Breaking the T2/3 Barrier for Sequential Calibration

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 under the standard ℓ1 calibration error metric, which was initially studied by Foster & Vohra (1998). They derived an algorithm with O(T2/3) calibration error after T time steps, and showed a lower bound of Omega(T1/2). These bounds remained stagnant for two decades, until Qiao & Valiant (2021) improved the lower bound to Omega(T0.528) 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(T2/3) 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 upper bound for the SPR game, which implies, via our equivalence, a forecasting algorithm with calibration error O(T2/3 - 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(T0.54389). Our lower bound is obtained by an oblivious adversary, marking the first ω(T1/2) calibration lower bound for oblivious adversaries.