Global Convergence Rate of Proximal Incremental Aggregated Gradient Methods

TL;DR

This paper establishes the first convergence rate analysis for the proximal incremental aggregated gradient (PIAG) method with deterministic update order, showing linear convergence under certain step size conditions in strongly convex optimization.

Contribution

It provides the first deterministic order convergence rate analysis for the PIAG method, improving understanding of its efficiency in convex optimization.

Findings

PIAG converges globally with a linear rate under small enough step size.

Explicit convergence rate and step size are derived.

Results improve the known condition number dependence for incremental gradient methods.

Abstract

We focus on the problem of minimizing the sum of smooth component functions (where the sum is strongly convex) and a non-smooth convex function, which arises in regularized empirical risk minimization in machine learning and distributed constrained optimization in wireless sensor networks and smart grids. We consider solving this problem using the proximal incremental aggregated gradient (PIAG) method, which at each iteration moves along an aggregated gradient (formed by incrementally updating gradients of component functions according to a deterministic order) and taking a proximal step with respect to the non-smooth function. While the convergence properties of this method with randomized orders (in updating gradients of component functions) have been investigated, this paper, to the best of our knowledge, is the first study that establishes the convergence rate properties of the PIAG method for any deterministic order. In particular, we show that the PIAG algorithm is globally convergent with a linear rate provided that the step size is sufficiently small. We explicitly identify the rate of convergence and the corresponding step size to achieve this convergence rate. Our results improve upon the best known condition number dependence of the convergence rate of the incremental aggregated gradient methods used for minimizing a sum of smooth functions.