A sequence of predictions is calibrated if and only if it induces no swap regret to all down-stream decision tasks. We study the Maximum Swap Regret (MSR) of predictions for binary events: the swap regret maximized over all downstream tasks with bounded payoffs. Previously, the best online prediction algorithm for minimizing MSR is obtained by minimizing the K1 calibration error, which upper bounds MSR up to a constant factor. However, recent work (Qiao and Valiant, 2021) gives an ${\Omega}(T^{0.528})$ lower bound for the worst-case expected $K_1$ calibration error incurred by any randomized algorithm in T rounds, presenting a barrier to achieving better rates for MSR. Several relaxations of MSR have been considered to overcome this barrier, via external regret (Kleinberg et al., 2023) and regret bounds depending polynomially on the number of actions in downstream tasks (Noarov et al., 2023; Roth and Shi, 2024). We show that the barrier can be surpassed without any relaxations: we give an efficient randomized prediction algorithm that guarantees $O(\sqrt{T}logT)$ expected MSR. We also discuss the economic utility of calibration by viewing MSR as a decision-theoretic calibration error metric and study its relationship to existing metrics.

翻译：摘要：若预测序列对所有下游决策任务均不产生交换遗憾，则该序列是校准的。我们研究了二元事件预测的最大交换遗憾问题：即下游有界收益任务中交换遗憾的最大化。先前，最小化MSR的最优在线预测算法通过最小化K1校准误差实现，该误差在常系数范围内是MSR的上界。然而，近期研究（Qiao和Valiant, 2021）指出，任何随机化算法在T轮中最坏情况下的期望K1校准误差存在下界${\Omega}(T^{0.528})$，这为改进MSR的收敛速率设置了障碍。为突破该障碍，已有工作通过外部遗憾（Kleinberg等，2023）以及依赖下游任务动作数多项式函数关系的遗憾界（Noarov等，2023；Roth和Shi，2024）提出了MSR的若干松弛方案。我们证明无需任何松弛即可超越该障碍：提出一种高效随机化预测算法，可实现$O(\sqrt{T}logT)$的期望MSR。此外，通过将MSR视为决策论校准误差度量，我们讨论了校准的经济效用，并研究其与现有度量的关系。