Distributionally Constrained Black-Box Stochastic Gradient Estimation and Optimization

Constructing stochastic gradient estimators and descent algorithms is a core problem in stochastic optimization. In the black-box setting, namely when only noisy function evaluations are available, such gradient estimators are typically constructed via finite-differencing. In this regard, there have been schemes that aim to obtain gradient estimators for potentially many dimensions simultaneously via only few sample observations or simulation runs. However, for problems with probability simplex constraints, which arise in a range of applications from distributionally robust analysis to inverse model calibration, these schemes run into challenges one way or another when attempting to balance bias-variance in a constraint-compatible manner. This motivates us to create a new design framework for random perturbation generators and estimation schemes that bypass these challenges. Our culminated class of estimators, which is based on the Dirichlet mixtures, is demonstrably effective in distributionally constrained gradient estimation and optimization under various black-box settings.