High-Dimensional Calibration from Swap Regret

We study the online calibration of multi-dimensional forecasts over an arbitrary convex set P Rd relative to an arbitrary norm k k. We connect this with the problem of external regret minimization for online linear optimization, showing that if it is possible to guarantee O( ρT) worst-case regret after T rounds when actions are drawn from P and losses are drawn from the dual k k unit norm ball, then it is also possible to obtain -calibrated forecasts after T = exp(O(ρ/2)) rounds.