Breaking the $T^{2/3}$ Barrier for Sequential Calibration

TL;DR

This paper improves the upper bound for online binary sequence calibration error from $O(T^{2/3})$ to $O(T^{2/3 - ext{some } ext{positive} ext{ } ext{epsilon}})$, and strengthens the lower bound to $ ext{Omega}(T^{0.54389})$, advancing understanding of calibration limits.

Contribution

It introduces the sign preservation with reuse (SPR) game, establishing a bidirectional link with calibration, and uses it to improve both upper and lower bounds for calibration error.

Findings

First improvement of the $O(T^{2/3})$ upper bound on calibration error.

Established a new lower bound of $ ext{Omega}(T^{0.54389})$ for oblivious adversaries.

Introduced the SPR game and proved its equivalence with calibrated forecasting.

Abstract

A set of probabilistic forecasts is calibrated if each prediction of the forecaster closely approximates the empirical distribution of outcomes on the subset of timesteps where that prediction was made. We study the fundamental problem of online calibrated forecasting of binary sequences, which was initially studied by Foster & Vohra (1998). They derived an algorithm with $O(T^{2/3})$ calibration error after $T$ time steps, and showed a lower bound of $\Omega(T^{1/2})$. These bounds remained stagnant for two decades, until Qiao & Valiant (2021) improved the lower bound to $\Omega(T^{0.528})$ by introducing a combinatorial game called sign preservation and showing that lower bounds for this game imply lower bounds for calibration. In this paper, we give the first improvement to the $O(T^{2/3})$ upper bound on calibration error of Foster & Vohra. We do this by introducing a variant of Qiao & Valiant's game that we call sign preservation with reuse (SPR). We prove that the relationship between SPR and calibrated forecasting is bidirectional: not only do lower bounds for SPR translate into lower bounds for calibration, but algorithms for SPR also translate into new algorithms for calibrated forecasting. We then give an improved \emph{upper bound} for the SPR game, which implies, via our equivalence, a forecasting algorithm with calibration error $O(T^{2/3 - \varepsilon})$ for some $\varepsilon > 0$, improving Foster & Vohra's upper bound for the first time. Using similar ideas, we then prove a slightly stronger lower bound than that of Qiao & Valiant, namely $\Omega(T^{0.54389})$. Our lower bound is obtained by an oblivious adversary, marking the first $\omega(T^{1/2})$ calibration lower bound for oblivious adversaries.