Optimal Multiclass U-Calibration Error and Beyond
CoRR(2024)
摘要
We consider the problem of online multiclass U-calibration, where a
forecaster aims to make sequential distributional predictions over K classes
with low U-calibration error, that is, low regret with respect to all bounded
proper losses simultaneously. Kleinberg et al. (2023) developed an algorithm
with U-calibration error O(K√(T)) after T rounds and raised the open
question of what the optimal bound is. We resolve this question by showing that
the optimal U-calibration error is Θ(√(KT)) – we start with a
simple observation that the Follow-the-Perturbed-Leader algorithm of Daskalakis
and Syrgkanis (2016) achieves this upper bound, followed by a matching lower
bound constructed with a specific proper loss (which, as a side result, also
proves the optimality of the algorithm of Daskalakis and Syrgkanis (2016) in
the context of online learning against an adversary with finite choices). We
also strengthen our results under natural assumptions on the loss functions,
including Θ(log T) U-calibration error for Lipschitz proper losses,
O(log T) U-calibration error for a certain class of decomposable proper
losses, U-calibration error bounds for proper losses with a low covering
number, and others.
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