Optimal rate for sequential marginal calibration

Determine the optimal minimax asymptotic rate, as a function of the horizon T, at which any online forecasting algorithm can guarantee expected calibration error for marginal (prediction-independent) calibration against arbitrary/adversarial outcome sequences. Concretely, establish the exact scaling of Err_T = sum_{v in {p^1,...,p^T}} |sum_{t: p^t = v} (p^t - y^t)| achievable in the worst case.

Background

The paper reviews the status of online marginal calibration, where expected calibration error is measured by aggregating empirical biases over distinct prediction values. Prior work showed o(T) is achievable, with long-standing upper bounds of O(T^{2/3}), and recent progress improving this to O(T^{2/3−ε}) along with lower bounds around T^{0.54}. Despite these advances, the precise optimal asymptotic rate remains undetermined.

This open question is independent of the authors’ main multicalibration results, but it contextualizes their separation between multicalibration and marginal calibration. Pinning down the exact rate for marginal calibration would resolve a decades-long uncertainty in the field and clarify baselines for reductions from multicalibration to marginal calibration.