Uniform exponential convergence of SAA with AMIS and asymptotics of its optimal value

TL;DR

This paper establishes uniform exponential convergence rates for SAA with AMIS and analyzes the asymptotic behavior of its optimal value using advanced probabilistic tools.

Contribution

It introduces a new exponential convergence rate for SAA with AMIS and derives asymptotics of the optimal value via a functional CLT and the Delta theorem.

Findings

New exponential convergence rate for SAA with AMIS

Asymptotic distribution of the optimal value derived

Utilizes a concentration inequality for martingale differences

Abstract

We discuss in this paper uniform exponential convergence of sample average approximation (SAA) with adaptive multiple importance sampling (AMIS) and asymptotics of its optimal value. Using a concentration inequality for bounded martingale differences, we obtain a new exponential convergence rate. To study the asymptotics, we first derive an important functional central limit theorem (CLT) for martingale difference sequences. Subsequently, exploiting this result with the Delta theorem, we prove the asymptotics of optimal values for SAA with AMIS.