Projected gradient ascent describes a form of no-regret learning algorithm that is known to converge to a coarse correlated equilibrium. Recent results showed that projected gradient ascent often finds the Nash equilibrium, even in situations where the set of coarse correlated equilibria is very large. We introduce semicoarse correlated equilibria, a solution concept that refines coarse correlated equilibria for the outcomes of gradient dynamics, while remaining computationally tractable through linear programming representations. Our theoretical analysis of the discretised Bertrand competition mirrors those recently established for mean-based learning in first-price auctions. With at least two firms of lowest marginal cost, Nash equilibria emerge as the only semicoarse equilibria under concavity conditions on firm profits. In first-price auctions, the granularity of the bid space affects semicoarse equilibria, but finer granularity for lower bids also induces convergence to Nash equilibria. Unlike previous work that aims to prove convergence to a Nash equilibrium that often relies on epoch based analysis and probability theoretic machinery, our LP-based duality approach enables a simple and tractable analysis of equilibrium selection under gradient-based learning.

翻译：投影梯度上升法描述了一种已知收敛于粗相关均衡的无悔学习算法。近期研究结果表明，即使粗相关均衡集规模极大时，投影梯度上升法仍常能收敛至纳什均衡。本文引入半粗相关均衡这一解概念，该概念在保持线性规划表示计算可处理性的同时，对梯度动态的结果细化了粗相关均衡。我们对离散化伯特兰竞争的理论分析，与近期在一价拍卖中基于均值学习所建立的理论具有相似性。在至少存在两家最低边际成本企业的条件下，当企业利润满足凹性条件时，纳什均衡成为唯一的半粗均衡。在一价拍卖中，投标空间的粒度会影响半粗均衡，但对较低投标采用更精细的粒度同样会引导系统收敛至纳什均衡。与以往依赖基于轮次分析和概率论工具来证明纳什均衡收敛性的研究不同，我们基于线性规划的对偶方法为梯度学习下的均衡选择提供了简洁且易于处理的分析框架。