Calibeating for general proper losses: A Bregman divergence approach

TL;DR

This paper develops a unified Bregman divergence framework for calibeating across a broad class of proper losses, including Tsallis and Lipschitz losses, achieving regret bounds and new theoretical insights.

Contribution

It extends calibeating to a large family of proper losses using Bregman divergence, providing U-calibration results and a novel regret equality for Be The Regularized Leader.

Findings

Achieves logarithmic regret for Tsallis and related losses.

Provides a new regret equality for Be The Regularized Leader.

Offers dimension-dependent regret bounds for a broad class of proper losses.

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 $\alpha$-Tsallis losses (for $\alpha \in [1, 2]$) 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.