Fundamental limits for weighted empirical approximations of tilted distributions

TL;DR

This paper analyzes the asymptotic efficiency of importance sampling for generating samples from tilted distributions, revealing a polynomial versus super-polynomial sample complexity dichotomy based on distribution bounds.

Contribution

It provides a sharp characterization of the accuracy of self-normalized importance sampling for tilted distributions, highlighting a fundamental complexity dichotomy.

Findings

Polynomial sample complexity for bounded distributions

Super-polynomial sample complexity for unbounded distributions

Sharp asymptotic accuracy characterization

Abstract

Consider the task of generating samples from a tilted distribution of a random vector whose underlying distribution is unknown, but samples from it are available. This finds applications in fields such as finance and climate science, and in rare event simulation. In this article, we discuss the asymptotic efficiency of a self-normalized importance sampler of the tilted distribution. We provide a sharp characterization of its accuracy, given the number of samples and the degree of tilt. Our findings reveal a surprising dichotomy: while the number of samples needed to accurately tilt a bounded random vector increases polynomially in the tilt amount, it increases at a super polynomial rate for unbounded distributions.