English

High dimensional online calibration in polynomial time

Machine Learning 2025-04-15 v1 Data Structures and Algorithms Computer Science and Game Theory Machine Learning

Abstract

In online (sequential) calibration, a forecaster predicts probability distributions over a finite outcome space [d][d] over a sequence of TT days, with the goal of being calibrated. While asymptotically calibrated strategies are known to exist, they suffer from the curse of dimensionality: the best known algorithms require exp(d)\exp(d) days to achieve non-trivial calibration. In this work, we present the first asymptotically calibrated strategy that guarantees non-trivial calibration after a polynomial number of rounds. Specifically, for any desired accuracy ϵ>0\epsilon > 0, our forecaster becomes ϵ\epsilon-calibrated after T=dO(1/ϵ2)T = d^{O(1/\epsilon^2)} days. We complement this result with a lower bound, proving that at least T=dΩ(log(1/ϵ))T = d^{\Omega(\log(1/\epsilon))} rounds are necessary to achieve ϵ\epsilon-calibration. Our results resolve the open questions posed by [Abernethy-Mannor'11, Hazan-Kakade'12]. Our algorithm is inspired by recent breakthroughs in swap regret minimization [Peng-Rubinstein'24, Dagan et al.'24]. Despite its strong theoretical guarantees, the approach is remarkably simple and intuitive: it randomly selects among a set of sub-forecasters, each of which predicts the empirical outcome frequency over recent time windows.

Keywords

Cite

@article{arxiv.2504.09096,
  title  = {High dimensional online calibration in polynomial time},
  author = {Binghui Peng},
  journal= {arXiv preprint arXiv:2504.09096},
  year   = {2025}
}
R2 v1 2026-06-28T22:55:45.372Z