High-Dimensional Calibration from Swap Regret

We study online calibration of multi-dimensional forecasts over an arbitrary convex set P \subset \mathbb{R}^d relative to an arbitrary norm |\cdot|. We connect this to external regret minimization for online linear optimization (OLO): if one can guarantee O(\sqrt{ρT}) worst-case regret after T rounds when actions are drawn from P and losses from the dual |\cdot|_* unit norm ball, then one can obtain ε-calibrated forecasts after T = \exp(\tilde O(ρ/ε^2)) rounds. When P is the d-dimensional simplex and |\cdot| is the \ell_1-norm, the O(\sqrt{T\log d}) experts regret bound yields ε-calibrated forecasts after T = \exp(\tilde O(\log d/ε^2)) = d^{\tilde O(1/ε^2)} 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 |\cdot|. 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 εT \ell_1-calibration error over the d-dimensional simplex requires T \geq \exp(poly(1/ε)) (assuming d \geq poly(1/ε)). This strengthens the corresponding d^{Ω(\log(1/ε))} lower bound of Peng (2025), and shows that an exponential dependence on 1/ε is necessary.