Model Releases
High-dimensional online calibration from harmonic weights
arXiv:2610.07740v1 Announce Type: cross Abstract: We study the online calibration of multidimensional forecasts over an arbitrary convex set YsubseteqR^d relative to an arbitrary error norm |dot|_{L}.
arXiv:2610.07740v1 Announce Type: cross Abstract: We study the online calibration of multidimensional forecasts over an arbitrary convex set YsubseteqR^d relative to an arbitrary error norm |dot|_{L}. For forecasting d binary outcomes simultaneously (Y=[0,1]^d), we give the first algorithm that achieves arepsilon-calibration in a number of rounds that is polynomial in d for every fixed accuracy. It requires d^{O(1/arepsilon)} rounds, exponentially improving the dimension dependence of previous bounds. For multi-class forecasting (Y=Delta_d), we obtain the same d^{O(1/arepsilon)} rate, improving the d^{widetilde{O}(1/arepsilon^2)} bounds of Peng and Fishelson et al. Our algorithm is simple: on each round, it outputs a harmonically weighted distribution over harmonically smoothed past outcomes. The same algorithm works for every forecast set and norm. More generally, it achieves arepsilon-calibration after exp(O(gamma(Y,L)/arepsilon)) rounds, where gamma(Y,L) is a geometric parameter defined by a matrix discrepancy problem. The harmonic weights are motivated by the fact that the discrete Hilbert transform matrix achieves the optimal discrepancy up to a universal constant, simultaneously for every L. This optimality result may be of independent interest.
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Source: arXiv cs.LG | 2026-10-07