Adaptive Importance Sampling via Stochastic Convex Programming

TL;DR

This paper introduces an adaptive importance sampling method leveraging convexity of variance in exponential families, leading to asymptotically optimal estimators for Monte Carlo integration.

Contribution

It demonstrates the convexity of variance in importance sampling and develops an adaptive algorithm that optimizes the sampling distribution.

Findings

Variance of importance sampling estimator is convex in natural parameters.

Proposed method achieves asymptotic optimality in variance.

Algorithm improves sampling efficiency over traditional methods.

Abstract

We show that the variance of the Monte Carlo estimator that is importance sampled from an exponential family is a convex function of the natural parameter of the distribution. With this insight, we propose an adaptive importance sampling algorithm that simultaneously improves the choice of sampling distribution while accumulating a Monte Carlo estimate. Exploiting convexity, we prove that the method's unbiased estimator has variance that is asymptotically optimal over the exponential family.