Discretize-Optimize vs. Optimize-Discretize for Time-Series Regression and Continuous Normalizing Flows
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We compare the discretize-optimize (Disc-Opt) and optimize-discretize (Opt-Disc) approaches for time-series regression and continuous normalizing flows using neural ODEs. Neural ODEs, first described in Chen et al. (2018), are ordinary differential equations (ODEs) with neural network components; these models have competitively solved a variety of machine learning applications. Training a neural ODE can be phrased as an optimal control problem where the neural network weights are the controls and the hidden features are the states. Every iteration of gradient-based training involves solving an ODE forward in time and another backward in time, which can require large amounts of computation, time, and memory. Gholami et al. (2019) compared the Opt-Disc and Disc-Opt approaches for neural ODEs arising as continuous limits of residual neural networks used in image classification tasks. Their findings suggest that Disc-Opt achieves preferable performance due to the guaranteed accuracy of gradients. In this paper, we extend this comparison to neural ODEs applied to time-series regression and continuous normalizing flows (CNFs). Time-series regression and CNFs differ from classification in that the actual ODE model is needed in the prediction and inference phase, respectively. Meaningful models must also satisfy additional requirements, e.g., the invertibility of the CNF. As the continuous model satisfies these requirements by design, Opt-Disc approaches may appear advantageous. Through our numerical experiments, we demonstrate that with careful numerical treatment, Disc-Opt methods can achieve similar performance as Opt-Disc at inference with drastically reduced training costs. Disc-Opt reduced costs in six out of seven separate problems with training time reduction ranging from 39 one case, Disc-Opt reduced training from nine days to less than one day.
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