统一解决在线校准预测问题，实现最优误差率与校准性兼顾
Calibeating Made Simple
- 将校准优化转化为已有在线学习框架，适用于多种损失函数
- 首次在二分类场景下实现最优 $O(\log T)$ 校准误差与完全校准并存
- 为混合损失和有界损失提供新的最优校准率，适合高精度预测应用
本文研究在线校准预测（calibeating）问题，目标是通过后处理外部预测来最小化累积损失，并匹配基于信息量的基准。不同于以往针对特定损失函数的分析，本文将校准优化归约为现有的在线学习技术，从而获得对一般恰当损失的通用结果。首先，证明校准优化在极小极大意义下等价于后悔最小化，恢复了Foster and Hart [FH23]对Brier损失和对数损失的$O(\log T)$校准率及其最优性，并推导出可混合损失与一般有界损失的新最优校准率。其次，证明多校准优化等价于校准优化与经典专家问题的组合，由此得到可混合损失（包括Brier和对数损失）及一般有界损失的新最优多校准率。最后，给出同时实现校准与校准优化的全新边界，对二分类情形，首次构造出在保持最优$O(\log T)$校准误差的同时完全校准的算法。
原文摘要 · Abstract (English)
We study calibeating, the problem of post-processing external forecasts online to minimize cumulative losses and match an informativeness-based benchmark. Unlike prior work, which analyzed calibeating for specific losses with specific arguments, we reduce calibeating to existing online learning techniques and obtain results for general proper losses. More concretely, we first show that calibeating is minimax-equivalent to regret minimization. This recovers the $O(\log T)$ calibeating rate of Foster and Hart [FH23] for the Brier and log losses and its optimality, and yields new optimal calibeating rates for mixable losses and general bounded losses. Second, we prove that multi-calibeating is minimax-equivalent to the combination of calibeating and the classical expert problem. This yields new optimal multi-calibeating rates for mixable losses, including Brier and log losses, and general bounded losses. Finally, we obtain new bounds for achieving calibeating and calibration simultaneously for the Brier loss. For binary predictions, our result gives the first calibrated algorithm that at the same time also achieves the optimal $O(\log T)$ calibeating rate.
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