Robust empirical risk minimization via Newton's method

TL;DR

This paper introduces a robust Newton's method for empirical risk minimization that uses robust estimators for gradients and Hessians, achieving faster convergence in high-dimensional and contaminated data scenarios.

Contribution

It develops a new robust Newton's method with convergence guarantees and practical algorithms for high-dimensional, contaminated, or heavy-tailed data.

Findings

Proves convergence of the robust Newton method to a small neighborhood of the true minimizer.

Demonstrates quadratic convergence rates similar to classical Newton's method under robustness.

Provides an algorithm suitable for high-dimensional problems with contaminated data.

Abstract

A new variant of Newton's method for empirical risk minimization is studied, where at each iteration of the optimization algorithm, the gradient and Hessian of the objective function are replaced by robust estimators taken from existing literature on robust mean estimation for multivariate data. After proving a general theorem about the convergence of successive iterates to a small ball around the population-level minimizer, consequences of the theory in generalized linear models are studied when data are generated from Huber's epsilon-contamination model and/or heavytailed distributions. An algorithm for obtaining robust Newton directions based on the conjugate gradient method is also proposed, which may be more appropriate for high-dimensional settings, and conjectures about the convergence of the resulting algorithm are offered. Compared to robust gradient descent, the proposed algorithm enjoys the faster rates of convergence for successive iterates often achieved by second-order algorithms for convex problems, i.e., quadratic convergence in a neighborhood of the optimum, with a stepsize that may be chosen adaptively via backtracking linesearch.