Fast convergence of generalized forward-backward algorithms for structured monotone inclusions

TL;DR

This paper introduces an accelerated forward-backward algorithm with inertial and correction terms, achieving faster convergence rates for solving structured monotone inclusions and saddle point problems in Hilbert spaces.

Contribution

It develops a modified forward-backward method with inertial acceleration, proving weak convergence with improved rates, and extends the approach to more general monotone inclusions and primal-dual problems.

Findings

Achieves convergence rate of o(n^{-2}) for the iterates.

Extends to primal-dual algorithms for saddle point problems.

Demonstrates faster convergence compared to classical methods.

Abstract

In this paper, we develop rapidly convergent forward-backward algorithms for computing zeroes of the sum of finitely many maximally monotone operators. A modification of the classical forward-backward method for two general operators is first considered, by incorporating an inertial term (closed to the acceleration techniques introduced by Nesterov), a constant relaxation factor and a correction term. In a Hilbert space setting, we prove the weak convergence to equilibria of the iterates $(x_n)$, with worst-case rates of $ o(n^{-2})$ in terms of both the discrete velocity and the fixed point residual, instead of the classical rates of ${\cal O}(n^{-1})$ established so far for related algorithms. Our procedure is then adapted to more general monotone inclusions and a fast primal-dual algorithm is proposed for solving convex-concave saddle point problems.