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This arXiv paper gives an online learning algorithm for multiclass multicalibration whose error adapts to the hidden structure of the outcome sequence. If the outcome means lie near a low-dimensional affine subspace, the error bound improves, and the authors claim it is minimax optimal up to polylog factors. Readers working on calibrated forecasting or uncertainty quantification should note that the guarantees are theoretical and that the stochastic special case, $\widetilde O(\sqrt{nT})$, is the most practically relevant result.
Instance-Adaptive Online Multiclass Multicalibration
arXiv:2610.15954v1 Announce Type: new Abstract: We study online multiclass multicalibration over $T$ rounds, and give an algorithm with beyond-worst-case guarantees. With $n$ possible outcomes, forecasts lie in a simplex of dimension $d=n-1$. In the worst case, cumulative $\ell_1$ multicalibration error scales with $\widetilde\Theta(T^{(d+1)/(d+2)})$, which becomes untenable once $d$ exceeds a small constant. Our algorithm adapts to the geometry of the realized sequence: if the sequence of adversarially selected outcome means lies in an affine subspace of dimension $s$, it obtains error $\widetilde O(n^{1/(s+2)}T^{(s+1)/(s+2)})$ with high probability. The algorithm adapts to this structure without knowing the subspace or its dimension. For $T\ge n\ge2$, this rate is minimax optimal up to polylogarithmic factors. For a stochastic instance with a fixed conditional mean, the bound specializes to $\widetilde O(\sqrt{nT})$, giving a square-root dependence on the horizon.
Abstract:We study online multiclass multicalibration over $T$ rounds, and give an algorithm with beyond-worst-case guarantees. With $n$ possible outcomes, forecasts lie in a simplex of dimension $d=n-1$. In the worst case, cumulative $\ell_1$ multicalibration error scales with $\widetilde\Theta(T^{(d+1)/(d+2)})$, which becomes untenable once $d$ exceeds a small constant. Our algorithm adapts to the geometry of the realized sequence: if the sequence of adversarially selected outcome means lies in an affine subspace of dimension $s$, it obtains error $\widetilde O(n^{1/(s+2)}T^{(s+1)/(s+2)})$ with high probability. The algorithm adapts to this structure without knowing the subspace or its dimension. For $T\ge n\ge2$, this rate is minimax optimal up to polylogarithmic factors. For a stochastic instance with a fixed conditional mean, the bound specializes to $\widetilde O(\sqrt{nT})$, giving a square-root dependence on the horizon.
Machine Learning (cs.LG)
arXiv:2610.15954 [cs.LG]
(or arXiv:2610.15954v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2610.15954 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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