Optimal Lower Bounds for Online Multicalibration

2026-01-08

arXiv.org

AI Summary

The information-theoretic limits of online multicalibration remain unclear, particularly whether it is fundamentally distinct from marginal calibration. This work addresses this gap by establishing the first tight $\widetilde{\Omega}(T^{2/3})$ lower bound on calibration error in the online setting. The analysis considers two regimes—depending on whether the group functions are allowed to depend on the learner’s predictions—and leverages adversarial constructions together with orthogonal function systems to derive the bound. This result matches existing upper bounds up to logarithmic factors, thereby characterizing the precise theoretical limit of online multicalibration. Moreover, it provides an information-theoretic proof that online multicalibration is inherently harder than marginal calibration, establishing a fundamental separation between the two notions.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Game Theory and Economic Paradigms: Adversarial Learning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• User Modeling, Personalization and Recommendation: User modeling for targeted and personalized online advertising

Abstract

We prove tight lower bounds for online multicalibration, establishing an information-theoretic separation from marginal calibration. In the general setting where group functions can depend on both context and the learner's predictions, we prove an $\Omega(T^{2/3})$ lower bound on expected multicalibration error using just three disjoint binary groups. This matches the upper bounds of Noarov et al. (2025) up to logarithmic factors and exceeds the $O(T^{2/3-\varepsilon})$ upper bound for marginal calibration (Dagan et al., 2025), thereby separating the two problems. We then turn to lower bounds for the more difficult case of group functions that may depend on context but not on the learner's predictions. In this case, we establish an $\widetilde{\Omega}(T^{2/3})$ lower bound for online multicalibration via a $\Theta(T)$-sized group family constructed using orthogonal function systems, again matching upper bounds up to logarithmic factors.

Problem

• online multicalibration
• lower bounds
• marginal calibration
• group functions
• calibration error

Innovation

• online multicalibration
• lower bounds
• information-theoretic separation
• orthogonal function systems
• calibration error