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 $\mathrm{\Theta }(\log T)$ 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 $\mathrm{\Theta }\left(\frac{1}{T}\right)$. 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 $O(\frac{{\log }^{3}T}{T})$, 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.

Funding: This work is generously supported by the National Science Foundation [Grant CCF-2106508]. This work is also partially funded by a grant from the European Research Council under the Synergy program and the 2024 New York University Center for Global Economy and Business grant.