Importance Sampling for Minimization of Tail Risks: A Tutorial

This tutorial provides an introductory overview of how one may use importance sampling to drastically reduce the sample requirements in solving stochastic optimization and elementary simulation optimization problems incorporating tail risk measures. Sample average approximations, while appealing due to their universality in use, require a large number of samples due to the rarity with which relevant tail events get observed. Importance Sampling is among the most potent methods for reducing the sample requirements in estimating rare event probabilities. Can importance sampling be used with similar effectiveness for solving optimization formulations (involving rare events) as well, and if so, what are the key ingredients required to operationalize this idea? Focusing on these questions, this tutorial aims to demonstrate (i) how to arrive at an effective change of measure prescription at every decision, and (ii) the prominent techniques available for integrating such a prescription within a solution paradigm for optimization.