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consistent surrogate losses

Consistent surrogate losses are computationally tractable proxy loss functions used in machine learning optimization that theoretically guarantee that minimizing the proxy loss leads to minimizing the true target objective. Because primary performance metrics, such as zero-one classification loss or discrete selection penalties, are often discontinuous, non-convex, and computationally intractable to optimize directly, learning algorithms train models using continuous surrogate losses instead. A surrogate loss is considered consistent, typically through properties such as Bayes consistency or hypothesis-set consistency, when minimizing its expected risk ensures that the resulting predictor converges toward optimal performance under the original true task metric, ensuring that optimizing the proxy effectively solves the underlying learning problem.

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Two-Stage Learning to Defer with Multiple Experts

Two-Stage Learning to Defer with Multiple Experts

Anqi Mao, Christopher Mohri, Mehryar Mohri, Yutao Zhong

Why you should read this

Proposes a theoretically grounded framework for two-stage learning to defer with multiple experts by designing surrogate losses with rigorous H\mathcal{H}-consistency guarantees that enable efficient post-hoc deferral without retraining the underlying predictor.

We study a two-stage scenario for learning to defer with multiple experts, which is crucial in practice for many applications. In this scenario, a predictor is derived in a first stage by training with a common loss function such as cross-entropy. In the second stage, a deferral function is learned to assign the most suitable expert to each input. We design a new family of surrogate loss functions for this scenario both in the score-based and the predictor-rejector settings and prove that they are supported by H-consistency bounds, which implies their Bayes-consistency. Moreover, we show that, for a constant cost function, our two-stage surrogate losses are realizable H-consistent. While the main focus of this work is a theoretical analysis, we also report the results of several experiments on CIFAR-10 and SVHN datasets.

Added

2026-09-26