An Elementary Predictor Obtaining $2\sqrt{T}+1$ Distance to Calibration
This work addresses a theoretical gap in online calibration for machine learning, offering a practical solution to a problem where existing methods like ECE fail.
The authors resolved an open problem by providing an explicit, efficient, deterministic algorithm that achieves a distance to calibration error of at most 2√T + 1 in the adversarial online setting, improving upon a prior non-constructive O(√T) bound.
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 $2\sqrt{T}+1$.