Improved and Oracle-Efficient Online $\ell_1$-Multicalibration

TL;DR

This paper introduces a direct, improved method for online -multicalibration that is oracle-efficient and achieves faster convergence rates in adversarial settings, advancing calibration guarantees across multiple groups.

Contribution

It proposes a novel reduction to online linear-product optimization, yielding improved -multicalibration rates and an oracle-efficient approach for large or infinite group families.

Findings

Achieves -multicalibration rate of ^{-1/3} with a linearized OLPO approach.

Develops an oracle-efficient method with rate ^{-1/4} for large group families.

Extends framework to infinite group families using Lipschitz properties.

Abstract

We study \emph{online multicalibration}, a framework for ensuring calibrated predictions across multiple groups in adversarial settings, across $T$ rounds. Although online calibration is typically studied in the $\ell_1$ norm, prior approaches to online multicalibration have taken the indirect approach of obtaining rates in other norms (such as $\ell_2$ and $\ell_{\infty}$) and then transferred these guarantees to $\ell_1$ at additional loss. In contrast, we propose a direct method that achieves improved and oracle-efficient rates of $\widetilde{\mathcal{O}}(T^{-1/3})$ and $\widetilde{\mathcal{O}}(T^{-1/4})$ respectively, for online $\ell_1$-multicalibration. Our key insight is a novel reduction of online \(\ell_1\)-multicalibration to an online learning problem with product-based rewards, which we refer to as \emph{online linear-product optimization} ($\mathtt{OLPO}$). To obtain the improved rate of $\widetilde{\mathcal{O}}(T^{-1/3})$, we introduce a linearization of $\mathtt{OLPO}$ and design a no-regret algorithm for this linearized problem. Although this method guarantees the desired sublinear rate (nearly matching the best rate for online calibration), it is computationally expensive when the group family \(\mathcal{H}\) is large or infinite, since it enumerates all possible groups. To address scalability, we propose a second approach to $\mathtt{OLPO}$ that makes only a polynomial number of calls to an offline optimization (\emph{multicalibration evaluation}) oracle, resulting in \emph{oracle-efficient} online \(\ell_1\)-multicalibration with a rate of $\widetilde{\mathcal{O}}(T^{-1/4})$. Our framework also extends to certain infinite families of groups (e.g., all linear functions on the context space) by exploiting a $1$-Lipschitz property of the \(\ell_1\)-multicalibration error with respect to \(\mathcal{H}\).