đ¤ AI Summary
This study addresses the long-standing stagnation of the sequential calibration error exponent at $O(T^{2/3})$ without explicit constants, which has hindered theoretical progress for over two decades. To overcome this limitation, this work proposes a two-stage recursive labeling strategy combined with an optimization reduction approach, thereby refining the equivalence between sign-preserving games and calibration. By integrating logarithmic instance reductions with explicit parameter composition techniques, it achieves a fundamental theoretical breakthrough. The primary contribution is the first derivation of a calibration error exponent strictly below $2/3$, establishing a new bound of $O(T^{0.662942288})$. This result significantly advances the theory of sequential online calibration by resolving a key open problem that has persisted for more than twenty years.
đ Abstract
Probability forecasts are calibrated when predicted probabilities match empirical outcome frequencies: among events assigned a probability $p$, we'd hope that the fraction of positive outcomes is close to $p$. We study the problem of sequential forecasting of binary outcomes. The classical $O(T^{2/3})$ bound on expected cumulative $\ell_1$-calibration error established by Foster and Vohra stood for over two decades until Dagan et al. reduced the exponent $2/3$ by an unspecified constant.
We establish a new two-phase recursive labeling strategy for the sign-preservation-with-reuse game that yields the bound $O(n^ιt^β)$ for all choices of space and time. We then sharpen the reduction from upper bounds on sign preservation to calibration by modifying the equivalence of Dagan et al. to use only $O(\log T)$ instances of the sign-preservation-with-reuse game. This lets us establish an explicit bound of $O(T^{0.662942288})$, the first explicit exponent below $2/3$ for sequential calibration, by combining both improvements and choosing explicit feasible parameters.