Consistent Estimators for Learning to Defer to an Expert

Hussein Mozannar; David Sontag

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Consistent Estimators for Learning to Defer to an Expert

Hussein Mozannar, David Sontag

arXiv:2006.01862v3Jun 2, 2020cs.LGcs.HCstat.ML

TL;DR

The paper addresses how learning systems can jointly predict or defer decisions to downstream experts, whose decisions or deferral costs are available as training information. It reduces expert deferral to cost-sensitive learning and proposes a consistent convex surrogate loss, achieving better accuracy with better sample complexity and computational cost than confidence-score baselines across image and text tasks.

Problem
Learning algorithms often ignore downstream expert decision makers, creating the question of how predictors should decide whether to predict or defer using expert information.
Method
The approach jointly learns a classifier and rejector through a reduction to multiclass cost-sensitive learning with a novel convex consistent surrogate loss that requires minimal architectural changes.
Results
Across image and text classification tasks, the approach achieves better accuracy than confidence-score baselines with better sample complexity and computational cost.
Takeaways & Limitations
Learning to defer can integrate expert strengths and weaknesses into prediction and deferral decisions rather than relying on independently learned confidence scores.
Takeaways & Limitations
Confidence-score approaches can fail when classifier and rejector searches are restricted to limited function classes, motivating joint learning in the deferral setting.

Abstract

from arXiv · show

Learning algorithms are often used in conjunction with expert decision makers in practical scenarios, however this fact is largely ignored when designing these algorithms. In this paper we explore how to learn predictors that can either predict or choose to defer the decision to a downstream expert. Given only samples of the expert's decisions, we give a procedure based on learning a classifier and a rejector and analyze it theoretically. Our approach is based on a novel reduction to cost sensitive learning where we give a consistent surrogate loss for cost sensitive learning that generalizes the cross entropy loss. We show the effectiveness of our approach on a variety of experimental tasks.

1 Introduction

The paper develops theoretically sound models that either predict or defer to downstream experts, aiming to adapt to expert strengths while requiring only expert decisions or deferral costs. It introduces a consistent surrogate-loss approach and evaluates it across image and text tasks.

The framework targets machine-learning systems that complement human decision makers by predicting some cases and deferring others.Motivating applications include healthcare, risk assessment, and content moderation.
The learned model should adapt to the expert to outperform deploying either the model or expert alone.This is especially relevant when data or model capacity are limited, or when experts possess private side-information.
The proposed procedure jointly learns a classifier and a rejector that decides whether the classifier or expert should predict.
The paper reduces expert deferral to cost-sensitive learning and proposes a novel convex consistent surrogate loss that requires minimal changes to existing architectures.The approach adds an output layer and changes the loss function, with minimal to no added computational costs.
Experiments evaluate the method and literature baselines on image and text classification tasks, including CIFAR, hate-speech detection, and chest-X-ray classification.

2 Related Work

The paper builds on rejection learning and emerging expert-deferral work, distinguishing its approach by jointly adapting the predictor to the downstream expert rather than relying on fixed confidence estimates or restricted procedures.

Rejection learning studies the trade-off between prediction accuracy and rejection rate, typically using a constant deferral cost.
Prior expert-deferral work includes methods that do not adapt the model to the expert, require expert uncertainty estimates, or use procedures feasible only for small datasets.
Other related approaches learn mixtures of experts, defer using estimated confidence, or study settings where experts accept or reject classifier decisions.
Selective classification constrains rejection probability without assuming a downstream expert, while active learning considers weak and strong labelers.

3 Problem Formulation

The problem is to learn a system that predicts labels or defers to an expert with additional information, while accounting for the target-dependent costs of either action.

The learner predicts target Y from X and may query an expert M with access to domain Z containing additional information.
The system loss combines classifier cost when predicting with expert cost when deferring.
The proposed decomposition learns a classifier h and rejector r, where r selects prediction or expert deferral.
The formulation generalizes rejection learning because a constant expert cost makes the objectives coincide.
For the main theoretical analysis, classifier and expert costs are treated as misclassification error, with additive query costs also supported.
Figure 1 depicts the rejector choosing between classifier h(x) and expert M(z), after which the selected decision incurs its corresponding cost.
The setup can be cast as cost-sensitive learning over an augmented action space containing K predictions plus deferral.The learned predictor minimizes expected cost over these K+1 actions using samples of instances, labels, expert decisions, and required costs.

4 Proposed Surrogate Loss

The paper constructs a convex surrogate for expert deferral through cost-sensitive learning and Bayes-optimal analysis, proving consistency while addressing failures of prior multiclass rejection surrogates.

Existing system loss is non-convex and hard to optimize, motivating convex upper-bounding surrogate losses.
Prior rejection-learning surrogates were consistent only for binary labels and become inconsistent in the multiclass setting.
The proposed cost-sensitive surrogate eLCE generalizes cross entropy for multiclass misclassification and is convex and consistent.
For 0-1 system loss, deferral is represented as an additional class so multiclass misclassification matches the expert-deferral objective.
The Bayes solution predicts with the best classifier and defers when its expected error exceeds the expert’s expected error.
The surrogate LCE is convex, upper bounds the system loss, and is consistent, with minimizers agreeing with the Bayes solution.
The loss can be interpreted as cross entropy on expert-error examples and as a competition between predicting and deferring when the expert is correct.

5 Theoretical analysis

The theoretical analysis shows why classifier and rejector must be learned jointly, and characterizes consistency and generalization properties of mixture-based and empirical approaches.

5.1 Failure of Confidence Scores Method: The confidence-score method can fail because independently learning the classifier prevents adaptation to the expert’s strengths and weaknesses in restricted hypothesis classes.The paper uses a two-group example where joint behavior can fit one group while deferring on another, whereas independent classifier learning may fit neither effectively.
5.1 Failure of Confidence Scores Method: Jointly learning a classifier and rejector can attain the Bayes solution within the hypothesis space when the classifier fits one group and the rejector separates the groups.In the example, the classifier fits group A = 0 and the rejector separates groups when that boundary is linear.
5.2 Inconsistency of mixtures of experts loss and Realizable-consistency: The mixture-of-experts loss extends prior work to multiclass prediction, but it is realizable consistent for scaling-closed classes rather than classification consistent.The stated class family includes linear models and neural networks.
5.2 Inconsistency of mixtures of experts loss and Realizable-consistency: The mixture loss is not jointly convex in classifier and rejector scores, so efficient optimization is unclear despite its realizable consistency.The paper also notes that replacing the classifier loss with a zero-one term would restore consistency at the cost of convexity and differentiability.
5.3 Generalization Bound For Joint Learning: Joint optimization is critical because the system-loss minimizer for the classifier need not minimize the classifier’s standalone misclassification loss.The empirical procedure directly approximates population minimizers over the joint classifier–rejector hypothesis space.
5.3 Generalization Bound For Joint Learning: The empirical joint-learning procedure is controlled by the Rademacher complexities of both model classes and the expert’s error.The bound recovers the rejection-learning result when the expert is always correct and indicates that expert deferral is more sample intensive than rejection learning.

6 Experiments

Experiments show that the proposed deferral methods generally outperform confidence-based and oracle-style baselines across synthetic, image, text, and clinical tasks, especially with limited data or non-uniform expert behavior. The method also supports favorable system accuracy, coverage, and ranking performance, while fairness outcomes depend on the combined system.

Synthetic Data: 6.39 average accuracy points over the confidence baseline and 0.22 positive difference versus the oracle were achieved with α = 0.With α ≠ 0, the method matched the confidence baseline because the model class could not both fit the target and adapt to the expert.
CIFAR-10: 1.01% and 1.94 average accuracy advantages over Confidence and LearnedOracle, respectively, were obtained for the expert-classification experiment.The method also compared classifier accuracy on non-deferred examples against coverage, while another setting reported a 1.60 test-accuracy advantage for 30 ≤ k ≤ 90.
CIFAR10H and limited expert data: 1.2 system-accuracy points and 3.1 classifier-accuracy points on non-deferred examples separated the proposed method from the confidence baseline with limited expert data.The comparison used CIFAR10H and evaluated imputation-based and two-step training variants.
Hate Speech and Offensive Language Detection: 92.91±0.17 system accuracy was achieved by L1CE, compared with 92.42±0.40 for Confidence and 93.22±0.11 for Oracle in the AAE biased-expert setting.The authors report similar accuracy and coverage overall, with the oracle only slightly better across the three experts.
CheXpert and further analysis: The method’s AU-ROC and AU-PR curves dominated baselines across coverage for the toy-expert clinical tasks, while performance gains increased with expert non-uniformity.As training data increased, the method continued improving while baselines began to saturate; on Cardiomegaly, its non-deferred AU-ROC exceeded ModelConfidence by at least 0.2 below 50% coverage.

7 Conclusion

The paper develops a framework in which a model predicts or defers to an expert, using a theoretically analyzed surrogate-loss approach and experiments across image and text tasks.

The framework lets a learning model choose between predicting and deferring to an expert.
The method reduces expert deferral to multiclass cost-sensitive learning and uses a novel surrogate loss.
Experiments on image and text classification show better accuracy than confidence-score baselines, with better sample complexity and computational cost.
Future work includes limited expert data, biased expert data, and simultaneous deferral to multiple experts.

A.1 General implementation

The implementation adds a deferral output to the model and trains it with a surrogate loss, while a rejector determines whether prediction or expert deferral is used.

The method represents prediction with a classifier and deferral with a rejector function.
Practical integration adds one output unit representing deferral and replaces ordinary cross entropy with LCE.
The implementation computes probabilities with softmax and uses expert agreement labels in the deferral loss.
A two-stage validation strategy trains initially with α = 1, then fine-tunes across α values and selects the best threshold.
At inference, the rejector compares the deferral output with the maximum class score using the selected threshold τ*.
System performance measures the combined machine-expert system, whereas classifier performance assumes no deferral.

B.1 CIFAR-10

On CIFAR-10 and related expert settings, the method improves system performance by adapting deferral to model and expert confidence, especially with limited data, while trading off coverage.

The experiments use Wide Residual Networks trained with SGD for 200 epochs, with early stopping providing no difference in test accuracy.
The method’s expert accuracy on deferred examples exceeds other baselines except at k = 1, 2, where coverage is very high.
The method considers both expert and model confidence, unlike LearnedOracle, which focuses only on regions where the expert is correct.
For α < 1, the model favors deferral when the expert is correct; α = 1 compares expert and model confidence consistently with the Bayes rejector.
On CIFAR-100, the method dominates the baseline across expert competence levels and achieves better expert and classifier accuracy but lower coverage.

B.4 Hate Speech experiments

The hate-speech experiments use Adam-trained models and report comparable performance across the proposed method and baselines, with system discrimination included among the metrics.

Models are trained with Adam for 15 epochs, selecting the best model on a validation set.
The proposed method and baselines achieve comparable results on the hate-speech detection task.
The evaluation reports system accuracy, classifier accuracy, coverage, and discrimination, including group-biased expert variants.

B.5 Baseline Implementation

The baseline directly optimizes a non-convex machine-expert loss using separately trained classifier and rejector components, with expert predictions represented from final labels. On CIFAR-10, this baseline often fails to defer appropriately, while the optimal systems emerge during intermediate training.

Baseline method: The [MPZ18] baseline uses a classifier h and binary rejector r to choose between classifier and expert losses.Its loss is non-convex in r and is optimized using a Concrete relaxation with two implementation outputs, r0 and r1.
Baseline method: The classifier is trained independently of the rejector by stopping gradients from r from backpropagating through h.Thus, h is trained concurrently with r using cross entropy rather than adapting to the expert.
Implementation details: In the CIFAR-10 setup, expert decisions are final labels, so the expert loss assigns −log(1−ϵ) to correct labels and −log(1/|Y|) to incorrect labels with ϵ = 10^-12.Training a network to model the expert’s prediction failed because the expert’s random behavior created substantial label noise.
CIFAR-10 results: On CIFAR-10, the baseline almost never defers for expert k < 8, while for k = 8, 9 its rejector never defers at the end of training.For k = 8, 9, the optimal system occurs around 100 epochs rather than at the end of 200 epochs.
CIFAR-10 results: The optimal systems for k = 8, 9 achieve 46.27 and 40.22 coverage, 98.81 and 98.89 expert accuracy on deferred examples, and 89.38 and 89.40 classifier accuracy on non-deferred examples, respectively.The classifier alone reaches approximately 86 classification accuracy on the full validation set for both experts.
CIFAR-10 results: The observed failure arises because the classifier loss approaches zero while the expert loss remains either 0 or −log(0.1), making never deferring easier for the rejector.At initial epochs, the still-learning classifier produces a non-trivial rejector, but coverage grows toward 100% during training.

B.6 CheXpert Experiments

This section develops a binary-label surrogate extension based on convex upper bounds for the classifier and deferral decisions. Consistency requires an input-dependent β rather than the constant β used in standard rejection learning.

Experimental analysis: Figure 11 plots classifier AU-ROC on non-deferred examples against coverage for systems trained with 100% and 10% of the training data.The caption attributes low-coverage noise to the reduced data size.
Binary surrogate: The binary extension represents both classifier h and rejector r as real-valued functions and defers when r(x) ≤ 0.It allows expert loss lexp(x, y, m) = max(c, I_m≠y), making ordinary rejection learning a special case.
Binary surrogate: Convex functions φ(−u) and ψ(−u) upper-bound the indicator I_u≤0, producing convex surrogate objectives for the classifier and rejector.The resulting right-hand sides are convex in both h and r.
Consistency: Consistency requires β to vary with x; with α = 1 and β = q, the proposed loss LSH is consistent.The paper defines q through the expert’s conditional behavior and gives an input-dependent choice via c(x).
Bayes analysis: The Bayes-optimal classifier component has the same sign as the minimizer of the exponential loss and therefore the same sign as hB(x).The rejector’s optimal sign matches rB(x) under the stated condition involving expert error and the cost parameter.

C.1.2 Multiclass setting

The multiclass construction introduces convex surrogate losses for cost-sensitive expert deferral and establishes their consistency with the Bayes-optimal classifier and rejector. The analysis also contrasts this with a mixed loss that is only realizable-consistent under restricted classes.

Cost-sensitive surrogate: Proposition 1 states that eLCE is convex and consistent for the associated cost-sensitive loss.The loss generalizes cross entropy when costs correspond to multiclass misclassification.
Bayes solution: The multiclass Bayes classifier hB predicts the class with maximal conditional probability, while deferral is optimal when the expert’s expected loss is lower than the classifier’s error.This characterizes the pointwise minimizers of the original system loss.
Cost-sensitive surrogate: Theorem 2 states that LCE is a convex upper bound of L0−1 and is consistent.Its minimizer is attained at the Bayes solution under the theorem’s stated conditions.
Bayes solution: For LCE, the optimal classifier agrees with the Bayes solution and the optimal rejector defers exactly when P(Y = M|X = x) ≥ max_y∈Y η_y(x).The rejector therefore compares the expert’s correctness probability with the classifier’s best class probability.
Comparison with mixed loss: The mixed loss Lmix is realizable (H, R)-consistent for classes closed under scaling but is not classification consistent.Its optimal rejector instead compares the entropy of hB(x) with the expert’s error probability, which does not always match rB(x).
Generalization: The empirical analysis derives a high-probability generalization bound for empirical minimizers using Rademacher complexity and concentration inequalities.The final bound is obtained with probability at least 1 − δ.