发表机构
Institute for Advanced Study; Google Research; Courant Institute of Mathematical Sciences（高等研究院; 谷歌研究院; 库朗数学科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种基于调和权重的简单在线校准算法，对任意凸集和范数实现高维预测的ε-校准，轮数在维度上多项式，显著改善已有界。
AI 中文摘要
我们研究了在任意凸集 $Y\subseteq\mathbb{R}^d$ 上、相对于任意误差范数 $\\|\cdot\\|_{L}$ 的多维预测的在线校准问题。对于同时预测 $d$ 个二元结果（$Y=[0,1]^d$）的情况，我们给出了第一个在固定精度下轮数关于 $d$ 为多项式的算法，该算法实现 $\varepsilon$-校准所需的轮数为 $d^{O(1/\varepsilon)}$，在维度依赖性上指数级地改善了之前的界。对于多类预测（$Y=\Delta_d$），我们获得了相同的 $d^{O(1/\varepsilon)}$ 速率，改进了 Peng 和 Fishelson 等人的 $d^{\widetilde{O}(1/\varepsilon^2)}$ 界。我们的算法很简单：在每一轮，它输出一个基于调和平滑的历史结果的调和加权分布。同一算法适用于任意预测集和范数。更一般地，它在 $\exp(O(\gamma(Y,L)/\varepsilon))$ 轮后实现 $\varepsilon$-校准，其中 $\gamma(Y,L)$ 是由矩阵差异问题定义的几何参数。调和权重的动机源于离散希尔伯特变换矩阵在每一个 $L$ 上同时达到最优差异（相差一个通用常数）。这一最优性结果可能具有独立的意义。
英文摘要
We study the online calibration of multidimensional forecasts over an arbitrary convex set $Y\subseteq\mathbb{R}^d$ relative to an arbitrary error norm $\|\cdot\|_{L}$. For forecasting $d$ binary outcomes simultaneously ($Y=[0,1]^d$), we give the first algorithm that achieves $\varepsilon$-calibration in a number of rounds that is polynomial in $d$ for every fixed accuracy. It requires $d^{O(1/\varepsilon)}$ rounds, exponentially improving the dimension dependence of previous bounds. For multi-class forecasting ($Y=Δ_d$), we obtain the same $d^{O(1/\varepsilon)}$ rate, improving the $d^{\widetilde{O}(1/\varepsilon^2)}$ bounds of Peng and Fishelson et al. Our algorithm is simple: on each round, it outputs a harmonically weighted distribution over harmonically smoothed past outcomes. The same algorithm works for every forecast set and norm. More generally, it achieves $\varepsilon$-calibration after $\exp(O(γ(Y,L)/\varepsilon))$ rounds, where $γ(Y,L)$ is a geometric parameter defined by a matrix discrepancy problem. The harmonic weights are motivated by the fact that the discrete Hilbert transform matrix achieves the optimal discrepancy up to a universal constant, simultaneously for every $L$. This optimality result may be of independent interest.
