Instance-Adaptive Online Multicalibration

Making calibrated online predictions is a central challenge in modern AI systems. Much of the existing literature focuses on fully adversarial environments where outcomes may be arbitrary, leading to conservative algorithms that can perform suboptimally in more benign settings, such as when outcomes are nearly stationary. This gap raises a natural question: can we design online prediction algorithms whose calibration error automatically adapts to the degree of non-stationarity in the environment, smoothly interpolating between i.i.d. and adversarial regimes? We answer this question in the affirmative and develop a suite of algorithms that achieve adaptive calibration guarantees under multiple calibration measures. Specifically, with $T$ being the number of rounds, $K$ being the unknown number of i.i.d. segments of the environment, and $C\in[0,T]$ being another unknown non-stationary measure defined as the minimal $\ell_1$ deviation of the mean outcomes, our algorithms attain $\widetilde{O}(\min\{\sqrt{T}+(TC)^{\frac{1}{3}}, \sqrt{KT}\})$ for $\ell_1$ calibration error and $\widetilde{O}(\min\{(1+C)^{\frac{1}{3}}, K\})$ for both $\ell_2$ and pseudo KL calibration error. These bounds match the optimal rates in the stationary case ($C=0$ and $K=1$) and recover known guarantees in the fully adversarial regime ($C, K=Ω(T)$). Our approach builds on and extends prior work [Hu et al., 2026, Luo et al., 2025], introducing an epoch-based scheduling together with a novel non-uniform partition of the prediction space that allocates finer resolution near the underlying ground truth.

We study the minimax sample complexity of multicalibration in the batch setting. A learner observes $n$ i.i.d. samples from an unknown distribution and must output a (possibly randomized) predictor whose population multicalibration error, measured by Expected Calibration Error (ECE), is at most $\varepsilon$ with respect to a given family of groups. For every fixed $κ> 0$, in the regime $|G|\le \varepsilon^{-κ}$, we prove that $\widetildeΘ(\varepsilon^{-3})$ samples are necessary and sufficient, up to polylogarithmic factors. The lower bound holds even for randomized predictors, and the upper bound is realized by a randomized predictor obtained via an online-to-batch reduction. This separates the sample complexity of multicalibration from that of marginal calibration, which scales as $\widetildeΘ(\varepsilon^{-2})$, and shows that mean-ECE multicalibration is as difficult in the batch setting as it is in the online setting, in contrast to marginal calibration which is strictly more difficult in the online setting. In contrast we observe that for $κ= 0$, the sample complexity of multicalibration remains $\widetildeΘ(\varepsilon^{-2})$ exhibiting a sharp threshold phenomenon. More generally, we establish matching upper and lower bounds, up to polylogarithmic factors, for a weighted $L_p$ multicalibration metric for all $1 \le p \le 2$, with optimal exponent $3/p$. We also extend the lower-bound template to a regular class of elicitable properties, and combine it with the online upper bounds of Hu et al. (2025) to obtain matching bounds for calibrating properties including expectiles and bounded-density quantiles.

We study the online binary sequential calibration problem. A recent breakthrough by \citet{dagan2024breaking} overcomes the classical $T^{2/3}$ barrier for calibration error. Building on this result, we present an efficient randomized forecaster that achieves an expected calibration error $O(T^{2/3-\varepsilon})$ for some constant $\varepsilon>0$. Our forecaster combines the \textsc{SPR-Calibration} procedure \citep{dagan2024breaking} with an outer Blackwell-style correction layer. The \textsc{SPR-Calibration} procedure controls calibration with respect to a surrogate sequence of conditional-mean estimates, while the correction layer controls the additional error incurred when these surrogates are used to approximate the true outcomes. The analysis decomposes the total calibration error into the surrogate calibration error and the residual discrepancy between the surrogate sequence and the true outcomes. The former is bounded by the \textsc{SPR-Calibration} guarantee in \citet{dagan2024breaking}, and the latter is controlled using a quadratic potential argument together with the sparsity of the \textsc{SPR-Calibration} forecaster.