发表机构
Edison Academy Magnet School; Institute for Advanced Study（爱迪生学院磁石学校; 普林斯顿高等研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对序贯校准问题，提出两阶段递归标记策略与改进的归约方法，首次给出低于经典 $2/3$ 指数的显式界 $O(T^{0.662942288})$。
AI 中文摘要
概率预测在预测概率与经验结果频率匹配时是校准的：在被赋予概率 $p$ 的事件中，我们希望正面结果的比例接近 $p$。我们研究二元结果的序贯预测问题。Foster 和 Vohra 建立的期望累积 $\ell_1$-校准误差的经典 $O(T^{2/3})$ 界在二十多年内未被改进，直到 Dagan 等人通过一个未指定的常数降低了指数 $2/3$。我们为带重用的符号保持博弈建立了一种新的两阶段递归标记策略，该策略对所有空间和时间选择都给出界 $O(n^{\alpha}t^{\beta})$。然后，我们通过修改 Dagan 等人的等价性，仅使用 $O(\log T)$ 次符号保持博弈实例，强化了从符号保持上界到校准的归约。结合这两项改进并选择显式可行参数，我们得以建立 $O(T^{0.662942288})$ 的显式界，这是序贯校准中第一个低于 $2/3$ 的显式指数。
英文摘要
Probability forecasts are calibrated when predicted probabilities match empirical outcome frequencies: among events assigned a probability $p$, we'd hope that the fraction of positive outcomes is close to $p$. We study the problem of sequential forecasting of binary outcomes. The classical $O(T^{2/3})$ bound on expected cumulative $\ell_1$-calibration error established by Foster and Vohra stood for over two decades until Dagan et al. reduced the exponent $2/3$ by an unspecified constant. We establish a new two-phase recursive labeling strategy for the sign-preservation-with-reuse game that yields the bound $O(n^αt^β)$ for all choices of space and time. We then sharpen the reduction from upper bounds on sign preservation to calibration by modifying the equivalence of Dagan et al. to use only $O(\log T)$ instances of the sign-preservation-with-reuse game. This lets us establish an explicit bound of $O(T^{0.662942288})$, the first explicit exponent below $2/3$ for sequential calibration, by combining both improvements and choosing explicit feasible parameters.
