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We consider the convex-concave saddle point problem $\min_{x}\max_{y}f(x)+y^\top A x-g(y)$ where $f$ is smooth and convex and $g$ is smooth andstrongly convex. We prove that if the coupling matrix $A$ has full column rank,the vanilla primal-dual gradient method can achieve linear convergence even if$f$ is not strongly convex. Our result generalizes previous work which eitherrequires $f$ and $g$ to be quadratic functions or requires proximal mappingsfor both $f$ and $g$. We adopt a novel analysis technique that in eachiteration uses a "ghost" update as a reference, and show that the iterates inthe primal-dual gradient method converge to this "ghost" sequence. Using thesame technique we further give an analysis for the primal-dual stochasticvariance reduced gradient (SVRG) method for convex-concave saddle pointproblems with a finite-sum structure.

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