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cost-sensitive softmax

A cost-sensitive softmax is a variant of the standard softmax and cross-entropy formulation in machine learning that incorporates non-uniform penalties, weights, or costs associated with different decision outcomes. Unlike the conventional softmax function, which treats all misclassifications symmetrically and optimizes for uniform accuracy, cost-sensitive softmax adjusts the output probabilities or surrogate loss calculations according to explicit cost structures. These costs can represent asymmetric misclassification impacts, class-imbalance penalties, or task-specific costs such as abstaining from a decision or deferring a prediction to an external expert. By integrating these differential penalties directly into the training objective, the model is trained to minimize total expected operational cost rather than simply minimizing standard classification error.

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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.

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2026-09-26