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# Quadratic convergence of smoothing Newton's Method for 0/1 loss optimization

Zhou, Shenglong, Pan, Lili, Xiu, Naihua and Qi, Hou-Duo (2021) Quadratic convergence of smoothing Newton's Method for 0/1 loss optimization. SIAM Journal on Optimization, 1-28. (In Press)

Record type: Article

## Abstract

It has been widely recognized that the $0/1$-loss function is one of the most natural choices for modelling classification errors, and it has a wide range of applications including support vector machines and $1$-bit compressed sensing.
Due to the combinatorial nature of the $0/1$-loss function, methods based on convex relaxations or smoothing approximations have dominated the existing research and are often able to provide approximate solutions of good quality.
However, those methods are not optimizing the $0/1$-loss function directly and hence no optimality has been established for the original problem. This paper aims to study the optimality conditions of the $0/1$ function minimization, and for the first time to develop Newton's method that directly optimizes the $0/1$ function with a local quadratic convergence under reasonable conditions. Extensive numerical experiments demonstrate its superior performance as one would expect from Newton-type methods.

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Accepted/In Press date: 5 September 2021
Keywords: $0/1$-loss function, optimality conditions, Newton's method, locally quadratic convergence

## Identifiers

Local EPrints ID: 451313
URI: http://eprints.soton.ac.uk/id/eprint/451313
ISSN: 1052-6234
PURE UUID: 482013dc-2f15-41c7-a6a1-cb59587b5da2
ORCID for Shenglong Zhou: orcid.org/0000-0003-2843-1614
ORCID for Hou-Duo Qi: orcid.org/0000-0003-3481-4814

## Catalogue record

Date deposited: 20 Sep 2021 16:32

## Contributors

Author: Shenglong Zhou
Author: Lili Pan
Author: Naihua Xiu
Author: Hou-Duo Qi

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