Robust empirical risk minimization via Newton's method
This addresses robust optimization for data with contamination or heavy tails, offering faster convergence than robust gradient descent, though it appears incremental as it adapts existing robust estimators to a known method.
The paper tackles robust empirical risk minimization by proposing a Newton's method variant that uses robust estimators for gradient and Hessian, achieving quadratic convergence near the optimum in convex problems.
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.