Stochastic Optimization with Optimal Importance Sampling

Importance Sampling (IS) is a widely used variance reduction technique for enhancing the efficiency of Monte Carlo methods, particularly in rare-event simulation and related applications. Despite its effectiveness, the performance of IS is highly sensitive to the choice of the proposal distribution and often requires stochastic calibration. While the design and analysis of IS have been extensively studied in estimation settings, applying IS within stochastic optimization introduces a fundamental challenge: the decision variable and the importance sampling distribution are mutually dependent, creating a circular optimization structure. This interdependence complicates both convergence analysis and variance control. We consider convex stochastic optimization problems with linear constraints and propose a single-loop stochastic approximation algorithm, based on a joint variant of Nesterov's dual averaging, that jointly updates the decision variable and the importance sampling distribution, without time-scale separation or nested optimization. The method is globally convergent and achieves minimal asymptotic variance among stochastic gradient schemes, matching the performance of an oracle sampler adapted to the optimal solution.