Root finding via DARTS — Dynamic Adaptive Random Target Shooting

Consider multi-dimensional root finding when the equations are available only implicitly via a Monte Carlo simulation oracle that for any solution returns a vector of point estimates. We develop DARTS, a stochastic-approximation algorithm that makes quasi-Newton moves to a new solution whenever the current sample size is large compared to the estimated quality of the current solution and estimated sampling error. We show that DARTS converges in a certain precise sense, and discuss reasons to expect substantial computational efficiencies over traditional stochastic approximation variations.