Optimal rates for online calibration

Determine the optimal convergence rates, as functions of T and the prediction dimension d, at which calibration error can be driven to zero in adversarial online settings.

Background

Calibration ensures forecasts are unbiased conditional on their own values and guarantees diminishing swap regret for agents who best respond to them. However, known algorithms achieve rates like T^{2/3} in one dimension and rates that degrade exponentially with dimension, and there is a one-dimensional lower bound ruling out O(\sqrt{T}) rates.

The authors highlight that the exact optimal rates for online calibration remain an important open question, motivating their pursuit of alternative forecasting notions that yield stronger guarantees for downstream agents.