We give a randomized online algorithm that guarantees near-optimal $\widetilde O(\sqrt T)$ expected swap regret against any sequence of $T$ adaptively chosen Lipschitz convex losses on the unit interval. This improves the previous best bound of $\widetilde O(T^{2/3})$ and answers an open question of Fishelson et al. [2025b]. In addition, our algorithm is efficient: it runs in $\mathsf{poly}(T)$ time. A key technical idea we develop to obtain this result is to discretize the unit interval into bins at multiple scales of granularity and simultaneously use all scales to make randomized predictions, which we call multi-scale binning and may be of independent interest. A direct corollary of our result is an efficient online algorithm for minimizing the calibration error for general elicitable properties. This result does not require the Lipschitzness assumption of the identification function needed in prior work, making it applicable to median calibration, for which we achieve the first $\widetilde O(\sqrt T)$ calibration error guarantee.

翻译：我们提出了一种随机在线算法，该算法针对单位区间上任意$T$个自适应选择的Lipschitz凸损失序列，保证具有近似最优的$\widetilde O(\sqrt T)$期望交换遗憾。这一结果改进了先前最佳的$\widetilde O(T^{2/3})$界限，并回答了Fishelson等人[2025b]提出的一个开放性问题。此外，我们的算法是高效的：其运行时间为$\mathsf{poly}(T)$。为达成此结果，我们开发的一个关键技术思想是将单位区间离散化为多个粒度尺度的区间，并同时利用所有尺度进行随机预测，我们称之为多尺度分箱法，该方法可能具有独立的研究价值。我们结果的一个直接推论是，为一般可引出性质提供了一种高效的在线算法以最小化校准误差。该结果无需先前工作中所需的识别函数的Lipschitz性假设，从而可应用于中位数校准，对此我们首次实现了$\widetilde O(\sqrt T)$的校准误差保证。