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).

翻译：我们引入了一个简单而一般的在线学习框架, 在每个回合中, 一个适应性对手都会引入一个新的游戏, 包括学习者的行动空间, 对手的行动空间, 以及一个矢量值客观功能, 在每个坐标中都是 convex- concave。学习者和对手然后在这个游戏中玩。学习者的目标是玩, 以便最大限度地减少累积矢量价值损失的最大协调。由此产生的一分球游戏不是 convex- concove, 所以迷你max 定理不适用。然而, 我们给出一个简单的算法, 它可以与对手首先宣布其行动的设置进行竞争, 并且以最佳的方式减少遗憾。我们展示了我们简单框架的力量, 利用它来形成最佳的界限和算法, 跨越各个领域。这包括不后悔的学习: 我们可以恢复最佳的算法和界限, 以尽量减少外部的遗憾、内部遗憾、适应性的遗憾、多组的遗憾、后遗悔, 以及睡眠专家设置中的一种遗憾概念。然而, 我们用它来得出一个黑威尔的变异的模型, 最终和度度度和多度的度度度度度度的度的的的度, 我们的, 我们的能够的的的的的的的的的的的的和的的的的的的的的的的的的的的和的的的的的的的的的的的的的和的的的的的的的的的的的的的的的的的的的的的的的的的的的的的的的的的的的的的的的和的的的的的的的的的的的的的的的的的的的的的的的的的的的的的的的的的