Last-Iterate Convergence of Single-Loop Stochastic Methods for Constrained Convex-Concave Minimax Problems

In this paper, we study last-iterate convergence in stochastic constrained convex--concave minimax optimization. A key difficulty is that the last iterates of vanilla stochastic extragradient (S-EG) and stochastic optimistic gradient descent--ascent (S-OGDA) can fail to converge in the presence of gradient noise, even for simple bilinear problems. To address this issue, we regularize the original convex--concave problem into a strongly convex--strongly concave one. Applying S-EG and S-OGDA to the regularized problem gives two simple single-loop first-order methods, which we call perturbed S-EG and perturbed S-OGDA. By carefully choosing the regularization parameter and balancing the resulting regularization bias, stochastic error, and optimization error, we prove that both methods achieve a last-iterate convergence rate of $\mathcal{O}(T^{-1/4+\varepsilon})$ for any $\varepsilon>0$ in terms of the primal--dual gap. This improves upon the best-known $\tilde{\mathcal{O}}(T^{-1/7})$ rate under comparable constrained settings. Moreover, for unconstrained problems, we establish almost sure convergence under a single-timescale stepsize scheme, whereas prior almost sure convergence results typically rely on two-timescale stepsizes and are limited to S-EG.