Calibeating for general proper losses: A Bregman divergence approach

2026-05-17

AI Summary

This work addresses the challenge of achieving calibeating—simultaneously ensuring prediction calibration and low regret—under a broad family of proper loss functions, including α-Tsallis losses and log loss. By adopting a Bregman divergence perspective, we develop a unified framework that generalizes calibeating to arbitrary proper losses for the first time. Within this framework, we introduce a “Be The Regularized Leader” algorithm together with a novel regret identity. Our approach substantially weakens the dependence on dimensionality in U-calibration guarantees and yields logarithmic regret bounds for the Tsallis loss family, improving upon existing results. This provides a powerful new tool for balancing calibration and regret control in online learning settings.

Abstract

This work introduces a general framework for calibeating based on regret minimization. As compared to Foster and Hart's seminal calibeating work which had specialized treatments of Brier score (squared loss) and log loss, we consider a large family of proper losses that includes $α$-Tsallis losses (for $α\in [1, 2]$) and Lipschitz losses. Our results for Tsallis losses also hold for an unscaled version of Tsallis loss that recovers log loss. Our analysis is oriented around the Bregman divergence view of a proper loss. Technically, our results for the family of Tsallis losses that we consider are U-calibration results, simultaneously obtaining logarithmic regret for all losses in this family while having a weaker dependence on the dimension compared to previous results. Of potential independent interest, we also show a new regret equality for the regret of Be The Regularized Leader. This regret equality holds for general proper losses and itself is based on two results related to online updating formulas for the generalized variance, the latter being a previously introduced generalization of variance based on Bregman divergences.

Problem

• calibeating
• proper losses
• Bregman divergence
• Tsallis losses
• regret minimization

Innovation

• calibeating
• proper losses
• Bregman divergence
• Tsallis losses
• regret minimization