Online Multiobjective Minimax Optimization and Applications

We introduce a simple but general online learning framework, in which at every round, an adaptive adversary introduces a new game, consisting of an action space for the learner, an action space for the adversary, and a vector valued objective function that is convex-concave in every coordinate. The learner and the adversary then play in this game. The learner's goal is to play so as to minimize the maximum coordinate of the cumulative vector-valued loss. The resulting one-shot game is not convex-concave, and so the minimax theorem does not apply. Nevertheless, we give a simple algorithm that can compete with the setting in which the adversary must announce their action first, with optimally diminishing regret. We demonstrate the power of our simple framework by using it to derive optimal bounds and algorithms across a variety of domains. This includes no regret learning: we can recover optimal algorithms and bounds for minimizing external regret, internal regret, adaptive regret, multigroup regret, subsequence regret, and a notion of regret in the sleeping experts setting. Next, we use it to derive a variant of Blackwell's Approachability Theorem, which we term "Fast Polytope Approachability". Finally, we are able to recover recently derived algorithms and bounds for online adversarial multicalibration and related notions (mean-conditioned moment multicalibration, and prediction interval multivalidity).