Online Stochastic Optimization of SDEs

We develop a new online algorithm for optimizing over the stationary distribution of stochastic differential equation (SDE) models. The algorithm optimizes over the parameters in the multi-dimensional SDE model in order to minimize the distance between the model's stationary distribution and the target statistics. We rigorously prove convergence for linear SDE models and present numerical results for nonlinear examples. The proof requires analysis of the fluctuations of the parameter evolution around the unbiased descent direction under the stationary distribution. Bounds on the fluctuations are challenging to obtain due to the online nature of the algorithm (e.g., the stationary distribution will continuously change as the parameters change). We prove bounds on a new class of Poisson partial differential equations, which are then used to analyze the parameter fluctuations in the algorithm. This presentation is based upon research with Ziheng Wang.