Improved and Oracle-Efficient Online $ell_1$-Multicalibration

2025-05-23

AI Summary

This paper studies ℓ₁-multi-calibration across multiple demographic groups in adversarial online settings, aiming to achieve strong calibration guarantees simultaneously for a large (even infinite) family of sensitive groups—e.g., all linear functions—at each prediction round. To overcome the inefficiency of prior approaches that rely on indirect ℓ₂/ℓ∞-norm conversions, we propose the first direct optimization paradigm for ℓ₁-multi-calibration, formulated as Online Linear Product Optimization (OLPO). Theoretically, we establish an optimal calibration error bound of $widetilde{mathcal{O}}(T^{-1/3})$. Algorithmically, we design the first oracle-efficient method achieving a convergence rate of $widetilde{mathcal{O}}(T^{-1/4})$. Our framework unifies theoretical optimality, computational tractability, and scalability to rich group families.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Learning to Search
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: User modeling for targeted and personalized online advertising

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 becomes 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}).

Problem

• Achieving improved online multicalibration in adversarial settings
• Direct method for efficient ell_1-multicalibration rates
• Scalable oracle-efficient approach for large group families

Innovation

• Direct online $ell_1$-multicalibration method
• Reduction to online linear-product optimization
• Oracle-efficient polynomial-time scalable approach