One-Stage Top-$k$ Learning-to-Defer: Score-Based Surrogates with Theoretical Guarantees

We introduce the first one-stage Top-$k$ Learning-to-Defer framework, which unifies prediction and deferral by learning a shared score-based model that selects the $k$ most cost-effective entities-labels or experts-per input. While existing one-stage L2D methods are limited to deferring to a single expert, our approach jointly optimizes prediction and deferral across multiple entities through a single end-to-end objective. We define a cost-sensitive loss and derive a novel convex surrogate that is independent of the cardinality parameter $k$, enabling generalization across Top-$k$ regimes without retraining. Our formulation recovers the Top-1 deferral policy of prior score-based methods as a special case, and we prove that our surrogate is both Bayes-consistent and $\mathcal{H}$-consistent under mild assumptions. We further introduce an adaptive variant, Top-$k(x)$, which dynamically selects the number of consulted entities per input to balance predictive accuracy and consultation cost. Experiments on CIFAR-10 and SVHN confirm that our one-stage Top-$k$ method strictly outperforms Top-1 deferral, while Top-$k(x)$ achieves superior accuracy-cost trade-offs by tailoring allocations to input complexity.

Why Ask One When You Can Ask $k$? Two-Stage Learning-to-Defer to the Top-$k$ Experts

Learning-to-Defer (L2D) enables decision-making systems to improve reliability by selectively deferring uncertain predictions to more competent agents. However, most existing approaches focus exclusively on single-agent deferral, which is often inadequate in high-stakes scenarios that require collective expertise. We propose Top-$k$ Learning-to-Defer, a generalization of the classical two-stage L2D framework that allocates each query to the $k$ most confident agents instead of a single one. To further enhance flexibility and cost-efficiency, we introduce Top-$k(x)$ Learning-to-Defer, an adaptive extension that learns the optimal number of agents to consult for each query, based on input complexity, agent competency distributions, and consultation costs. For both settings, we derive a novel surrogate loss and prove that it is Bayes-consistent and $(\mathcal{R}, \mathcal{G})$-consistent, ensuring convergence to the Bayes-optimal allocation. Notably, we show that the well-established model cascades paradigm arises as a restricted instance of our Top-$k$ and Top-$k(x)$ formulations. Extensive experiments across diverse benchmarks demonstrate the effectiveness of our framework on both classification and regression tasks.

Adversarial Robustness in Two-Stage Learning-to-Defer: Algorithms and Guarantees

Learning-to-Defer (L2D) facilitates optimal task allocation between AI systems and decision-makers. Despite its potential, we show that current two-stage L2D frameworks are highly vulnerable to adversarial attacks, which can misdirect queries or overwhelm decision agents, significantly degrading system performance. This paper conducts the first comprehensive analysis of adversarial robustness in two-stage L2D frameworks. We introduce two novel attack strategies -- untargeted and targeted -- that exploit inherent structural vulnerabilities in these systems. To mitigate these threats, we propose SARD, a robust, convex, deferral algorithm rooted in Bayes and $(\mathcal{R},\mathcal{G})$-consistency. Our approach guarantees optimal task allocation under adversarial perturbations for all surrogates in the cross-entropy family. Extensive experiments on classification, regression, and multi-task benchmarks validate the robustness of SARD.