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无遗憾学习算法，已知其收敛于粗相关均衡。近期研究结果表明，即使在粗相关均衡集非常大的情况下，投影梯度上升法也常常能找到纳什均衡。我们引入了半粗相关均衡这一解概念，该概念针对梯度动态的结果细化了粗相关均衡，同时通过线性规划表示保持计算可处理性。我们对离散化伯特兰竞争的理论分析，与近期针对一价拍卖中基于均值学习所建立的分析相呼应。在至少存在两家边际成本最低的企业且企业利润满足凹性条件的情况下，纳什均衡成为唯一的半粗均衡。在一价拍卖中，出价空间的粒度会影响半粗均衡，但对较低出价采用更细的粒度也会促使收敛到纳什均衡。与以往旨在证明收敛到纳什均衡（通常依赖于基于轮次的分析和概率论工具）的研究不同，我们基于线性规划的对偶方法能够对基于梯度学习的均衡选择进行简单且可处理的分析。