Optimal Lower Bounds for Online Multicalibration

TL;DR

This paper establishes tight lower bounds for online multicalibration, demonstrating an information-theoretic separation from marginal calibration and matching known upper bounds up to logarithmic factors.

Contribution

It provides the first tight lower bounds for online multicalibration in various settings, clarifying the fundamental difficulty of the problem.

Findings

Proves an $oldsymbol{ ext{Omega}}(T^{2/3})$ lower bound for group functions depending on context and predictions.

Establishes an $oldsymbol{ ext{Omega}}( ilde{T}^{2/3})$ lower bound for context-dependent group functions without predictions.

Matches upper bounds up to logarithmic factors, confirming the optimality of existing algorithms.

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 an $O(\log^3 T)$-sized group family constructed from an orthonormal basis, again matching upper bounds up to logarithmic factors.