  Overparameterized stochastic differential equation (SDE) models have achieved
remarkable success in various complex environments, such as PDE-constrained
optimization, stochastic control and reinforcement learning, financial
engineering, and neural SDEs. These models often feature system evolution
coefficients that are parameterized by a high-dimensional vector $\theta \in
\mathbb{R}^n$, aiming to optimize expectations of the SDE, such as a value
function, through stochastic gradient ascent. Consequently, designing efficient
gradient estimators for which the computational complexity scales well with $n$
is of significant interest. This paper introduces a novel unbiased stochastic
gradient estimator--the generator gradient estimator--for which the computation
time remains stable in $n$. In addition to establishing the validity of our
methodology for general SDEs with jumps, we also perform numerical experiments
that test our estimator in linear-quadratic control problems parameterized by
high-dimensional neural networks. The results show a significant improvement in
efficiency compared to the widely used pathwise differentiation method: Our
estimator achieves near-constant computation times, increasingly outperforms
its counterpart as $n$ increases, and does so without compromising estimation
variance. These empirical findings highlight the potential of our proposed
methodology for optimizing SDEs in contemporary applications.
