Adaptive Importance Sampling for Efficient Stochastic Root Finding and Quantile Estimation

Stochastic root-finding problems are fundamental in the fields of operations research and data science. However, when the root-finding problem involves rare events, crude Monte Carlo can be prohibitively inefficient. Importance sampling (IS) is a commonly used approach, but selecting a good IS parameter requires knowledge of the problem’s solution, which creates a circular challenge. In “Adaptive Importance Sampling for Efficient Stochastic Root Finding and Quantile Estimation,” He, Jiang, Lam, and Fu propose an adaptive IS approach to untie this circularity. The adaptive IS simultaneously estimates the root and the IS parameters, and can be embedded in sample average approximation–type algorithms and stochastic approximation–type algorithms. They provide theoretical analysis on strong consistency and asymptotic normality of the resulting estimators, and show the benefit of adaptivity from a worst-case perspective. They also provide specialized analyses on extreme quantile estimation under milder conditions.