Error Bounds, Quadratic Growth, and Linear Convergence of Proximal Methods

The proximal gradient algorithm for minimizing the sum of a smooth and 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 and quadratic 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.