Linear Convergence of the Primal-Dual Gradient Method for Convex-Concave Saddle Point Problems without Strong Convexity
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We consider the convex-concave saddle point problem _x_y f(x)+y^ A x-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 primal-dual 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.
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