High-Dimensional Calibration from Swap Regret

We study online calibration of multi-dimensional forecasts over an arbitrary convex set P subset ℝd relative to an arbitrary norm |·|. We connect this to external regret minimization for online linear optimization (OLO): if one can guarantee O(√ρ T) worst-case regret after T rounds when actions are drawn from P and losses from the dual |·|_* unit norm ball, then one can obtain ε-calibrated forecasts after T = exp(tilde O(ρ/ε2)) rounds. When P is the d-dimensional simplex and |·| is the ℓ1-norm, the O(√Tlog d) experts regret bound yields ε-calibrated forecasts after T = exp(tilde O(log d/ε2)) = dtilde O(1/ε2) rounds, recovering a recent result of Peng (2025). Interestingly, our algorithm obtains this guarantee without requiring access to any online linear optimization subroutine or knowledge of the optimal rate ρ -- in fact, our algorithm is identical for every setting of P and |·|. Instead, we show that the optimal regularizer for the above OLO problem can be used to upper bound the above calibration error by a swap regret, which we then minimize by running the recent TreeSwap algorithm (Dagan et al., 2024; Peng and Rubinstein, 2024) with Follow-The-Leader as a subroutine. The resulting algorithm is highly efficient and plays a distribution over simple averages of past observations in each round. Finally, we prove that any online calibration algorithm that guarantees ε T ℓ1-calibration error over the d-dimensional simplex requires T ≥ exp(poly(1/ε)) (assuming d ≥ poly(1/ε)). This strengthens the corresponding dOmega(log(1/ε)) lower bound of Peng (2025), and shows that an exponential dependence on 1/ε is necessary.