Adaptive, Doubly Optimal No-Regret Learning in Strongly Monotone and Exp-Concave Games with Gradient Feedback

Online gradient descent (OGD) is well-known to be doubly optimal under strong convexity or monotonicity assumptions: (1) in the single-agent setting, it achieves an optimal regret of Θ ( log T ) for strongly convex cost functions, and (2) in the multiagent setting of strongly monotone games with each agent employing OGD, we obtain last-iterate convergence of the joint action to a unique Nash equilibrium at an optimal rate of Θ ( 1 T ) . Whereas these finite-time guarantees highlight its merits, OGD has the drawback that it requires knowing the strong convexity/monotonicity parameters. In this paper, we design a fully adaptive OGD algorithm, AdaOGD , that does not require a priori knowledge of these parameters. In the single-agent setting, our algorithm achieves O ( log 2 ( T ) ) regret under strong convexity, which is optimal up to a log factor. Further, if each agent employs AdaOGD in strongly monotone games, the joint action converges in a last-iterate sense to a unique Nash equilibrium at a rate of O ( log 3 T T ) , again optimal up to log factors. We illustrate our algorithms in a learning version of the classic newsvendor problem, in which, because of lost sales, only (noisy) gradient feedback can be observed. Our results immediately yield the first feasible and near-optimal algorithm for both the single-retailer and multiretailer settings. We also extend our results to the more general setting of exp-concave cost functions and games, using the online Newton step algorithm.