Statistical Robustness of Empirical Risks in Machine Learning

This paper studies convergence of empirical risks in reproducing kernel Hilbert spaces (RKHS). A conventional assumption in the existing research is that empirical training data are generated by the unknown true probability distribution but this may not be satisfied in some practical circumstances. Consequently the existing convergence results may not provide a guarantee as to whether the empirical risks are reliable or not when the data are potentially corrupted (generated by a distribution perturbed from the true). In this paper, we fill out the gap from robust statistics perspective (Kra center dot tschmer, Schied and Za center dot hle (2012); Kra center dot tschmer, Schied and Za center dot hle (2014); Guo and Xu (2020)). First, we derive moderate sufficient conditions under which the expected risk changes stably (continuously) against small perturbation of the probability distributions of the underlying random variables and demonstrate how the cost function and kernel affect the stability. Second, we examine the difference between laws of the statistical estimators of the expected optimal loss based on pure data and contaminated data using Prokhorov metric and Kantorovich metric, and derive some asymptotic qualitative and non-asymptotic quantitative statistical robustness results. Third, we identify appropriate metrics under which the statistical estimators are uniformly asymptotically consistent. These results provide theoretical grounding for analysing asymptotic convergence and examining reliability of the statistical estimators in a number of regression models.