Quadratic Convergence of Smoothing Newton's Method for 0/1 Loss Optimization

TL;DR

This paper introduces a Newton's method that directly optimizes the 0/1 loss function, achieving quadratic convergence and outperforming existing approximation-based methods in classification error minimization.

Contribution

It develops the first Newton's method for direct 0/1 loss optimization with proven local quadratic convergence under reasonable conditions.

Findings

Demonstrates superior performance in numerical experiments

Achieves quadratic convergence rate

Outperforms existing approximation methods

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.ions. Extensive numerical experiments demonstrate its superior performance as one would expect from Newton-type methods.