A set of probabilistic forecasts is calibrated if each prediction of the forecaster closely approximates the empirical distribution of outcomes on the subset of timesteps where that prediction was made. We study the fundamental problem of online calibrated forecasting of binary sequences, which was initially studied by Foster & Vohra (1998). They derived an algorithm with $O(T^{2/3})$ calibration error after $T$ time steps, and showed a lower bound of $\Omega(T^{1/2})$. These bounds remained stagnant for two decades, until Qiao & Valiant (2021) improved the lower bound to $\Omega(T^{0.528})$ by introducing a combinatorial game called sign preservation and showing that lower bounds for this game imply lower bounds for calibration. In this paper, we give the first improvement to the $O(T^{2/3})$ upper bound on calibration error of Foster & Vohra. We do this by introducing a variant of Qiao & Valiant's game that we call sign preservation with reuse (SPR). We prove that the relationship between SPR and calibrated forecasting is bidirectional: not only do lower bounds for SPR translate into lower bounds for calibration, but algorithms for SPR also translate into new algorithms for calibrated forecasting. We then give an improved \emph{upper bound} for the SPR game, which implies, via our equivalence, a forecasting algorithm with calibration error $O(T^{2/3 - \varepsilon})$ for some $\varepsilon > 0$, improving Foster & Vohra's upper bound for the first time. Using similar ideas, we then prove a slightly stronger lower bound than that of Qiao & Valiant, namely $\Omega(T^{0.54389})$. Our lower bound is obtained by an oblivious adversary, marking the first $\omega(T^{1/2})$ calibration lower bound for oblivious adversaries.

翻译：如果预测者作出的每个预测都紧密逼近在其作出该预测的时间步子集上结果的经验分布，则称一组概率预测是校准的。我们研究了二元序列在线校准预测这一基本问题，该问题最初由 Foster 和 Vohra (1998) 研究。他们推导出一种算法，在 $T$ 个时间步后具有 $O(T^{2/3})$ 的校准误差，并给出了 $\Omega(T^{1/2})$ 的下界。这些界限停滞了二十年，直到 Qiao 和 Valiant (2021) 通过引入一个称为符号保持的组合博弈，并证明该博弈的下界意味着校准的下界，从而将下界改进为 $\Omega(T^{0.528})$。在本文中，我们首次改进了 Foster 和 Vohra 的 $O(T^{2/3})$ 校准误差上界。我们通过引入 Qiao 和 Valiant 博弈的一个变体来实现这一点，我们称之为带重用的符号保持（SPR）。我们证明了 SPR 与校准预测之间的关系是双向的：不仅 SPR 的下界可以转化为校准的下界，而且 SPR 的算法也可以转化为校准预测的新算法。然后，我们给出了 SPR 博弈的一个改进的 \emph{上界}，这通过我们的等价关系意味着存在一个校准误差为 $O(T^{2/3 - \varepsilon})$（对于某个 $\varepsilon > 0$）的预测算法，首次改进了 Foster 和 Vohra 的上界。使用类似的思想，我们随后证明了比 Qiao 和 Valiant 的结果稍强的下界，即 $\Omega(T^{0.54389})$。我们的下界是由一个遗忘型对手获得的，这标志着针对遗忘型对手的首个 $\omega(T^{1/2})$ 校准下界。