High dimensional online calibration in polynomial time

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.