Non-Asymptotic Convergence Analysis of Inexact Gradient Methods for Machine Learning Without Strong Convexity

TL;DR

This paper proves non-asymptotic linear convergence of inexact gradient methods for structured convex optimization problems without requiring strong convexity, under the condition that gradient errors decrease linearly.

Contribution

It establishes non-asymptotic linear convergence results for IGMs in non-strongly convex settings with linearly decreasing errors, expanding theoretical understanding.

Findings

Convergence holds for least squares and logistic regression.

Linear error decay leads to non-asymptotic convergence.

Applicable to structured convex optimization problems.

Abstract

Many recent applications in machine learning and data fitting call for the algorithmic solution of structured smooth convex optimization problems. Although the gradient descent method is a natural choice for this task, it requires exact gradient computations and hence can be inefficient when the problem size is large or the gradient is difficult to evaluate. Therefore, there has been much interest in inexact gradient methods (IGMs), in which an efficiently computable approximate gradient is used to perform the update in each iteration. Currently, non-asymptotic linear convergence results for IGMs are typically established under the assumption that the objective function is strongly convex, which is not satisfied in many applications of interest; while linear convergence results that do not require the strong convexity assumption are usually asymptotic in nature. In this paper, we combine the best of these two types of results and establish---under the standard assumption that the gradient approximation errors decrease linearly to zero---the non-asymptotic linear convergence of IGMs when applied to a class of structured convex optimization problems. Such a class covers settings where the objective function is not necessarily strongly convex and includes the least squares and logistic regression problems. We believe that our techniques will find further applications in the non-asymptotic convergence analysis of other first-order methods.