Optimal No-Regret Learning in Strongly Monotone Games with Bandit Feedback
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We consider online no-regret learning in unknown games with bandit feedback, where each agent only observes its reward at each time – determined by all players' current joint action – rather than its gradient. We focus on the class of smooth and strongly monotone games and study optimal no-regret learning therein. Leveraging self-concordant barrier functions, we first construct an online bandit convex optimization algorithm and show that it achieves the single-agent optimal regret of Θ̃(√(T)) under smooth and strongly-concave payoff functions. We then show that if each agent applies this no-regret learning algorithm in strongly monotone games, the joint action converges in last iterate to the unique Nash equilibrium at a rate of Θ̃(1/√(T)). Prior to our work, the best-know convergence rate in the same class of games is O(1/T^1/3) (achieved by a different algorithm), thus leaving open the problem of optimal no-regret learning algorithms (since the known lower bound is Ω(1/√(T))). Our results thus settle this open problem and contribute to the broad landscape of bandit game-theoretical learning by identifying the first doubly optimal bandit learning algorithm, in that it achieves (up to log factors) both optimal regret in the single-agent learning and optimal last-iterate convergence rate in the multi-agent learning. We also present results on several simulation studies – Cournot competition, Kelly auctions, and distributed regularized logistic regression – to demonstrate the efficacy of our algorithm.
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