Instance-Adaptive Online Multicalibration

We study online multicalibration beyond the worst-case. We give a single, efficient algorithm which dynamically interpolates between benign and worst-case sequences by adaptively refining a dyadic grid of prediction values. Its error is controlled by the number of leaves in the refinement tree. Our analysis recovers the known widetilde O(T2/3) worst-case-optimal rate for online multicalibration, while simultaneously automatically adapting to easier instances: in the marginal stochastic setting it obtains a rate of widetilde O(√ T), and for piecewise-stationary means with J segments its rate is widetilde O(√JT). More generally, the rate depends on a threshold-complexity measure of the predictable mean process relative to the group family. We show that this dependence is tight up to logarithmic factors.