A Note on the Convergence of the OGAProx

Abstract

In this note, we consider the Optimistic Gradient Ascent-Proximal Point Algorithm (OGAProx) proposed by Bo{\c{t}}, Csetnek, and Sedlmayer for solving a saddle-point problem associated with a convex-concave function constructed by a nonsmooth coupling function and one regularizing function. We first provide a counterexample to show that the convergence of the minimax gap function, evaluated at the ergodic sequences, is insufficient to demonstrate the convergence of the function values evaluated at the ergodic sequences. Then under the same assumptions used by Bo{\c{t}} et al.\,for proving the convergence of the minimax gap function, we present convergence results for the function values evaluated at the ergodic sequences generated by the OGAProx with convergence rates of order $\mathcal{O}\left(\frac{1}{k}\right)$, $\mathcal{O}\left(\frac{1}{k^{2}}\right)$, and $\mathcal{O}\left(\theta^{k}\right)$ with $\theta \in (0,1)$ for the associated convex-concave coupling function being convex-concave, convex-strongly concave, and strongly convex-strongly concave, respectively.