Improved bounds for calibration via stronger sign preservation games

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. We introduce a strengthening 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. In particular, any strategy that improves the trivial upper bound for the value of the SPR game would imply a forecasting algorithm with calibration error exponent less than 2/3, 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.