Efficient and Accurate Gradients for Neural SDEs
Authors: Patrick Kidger, James Foster, Xuechen (Chen) Li, Terry Lyons

Neural SDEs combine many of the best qualities of both RNNs and SDEs, and as such are a natural choice for modelling many types of temporal dynamics. They offer memory efficiency, high-capacity function approximation, and strong priors on model space. Neural SDEs may be trained as VAEs or as GANs; in either case it is necessary to backpropagate through the SDE solve. In particular this may be done by constructing a backwards-in-time SDE whose solution is the desired parameter gradients. However, this has previously suffered from severe speed and accuracy issues, due to high computational complexity, numerical errors in the SDE solve, and the cost of reconstructing Brownian motion. Here, we make several technical innovations to overcome these issues. First, we introduce the \textit{reversible Heun method}: a new SDE solver that is algebraically reversible -- which reduces numerical gradient errors to almost zero, improving several test metrics by substantial margins over state-of-the-art. Moreover it requires half as many function evaluations as comparable solvers, giving up to a $1.98\times$ speedup. Next, we introduce the \textit{Brownian interval}. This is a new and computationally efficient way of exactly sampling \textit{and reconstructing} Brownian motion; this is in contrast to previous reconstruction techniques that are both approximate and relatively slow. This gives up to a $10.6\times$ speed improvement over previous techniques. After that, when specifically training Neural SDEs as GANs (Kidger et al. 2021), we demonstrate how SDE-GANs may be trained through careful weight clipping and choice of activation function. This reduces computational cost (giving up to a $1.87\times$ speedup), and removes the truncation errors of the double adjoint required for gradient penalty, substantially improving several test metrics. Altogether these techniques offer substantial improvements over the state-of-the-art, with respect to both training speed and with respect to classification, prediction, and MMD test metrics. We have contributed implementations of all of our techniques to the \texttt{torchsde} library to help facilitate their adoption.