A globally convergent incremental Newton method

TL;DR

This paper introduces an incremental Newton method tailored for large-scale, distributed, strongly convex optimization problems, demonstrating global convergence and linear rates under specific conditions.

Contribution

The paper develops a new incremental Newton method with proven global convergence and linear convergence rates under a gradient growth condition, extending analysis to related algorithms.

Findings

Method is globally convergent with variable stepsize.

Achieves linear convergence under gradient growth condition.

Extension to incremental Gauss-Newton algorithm with similar convergence properties.

Abstract

Motivated by machine learning problems over large data sets and distributed optimization over networks, we develop and analyze a new method called incremental Newton method for minimizing the sum of a large number of strongly convex functions. We show that our method is globally convergent for a variable stepsize rule. We further show that under a gradient growth condition, convergence rate is linear for both variable and constant stepsize rules. By means of an example, we show that without the gradient growth condition, incremental Newton method cannot achieve linear convergence. Our analysis can be extended to study other incremental methods: in particular, we obtain a linear convergence rate result for the incremental Gauss-Newton algorithm under a variable stepsize rule.