Optimal Lower Bounds for Online Multicalibration

TL;DR

This work establishes tight lower bounds for online multicalibration, demonstrating an information-theoretic separation from marginal calibration. It proves an $\Omega(T^{2/3})$ lower bound for the general case with prediction-dependent group functions (even for just three disjoint binary groups), matching existing upper bounds up to logarithmic factors. It further proves a $\tilde{\Omega}(T^{2/3})$ lower bound for prediction-independent groups when the group family has size $|G|=\Theta(T)$, again aligning with known upper bounds. Collectively, the results separate the computational and statistical complexity of multicalibration from marginal calibration and delineate two distinct regimes (prediction-dependent vs prediction-independent) with matching rates up to polylog factors. The constructions rely on a three-group hard instance for the general case and a Walsh/Hadamard–based orthogonal group framework for the prediction-independent case, highlighting the central roles of honest vs dishonest predictions, martingale analysis, and adaptive bucketing in the lower-bound landscape.

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 $Ω(T^{2/3})$ 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 $\widetildeΩ(T^{2/3})$ lower bound for online multicalibration via a $Θ(T)$-sized group family constructed using orthogonal function systems, again matching upper bounds up to logarithmic factors.

Figures (1)

• Figure 1: Summary of regimes and rates. We study online adversarial multicalibration. For general prediction-dependent groups $g(x,v)$ we prove an optimal $\Omega(T^{2/3})$ lower bound, separating multicalibration from marginal calibration. For prediction-independent groups $g(x)$, constant-sized families reduce to marginal calibration up to a $2^{|G|}$ factor, precluding a separation from marginal calibration. For a group family of size $|G|=\Theta(T)$ we again prove an optimal $\tilde{\Omega}(T^{2/3})$ lower bound.

Theorems & Definitions (72)

• Definition 1: Group functions
• Definition 2: Binary and prediction-independent groups
• Definition 3: Empirical Bias and multicalibration error
• Theorem 1: Prediction-dependent lower bound
• Definition 4: Hard distribution $\mathcal{D}_{T,m}$
• Proposition 1: Dense martingale transform deviation
• proof
• Lemma 1: Burkholder--Rosenthal Inequality burkholder1973distribution, Theorem 21.1
• Definition 5: Big deviations and $\eta$-honest rounds
• Lemma 2: Reduction to deterministic predictors