Classic but Everlasting: Traditional Gradient-Based Algorithms Converge Fast Even in Time-Varying Multi-Player Games

Last-iterate convergence behaviours of well-known algorithms are intensively investigated in various games, such as two-player bilinear zero-sum games.However, most known last-iterate convergence properties rely on strict settings where the underlying games must have time-invariant payoffs.Besides, the limited known attempts on the games with time-varying payoffs are in two-player bilinear time-varying zero-sum games and strictly monotone games. By contrast, in other time-varying games, the last-iterate behaviours of two classic algorithms, i.e., extra gradient (EG) and optimistic gradient (OG) algorithms, still lack research, especially the convergence rates in multi-player games.In this paper, we investigate the last-iterate behaviours of EG and OG algorithms for convergent perturbed games, which extend upon the usual model of time-invariant games and incorporate external factors, such as vanishing noises.Using the recently proposed notion of the tangent residual (or its modifications) as the potential function of games and the measure of proximity to the Nash equilibrium, we prove that the last-iterate convergence rates of EG and OG algorithms for perturbed games on bounded convex closed sets are $O({1}/{\sqrt{T}})$ if such games converge to monotone games at rates fast enough and that such a result holds true for certain unconstrained perturbed games. With this result, we address an open questionasking for the last-iterate convergence rate of EG and OG algorithms in constrained and time-varying settings. The above convergence rates are similar to known tight results on corresponding time-invariant games.