Learning to Defer (L2D) in Human–AI Collaboration

Updated 11 July 2026

Learning to Defer (L2D) is a framework for human–AI collaboration that decides when to predict or defer based on comparative estimates of accuracy.
It employs surrogate loss functions and calibration strategies to effectively balance automated predictions with expert inputs.
Extensions of L2D include multi-expert, cost-sensitive, and sequential formulations that address operational constraints and dynamic human performance.

Learning to Defer (L2D) is a family of methods for human–AI collaborative decision-making in which a predictive system learns, for each input, whether to output its own prediction or to hand the case to a human or other expert. In contemporary work it is treated as the dominant formal framework for human–AI collaboration in classification tasks, and its scope has expanded from single-expert abstention to multiple experts, cost-sensitive objectives, workload constraints, adaptive expert populations, and sequential decision-making under changing human performance (Alves et al., 2024, Leitão et al., 2022).

1. Formal problem statement

The standard L2D setup introduces an input $x \in \mathcal{X}$ , a true label $y \in \mathcal{Y}$ , a classifier prediction $\hat{y}_C(x)$ , a human or expert prediction $\hat{y}_H(x)$ , and a deferral decision. In the single-expert case, the decision variable is often $d(x)\in\{0,1\}$ , where $d(x)=0$ means the model predicts and $d(x)=1$ means the system defers. In the multi-expert case, experts are indexed by $j\in\{1,\dots,J\}$ , and the rejector becomes $r(x)\in\{0,1,\dots,J\}$ , where $r(x)=0$ means the classifier decides and $y \in \mathcal{Y}$ 0 means deferral to expert $y \in \mathcal{Y}$ 1 (Alves et al., 2024).

A canonical multi-expert objective is the expected $y \in \mathcal{Y}$ 2– $y \in \mathcal{Y}$ 3 deferral loss

$y \in \mathcal{Y}$ 4

This loss makes the system-level objective explicit: the model is penalized when it predicts and is wrong, while expert $y \in \mathcal{Y}$ 5 is penalized when the system defers to that expert and the expert is wrong (Verma et al., 2022).

The Bayes-optimal policy compares the classifier’s best attainable correctness probability with the experts’ correctness probabilities. In the single-expert formulation reproduced in later surveys, the optimal rejector defers when $y \in \mathcal{Y}$ 6 exceeds $y \in \mathcal{Y}$ 7; in the multi-expert formulation, the optimal system chooses the best expert if that expert is more likely to be correct than the classifier, and otherwise predicts with the classifier (Leitão et al., 2022). This is the central conceptual distinction between L2D and ordinary abstention: the defer decision is not based solely on model uncertainty, but on a comparative estimate of model and expert performance.

2. Surrogate losses, calibration, and statistical properties

Because the target objective is a discontinuous $y \in \mathcal{Y}$ 8– $y \in \mathcal{Y}$ 9 loss, most L2D methods optimize surrogate losses. Two parameterizations have become especially prominent in the literature: softmax-based surrogates and one-vs-all (OvA) surrogates. For multiple experts, both parameterizations admit consistent surrogate losses, extending earlier single-expert constructions to augmented output spaces that include deferral actions to individual experts (Verma et al., 2022).

A major technical issue is calibration of expert-correctness probabilities. In multiclass L2D, the system must estimate quantities of the form $\hat{y}_C(x)$ 0, because these drive the Bayes-optimal defer decision. Calibration work showed that the widely used softmax formulation is not calibrated with respect to expert correctness and is not even guaranteed to produce valid probabilities due to a degenerate parameterization. In particular, if the softmax reject probability is $\hat{y}_C(x)$ 1, the implied estimate

$\hat{y}_C(x)$ 2

can exceed $\hat{y}_C(x)$ 3 whenever $\hat{y}_C(x)$ 4, which makes it invalid as a probability estimate (Verma et al., 2022). The OvA alternative replaces this construction with independently trained binary heads and yields calibrated probabilities of expert correctness while remaining a consistent surrogate for multiclass L2D.

In the multi-expert setting, calibration and estimation become more delicate. The multi-expert calibration study showed that the softmax-based loss causes mis-calibration to propagate between the estimates $\hat{y}_C(x)$ 5, whereas the OvA-based loss does not, although practical trade-offs remain (Verma et al., 2022). This matters because mis-calibrated correctness estimates distort not only the rejector but also downstream procedures such as expert ranking and multi-expert ensembling.

A separate line of work has shown that multi-expert L2D is statistically harder than single-expert L2D. Recent theory identifies an intrinsic expert identifiability problem: with multiple experts, classifier underfitting becomes inherent rather than incidental, because the system must learn not just whether to defer, but which expert to trust from a diverse pool. The proposed PiCCE method addresses this by adaptively identifying a reliable expert based on empirical evidence, reducing multi-expert L2D to a single-expert-like learning problem and proving consistency as well as recovery of class probabilities and expert accuracies (Liu et al., 19 Feb 2026).

3. Assumptions, failure modes, and critiques

L2D is often presented as a principled routing framework, but surveys of human–AI collaboration have emphasized that the standard formulation rests on demanding assumptions. A central requirement is the availability of human predictions for every training instance; in multi-expert settings this can strengthen into requiring predictions from every expert on every instance. In many operational domains, however, each case is reviewed by only one analyst, clinician, or moderator, making the required training matrix unavailable by design (Leitão et al., 2022).

A second assumption is that ground-truth labels are available independently of the human decision-maker. This fails in selective-label settings such as credit approval, bail decisions, and fraud screening, where outcomes are observed only if a preceding human or automated action allows them to materialize. Under selective labels, the standard L2D loss is no longer directly observable for all instances, because the true outcome depends on the action itself (Leitão et al., 2022).

A third recurring limitation is that classical L2D is usually pointwise rather than operational. Standard formulations decide on each instance in isolation and typically ignore capacity management, expert absence, scheduling, or fatigue. The same critique applies to many fairness-aware extensions: while L2D can optimize group-wise metrics for the combined system, the routing policy may still be unrealistic if it presumes unlimited human capacity or static expert behavior (Leitão et al., 2022).

These critiques motivate a broader interpretation of L2D as one component of human–AI collaboration rather than a complete deployment solution. This suggests that L2D should be understood alongside selective prediction, reject-option classification, counterfactual evaluation under selective labels, and workflow design, rather than as a universally sufficient abstraction.

4. Generalizations beyond fixed single-expert deferral

Several recent strands of work broaden L2D far beyond its original “predict or defer to one fixed expert” form.

Extension	Core idea	Representative paper
Multi-expert L2D	Learn consistent softmax and OvA surrogates for choosing among several experts	(Verma et al., 2022)
Population L2D	Adapt to never-before-seen experts from a small context set via meta-learning	(Tailor et al., 2024)
Limited-demonstration population L2D	Use expert embeddings and pseudo-label generation to train with only a few demonstrations per expert	(Ramgolam et al., 22 Oct 2025)
Two-stage Top- $\hat{y}_C(x)$ 6 L2D	Defer each query to the $\hat{y}_C(x)$ 7 most cost-effective experts, with adaptive Top- $\hat{y}_C(x)$ 8 selection	(Montreuil et al., 17 Apr 2025)
One-stage Top- $\hat{y}_C(x)$ 9 L2D	Learn a shared score-based model with a $\hat{y}_H(x)$ 0-independent convex surrogate	(Montreuil et al., 15 May 2025)
Training-free multiple-expert deferral	Use conformal prediction and segregativity instead of learning a deferral model	(Bary et al., 16 Sep 2025)

Population-based L2D is a particularly important change in problem formulation. Instead of assuming a fixed, well-identified expert, population L2D treats the available expert as sampled from a distribution and conditions the rejector on a small context set that characterizes the current expert. Meta-learning then allows rapid adaptation to experts never seen during training, including model-based variants that use attention to retrieve context points similar to the current test point (Tailor et al., 2024). A related data-efficiency extension uses context-aware expert embeddings to generate pseudo-labels at scale and to enable on-the-fly adaptation to new experts with only a few demonstrations, thereby addressing a major training bottleneck for population-level deployment (Ramgolam et al., 22 Oct 2025).

Top- $\hat{y}_H(x)$ 1 L2D changes the action space itself. Rather than selecting exactly one decision-maker, Top- $\hat{y}_H(x)$ 2 frameworks defer to the $\hat{y}_H(x)$ 3 most cost-effective experts for a given input, and Top- $\hat{y}_H(x)$ 4 learns the cardinality as a function of input complexity and consultation cost. The two-stage version strictly generalizes classical L2D and recovers model cascades as a special case, while the one-stage version unifies prediction and deferral in a single score-based model with a cost-sensitive loss and a convex surrogate independent of $\hat{y}_H(x)$ 5 (Montreuil et al., 17 Apr 2025, Montreuil et al., 15 May 2025).

Not all recent work retains a learned deferral policy. A training-free alternative replaces learned routing with conformal prediction: the model predicts when its own label set is singleton and otherwise defers to the most discriminative expert according to a segregativity criterion computed from expert confusion patterns. This line of work is explicitly framed as a retraining-free alternative to L2D when expert composition changes frequently (Bary et al., 16 Sep 2025).

5. Cost sensitivity, capacity constraints, and operationalization

A major practical shift in the literature is the move from symmetric error minimization to cost-sensitive, capacity-aware deferral. Real deployments rarely treat false positives and false negatives equally, and they almost never have unlimited human review capacity. The DeCCaF framework addresses exactly these gaps by introducing deferral under cost and capacity constraints, using supervised learning to model the probability of human error from only one expert prediction per instance and then applying constraint programming to globally minimize expected misclassification cost subject to workload limitations (Alves et al., 2024).

In the DeCCaF formulation, the classifier estimates $\hat{y}_H(x)$ 6, a Human Expertise Model estimates each expert’s probability of correctness, and a global assigner solves a batch-level optimization problem under per-expert capacities. This differs from classic pointwise L2D in two ways: the training objective is explicitly cost-sensitive, and the routing decision is global rather than local. In cost-sensitive fraud detection scenarios with teams of 9 synthetic fraud analysts and individual work-capacity constraints, DeCCaF achieved an average $\hat{y}_H(x)$ 7 reduction in misclassification cost relative to the baselines reported in that study (Alves et al., 2024).

The same operational emphasis appears in dataset design. FiFAR was introduced to provide a public benchmark for L2D in fraud detection, combining a synthetic bank account fraud dataset with the predictions of 50 highly complex and varied synthetic fraud analysts and a realistic definition of human work capacity constraints. The benchmark supports evaluation under 300 distinct testing scenarios and was explicitly motivated by the absence of public L2D datasets that reflect expert heterogeneity, limited human availability, and domain-specific cost structures (Alves et al., 2023).

Taken together, these developments shift L2D from a purely confidence-based abstention mechanism toward a constrained resource-allocation problem. This suggests that in high-stakes deployments the central object is not merely an instancewise defer score, but a policy for assigning limited human attention under asymmetric costs, varying expert skill, and changing operational budgets.

6. Sequential environments, domain-specific reformulations, and current frontiers

The static formulation of L2D is increasingly being replaced by sequential and nonstationary variants. In online L2D, data arrive as a stream, the available expert pool may vary across rounds, and feedback can be bandit rather than full-information. Recent work introduced the first online L2D algorithm for multiclass classification with bandit feedback and a dynamically varying pool of experts, with regret guarantees of $\hat{y}_H(x)$ 8 in general and $\hat{y}_H(x)$ 9 under a low-noise condition (Duy et al., 12 May 2026).

Another major frontier is human performance dynamics. Fatigue-aware L2D models human accuracy as a function of cumulative workload rather than as a constant. FALCON formulates L2D as a constrained Markov decision process whose state includes task features and cumulative human workload, uses psychologically grounded fatigue curves, and optimizes under human–AI cooperation budgets with PPO-Lagrangian. Across multiple datasets it outperformed static L2D baselines across coverage levels, generalized zero-shot to unseen experts with different fatigue patterns, and demonstrated the advantage of adaptive human–AI collaboration over AI-only or human-only decision-making when coverage lies strictly between $d(x)\in\{0,1\}$ 0 and $d(x)\in\{0,1\}$ 1 (Zhang et al., 1 Apr 2026).

L2D is also being reformulated in domain-specific ways. In concept bottleneck models, Deferring CBMs treat intervention as a deferral decision and derive a consistent L2D loss so that a concept-based model can learn when an intervention is needed while explaining why defer occurs on the final task (Pugnana et al., 20 Mar 2025). In causal discovery, L2D-CD adapts L2D to pairwise causal direction tasks, learning when to trust a numerical causal discovery algorithm and when to defer to expert recommendations derived from textual metadata; the method outperformed either source in isolation on the Tübingen pairs setting and identified domains where the expert was strong or weak (Clivio et al., 18 Feb 2025).

Across these lines of work, the central research question remains stable: estimate comparative competence well enough to route each decision to the most suitable decision-maker. What has changed is the recognition that competence is not static, not always fully observed, often cost-sensitive, and sometimes only inferable from sparse demonstrations, context sets, or workload trajectories. In that sense, contemporary L2D is no longer just a reject-option variant; it is an increasingly general framework for calibrated, constrained, and adaptive allocation of decisions across heterogeneous human–AI systems.