Empirical Variance Minimization with Applications in Variance Reduction and Optimal Control

TL;DR

This paper develops empirical variance minimization techniques with sharp bounds, enabling fast convergence rates for variance reduction and optimal control in complex functional classes.

Contribution

It introduces non-asymptotic bounds for empirical variance minimization and demonstrates their effectiveness in variance reduction and optimal control applications.

Findings

Sharp non-asymptotic bounds for excess variance

Fast convergence rates under certain restrictions

Optimal non-parametric rates in non-Donsker regimes

Abstract

We study the problem of empirical minimization for variance-type functionals over functional classes. Sharp non-asymptotic bounds for the excess variance are derived under mild conditions. In particular, it is shown that under some restrictions imposed on the functional class fast convergence rates can be achieved including the optimal non-parametric rates for expressive classes in the non-Donsker regime under some additional assumptions. Our main applications include variance reduction and optimal control.