Estimating the geometric median in Hilbert spaces with stochastic gradient algorithms: $L^{p}$ and almost sure rates of convergence

TL;DR

This paper analyzes the convergence properties of stochastic gradient algorithms for estimating the geometric median in high-dimensional Hilbert spaces, providing $L^{p}$ and almost sure rates, and establishing optimal quadratic mean convergence rates.

Contribution

It offers a detailed study of the asymptotic behavior and convergence rates of recursive stochastic gradient estimators for the geometric median in Hilbert spaces, including optimal mean convergence.

Findings

Derived $L^{p}$ convergence rates for the estimators.

Established almost sure convergence rates.

Identified the optimal quadratic mean convergence rate for the averaged algorithm.

Abstract

The geometric median, also called $L^{1}$-median, is often used in robust statistics. Moreover, it is more and more usual to deal with large samples taking values in high dimensional spaces. In this context, a fast recursive estimator has been introduced by Cardot, Cenac and Zitt. This work aims at studying more precisely the asymptotic behavior of the estimators of the geometric median based on such non linear stochastic gradient algorithms. The $L^{p}$ rates of convergence as well as almost sure rates of convergence of these estimators are derived in general separable Hilbert spaces. Moreover, the optimal rate of convergence in quadratic mean of the averaged algorithm is also given.