Importance Sampling for Minimization of Tail Risks: A Tutorial

TL;DR

This tutorial explains how importance sampling can be effectively used to minimize tail risks in stochastic optimization, reducing the sample size needed for accurate risk estimation.

Contribution

It introduces the key components for applying importance sampling to tail risk optimization, including change of measure and integration techniques.

Findings

Importance sampling reduces sample requirements for tail risk estimation.

Methodology for incorporating importance sampling into stochastic optimization.

Guidelines for selecting importance sampling measures in risk minimization.

Abstract

This paper provides an introductory overview of how one may employ importance sampling effectively as a tool for solving stochastic optimization formulations incorporating tail risk measures such as Conditional Value-at-Risk. Approximating the tail risk measure by its sample average approximation, while appealing due to its simplicity and universality in use, requires a large number of samples to be able to arrive at risk-minimizing decisions with high confidence. This is primarily due to the rarity with which the relevant tail events get observed in the samples. In simulation, Importance Sampling is among the most prominent methods for substantially reducing the sample requirement while estimating probabilities of rare events. Can importance sampling be used for optimization as well? If so, what are the ingredients required for making importance sampling an effective tool for optimization formulations involving rare events? This tutorial aims to provide an introductory overview of the two key ingredients in this regard, namely, (i) how one may arrive at an importance sampling change of measure prescription at every decision, and (ii) the prominent techniques available for integrating such a prescription within a solution paradigm for stochastic optimization formulations.