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：我们研究一个序列二元预测场景，其中预测者通过校准距离进行评估。校准距离定义为预测值与事后完全校准的预测集合之间的$L_1$距离。这类似于Błasiok、Gopalan、Hu和Nakkiran（STOC 2023）最近为离线场景提出的校准度量。校准距离是偏离完美校准的一种自然直观的度量，且满足Lipschitz连续性性质——该性质在许多常用校准度量（如$L_1$校准误差及其变体）中并不成立。我们证明存在一种预测算法，能在对抗性选择的$T$个二元结果序列上，实现期望$O(\sqrt{T})$的校准距离。该上界证明的核心是一个结构性结论：校准距离可通过下校准距离精确逼近，后者是前者的连续松弛形式。随后我们证明，通过简单的极小极大论证和向Lipschitz类在线学习的规约，可以实现$O(\sqrt{T})$的下校准距离。在下界方面，我们证明即使对手输出独立随机比特序列并具备提前停止能力（即停止生成随机比特并在剩余步骤中输出相同比特），$\Omega(T^{1/3})$的校准距离也是不可避免的。值得注意的是，若不具备这种提前停止能力，预测者可以实现$\mathrm{polylog}(T)$的极小校准距离。