Statistical learning with indirect observations

TL;DR

This paper investigates statistical learning with indirect, contaminated data, deriving fast convergence rates for empirical risk minimizers using regularization, thus extending understanding of learning performance under measurement errors.

Contribution

It introduces a framework for analyzing statistical learning with indirect observations and establishes fast convergence rates for regularized empirical risk minimizers.

Findings

Fast convergence rates comparable to direct data cases.

Regularization methods effectively handle indirect measurements.

Insights into the impact of measurement errors on learning rates.

Abstract

Let $(X,Y)\in\mathcal{X}\times \mathcal{Y}$ be a random couple with unknown distribution $P$. Let $\GG$ be a class of measurable functions and $\ell$ a loss function. The problem of statistical learning deals with the estimation of the Bayes: $$g^*=\arg\min_{g\in\GG}\E_P \ell(g(X),Y). $$ In this paper, we study this problem when we deal with a contaminated sample $(Z_1,Y_1),..., (Z_n,Y_n)$ of i.i.d. indirect observations. Each input $Z_i$, $i=1,...,n$ is distributed from a density $Af$, where $A$ is a known compact linear operator and $f$ is the density of the direct input $X$. We derive fast rates of convergence for empirical risk minimizers based on regularization methods, such as deconvolution kernel density estimators or spectral cut-off. These results are comparable to the existing fast rates in Koltchinskii for the direct case. It gives some insights into the effect of indirect measurements in the presence of fast rates of convergence.