High-dimensional online calibration from harmonic weights

2026-10-06

AI Summary

This study addresses the problem of exponential growth in the number of rounds with respect to dimensionality in high-dimensional online calibration. To overcome this, it proposes a general calibration algorithm based on harmonic weights, applicable to arbitrary convex sets and error norms. Methodologically, by leveraging the optimal discrepancy properties of discrete Hilbert transform matrices alongside geometric parameter analysis, this work achieves ε-calibration with polynomial dependence on the dimensionality for the first time, substantially tightening existing theoretical bounds. Consequently, the proposed approach reduces the number of rounds required for both binary and multi-class prediction to d^O(1/ε), significantly outperforming prior methods. Overall, this research establishes an efficient theoretical and algorithmic foundation for high-dimensional online calibration.

Abstract

We study the online calibration of multidimensional forecasts over an arbitrary convex set $Y\subseteq\mathbb{R}^d$ relative to an arbitrary error norm $\|\cdot\|_{L}$. For forecasting $d$ binary outcomes simultaneously ($Y=[0,1]^d$), we give the first algorithm that achieves $\varepsilon$-calibration in a number of rounds that is polynomial in $d$ for every fixed accuracy. It requires $d^{O(1/\varepsilon)}$ rounds, exponentially improving the dimension dependence of previous bounds. For multi-class forecasting ($Y=Δ_d$), we obtain the same $d^{O(1/\varepsilon)}$ rate, improving the $d^{\widetilde{O}(1/\varepsilon^2)}$ 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 $\varepsilon$-calibration after $\exp(O(γ(Y,L)/\varepsilon))$ rounds, where $γ(Y,L)$ 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.

Problem

• online calibration
• multidimensional forecasts
• high-dimensional
• epsilon-calibration
• binary outcomes

Innovation

• Online Calibration
• Harmonic Weights
• High-dimensional Forecasting
• Matrix Discrepancy
• Discrete Hilbert Transform