Instance-Adaptive Online Multicalibration

TL;DR

This paper introduces an adaptive online multicalibration algorithm that automatically adjusts to different data complexities, achieving optimal rates in worst-case and easier scenarios.

Contribution

It presents a single efficient algorithm that interpolates between worst-case and easier instances, with error bounds depending on data complexity measures.

Findings

Recovers the known worst-case rate of  O(T^{2/3})

Achieves  O(\u221a T) in stochastic settings

Adapts to piecewise-stationary means with J segments

Abstract

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 $\widetilde O(T^{2/3})$ worst-case-optimal rate for online multicalibration, while simultaneously automatically adapting to easier instances: in the marginal stochastic setting it obtains a rate of $\widetilde O(\sqrt T)$, and for piecewise-stationary means with $J$ segments its rate is $\widetilde O(\sqrt{JT})$. 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.