Instance-Adaptive Online Multicalibration

Abstract: We study online multicalibration beyond the worst-case. We give a single, efficient algorithm which dynamically interpolates between benign and worst-case sequences by adaptively refining a dyadic grid of prediction values. Its error is controlled by the number of leaves in the refinement tree. Our analysis recovers the known $\widetilde O(T^{2/3})$ worst-case-optimal rate for online multicalibration, while simultaneously automatically adapting to easier instances: in the marginal stochastic setting it obtains a rate of $\widetilde O(\sqrt T)$, and for piecewise-stationary means with $J$ segments its rate is $\widetilde O(\sqrt{JT})$. More generally, the rate depends on a threshold-complexity measure of the predictable mean process relative to the group family. We show that this dependence is tight up to logarithmic factors.

• The paper presents a novel adaptive algorithm that achieves instance-dependent multicalibration rates, interpolating from O(√T) in benign cases to O(T^(2/3)) in adversarial scenarios.
• It leverages adaptive dyadic partitioning and a confidence-rated sleeping experts scheme to dynamically split prediction intervals based on play-count thresholds.
• The analysis, including matching lower bounds and corollaries, shows that the method attains tight error guarantees up to logarithmic factors across diverse environments.

Instance-Adaptive Online Multicalibration: An Expert Analysis

Problem Motivation and Background

Online calibration is a foundational property for probabilistic forecasting: the empirical frequency of an event conditioned on a forecasted probability must match the forecast itself. The synthesis between calibration and sequential decision-making has been achieved even in adversarial environments through seminal results, but classical online calibration rates exhibit a clear gap between stochastic (statistical) and adversarial (worst-case) environments, with the former scaling as $O(\sqrt{T})$ in time ($T$) and the latter as $O(T^{2/3})$ [foster1998asymptotic, qiao2021stronger, dagan2025breaking].

Multicalibration extends calibration guarantees to hold simultaneously across a potentially rich collection of subgroups, imposing much more stringent requirements than marginal calibration. State-of-the-art online multicalibration algorithms achieve worst-case $O(T^{2/3})$ regret but offer no adaptation to instance structure or distributional regularity [gupta2022online, noarov2023high]. The chief question addressed by this work is whether a unified algorithm can achieve instance-adaptive bounds—recovering $\tilde O(\sqrt{T})$ calibration for stochastic instances while maintaining $O(T^{2/3})$ in the adversarial case, and interpolating smoothly in between based on intrinsic instance complexity.

Algorithmic Framework: Adaptive Dyadic Partitioning

The core algorithm operates by dynamically maintaining a dyadic grid over the prediction space $[0,1]$. Rather than fixing prediction bins a priori, intervals are adaptively split based on play-count thresholds derived from the tradeoff between discretization and online learning regret. At every round, the algorithm outputs a distribution over the active bin midpoints. When the weight assigned to a bin exceeds $O(\log T / w^2)$ (with $w$ the bin width), the interval is subdivided. The resulting tree structure captures the adaptive partitioning of the forecast space.

Crucially, a confidence-rated sleeping experts scheme (using AdaNormalHedge [luo2015achieving]) is used for group-weighted bias control, monitoring directional errors for each group-interval pair and every contiguous block. Forecast selection at every step is achieved by a minimax-linear program, ensuring no convex combination of group-interval biases can have positive "one-step bias" for any realized outcome. This is formulated as a small linear program (with two constraints per round), enabling efficient implementation.

Adaptive Calibration Rate and Complexity Measures

The main theoretical innovation concerns analyzing calibration error in terms of an adaptive, data-dependent measure of complexity. The classical worst-case regime ($O(T^{2/3})$) is understood as arising when the number of active leaves in the partition tree is $T$0.

However, for more "benign" sequences, much finer control is possible. In the marginal calibration setting, error adapts to the drift complexity $T$1, which sums over dyadic scales the minimum number of contiguous blocks on which the mean outcome is essentially constant at that scale. Piecewise-stationary environments with $T$2 stationary segments yield bounds of $T$3. In the multicalibration context, a new notion—multiscale threshold complexity $T$4—is defined. It quantifies the minimal total cost (number of thresholds times representation norm in the group span) for threshold representations of the predictable mean process at each scale.

The analysis shows, for any family of binary groups $T$5 and predicted mean process $T$6, that the calibration error scales as

$T$7

whenever $T$8 is small, and never exceeds $T$9.

The proof architecture systematically decomposes regret and discretization error per interval, links the frequency of interval splitting to group-threshold representability, and leverages martingale deviation/peeling arguments for high-probability bounds.

Main Theorem and Corollaries

The central result states, with high probability over the data/adversary sequence, that the dynamic-bin algorithm achieves multicalibration error at most

$O(T^{2/3})$0

[(2605.09273), Theorem 1].

Notable immediate corollaries include:

• In the stochastic setting or $O(T^{2/3})$1-piecewise stationary environments without groups, the bound specializes to $O(T^{2/3})$2.
• When the mean process is approximable by a $O(T^{2/3})$3-level score with group representations of cost $O(T^{2/3})$4, error scales as $O(T^{2/3})$5.
• In the adversarial regime, tight $O(T^{2/3})$6 is recovered.

Furthermore, the analysis establishes matching lower bounds (parameterized Walsh lower bound), showing that the dependence on threshold complexity is tight up to polylogarithmic factors across the entire interpolation spectrum.

Technical Strengths and Claims

The algorithm dynamically interpolates between worst-case and benign environments without prior knowledge of the environment's structure or explicit specification of complexity parameters. The adaptivity is algorithmic and requires no tuning. The proof demonstrates that instance complexity is governed not by total variation or total drift per se but by the "threshold representability" of the mean process with respect to the group structure and the scale of discretization—aligning with recent progress in sequential prediction and multicalibration complexity theory [collina2026optimal, gupta2022online].

The authors explicitly prove that this adaptive rate is tight up to logarithmic factors. That is, for any threshold complexity budget $O(T^{2/3})$7, there exist group families and instance sequences attaining expected error $O(T^{2/3})$8.

Implications and Prospective Extensions

Practically, this algorithm and analysis enable sequential forecasters to provide sharp fairness/calibration guarantees that reflect the realized regularity of the time series and subgroup structure, rather than suffering the pessimism of worst-case rates. This is particularly germane for deploying sequential models with complex intersectional fairness desiderata, where prior approaches would be overly conservative.

Theoretically, the shift to instance-adaptive regret provides a powerful lens on the gap between stochastic and adversarial learning, clarifies the role of multiscale representation, and prompts further investigation of instance-optimal algorithms for other multi-group and distributionally robust objectives (cf. omniprediction, swap regret for general properties, etc.).

It is anticipated that the general strategy—adaptive discretization governed by complexity-based splitting, sleeping expert-based bias control, and tight martingale tail control—will find significant application in both multicalibration and broader sequential fair learning domains. Of particular interest is further extending to continuous or infinite group families with computationally efficient or oracle-efficient procedures [garg2024oracle, ghuge2025improved, farina2026efficient], and potential integration in practical decision-making systems.

Conclusion

The paper establishes an efficient, adaptive algorithm for online multicalibration that achieves instance-dependent rates, seamlessly spanning the spectrum from the stochastic ($O(T^{2/3})$9) to the adversarial ($O(T^{2/3})$0) regime. The sharpness of the analysis, technical contributions in adaptive discretization and complexity quantification, and unification of disparate regimes mark substantial progress in the theory of sequential calibration/fairness (2605.09273). This framework offers a flexible, theoretically justified foundation for trustworthy prediction in dynamic and heterogeneous environments.