High dimensional online calibration in polynomial time

• URL: http://arxiv.org/abs/2504.09096v1
• Date: Sat, 12 Apr 2025 06:28:05 GMT
• Title: High dimensional online calibration in polynomial time
• Authors: Binghui Peng,
• Abstract summary: In online (sequential) calibration, a forecaster predicts probability distributions over a finite outcome space $[d]$ over a sequence of $T$ days.<n>Best known algorithms require $exp(d)$ days to achieve non-trivial calibration.<n>We present the firstally calibrated strategy that guarantees non-trivial algorithm calibration after a number of rounds.
• Score: 17.45683822446751
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In online (sequential) calibration, a forecaster predicts probability distributions over a finite outcome space $[d]$ over a sequence of $T$ 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 $\exp(d)$ 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 $\epsilon > 0$, our forecaster becomes $\epsilon$-calibrated after $T = d^{O(1/\epsilon^2)}$ days. We complement this result with a lower bound, proving that at least $T = d^{\Omega(\log(1/\epsilon))}$ rounds are necessary to achieve $\epsilon$-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.

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