Error bounds, quadratic growth, and linear convergence of proximal methods

Abstract

The proximal gradient algorithm for minimizing the sum of a smooth and a nonsmooth convex function often converges linearly even without strong convexity. One common reason is that a multiple of the step length at each iteration may linearly bound the "error" -- the distance to the solution set. We explain the observed linear convergence intuitively by proving the equivalence of such an error bound to a natural quadratic growth condition. Our approach generalizes to linear convergence analysis for proximal methods (of Gauss-Newton type) for minimizing compositions of nonsmooth functions with smooth mappings. We observe incidentally that short step-lengths in the algorithm indicate near-stationarity, suggesting a reliable termination criterion.