A celebrated result in the interface of online learning and game theory guarantees that the repeated interaction of no-regret players leads to a coarse correlated equilibrium (CCE) -- a natural game-theoretic solution concept. Despite the rich history of this foundational problem and the tremendous interest it has received in recent years, a basic question still remains open: how many iterations are needed for no-regret players to approximate an equilibrium? In this paper, we establish the first computational lower bounds for that problem in two-player (general-sum) games under the constraint that the CCE reached approximates the optimal social welfare (or some other natural objective). From a technical standpoint, our approach revolves around proving lower bounds for computing a near-optimal $T$-sparse CCE -- a mixture of $T$ product distributions, thereby circumscribing the iteration complexity of no-regret learning even in the centralized model of computation. Our proof proceeds by extending a classical reduction of Gilboa and Zemel [1989] for optimal Nash to sparse (approximate) CCE. In particular, we show that the inapproximability of maximum clique precludes attaining any non-trivial sparsity in polynomial time. Moreover, we strengthen our hardness results to apply in the low-precision regime as well via the planted clique conjecture.

翻译：在线学习与博弈论交叉领域的一个著名结果表明，无遗憾玩家的重复互动会形成粗相关均衡（CCE）——一种自然的博弈论解概念。尽管这一基础问题历史悠久且近年来备受关注，但一个基本问题仍未解决：无遗憾玩家需要多少次迭代才能逼近均衡？本文针对双人（一般和）博弈，在要求所达成的CCE逼近最优社会福利（或其他自然目标）的约束下，首次建立了该问题的计算下界。从技术角度看，我们的方法围绕证明计算近最优$T$-稀疏CCE（即$T$个乘积分布的混合）的下界展开，从而界定了即使在集中计算模型中无遗憾学习的迭代复杂度。我们的证明通过扩展Gilboa和Zemel[1989]关于最优纳什均衡到稀疏（近似）CCE的经典归约来实现。特别地，我们证明最大团问题的不可逼近性排除了在多项式时间内获得任何非平凡稀疏度的可能性。此外，我们通过植入团猜想将硬度结果强化至适用于低精度场景。