Scalable Gradients for Stochastic Differential Equations

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Xuechen Li, Ting-Kam Leonard Wong, Ricky T. Q. Chen, David Duvenaud ;
Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:3870-3882, 2020.

Abstract

The adjoint sensitivity method scalably computes gradients of solutions to ordinary differential equations. We generalize this method to stochastic differential equations, allowing time-efficient and constant-memory computation of gradients with high-order adaptive solvers. Specifically, we derive a stochastic differentialequation whose solution is the gradient, a memory-efficient algorithm for cachingnoise, and conditions under which numerical solutions converge. In addition, we combine our method with gradient-based stochastic variational inference for latent stochastic differential equations. We use our method to fit stochastic dynamics defined by neural networks, achieving competitive performance ona 50-dimensional motion capture dataset.

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