We study a sequential binary prediction setting where the forecaster is evaluated in terms of the calibration distance, which is defined as the $L_1$ distance between the predicted values and the set of predictions that are perfectly calibrated in hindsight. This is analogous to a calibration measure recently proposed by B{\l}asiok, Gopalan, Hu and Nakkiran (STOC 2023) for the offline setting. The calibration distance is a natural and intuitive measure of deviation from perfect calibration, and satisfies a Lipschitz continuity property which does not hold for many popular calibration measures, such as the $L_1$ calibration error and its variants. We prove that there is a forecasting algorithm that achieves an $O(\sqrt{T})$ calibration distance in expectation on an adversarially chosen sequence of $T$ binary outcomes. At the core of this upper bound is a structural result showing that the calibration distance is accurately approximated by the lower calibration distance, which is a continuous relaxation of the former. We then show that an $O(\sqrt{T})$ lower calibration distance can be achieved via a simple minimax argument and a reduction to online learning on a Lipschitz class. On the lower bound side, an $\Omega(T^{1/3})$ calibration distance is shown to be unavoidable, even when the adversary outputs a sequence of independent random bits, and has an additional ability to early stop (i.e., to stop producing random bits and output the same bit in the remaining steps). Interestingly, without this early stopping, the forecaster can achieve a much smaller calibration distance of $\mathrm{polylog}(T)$.

翻译：我们研究了一个序列二元预测设置，其中预测者根据校准距离进行评估，该距离定义为预测值与事后完全校准的预测集之间的$L_1$距离。这类似于B{\l}asiok、Gopalan、Hu和Nakkiran（STOC 2023）近期针对离线设置提出的校准度量。校准距离是衡量偏离完美校准的自然且直观的指标，并满足Lipschitz连续性性质，而许多流行的校准度量（如$L_1$校准误差及其变体）并不具备这一性质。我们证明存在一种预测算法，在对抗性选择的$T$个二元结果序列上，期望校准距离可达到$O(\sqrt{T})$。该上界核心是一个结构结果，表明校准距离可由低校准距离精确近似，后者是前者的连续松弛。随后我们证明，通过简单的极小极大论证并归约到Lipschitz类上的在线学习，可实现$O(\sqrt{T})$的低校准距离。在下界方面，我们表明$\Omega(T^{1/3})$的校准距离是不可避免的，即使对手输出独立随机比特序列，并具备提前停止的额外能力（即停止产生随机比特并在剩余步骤中输出相同比特）。有趣的是，若无此提前停止，预测者可实现$\mathrm{polylog}(T)$这一小得多的校准距离。