Stochastic approximation algorithms for superquantiles estimation

TL;DR

This paper develops and analyzes two stochastic approximation algorithms for estimating superquantiles, establishing their convergence properties and asymptotic behavior through theoretical proofs and numerical experiments.

Contribution

It introduces two two-time-scale stochastic approximation algorithms for superquantile estimation and provides their convergence analysis and asymptotic properties.

Findings

Almost sure convergence of estimators

Quadratic strong law and law of iterated logarithm proven

Numerical experiments validate theoretical results

Abstract

This paper is devoted to two different two-time-scale stochastic approximation algorithms for superquantile estimation. We shall investigate the asymptotic behavior of a Robbins-Monro estimator and its convexified version. Our main contribution is to establish the almost sure convergence, the quadratic strong law and the law of iterated logarithm for our estimates via a martingale approach. A joint asymptotic normality is also provided. Our theoretical analysis is illustrated by numerical experiments on real datasets.