We study the problem of online learning in Stackelberg games with side information between a leader and a sequence of followers. In every round the leader observes contextual information and commits to a mixed strategy, after which the follower best-responds. We provide learning algorithms for the leader which achieve $O(T^{1/2})$ regret under bandit feedback, an improvement from the previously best-known rates of $O(T^{2/3})$. Our algorithms rely on a reduction to linear contextual bandits in the utility space: In each round, a linear contextual bandit algorithm recommends a utility vector, which our algorithm inverts to determine the leader's mixed strategy. We extend our algorithms to the setting in which the leader's utility function is unknown, and also apply it to the problems of bidding in second-price auctions with side information and online Bayesian persuasion with public and private states. Finally, we observe that our algorithms empirically outperform previous results on numerical simulations.

翻译：我们研究了在领导者与一系列跟随者之间带有侧信息的斯塔克尔伯格博弈中的在线学习问题。每一轮中，领导者观察情境信息并承诺一个混合策略，随后跟随者做出最优反应。在赌博反馈下，我们为领导者提供了一种实现$O(T^{1/2})$后悔的学习算法，相比此前已知最优的$O(T^{2/3})$速率有所改进。我们的算法依赖于对效用空间中的线性情境赌博问题的归约：在每一轮中，线性情境赌博算法推荐一个效用向量，我们的算法将其求逆以确定领导者的混合策略。我们将算法扩展到领导者效用函数未知的情形，并将其应用于带侧信息的第二价格拍卖投标以及具有公开和私有状态的在线贝叶斯说服问题。最后，我们观察到，在数值模拟中，我们的算法在经验性能上优于以往结果。