Convergence Rates of Inexact Proximal-Gradient Methods for Convex Optimization

TL;DR

This paper analyzes how inexact computations in proximal-gradient methods affect convergence, showing that with decreasing errors, these methods retain optimal convergence rates for convex optimization problems.

Contribution

It establishes that both basic and accelerated inexact proximal-gradient methods achieve optimal convergence rates if errors decrease appropriately, extending their applicability.

Findings

Both methods achieve the same convergence rate as error-free cases.

Properly decreasing errors lead to effective optimization performance.

In experiments, methods perform as well or better than fixed error levels.

Abstract

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.