Exact and Inexact Subsampled Newton Methods for Optimization

TL;DR

This paper investigates subsampled Newton methods for stochastic optimization, analyzing their convergence, complexity, and practical performance in machine learning tasks like logistic regression.

Contribution

It introduces a superlinear convergence analysis for Newton-like methods with subsampled derivatives and evaluates an inexact Newton approach using conjugate gradient in this context.

Findings

Superlinear convergence achieved with proper gradient and Hessian accuracy coordination

Complexity analysis of inexact Newton method with Hessian sampling and CG

Preliminary results show promising performance on logistic regression tasks

Abstract

The paper studies the solution of stochastic optimization problems in which approximations to the gradient and Hessian are obtained through subsampling. We first consider Newton-like methods that employ these approximations and discuss how to coordinate the accuracy in the gradient and Hessian to yield a superlinear rate of convergence in expectation. The second part of the paper analyzes an inexact Newton method that solves linear systems approximately using the conjugate gradient (CG) method, and that samples the Hessian and not the gradient (the gradient is assumed to be exact). We provide a complexity analysis for this method based on the properties of the CG iteration and the quality of the Hessian approximation, and compare it with a method that employs a stochastic gradient iteration instead of the CG method. We report preliminary numerical results that illustrate the performance of inexact subsampled Newton methods on machine learning applications based on logistic regression.