Accelerated Primal-Dual Gradient Method for Smooth and Convex-Concave Saddle-Point Problems with Bilinear Coupling

In this paper we study the convex-concave saddle-point problem min(x) max(y) f(x)+ y(T)Ax- g(y), where f(x) and g(y) are smooth and convex functions. We propose an Accelerated Primal-Dual Gradient Method (APDG) for solving this problem, achieving (i) an optimal linear convergence rate in the strongly-convex-stronglyconcave regime, matching the lower complexity bound (Zhang et al., 2021), and (ii) an accelerated linear convergence rate in the case when only one of the functions f(x) and g(y) is strongly convex or even none of them are. Finally, we obtain a linearly convergent algorithm for the general smooth and convex-concave saddle point problem min(x) max(y) F(x, y) without the requirement of strong convexity or strong concavity.