Statistical Robustness of Kernel Learning Estimator with Respect to Data Perturbation

TL;DR

This paper investigates the statistical robustness of kernel learning estimators under data perturbations, providing both qualitative and quantitative analyses, including influence functions and variational inequalities, with practical examples.

Contribution

It extends robustness analysis to kernel learning estimators, deriving new conditions and methods for both convex and non-convex cost functions in RKHS.

Findings

Qualitative robustness of regularized empirical risk minimizer derived.

First-order optimality conditions for stationary solutions established.

Lipschitz continuity of solutions against distribution perturbations demonstrated.

Abstract

Inspired by the recent work [28] on the statistical robustness of empirical risks in reproducing kernel Hilbert space (RKHS) where the training data are potentially perturbed or even corrupted, we take a step further in this paper to investigate the statistical robustness of the kernel learning estimator (the regularized empirical risk minimizer or stationary point). We begin by deriving qualitative statistical robustness of the estimator of the regularized empirical risk minimizer for a broad class of convex cost functions when all of the training data are potentially perturbed under some topological structures, and then move on to consider the quantitative statistical robustness of the stationary solution for a specific case that the cost function is continuously differentiable but not necessarily convex. In the latter case, we derive the first-order optimality condition of the regularized expected risk minimization problem, which is essentially a stochastic variational inequality problem (SVIP) in RKHS, and then use the SVIP as a platform to investigate local and global Lipschitz continuity of the stationary solution against perturbation of the probability distribution under the Fortet-Mourier metric. A crucial assumption in the analysis is that the perturbed data are independent and identically distributed (iid). In some practical applications, this assumption may not be fulfilled when a small proportion of perceived data is seriously perturbed/contaminated. In this case, we use the influence function to investigate the impact of single data perturbation on the expected risk minimizer. Differing from [64, Chapter 10], we concentrate on constrained expected risk minimization problems. The research is essentially down to the derivation of the implicit function theorem of the SVIP in RKHS. Finally, we illustrate our theoretical analysis with a couple of academic examples.