Linear Convergence of the Primal-Dual Gradient Method for Convex-Concave Saddle Point Problems without Strong Convexity

We consider the convex-concave saddle point problem minx maxy f(x) + y>Ax− g(y) where f is smooth and convex and g is smooth and strongly convex. We prove that if the coupling matrix A has full column rank, the vanilla primaldual gradient method can achieve linear convergence even if f is not strongly convex. Our result generalizes previous work which either requires f and g to be quadratic functions or requires proximal mappings for both f and g. We adopt a novel analysis technique that in each iteration uses a “ghost” update as a reference, and show that the iterates in the primal-dual gradient method converge to this “ghost” sequence. Using the same technique we further give an analysis for the primal-dual stochastic variance reduced gradient (SVRG) method for convex-concave saddle point problems with a finite-sum structure.