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

• URL: http://arxiv.org/abs/2310.14085v4
• Date: Thu, 28 Mar 2024 19:37:02 GMT
• Title: Adaptive, Doubly Optimal No-Regret Learning in Strongly Monotone and Exp-Concave Games with Gradient Feedback
• Authors: Michael I. Jordan, Tianyi Lin, Zhengyuan Zhou,
• Abstract summary: Online gradient descent (OGD) is well known to be doubly optimal under strong convexity or monotonicity assumptions. In this paper, we design a fully adaptive OGD algorithm, textsfAdaOGD, that does not require a priori knowledge of these parameters.
• Score: 75.29048190099523
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $\Theta(\log T)$ for strongly convex cost functions; and (2) in the multi-agent 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 $\Theta(\frac{1}{T})$. While 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, \textsf{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 \textsf{AdaOGD} in strongly monotone games, the joint action converges in a last-iterate sense to a unique Nash equilibrium at a rate of $O(\frac{\log^3 T}{T})$, again optimal up to log factors. We illustrate our algorithms in a learning version of the classical newsvendor problem, where due to 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 multi-retailer settings. We also extend our results to the more general setting of exp-concave cost functions and games, using the online Newton step (ONS) algorithm.

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