Optimal Lower Bounds for Online Multicalibration

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.

• The paper presents an information-theoretic lower bound of Ω(T^(2/3)) for online multicalibration, establishing a strict separation from marginal calibration.
• It constructs adversarial instances using both prediction-dependent and prediction-independent group functions to force significant calibration error.
• The analysis leverages martingale deviation and orthogonal group techniques to demonstrate the unavoidable statistical complexity in achieving multicalibration.

Optimal Lower Bounds for Online Multicalibration

Introduction and Motivation

The paper "Optimal Lower Bounds for Online Multicalibration" (2601.05245) gives tight, information-theoretic lower bounds on the statistical complexity of online multicalibration. The focus is on the scenario where predictions must remain multicalibrated across potentially adversarial sequences, raising key questions about the separation between standard calibration (marginal calibration) and its stronger, subgroup-based variant (multicalibration). Until now, upper bounds on calibration error have been matched only by loose lower bounds, particularly failing to resolve whether multicalibration is strictly harder in the online setting.

This work resolves this by constructing adversarial instances that force any online algorithm to incur a calibration error at least $\Omega(T^{2/3})$ for multicalibration, even when the family of subgroups (groups) is small and simple, and up to logarithmic factors for larger, prediction-independent group families. Consequently, the paper establishes a genuine separation between the optimal rates for marginal calibration (recently improved to $O(T^{2/3-\epsilon})$ [Dagan et al., 2025]) and those for multicalibration.

Problem Setup and Definitions

In the online calibration framework, a predictor sequentially estimates probabilities $p^t$ for binary outcomes $y^t$, potentially under adversarial choice of both contexts and outcomes. Perfect calibration requires that the predicted probabilities match the empirical frequencies, not only marginally, but conditionally on the predictions themselves.

Multicalibration generalizes this by demanding calibration to hold simultaneously across many subgroups, specified by group functions $g$. The worst-case group calibration error over a collection $\mathcal{G}$ is:

$$\mathrm{MCerr}_T(\mathcal{G}) = \max_{g \in \mathcal{G}} \sum_v |B_T(v, g)|,$$

where $B_T(v, g)$ is the (group-weighted) empirical bias at prediction value $v$.

Two main settings are considered:

• Prediction-dependent groups: $g$ can depend on both context and the prediction value.
• Prediction-independent groups: $g$ depends only on context.

Historically, upper bounds for both marginal and multicalibration have hovered at $O(T^{2/3})$, but recent progress for marginal calibration has shown genuinely faster rates can be achieved, leaving open whether multicalibration can similarly benefit.

Main Results

Lower Bounds for Prediction-dependent Groups

The core result is the construction of an online prediction problem in which any predictor must incur a multicalibration error at least $\Omega(T^{2/3})$, with only three disjoint binary groups that depend on both context and the prediction.

• Construction: The adversary provides contexts cycling through a regular grid; outcomes are i.i.d. Bernoulli with means signaled by context.
• Group functions: Three groups detect over- and under-estimation (by the predictor) relative to the mean (context), and a "narrow" third group captures predictions close to honest.
• Analysis: If the predictor frequently makes large deviations from honesty, the first two groups detect this, forcing calibration error. If the predictor is mostly honest, the third group accumulates noise at the rate $\sim T^{2/3}$ due to lack of cancellation across many distinct prediction bins.

Numerical Rate

The lower bound matches the best known upper bounds for online multicalibration, $\tilde{O}(T^{2/3})$, obtained via Blackwell approachability-based and other multiobjective online learning methods [Noarov et al., 2023]. Notably, it also strictly exceeds the improved upper bound $O(T^{2/3-\epsilon})$ for marginal calibration [Dagan et al., 2025], thereby demonstrating a strict information-theoretic gap.

Lower Bounds for Prediction-independent Groups

For the more constrained case of prediction-independent groups, the paper proves that when the group family size grows linearly with $T$, the same $\tilde{\Omega}(T^{2/3})$ lower bound is unavoidable.

• Construction: The adversary employs a context and label generation scheme where group functions form an orthogonal system (Walsh/Hadamard), partitioning the context space.
• Key argument: Simultaneous small calibration error on all groups can only occur if the predictor's sequence is close to the honest (mean) predictor in $\ell_1$, which in turn forces a large number of distinct prediction bins.
• Martingale analysis: Ultimately, the random fluctuation (noise) in each bin cannot be reduced by adaptive strategies due to a "noise bucketing" theorem, ensuring that the total expected calibration error remains at least $\tilde{\Omega}(T^{2/3})$.

Separation by Group Family Size

If the group family is of constant size and prediction-independent, a reduction to marginal calibration is possible (by running one marginal calibrator per group intersection pattern). Thus, strict hardness only emerges when $|\mathcal{G}|$ scales polynomially with $T$. The paper further provides reductions and lower bounds that rule out improved rates even in scenarios where only logarithmically many marginal calibration oracles are available, thereby showing this tradeoff to be tight up to exponential scaling.

Technical Contributions

The analysis relies on intricate probabilistic and combinatorial techniques:

• Martingale deviation arguments: Core to establishing that the noise accumulation cannot be evaded by the forecaster, even when allowed to adaptively bucket random fluctuations.
• Orthogonal group families: The use of Hadamard and Walsh systems ensures that no prediction strategy can be close to honest in all directions, thereby distributing error mass and preventing "compression."
• Lower bounds hold for binary-valued groups: The constructions show that hardness does not require complex, weighted group functions—simple binary (indicator) groups suffice.

Additionally, the paper precisely characterizes regimes of prediction-independent group families where multicalibration reduces to marginal calibration, and where this is no longer the case.

Implications and Future Directions

These findings have far-reaching consequences:

• Impossibility of matching marginal calibration rates: The strict separation established here means that online multicalibration fundamentally requires higher calibration error than marginal calibration, even under benign group families.
• Algorithm design tradeoffs: Practitioners must account for this gap when requiring calibration across many subgroups, particularly in adversarial or high-frequency online settings.
• Reduction limits: The paper's oracle lower bound demonstrates that widely used sleeping-experts reductions or aggregation methods cannot provide rate-improvements unless exponentially many oracles are run—sharpening the picture of algorithmic barriers for multicalibration.

Potential future questions include:

• Determining precise minimax rates for intermediate group-family sizes (e.g., polylogarithmic in $T$).
• Understanding algorithmic strategies for special (structurally constrained) group collections where tighter rates might be achievable.
• Extending lower bounds to contextual or partial-information settings, or alternate adverse environments.

Conclusion

This paper definitively answers the open question about the hardness of online multicalibration, exhibiting optimal lower bounds that match the best upper bounds, and sharply contrasting the complexity of multicalibration with that of marginal calibration. Through detailed probabilistic and combinatorial constructions, it sets a rigorous foundation for understanding the combinatorial and statistical barriers in online group fairness, calibration, and related learning paradigms. The implications are both theoretical and practical, as they inform both impossibility results and future algorithmic developments in adaptive and fair online learning.