"Hey, that's not an ODE": Faster ODE Adjoints via Seminorms
This work addresses a computational bottleneck for researchers and practitioners using neural differential equations in tasks like time series and generative modeling, offering a simple code modification for significant speed-ups.
The paper tackled the inefficiency of training neural differential equations via the adjoint method by proposing a new seminorm to reduce unnecessary step rejections, resulting in a median improvement of 40% fewer function evaluations and up to 62% faster training.
Neural differential equations may be trained by backpropagating gradients via the adjoint method, which is another differential equation typically solved using an adaptive-step-size numerical differential equation solver. A proposed step is accepted if its error, \emph{relative to some norm}, is sufficiently small; else it is rejected, the step is shrunk, and the process is repeated. Here, we demonstrate that the particular structure of the adjoint equations makes the usual choices of norm (such as $L^2$) unnecessarily stringent. By replacing it with a more appropriate (semi)norm, fewer steps are unnecessarily rejected and the backpropagation is made faster. This requires only minor code modifications. Experiments on a wide range of tasks -- including time series, generative modeling, and physical control -- demonstrate a median improvement of 40% fewer function evaluations. On some problems we see as much as 62% fewer function evaluations, so that the overall training time is roughly halved.