Improved and Oracle-Efficient Online $\ell_1$-Multicalibration
Rohan Ghuge, Vidya Muthukumar, Sahil Singla
TL;DR
The paper tackles online $\ell_1$-multicalibration in adversarial settings and introduces a direct reduction to online linear-product optimization ($\mathtt{OLPO}$), enabling improved sublinear calibration rates. By linearizing $\mathtt{OLPO}$ and employing a halfspace oracle, the authors achieve a $\widetilde{\mathcal{O}}(T^{-1/3})$ rate for finite hypothesis classes and an oracle-efficient $\widetilde{\mathcal{O}}(T^{-1/4})$ rate for large or infinite classes via $\beta$-covers and Lipschitz properties. The framework further extends to infinite group families (e.g., polynomials, Lipschitz convex families) and provides concrete oracle-efficient algorithms under transductive or separated-context assumptions, with applications to polynomial regression and bounded Lipschitz convex functions. A central contribution is the reduction to $\mathtt{OLPO}$, together with a generalized FTPL-based oracle-efficient approach that requires only a single offline optimization per round, enhancing scalability for large $\mathcal{H}$ and enabling online omniprediction-type guarantees in practice.
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 \(\mathcal{H}\).