Finite-Time Last-Iterate Convergence for Multi-Agent Learning in Games
Tianyi Lin, Zhengyuan Zhou, Panayotis Mertikopoulos, Michael Jordan
Abstract
In this paper, we consider multi-agent learning via online gradient descent in a class of games called $\lambda$-cocoercive games, a fairly broad class of games that admits many Nash equilibria and that properly includes unconstrained strongly monotone games. We characterize the finite-time last-iterate convergence rate for joint OGD learning on $\lambda$-cocoercive games; further, building on this result, we develop a fully adaptive OGD learning algorithm that does not require any knowledge of problem parameter (e.g. cocoercive constant $\lambda$) and show, via a novel double-stopping time technique, that this adaptive algorithm achieves same finite-time last-iterate convergence rate as non-adaptive counterpart. Subsequently, we extend OGD learning to the noisy gradient feedback case and establish last-iterate convergence results–first qualitative almost sure convergence, then quantitative finite-time convergence rates– all under non-decreasing step-sizes. To our knowledge, we provide the first set of results that fill in several gaps of the existing multi-agent online learning literature, where three aspects–finite-time convergence rates, non-decreasing step-sizes, and fully adaptive algorithms have been unexplored before.
BibTeX
@InProceedings{pmlr-v119-lin20h,
title = {Finite-Time Last-Iterate Convergence for Multi-Agent Learning in Games},
author = {Lin, Tianyi and Zhou, Zhengyuan and Mertikopoulos, Panayotis and Jordan, Michael},
booktitle = {Proceedings of the 37th International Conference on Machine Learning},
pages = {6161--6171},
year = {2020},
editor = {III, Hal Daumé and Singh, Aarti},
volume = {119},
series = {Proceedings of Machine Learning Research},
month = {13--18 Jul},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v119/lin20h/lin20h.pdf},
url = {https://proceedings.mlr.press/v119/lin20h.html},
abstract = {In this paper, we consider multi-agent learning via online gradient descent in a class of games called $\lambda$-cocoercive games, a fairly broad class of games that admits many Nash equilibria and that properly includes unconstrained strongly monotone games. We characterize the finite-time last-iterate convergence rate for joint OGD learning on $\lambda$-cocoercive games; further, building on this result, we develop a fully adaptive OGD learning algorithm that does not require any knowledge of problem parameter (e.g. cocoercive constant $\lambda$) and show, via a novel double-stopping time technique, that this adaptive algorithm achieves same finite-time last-iterate convergence rate as non-adaptive counterpart. Subsequently, we extend OGD learning to the noisy gradient feedback case and establish last-iterate convergence results–first qualitative almost sure convergence, then quantitative finite-time convergence rates– all under non-decreasing step-sizes. To our knowledge, we provide the first set of results that fill in several gaps of the existing multi-agent online learning literature, where three aspects–finite-time convergence rates, non-decreasing step-sizes, and fully adaptive algorithms have been unexplored before.}
}