-expert work. Mao et al . [46] 2024b General L2D (𝐿general) Yes No Yes First to provide ℋ -Consistency Bounds in L2D, offering stronger, non-asymptotic guarantees for the multi-expert case. Mao et al . [49] 2025 Unified L2D (𝐿Ψ) Yes Yes Yes Unified all guarantees for the one-stage, multi-expert case. Introduced the first surrogate loss to satisfy all three theoretical properties in this setting. ** Realizable ℋ-consistent when ℋis closed under scaling, i.e., if ℎ∈ ℋ =⇒𝛼ℎ∈ ℋ ,∀𝛼∈R. 19 for deferral. While this created a broad family of potential surrogates, two practical issues with this framework arose. First, a calibration problem was identified by Verma and Nalisnick [70] , demonstrating that the score-based cross-entropy surrogate 𝐿𝐶𝐸 [ 57 ] leads to unbounded and poorly calibrated estimates of expert correctness, a problem their bounded One-vs-All (OvA) surrogate mitigated. The cause of this calibration issue was subsequently diagnosed by Cao et al . [8] , who proved that the unboundedness was not a flaw in the softmax function itself, but a fundamental consequence of using any standard, symmetric loss within the general framework of Charusaie et al . [9] . More importantly, their work re-contextualized the field by showing that the successful, bounded losses from both their own work and that of Verma and Nalisnick [70] could be understood as special instances of this unified framework, succeeding precisely because they are derived from novel, asymmetric base losses. This work solidified the understanding that achieving well-calibrated estimates requires a careful, asymmetric design of the underlying loss. Second, even with these well-calibrated, consistent losses, a further practical limitation was identified: underfitting induced by non-zero deferral costs. First noted by Narasimhan et al . [58] , the performance of these surrogates can degrade significantly when a non-zero fixed cost 𝑐0>0 is introduced for consulting an expert, such that the total cost is 𝑐(𝑥, 𝑦)=𝑐0+ 1 ℰ(𝑥)≠𝑦 . This issue was systematically addressed by Liu et al . [38] , who diagnosed that the general framework of Charusaie et al . [9] implicitly introduces a redundant label-smoothing term whenever 𝑐0>0 . This term flattens the training distribution and degrades the performance of the classifier component as the deferral cost increases. To solve this, they proposed a novel “label-smoothing-free” formulation 𝐿LSF which eliminates this harmful effect by applying the cost penalty in a targeted, non-uniform manner, using the model’s own intermediate prediction to guide the penalty. This provided a family of surrogates that maintain high performance in realistic, cost-sensitive scenarios. (2) The Challenge of Realizable ℋ -Consistency A turning point in the field came when Mozannar et al . [56] showed a significant limitation in all prior approaches. They demonstrate that the prior popular Bayes-consistent surrogates are not realizable ℋ -consistent (Def. 2). This property is of practical importance: it guarantees that if a perfect, zero-error predictor-rejector pair exists within the chosen hypothesis class (e.g., linear models), the learning algorithm is guaranteed to find it. The failure to satisfy this property means that even in a simple, noise-free setting where a perfect linear solution exists, these established surrogates could converge to a suboptimal solution with non-zero error. Mozannar et al . [56] make this abstract concern concrete with a synthetic counterexample, highlighting that Bayes-consistency alone is insufficient and motivating the search for surrogates with stronger guarantees. In the same work, Mozannar et al . [56] propose a new surrogate loss to address this gap, which they term the Realizable Surrogate ( 𝐿𝑅𝑆 ). The key innovation is to make the human’s correctness acts as a switch inside the logarithm of a softmax-like term, rather than as an external weight. This seemingly small change has a profound effect: the authors prove that 𝐿𝑅𝑆 is indeed realizable ℋ -consistent for any hypothesis class closed under scaling. However, this work left a theoretical question unanswered: while the new loss is realizable, it is not proven to be Bayes-consistent. (3) A Unified Framework with All Theoretical Guarantees The open question from Mozannar et al . [56] was recently resolved by Mao et al . [47] . Their work provides two key contributions that unify the theoretical landscape for single-expert L2D. First, they formally prove that the surrogate 𝐿𝑅𝑆 is, in fact, also Bayes-consistent, establishing 𝐿𝑅𝑆 as the first surrogate loss to satisfy both guarantees. More significantly, they introduced a broad, unified family of surrogate 20 losses, 𝐿RL2D , parameterized by a non-increasing function Ψ . This family is constructed by re-deriving the surrogate from first principles and takes the general form: 𝐿RL2D(ℎ, 𝑥, 𝑦)=𝑐(𝑥, 𝑦)𝑙comp(ℎ, 𝑥, 𝑦) + (1−𝑐(𝑥, 𝑦))˜ 𝑙comp(ℎ, 𝑥, 𝑦)(6) where 𝑙comp is a standard comp-sum loss (e.g., cross-entropy, MAE) that maximizes the correctclass score, and ˜ 𝑙comp modifies it to maximize the sum of the correct-class and deferral scores. Under the mild conditions on Ψ (non-increasing, Ψ2 3>0 , and lim𝑡→1Ψ(𝑡)=0 ) the formulation is Bayes-consistent, realizable ℋ -consistent, and admits explicit ℋ -consistency bounds (Def. 3). This yields a family of losses parameterized by the link Ψand cost 𝑐. 5.2.2 The Multi-Expert Problem: Generalization and Stronger Guarantees While the single-expert setting provides a crucial theoretical foundation, many real-world, practical applications involve routing decisions among a pool of multiple experts with diverse specializations and consultation costs. Early approaches to the multi-expert problem, such as the work by Hemmer et al . [27] , extended the existing one-stage framework by simultaneously learning a classifier and a multi-expert routing system, drawing inspiration from Mixture-ofExperts models [ 31 ]. However, these initial methods were often heuristic and lacked theoretical guarantees, a limitation later addressed by a new generation of research focused on developing provably consistent surrogate losses for the multi-expert setting. First-Generation Consistent Surrogates The first principled extension of consistent surrogate losses to the multi-expert setting was presented by Verma et al . [69] , who generalized the initial single-expert surrogate losses (i.e., the softmax and One-vs-All (OvA) formulations) to handle 𝐽 experts. Their multi-expert softmax surrogate takes the form: Φ𝐽 𝑆𝑀(𝑠, 𝑥, 𝑦, m)=−log exp 𝑠(𝑥, 𝑦) Í𝑎∈𝒴⊥exp 𝑠(𝑥, 𝑎)− 𝐽 Õ 𝑗=1 1 ℰ𝑗=𝑦log exp 𝑠(𝑥, ⊥𝑗) Í𝑎∈𝒴⊥exp 𝑠(𝑥, 𝑎)(7) Here, 𝒴⊥ is the augmented label space including 𝐽 distinct deferral options {⊥1, . . . , ⊥𝐽} , 𝑠(𝑥, 𝑎) is the score for action 𝑎 , and m={ℰ1(𝑥), . . . , ℰ𝐽(𝑥)} is the vector of 𝐽 expert predictions. Verma et al . [69] proved that this loss is Bayes-consistent (Def. 1) with the optimal multi-expert routing strategy. However, these first-generation multi-expert surrogates inherited the same theoretical limitation as their single-expert counterparts in that they were later shown to lack the stronger guarantee of realizable ℋ-consistency (Def. 2) by Mozannar et al. [56]. Principled Generalizations and ℋ -Consistency Bounds A significant theoretical step forward was made by Mao et al . [46] , who introduced a principled derivation for generalizing multiexpert surrogate losses. Rather than extending specific losses, they show how any multiclass surrogate ℓ that admits an ℋ -consistency bound (Def. 3) can be adapted to the multi-expert deferral task, and then prove that the resulting family inherits these bounds. These guarantees are hypothesis-set–specific and non-asymptotic, and the paper further gives the first finitesample learning bound for multi-expert L2D. Unlike earlier excess-risk bounds (i.e., classical surrogate-to-target bounds based on approximation error), ℋ -consistency bounds account for the hypothesis class and the minimizability gap, thereby implying Bayes-consistency while providing a more practical quantitative link between surrogate and target losses. The authors do not claim realizable ℋ-consistency for this framework. 21 A Unified Framework for Multi-Expert Deferral Unification of all three major consistency guarantees for the multi-expert setting was recently achieved in the framework of Mao et al . [49] . Their work introduces a novel family of surrogate losses, 𝐿Ψ , designed from first principles to satisfy all three theoretical properties simultaneously. They first derived an alternative formulation of the true deferral loss, then systematically replaced its indicator functions with smooth surrogates from the broad class of comp-sum losses (parameterized by a non-increasing function Ψ). This results in the following general form for their surrogate family: 𝐿Ψ(𝑠, 𝑥, 𝑦)= 𝐽 Õ 𝑗=1 𝑐𝑗(𝑥, 𝑦)+1−𝐽 Ψ 𝑒𝑠(𝑥,𝑦) Í𝑦′∈𝒴⊥𝑒𝑠(𝑥,𝑦′)!+ 𝐽 Õ 𝑗=1 [1−𝑐𝑗(𝑥, 𝑦)]Ψ 𝑒𝑠(𝑥,𝑦)+𝑒𝑠(𝑥,⊥𝑗) Í𝑦′∈𝒴⊥𝑒𝑠(𝑥,𝑦′)!(8) where 𝑠(𝑥, 𝑦) is the score for label 𝑦 , 𝑠(𝑥, ⊥𝑗) is the score for deferring to expert 𝑗 , 𝐽 is the number of experts, and 𝒴⊥ is the augmented label space. The key innovation lies in the final term, where the scores for the correct label 𝑦 and the correct deferral option ⊥𝑗 are summed in the numerator, a modification crucial for achieving realizability. Their main theoretical results prove that for specific choices of Ψ (e.g., corresponding to mean absolute error, where Ψ(𝑡)=1−𝑡 ), the resulting surrogate loss is simultaneously Bayes-consistent, realizable ℋ -consistent, and admits ℋ -consistency bounds. This work provides the most complete theoretical foundation to date for one-stage, multi-expert L2D, closing theoretical gaps and offering practitioners a principled and flexible framework with the strongest possible guarantees. 5.3 Optimization in the Two-Stage Setup In the two-stage setting (§4.2), we have a frozen, pre-trained predictor ℎ . The optimization challenge is to learn an optimal deferral mechanism (scorer 𝑠𝑑 or rejector 𝑟 ) that routes inputs between ℎ and the 𝐽 experts. The development of optimization strategies in this framework began with the work of Mao et al . [43] , who introduced the first consistent surrogates. Subsequent work by Mao et al. [49] addressed the additional theoretical gaps. Foundational Two-Stage Surrogates Mao et al . [43] proposed consistent surrogate losses for both the Score-Based (SB) and PredictorRejector (PR) formulations in the two-stage setting. We detail both below. Score-Based Formulation. Let 𝑠ℎ:𝒳 ×𝒴 → R denote per-class scores (e.g., logits or calibrated probabilities) produced by the predictor ℎ . The predicted label is ℎ(𝑥)=argmax𝑦∈𝒴 𝑠ℎ(𝑥, 𝑦) . In the SB approach, the goal is to learn a deferral scoring function 𝑠𝑑:𝒳 × [𝐽] → R . This is achieved by constructing a unified scoring function ¯ 𝑠𝑑:𝒳 × ({0}∪[𝐽]) → R , where the predictor ℎ(indexed by 0) contributes its confidence score: ¯ 𝑠𝑑(𝑥, 𝑗)=(max𝑦′∈𝒴 𝑠ℎ(𝑥, 𝑦′),if 𝑗=0 𝑠𝑑(𝑥, 𝑗),if 𝑗∈ [𝐽](9) The decision is action(𝑥)=argmax𝑗∈{0}∪[𝐽]¯ 𝑠𝑑(𝑥, 𝑗) . The surrogate loss, constructed using a standard multi-class surrogate ℓ2(e.g., logistic loss), is defined as: 𝐿ℎ(𝑠𝑑, 𝑥, 𝑦)= 1 ℎ(𝑥)=𝑦·ℓ2(¯ 𝑠𝑑, 𝑥, 0) + 𝐽 Õ 𝑗=1 ¯ 𝑐𝑗(𝑥, 𝑦) · ℓ2(¯ 𝑠𝑑, 𝑥, 𝑗).(10) 22 where ¯ 𝑐𝑗(𝑥, 𝑦)=1−𝑐𝑗(𝑥, 𝑦) is the expert reward. This loss encourages high scores for correct actions: it rewards the predictor if it is correct, and rewards expert 𝑗 weighted by their correctness ¯ 𝑐𝑗(𝑥, 𝑦). Predictor-Rejector Formulation. In the PR formulation, the rejector is generalized to 𝑟:𝒳 → R𝐽 . The system uses ℎ if 0<min𝑗∈[𝐽]𝑟𝑗(𝑥) ; otherwise, it defers to argmin𝑗∈[𝐽]𝑟𝑗(𝑥) . By defining an associated hypothesis ¯ 𝑟 where ¯ 𝑟(𝑥, 0)=0 and ¯ 𝑟(𝑥, 𝑗)=−𝑟𝑗(𝑥) , the surrogate loss is defined analogously: 𝐿ℎ(𝑟, 𝑥, 𝑦)= 1 ℎ(𝑥)=𝑦·ℓ2(¯ 𝑟, 𝑥, 0) + 𝐽 Õ 𝑗=1 ¯ 𝑐𝑗(𝑥, 𝑦) · ℓ2(¯ 𝑟, 𝑥, 𝑗).(11) Theoretical Guarantees and Limitations. Mao et al . [43] proved that both surrogates admit ℛ -consistency bounds (Def. 3) (implying Bayes-consistency (Def. 1)). However, they only established realizable ℛ -consistency (Def 2) for the restricted case of constant deferral costs (i.e., 𝑐𝑗(𝑥, 𝑦)=𝛽𝑗 ). This left a notable gap, as the guarantee did not hold for the common scenario where costs are based on expert misclassification error (𝑐𝑗(𝑥, 𝑦)= 1 ℰ𝑗(𝑥)≠𝑦). Achieving Realizable ℛ-Consistency in Two-Stage Deferral This limitation was recently addressed by Mao et al . [49] . They introduced a new family of surrogate losses designed specifically to achieve realizable ℛ -consistency for classification-errorbased costs in the two-stage setting. This framework simplifies the objective by focusing only on routing among a set of 𝐽 experts, assuming the pre-trained predictor ℎ is simply one of the available experts (e.g., ℎ=ℰ1 ). The goal is to learn a routing function 𝑟:𝒳 × [𝐽] → R to minimize the two-stage target deferral loss 𝐿tdef(𝑟, 𝑥, 𝑦)=Í𝐽 𝑗=1𝑐𝑗(𝑥, 𝑦) 1 𝑟(𝑥)=𝑗. Mao et al . [49] propose the following surrogate loss family, based on the comp-sum [ 44 ] structure parameterized by a decreasing function Ψ(e.g., Ψ(𝑢)=1−𝑢or Ψ(𝑢)=−log(𝑢)): 𝐿Ψ(𝑟, 𝑥, 𝑦)= 𝐽 Õ 𝑗=1©«Õ 𝑗′≠𝑗 𝑐𝑗′(𝑥, 𝑦) − 𝐽+2ª®¬Ψ 𝑒𝑟(𝑥,𝑗) Í𝑗′∈[𝐽]𝑒𝑟(𝑥,𝑗′)!.(12) This formulation applies a weighted penalty to the standard comp-sum loss for each expert 𝑗 , where the weight depends on the costs of the other experts. Unified Guarantees and Assumptions. If (i) the score class ℛ is closed under positive scaling, (ii) costs are classification-error costs as above, (iii) for every (𝑥, 𝑦) at most one expert is perfect (i.e., at most one 𝑗 has 𝑐𝑗(𝑥, 𝑦)=0 ), and (iv) lim𝑢→1−Ψ(𝑢)=0 , then 𝐿Ψ is realizable ℛ -consistent with respect to 𝐿tdef . Beyond realizability, Ψ surrogates admit ℛ -consistency bounds (hence Bayesconsistency) under symmetry and completeness of ℛ and the condition Í𝑗′≠𝑗𝑐𝑗′(𝑥, 𝑦) ≥ 𝐽−2 for all 𝑗, (𝑥, 𝑦)(equivalently, on any instance, at most two experts can be correct). The “at most one perfect expert” assumption enforces uniqueness of the zero-cost expert. It is plausible when experts are deliberately specialized with minimal overlap, but can be strong in settings where multiple experts often agree. The other requirements (scaling closure of ℛ and lim𝑢→1−Ψ(𝑢)=0 ) are mild and satisfied by standard score classes and common Ψ choices. 23 5.4 An Alternative Formulation: Dependent Bayes Optimality Although the majority of research has focused on developing surrogates that are consistent with the standard Bayes-optimal rule (Eq. 3), alternative research has emerged that questions the fundamental formulation of the problem itself, namely the statistical dependence between the model and expert predictions. While the standard optimality condition estimates and compares the independent marginal quantities for model and expert confidences, Wei et al . [71] introduce Dependent Bayes Optimality, which recasts the deferral decision as a comparison between two mutually exclusive events: the system should defer if and only if it is more probable that (1) the model is wrong, and the expert is correct, than that (2) the model is correct, and the expert is wrong. Based on this principle, the authors proposed a new surrogate, the Dependent CrossEntropy loss 𝐿⊥ DCE , which directly implements the deferral principle during training by directly encouraging a relative ordering of model prediction and deferral scores based on the observed dependence patterns in the training triplets. This approach bypasses the intermediate step of confidence estimation, offering a simpler and potentially more direct path to learning an effective deferral policy. 5.5 Key Considerations for Implementation This section is intended to equip L2D researchers with principles to inform design choices for building L2D systems that are reliable for the intended task, rather than identifying a single “best” loss. In short: pick the framework that fits your deployment constraints; select surrogates for calibration and guarantees; and evaluate on coverage/cost and reliability, not accuracy alone (Figure 7). Figure 7: Key Considerations for Implementation Choose by setting, not by headline accuracy. On common benchmarks, many consistent surrogates tie on system accuracy [ 49 ]. Prioritize calibration of expert-correctness,robustness to consultation costs, and theoretical guarantees (cf. Table 3). Match architecture to constraints. Use score-based setups for one-stage classification; predictor–rejector when doing two-stage,regression, or when experts are heterogeneous/black-box. One-stage (joint) rule of thumb. Start with bounded/asymmetric SB surrogates for calibration; when realizable ℋ -consistency or full ℋ -bounds matter, choose unified/realizable families (Table 3). Avoid label-smoothing side-effects if you include a non-zero base consultation cost. Two-stage (fixed predictor) rule of thumb. Use surrogates that admit ℋ -consistency bounds; when costs are error-based, prefer realizable two-stage objectives. This framework stays modular and resilient to expert/model changes. Alternative optimality (when to care). Dependent Bayes optimality is useful when model–expert dependence is itself the object of interest; otherwise the standard Bayes target suffices. Report what matters. Beyond accuracy: (i) Accuracy–Coverage curves; (ii) calibration of expertcorrectness (ECE or equivalent); (iii) cost/coverage sensitivity; (iv) workload/budget compliance; and a brief rationale for the chosen guarantees. 6 L2D Task Generalizations The standard L2D frameworks, as discussed in the preceding sections, were primarily developed for multi-class classification. However, L2D’s core principle of optimally routing between an automated model and an expert is a concept with far broader applicability. This section describes 24 recent efforts to generalize the L2D field beyond simple classification. We will examine how L2D has been reformulated for regression tasks with continuous outputs (§6.1), for multi-task problems requiring complex, structured predictions (§6.2), for settings where the goal is to select a committee of top agents rather than a single one (§6.3), and for sequential scenarios where the objective shifts from immediate accuracy to optimizing long-term outcomes (§6.4). Notation. We retain the global setup of §3.1 (Table 1). This section changes the task space and/or decision objects per subsection (regression, multi-task, top𝑘 , sequential, causal); each subsection below states its deviations explicitly. 6.1 Regression with Deferral Extension of L2D to regression tasks presents unique challenges due to the continuous nature of the output space, 𝒴 ⊂ R . In this setting, the predictor ℎ:𝒳 → R is a regression model, and the 𝐽experts ℰ1, . . . , ℰ𝐽are also regressors. The objective remains to minimize the system loss, but the classification error (e.g., 0-1 loss) is replaced by a standard regression loss 𝐿 (e.g., squared or absolute error), which is typically assumed to be bounded. This change has architectural consequences. As established in §3, the score-based formulation (§3.2) is fundamentally inapplicable to regression. The core mechanism of augmenting the label space 𝒴 with deferral options {⊥1, . . . , ⊥𝐽} and learning a unified scoring function over them breaks down, as it is intractable to simultaneously score every real value in the infinite space 𝒴 and the discrete deferral options. This incompatibility necessitates the adoption of the predictor-rejector (PR) architecture (§3.2), which naturally handles regression by decoupling the continuous output of the predictor ℎ from the selection of the decision-making expert. The first comprehensive framework for regression with multi-expert deferral, addressing both one-stage and two-stage scenarios, was introduced by Mao et al. [48]. Below, we detail the one-stage setup. The Generalized Predictor-Rejector Formulation for Regression To handle multiple experts in the regression context, the standard binary PR formulation (which decides only whether to predict or defer) must be generalized. In this framework, the rejector 𝑟 is defined as a multi-output scoring function, 𝑟:𝒳 × {0,1, . . . , 𝐽} → R . This function scores 𝐽+1 options: option 0 corresponds to the predictor ℎ , and options 1 through 𝐽 correspond to deferral to the respective experts. The system’s decision, denoted 𝑟∗(𝑥) , is determined by which expert receives the highest score: 𝑟(𝑥)=argmax𝑗∈{0,...,𝐽}𝑟(𝑥, 𝑗) . If 𝑟∗(𝑥)=0 , the system predicts ℎ(𝑥) ; otherwise, it defers to expert 𝑟∗(𝑥). The target deferral loss 𝐿def adapts the PR objective to this multi-expert regression setting: 𝐿def(ℎ, 𝑟, 𝑥, 𝑦)=𝐿(ℎ(𝑥), 𝑦) · 1 𝑟∗(𝑥)=0+ 𝐽 Õ 𝑗=1 𝑐𝑗(𝑥, 𝑦) · 1 𝑟∗(𝑥)=𝑗(13) where 𝐿 is the regression loss and 𝑐𝑗(𝑥, 𝑦) is the cost of deferring to expert 𝑗 (e.g., the expert’s own regression loss, potentially plus a base cost). A Predictor-Rejector Surrogate via Transformation Direct optimization of Eq. 13 is intractable. In deriving a tractable surrogate for the one-stage setting (where ℎ and 𝑟 are learned jointly), Mao et al . [48] employ a novel algebraic transformation of the target loss. The main insight is to rewrite 𝐿def to expose a structure that mimics a weighted multi-class classification problem over the 𝐽+1 choices. This reformulation allows for leveraging well-understood multi-class surrogate losses from classification, despite the underlying regression task. 25 Modeling from Incomplete Data: Sparsity-Aware and Probabilistic Approaches. A second strategy is to design models that inherently handle data scarcity without requiring a fully annotated dataset. A key example is sparsity-aware model design, which is particularly useful in multi-expert settings where annotations are sparse (e.g., “single-annotator sparsity”, where each instance is typically annotated by only one expert). The DeCCaF framework by Alves et al . [1] directly addresses this by training a unified “Human Expertise Model” (HEM). By conditioning the model output on both the instance features and expert unique identity, it learns a single, shared representation of team behavior from these sparse annotations. Other works have turned to more formal probabilistic models to handle missingness during the learning process. Nguyen et al . [59] frame the problem of missing expert annotations within a latent variable model, using the Expectation-Maximization (EM) algorithm to jointly infer the missing labels while simultaneously learning the L2D system. This provides a principled, end-to-end framework for training with incomplete data. In a different vein, the decoupled Bayesian approach of Strong et al . [67] avoids the need for expert annotations on the main training set entirely. It constructs an explicit, statistical representation of an expert’s class-wise performance from a small, separate “context set.” This expert model, which can be informed by priors, is then used directly at test-time to make deferral decisions, decoupling the expert modeling from the primary task training. 7.2 Handling Dynamic Expert Pools A significant limitation of classic L2D frameworks is their assumption of a fixed expert pool. These systems are trained on data from a specific, well-identified set of expert(s), and their deferral policies are inherently customized to this exact group. This is often untenable in practice. For example, in a hospital, clinicians have different shift patterns, and subspecialists may be intermittently available. To address this, a line of research has focused on enabling L2D systems to adapt to previously unseen experts at test-time. The central idea shared by works such as Tailor et al . [68] and Strong et al . [67] is to enable this adaptation by leveraging a small amount of data from the new expert, known as a context set. This set provides a few-shot demonstration of the new expert’s behavior. While both frameworks leverage this concept, they diverge fundamentally in how they represent expert behavior and structure their deferral mechanisms. Expert Representation and Deferral The key distinction between these approaches lies in how they model an expert from the context set. Tailor et al . [68] use a meta-learning framework with Neural Processes [ 22 ] to encode expert behavior into a learned implicit latent vector. The deferral mechanism is then additionally conditioned on this embedding, learning to associate different latent expert profiles with optimal deferral actions. The alternative approach by Strong et al . [67] uses a Bayesian statistical model to construct an explicit, interpretable representation of expert performance. This representation consists of metrics, such as the estimated per-class accuracy and the associated uncertainty. This enables an expert-agnostic deferral mechanism, where the deferral logic is learned based on the general, structural relationship between predictor confidence and expert quantified competence (via the aforementioned metrics), rather than being tied to specific expert identities or profiles seen during training. 32 Implications and Trade-offs These methods represent a step towards making L2D systems practical and deployable in non-static, real-world environments. The choice between these approaches introduces a key trade-off for system designers. Learned latent representations are useful for modeling complex, holistic behavioral patterns, but may be less interpretable and potentially less robust to outof-distribution experts. In contrast, structured statistical representations can offer greater robustness in low-data regimes, interpretability, and data efficiency, but might provide a more generalized, and thus potentially less detailed representation of expert behavior. Future work in this area will likely focus on bridging this gap, seeking methods that combine the expressive power of latent models with the robustness and interpretability of structured ones. 7.3 Integrating Policy Constraints into Deferral Frameworks Real-world L2D systems must often operate under policy constraints, such as fairness or budget limits, rather than only optimizing a single objective like accuracy. This has motivated a shift in L2D research from surrogate minimization to constrained optimization as described next. A key contribution is the unifying post-processing framework of Charusaie and Samadi [10] , which provides a general, theoretically-grounded method for finding a Bayes-optimal deferral policy that maximizes a primary objective while satisfying arbitrary constraints. Their two-stage methodology first uses standard models to estimate the necessary scores for the objective and constraints (e.g., model confidence, expert accuracy, fairness metrics). In the second stage, it leverages a generalization of the Neyman-Pearson lemma [ 35 ] to derive an optimal decision rule. This rule takes the form of a simple linear combination of the pre-computed scores: decision(𝑥)=argmax 𝜓0(𝑥) − Í𝑚 𝑖=1𝑘𝑖𝜓𝑖(𝑥) Here, 𝜓0(𝑥) represents the objective, 𝜓𝑖(𝑥) are costs for the 𝑚 constraints, and the trade-off parameters 𝑘𝑖 are tuned on a validation set to meet the desired constraint tolerances. The significance of this work is its generality, which moves the field beyond designing task-specific surrogate losses. The authors also prove that finding a deterministic solution is NP-Hard, providing strong justification for their use of a tractable and provably optimal randomized policy. 7.4 Enhancing L2D Safety The overall reliability of an L2D system hinges on the integrity of its core deferral mechanism. While much of the literature to date focuses on optimizing this mechanism under ideal data conditions, its performance can degrade significantly when faced with real-world complexities, compromising system safety and effectiveness. This has motivated a line of research aimed at enhancing the robustness of the deferral. These efforts address three primary failure modes: (1) the system’s vulnerability to deliberate, adversarial manipulation; (2) its inability to express uncertainty when the optimal allocation is ambiguous; and (3) the lack of interpretability or rationale for deferral decisions needed for effective oversight. Robustness Against Adversarial Attacks The first study of adversarial robustness in L2D, by Montreuil et al . [50] , reveals that systems are susceptible to attacks targeting the core allocation mechanism. Focusing on the practical two-stage setup (cf. §4.2) where only the rejector function 𝑟 is learned, an adversary’s goal is to apply a small perturbation to an input x to create 𝑥′ , manipulating the rejector’s output 𝑟(𝑥′) to cause a 33 suboptimal allocation. This can lead to consequences like increased costs or denial of service. The authors formalize two attack strategies: 1. Untargeted Attacks: The adversary crafts a perturbation to disrupt the optimal allocation, forcing the system to select any suboptimal agent and thereby degrading overall performance. 2. Targeted Attacks: The adversary forces the system to route a query to a specific, predetermined agent, which could enable fraud in a pay-per-query system. A Robust Deferral Framework To defend against these threats, the same work proposes SARD (Smooth Adversarial Regularized Deferral), an adversarial training algorithm for the two-stage task. The approach learns a robust rejector 𝑟 by minimizing the worst-case deferral loss within a small perturbation ball around each training input. This objective is the adversarial true deferral loss: ˜ ℓdef(𝑟, 𝑔, 𝑚, 𝑧)=sup 𝑥′∈𝐵𝑝(𝑥,𝛾) 𝐽 Õ 𝑗=0 𝑐𝑗(𝑔(𝑥), 𝑚𝑗(𝑥), 𝑧) · 1 𝑟(𝑥′)=𝑗(20) A key subtlety is that the costs 𝑐𝑗 are evaluated on the clean input 𝑥 , while the rejector’s decision is based on the perturbed input 𝑥′. As this loss is intractable, the SARD algorithm optimizes a smooth, regularized version of a tractable surrogate from a novel family of adversarial margin deferral surrogates. This framework has strong theoretical backing; the surrogates are proven to be both Bayes-consistent and ( ℛ , 𝒢 )-consistent, providing a formal guarantee that the learned policy is robust against these allocation attacks. Handling Uncertainty in the Deferral Decision Beyond robustness to external attacks, the safety of an L2D system also depends on the reliability of its rejector. As a predictive model, the rejector can be wrong, especially on ambiguous inputs, making a forced allocation decision a safety risk. To address this, Fang and Nalisnick [21] quantify rejector uncertainty using conformal prediction [ 2 ]. They construct a deferral set 𝒮(𝑥) that can return {ℎ} , {ℰ} , or {ℎ, ℰ} in the single-expert L2D setting (classifier ℎ vs. expert ℰ ). The set is calibrated (via split-conformal prediction) to marginally cover the indicator of expert correctness with level 1−𝛼 , so that P 1 {ℰ(𝑥)=𝑦} ∈ 𝒮(𝑥)≥1−𝛼 under the exchangeability assumption. When 𝒮(𝑥)={ℎ} the system acts with ℎ ; when 𝒮(𝑥)={ℰ} it defers; and when 𝒮(𝑥)={ℎ, ℰ} the rejector expresses uncertainty about the allocation. This uncertainty signal supports safer downstream protocols when allocation is ambiguous: 1. Abstention. If 𝒮(𝑥)={ℎ, ℰ} , the system abstains; otherwise it follows the singleton in 𝒮(𝑥) (act with ℎif {ℎ}, defer if {ℰ}). 2. Consensus prediction. If 𝒮(𝑥)={ℎ, ℰ} , query both ℎ and ℰ and output a prediction only when they agree; otherwise abstain. Interpretability in Deferral Decisions Beyond robustness to attacks and uncertainty, a third pillar of L2D safety is interpretability. For a human-AI team to function effectively, it is often not enough for the human to know that a case has been deferred; they must also understand why. This can be important in high-stakes domains, as an explanation for deferral can build trust, reveal model blind spots, and guide the expert attention to the most challenging aspects of a problem. Addressing this, work has 34 focused on integrating L2D with inherently interpretable model architectures, most notably Concept Bottleneck Models (CBMs) [33]. CBMs are models designed to be transparent by forcing them to first predict a set of humanunderstandable intermediate “concepts” before making a final prediction. Pugnana et al . [64] introduce Deferring Concept Bottleneck Models (DCBMs), a framework that equips CBMs with an L2D mechanism. The key innovation is that the system can learn to defer at two distinct levels: it can defer on the final task, or it can defer on one of the intermediate concepts that it finds ambiguous. This provides a granular, concept-level explanation for the deferral decision (e.g., “I am deferring this medical image case because I am uncertain about the ‘Glandular’ concept”). By modeling the problem as a composition of deferring systems, DCBMs can boost predictive performance while providing transparent reasons for human intervention, even when the human experts themselves are imperfect. Similarly, the DeCoDe framework by He et al . [26] also leverages CBMs to create an interpretable, concept-driven deferral system. DeCoDe extends this framework by introducing a third collaborative mode beyond the standard AI or human only decision: a “complementarity” mode where AI and human inputs are fused for a joint decision. The choice between these three modes is governed by a gating network that operates on the concept representations, ensuring that the entire decision-making process (whether to predict, defer, or collaborate) remains transparent. 7.5 Controlling Workload Distribution to Experts Table 4: Comparison of methods for controlling workload distribution to experts (L2D). Approach (Citation) Core Mechanism Benefits Limitations Cost-Regularized Training Zhang et al. [76] Additive penalty on collaboration cost (e.g., number of users engaged) in the training objective; a selector learns among modes (AI only, defer, or collaborate with 𝐽experts). End-to-end, supports multiuser deferral/complementarity and trades off accuracy vs. human involvement. No hard coverage/workload guarantees; tuning 𝜆 does not map directly to a target deferral rate; behavior depends on validation-time operating point. MILP (Offline) Mozannar et al. [56] Exact mixed-integer linear program that jointly optimizes a (linear) classifier and rejector with linear constraints (e.g., coverage). Provably optimal for the training objective; straightforward to impose coverage and other linear constraints (e.g., fairness). Computationally expensive; designed for linear (halfspace) models; typicallyusedon modest 𝑛, 𝑑 with precomputed features. EM (Train-Time) Nguyen et al. [59] Mixture-of-experts with constraints enforced in the E-step by bounding posterior assignment rates per expert (incl. AI). Principled, integrates with minibatch deep learning, allows explicit per-expert workload budgets. Requires specifying feasible per-expert lower/upper bounds; not an exact 0–1 optimizer like MILP. Post-Hoc Calibration Ponomarev [63] Calibrate a deferral score threshold on a held-out set to meet a required coverage, then apply per-sample at inference. Fast, model-agnostic, and supports “on-the-fly” decisions without batching. Heuristic with no formal guarantees; smallbudgetviolations can occur; performance depends on the calibration set. Early L2D systems primarily maximized human-AI team accuracy, but real deployments often face bounded human capacity. Unconstrained L2D can overload a small subset of experts while underutilizing others, motivating methods that explicitly control workload distribution. Broadly, these either treat workload as a soft penalty learned end-to-end or impose a hard budget that must be met. 35 Cost-based approaches: workload as a soft penalty A natural approach is to encode the cost of collaboration within the training loss. Zhang et al . [76] extend L2D to multiple users and joint complement/deferral (LECODU), defining a collaboration cost as the number of users engaged and optimizing ℒtotal =ℒtask +𝜆·cost(𝑔𝜙(𝑥)), where 𝑔𝜙(𝑥) selects a collaboration mode (AI alone, defer, or collaborate with 𝐽 experts). Varying 𝜆 smoothly trades accuracy for human effort, but there is no direct guarantee on coverage or per-expert workload, and the achieved operating point is typically chosen by validation. Related work has also noted that surrogate losses can be brittle in how well they complement the human, even without hard budgets, and may require careful tuning [56]. Budget-based approaches: workload as a hard constraint When operations require strict capacity limits, budget-based methods are preferable. These fall into: (i) exact offline optimization, (ii) stochastic train-time optimization, and (iii) post-hoc calibration. Exact offline optimization Mozannaretal .[56] show thelinear L2Dproblemiscomputationally hard in general but give an exact mixed integer linear programming (MILP) solution for (linear) classifier and rejector, to which linear constraints are easily added. Let 𝑟(𝑥) ∈ {0,1} denote deferral; a coverage constraint can be imposed as 1 𝑁Í𝑖𝑟(𝑥𝑖) ≤ 𝛽 , i.e., AI covers at least 1−𝛽 of inputs. On variables 𝑑𝑖=1{𝑟(𝑥𝑖)=1} this is equivalent to 1 𝑁Í𝑖(1−𝑑𝑖) ≥ 𝐶𝑟 , where 𝐶𝑟 is the required AI coverage. The MILP yields globally optimal training solutions under the stated assumptions but is computationally heavier and targeted to halfspaces. Stochastic train-time optimization via expectation–maximization (EM) Nguyen et al . [59] model L2D as a constrained mixture of multiple human experts plus the AI. Workload control is enforced during the E-step of EM by constraining posterior assignment rates 𝑞(𝑧𝑖) to lie within per-expert bounds 𝜺ℓ≤1 𝑁Í𝑖𝑞(𝑧𝑖) ≤ 𝜺𝒖 . This integrates naturally with deep learning, supports missing annotations, and trains a gate that respects budgeted workloads without requiring post-hoc thresholding on the test set. Post-hoc calibration heuristics. Ponomarev [63] propose a simple three-step recipe: (1) train any L2D model, (2) build a deferral score that ranks “AI vs. human,” and (3) on a held-out set, pick a threshold that maximizes accuracy subject to the desired AI coverage 𝐶𝑟 . At inference, defer iff the score falls below the calibrated threshold. This is lightweight and online, but lacks guarantees; empirical results show small but nonzero budget violations in some settings. Discussion Method choice hinges on operational needs (Table 4). Soft-penalty methods (e.g., 76 ) are attractive for multi-expert collaboration but provide no strict budget guarantees. MILP [ 56 ] offers exact control and provable optimality (in the stated linear setting), at higher computation. EM-based training [ 59 ] brings explicit workload budgets into end-to-end learning and avoids post-hoc dependence on test batches. Post-hoc calibration [ 63 ] is the most deployment-friendly when you only need an operating point, accepting approximate budget satisfaction. A promising direction is to combine EM-style train-time constraints with calibrated post-hoc tuning, or to pair MILP-style budget guarantees with richer (non-linear) models via learned surrogates. 36 8 Future Directions, Open Challenges and Concluding Remarks The theory for L2D is well developed, but (to the best of our knowledge) there are no published real-world case studies or deployments. Because L2D is a systems problem (routing among models and experts under budgets, latency, and fairness), a priority should be operational validation: rigorously designed studies that test L2D end-to-end under real constraints. This complements ongoing theoretical work and establishes whether and when L2D delivers practical benefit. These studies will also surface real failure modes (e.g., distribution shift, expert drift, budget mis-specification, miscalibrated expert-correctness, handoff/latency bottlenecks), strongly motivating the next generation of L2D theory and methods. Targeted agenda by taxonomy. • Methodological frameworks (§4). Default to two-stage when the predictor is frozen/API; train the rejector from weak/bandit feedback—no logits or dense expert labels required (§4.2); mitigate one-stage brittleness with “anti-specialization” regularizers and advisor-compatible heads that preserve competence on deferred regions. • Optimization & theory (§5). Makeguaranteesshift-robust (covariate/label/expert-population) while preserving calibration, budgets, and fairness (§5.2). Relax strong separability/modeling assumptions in multi-expert two-stage deferral (§5.3) (e.g., unique perfect expert or ≤2 correct experts), and develop surrogates/analyses that remain valid with overlapping experts, richer cost structures, and non-unique optima. Develop guarantees under model–expert dependence (§5.4): prove consistency/Bayes-consistency and calibration for dependence-aware surrogates (e.g., DCE), characterize identifiability and sample complexity for estimating joint events, and extend to multi-expert/two-stage, budgeted, and fairness-constrained settings with robustness to misspecification and selection bias. • Task generalizations (§6). Develop realistic task generalizations that enable real-world deployments of L2D in healthcare, finance, and other decision-critical settings. • Real-world adaptations (§7). Model the dynamics of human behavior and the operational context in which L2D operates, and design frameworks that adapt to these complexities. Conclusion L2D reframes reliability from confidence estimation to decision allocation: deciding who should act, for which input, under what costs and constraints. The field now has clear frameworks, robust objectives with meaningful guarantees, extensions beyond multiclass prediction, and an emerging toolkit for deployment under real-world constraints. We hope the taxonomy, synthesis, and guidance in this survey help researchers target the most impactful gaps and help practitioners build human-AI systems that are not only accurate, but calibrated, accountable, and safe. Acknowledgments This work was supported by the Engineering and Physical Sciences Research Council. JS is funded by the EPSRC Center for Doctoral Training in Health Data Science [EP/S02428X/1]. ES is funded by an EPSRC Doctoral Training Partnership [EP/W524311/1]. 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