Online estimation of the geometric median in Hilbert spaces : non asymptotic confidence balls

TL;DR

This paper develops non-asymptotic confidence balls for a recursive stochastic gradient algorithm estimating the geometric median in Hilbert spaces, providing improved convergence rates and exponential inequalities.

Contribution

It offers a detailed non-asymptotic analysis of a recursive estimator for the geometric median, including confidence balls and enhanced convergence bounds.

Findings

Derived non-asymptotic confidence balls for the estimator

Established improved $L^2$ convergence rates

Proved exponential inequalities for martingale terms

Abstract

Estimation procedures based on recursive algorithms are interesting and powerful techniques that are able to deal rapidly with (very) large samples of high dimensional data. The collected data may be contaminated by noise so that robust location indicators, such as the geometric median, may be preferred to the mean. In this context, an estimator of the geometric median based on a fast and efficient averaged non linear stochastic gradient algorithm has been developed by Cardot, C\'enac and Zitt (2013). This work aims at studying more precisely the non asymptotic behavior of this algorithm by giving non asymptotic confidence balls. This new result is based on the derivation of improved $L^2$ rates of convergence as well as an exponential inequality for the martingale terms of the recursive non linear Robbins-Monro algorithm.