Berry--Esseen Bounds for Multivariate Nonlinear Statistics with Applications to M-estimators and Stochastic Gradient Descent Algorithms

TL;DR

This paper develops a new multivariate concentration inequality to establish Berry--Esseen bounds for nonlinear statistics, improving understanding of their convergence behavior, with applications to M-estimators and stochastic gradient descent algorithms.

Contribution

It introduces a novel multivariate randomized concentration inequality and derives optimal Berry--Esseen bounds for complex nonlinear statistics, including M-estimators and SGD algorithms.

Findings

Established the best possible Berry--Esseen bounds for many multivariate nonlinear statistics.

Applied the bounds to analyze the convergence of M-estimators.

Provided bounds for averaged stochastic gradient descent algorithms.

Abstract

We establish a Berry--Esseen bound for general multivariate nonlinear statistics by developing a new multivariate-type randomized concentration inequality. The bound is the best possible for many known statistics. As applications, Berry--Esseen bounds for M-estimators and averaged stochastic gradient descent algorithms are obtained.