Blasiok et al. [2023] proposed distance to calibration as a natural measure of calibration error that unlike expected calibration error (ECE) is continuous. Recently, Qiao and Zheng [2024] gave a non-constructive argument establishing the existence of an online predictor that can obtain $O(\sqrt{T})$ distance to calibration in the adversarial setting, which is known to be impossible for ECE. They leave as an open problem finding an explicit, efficient algorithm. We resolve this problem and give an extremely simple, efficient, deterministic algorithm that obtains distance to calibration error at most $2\sqrt{T}$.

翻译：Blasiok 等人 [2023] 提出了校准距离作为校准误差的一种自然度量，与期望校准误差（ECE）不同，该度量是连续的。最近，Qiao 和 Zheng [2024] 通过非构造性论证证明了存在一个在线预测器，可以在对抗性设置中获得 $O(\sqrt{T})$ 的校准距离，而这对于 ECE 而言已知是不可能的。他们留下了一个开放问题：寻找一个显式、高效的算法。我们解决了这个问题，并给出一个极其简单、高效、确定性的算法，该算法获得的校准距离误差至多为 $2\sqrt{T}$。