# Calibeating for general proper losses: A Bregman divergence approach

arXiv: https://arxiv.org/abs/2605.17269 · HTML: https://mlsanity.com/paper/2605.17269/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 31 references that are in the map.
- First posted on arXiv 17 May 2026.

## Builds on (most cited first, 8 of 8)

- [Online Learning: A Modern Introduction Using Convex Optimization](https://mlsanity.com/paper/1912.13213/) (2019)
- ["Calibeating": Beating Forecasters at Their Own Game](https://mlsanity.com/paper/2209.04892/) (2022)
- [Optimal Multiclass U-Calibration Error and Beyond](https://mlsanity.com/paper/2405.19374/) (2024)
- [The fundamental nature of the log loss function](https://mlsanity.com/paper/1502.06254/) (2015)
- [Online Platt Scaling with Calibeating](https://mlsanity.com/paper/2305.00070/) (2023)
- [A Generalized Bias-Variance Decomposition for Bregman Divergences](https://mlsanity.com/paper/2511.08789/) (2025)
- [Calibeating Made Simple](https://mlsanity.com/paper/2603.22167/) (2026)
- [Robust confidence distributions from proper scoring rules](https://mlsanity.com/paper/2109.01219/) (2021)

## Abstract (arXiv)

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 ∈ [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.