Instance-Adaptive Online Multicalibration

We study online multicalibration beyond the worst-case. We give a single, efficient algorithm which dynamically interpolates between stochastic 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 $\tilde{O}(T^{2/3})$ worst-case-optimal rate for online multicalibration, while simultaneously automatically adapting to easier instances: in the marginal stochastic setting it obtains a rate of $\tilde{O}(\sqrt{T})$, and for piecewise-stationary means with $J$ segments its rate is $\tilde{O}(\sqrt{JT})$. 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.