Modified Fej\'er sequences and applications

TL;DR

This paper introduces modified Fejér sequences as a unifying framework in Hilbert spaces to analyze convergence rates of various optimization algorithms, including forward-backward splitting and Douglas-Rachford methods.

Contribution

It proposes a new concept of modified Fejér sequences that generalizes and unifies convergence analysis for multiple optimization algorithms.

Findings

Provides convergence rate proofs for several algorithms

Generalizes known results in optimization convergence analysis

Unifies analysis framework for different splitting methods

Abstract

In this note, we propose and study the notion of modified Fej\'{e}r sequences. Within a Hilbert space setting, we show that it provides a unifying framework to prove convergence rates for objective function values of several optimization algorithms. In particular, our results apply to forward-backward splitting algorithm, incremental subgradient proximal algorithm, and the Douglas-Rachford splitting method including and generalizing known results.