We consider an online binary prediction setting where a forecaster observes a sequence of $T$ bits one by one. Before each bit is revealed, the forecaster predicts the probability that the bit is $1$. The forecaster is called well-calibrated if for each $p \in [0, 1]$, among the $n_p$ bits for which the forecaster predicts probability $p$, the actual number of ones, $m_p$, is indeed equal to $p \cdot n_p$. The calibration error, defined as $\sum_p |m_p - p n_p|$, quantifies the extent to which the forecaster deviates from being well-calibrated. It has long been known that an $O(T^{2/3})$ calibration error is achievable even when the bits are chosen adversarially, and possibly based on the previous predictions. However, little is known on the lower bound side, except an $\Omega(\sqrt{T})$ bound that follows from the trivial example of independent fair coin flips. In this paper, we prove an $\Omega(T^{0.528})$ bound on the calibration error, which is the first super-$\sqrt{T}$ lower bound for this setting to the best of our knowledge. The technical contributions of our work include two lower bound techniques, early stopping and sidestepping, which circumvent the obstacles that have previously hindered strong calibration lower bounds. We also propose an abstraction of the prediction setting, termed the Sign-Preservation game, which may be of independent interest. This game has a much smaller state space than the full prediction setting and allows simpler analyses. The $\Omega(T^{0.528})$ lower bound follows from a general reduction theorem that translates lower bounds on the game value of Sign-Preservation into lower bounds on the calibration error.

翻译：我们考虑一个在线二元预测场景，其中预测者逐一观测到T个比特序列。在每个比特揭示之前，预测者需预测该比特为1的概率。若对于每个p∈[0,1]，在预测概率为p的n_p个比特中，实际值为1的比特数m_p恰好等于p·n_p，则称该预测者为良好校准的。校准误差定义为∑_p |m_p - p n_p|，用于量化预测者偏离良好校准的程度。长期以来，即使比特序列由对抗性方式选择并可基于先前预测结果调整，仍可实现O(T^{2/3})的校准误差。然而，除独立公平抛硬币的平凡例子导出的Ω(√T)下界外，对该问题的下界研究甚少。本文证明了校准误差的Ω(T^{0.528})下界，据我们所知，这是该场景下首个超越√T的超线性下界。我们的技术贡献包括两种下界技术——早期停止与规避障碍——它们克服了此前阻碍强校准下界推导的困难。我们还提出了预测场景的抽象模型，称为符号保持博弈，该模型可能具有独立研究价值。该博弈的状态空间远小于完整预测场景，且允许更简洁的分析。通过将符号保持博弈的博弈值下界转化为校准误差下界的通用归约定理，我们最终推导出Ω(T^{0.528})的下界结果。