Consistency Conditions for Differentiable Surrogate Losses

2025-05-19

AI Summary

This paper addresses the statistical calibration of non-polyhedral differentiable surrogate loss functions for discrete prediction tasks, aiming to alleviate the analytical difficulty inherent in conventional calibration analysis. Method: We introduce a convex differentiable generalization of the “indirect elicitation (IE)” condition and, for the first time, define the stronger “strong IE” condition. Contribution/Results: We rigorously prove that, in the one-dimensional convex differentiable setting, IE is equivalent to calibration; however, we construct counterexamples in higher dimensions showing this equivalence fails generally. Moreover, we establish that strong IE is necessary and sufficient for calibration of strongly convex differentiable surrogates. Integrating tools from convex analysis, differentiable optimization, and statistical learning theory, our work provides a novel theoretical framework and practical certification criteria for designing and analyzing calibrated differentiable surrogate losses.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

The statistical consistency of surrogate losses for discrete prediction tasks is often checked via the condition of calibration. However, directly verifying calibration can be arduous. Recent work shows that for polyhedral surrogates, a less arduous condition, indirect elicitation (IE), is still equivalent to calibration. We give the first results of this type for non-polyhedral surrogates, specifically the class of convex differentiable losses. We first prove that under mild conditions, IE and calibration are equivalent for one-dimensional losses in this class. We construct a counter-example that shows that this equivalence fails in higher dimensions. This motivates the introduction of strong IE, a strengthened form of IE that is equally easy to verify. We establish that strong IE implies calibration for differentiable surrogates and is both necessary and sufficient for strongly convex, differentiable surrogates. Finally, we apply these results to a range of problems to demonstrate the power of IE and strong IE for designing and analyzing consistent differentiable surrogates.

Problem

• Equivalence of IE and calibration for differentiable surrogates
• Failure of IE-calibration equivalence in higher dimensions
• Introduction of strong IE for consistent surrogate design

Innovation

• Equivalence of IE and calibration for differentiable losses
• Introduction of strong IE for higher dimensions
• Application to consistent surrogate design