Convergence Rates of Inexact Proximal-Gradient Methods for Convex Optimization

We consider the problem of optimizing the sum of a smooth convex function and a non-smooth convex function using proximal-gradient methods, where an error is present in the calculation of the gradient of the smooth term or in the proximity operator with respect to the non-smooth term. We show that both the basic proximal-gradient method and the accelerated proximal-gradient method achieve the same convergence rate as in the error-free case, provided that the errors decrease at appropriate rates.Using these rates, we perform as well as or better than a carefully chosen fixed error level on a set of structured sparsity problems.