Subsampled inexact Newton methods for minimizing large sums of convex functions

Author:

Bellavia Stefania1,Krejić Nataša2,Krklec Jerinkić Nataša2

Affiliation:

1. Department of Industrial Engineering, University of Florence, Viale Morgagni, Florence, Italy

2. Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Trg Dositeja Obradovića 4, Novi Sad, Serbia

Abstract

Abstract This paper deals with the minimization of a large sum of convex functions by inexact Newton (IN) methods employing subsampled functions, gradients and Hessian approximations. The conjugate gradient method is used to compute the IN step and global convergence is enforced by a nonmonotone line-search procedure. The aim is to obtain methods with affordable costs and fast convergence. Assuming strongly convex functions, R-linear convergence and worst-case iteration complexity of the procedure are investigated when functions and gradients are approximated with increasing accuracy. A set of rules for the forcing parameters and subsample Hessian sizes are derived that ensure local q-linear/q-superlinear convergence of the proposed method. The random choice of the Hessian subsample is also considered and convergence in the mean square, both for finite and infinite sums of functions, is proved. Finally, the analysis of global convergence with asymptotic R-linear rate is extended to the case of the sum of convex functions and strongly convex objective function. Numerical results on well-known binary classification problems are also given. Adaptive strategies for selecting forcing terms and Hessian subsample size, streaming out of the theoretical analysis, are employed and the numerical results show that they yield effective IN methods.

Funder

INdAM Research Group-Gruppo Nazionale per il Calcolo Scientifico

GNCS-Istituto Nazionale di Alta Matematica of Italy

Serbian Ministry of Education Science and Technological Development

Publisher

Oxford University Press (OUP)

Subject

Applied Mathematics,Computational Mathematics,General Mathematics

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