How to train your neural ODE: the world of Jacobian and kinetic regularization (2002.02798v3)

Abstract: Training neural ODEs on large datasets has not been tractable due to the necessity of allowing the adaptive numerical ODE solver to refine its step size to very small values. In practice this leads to dynamics equivalent to many hundreds or even thousands of layers. In this paper, we overcome this apparent difficulty by introducing a theoretically-grounded combination of both optimal transport and stability regularizations which encourage neural ODEs to prefer simpler dynamics out of all the dynamics that solve a problem well. Simpler dynamics lead to faster convergence and to fewer discretizations of the solver, considerably decreasing wall-clock time without loss in performance. Our approach allows us to train neural ODE-based generative models to the same performance as the unregularized dynamics, with significant reductions in training time. This brings neural ODEs closer to practical relevance in large-scale applications.

• The paper introduces a dual regularization strategy that combines kinetic regularization and Jacobian regularization to improve the efficiency of neural ODE training.
• It incorporates optimal transport principles to encourage straight-line energy-efficient trajectories and ensure smooth vector field dynamics.
• Empirical results demonstrate up to 2.8x faster training times while maintaining performance, making neural ODEs more practical for large-scale applications.

Neural ODEs: Jacobian and Kinetic Regularization for Efficient Training

In an important advancement for neural ordinary differential equations (neural ODEs), the paper "How to Train Your Neural ODE: the World of Jacobian and Kinetic Regularization" offers fresh insights into augmenting the practicality of neural ODEs in large-scale applications. The research addresses the often prohibitive training time of neural ODEs by introducing a novel method that integrates Jacobian and kinetic regularization.

Key Contributions

The paper focuses on resolving the efficiency bottleneck that occurs when neural ODEs are applied to extensive datasets. Training neural ODEs requires adaptive numerical solvers to ensure precision, often resulting in numerous layer evaluations. A theoretically-grounded regularization approach combining optimal transport with stability constraints is proposed to simplify dynamics without sacrificing solution quality.

Methodological Approach

The authors present a dual regularization strategy:

1. Kinetic Regularization: Drawing from optimal transport principles, this regularizer is positioned as the kinetic energy of the flow field. It encourages paths that follow straight lines, promoting constant speed for particle trajectories, which leads to computationally tractable dynamics.
2. Jacobian Regularization: This method regularizes the Frobenius norm of the Jacobian, ensuring the smoothness of the vector field and limiting the potential numerical errors that may arise from large Jacobian values. The regularization of the Jacobian is crucial for maintaining path regularity and tractable computation across data distributions, providing enhanced generalization capabilities, particularly for off-distribution data.

Empirical Validation

The effectiveness of these regularization strategies was validated using four datasets: MNIST, CIFAR10, downsampled ImageNet, and CelebA-HQ. The regularized neural ODE (RNODE) method demonstrated significant reductions in training time while maintaining performance metrics comparable to unregularized dynamics. For instance, RNODE achieved a log-likelihood similar to baseline FFJORD models but with dramatically reduced training times—approximately 2.8x faster on key datasets.

Implications

The implications of these findings are notable:

• Practical Relevance: By making neural ODEs computationally feasible for large datasets, the paper paves the way for broader application in fields requiring high-dimensional data processing, such as generative modeling and physical simulations.
• Training Flexibility: The ability to use either fixed-grid or adaptive solvers offers flexibility in resource allocation during training, potentially reducing the infrastructure costs associated with deploying large-scale models.
• Theoretical Alignment: By bridging connections between optimal transport theory and neural ODE architectures, the paper enhances our understanding of how continuous normalizing flows can be optimized for deep learning applications.

Future Directions

Looking forward, the proposed regularization frameworks may serve as a foundation for further explorations into other domains reliant on neural ODEs, possibly extending the approach to stochastic differential equations (SDEs) or other complex systems.

Developing finer regularization techniques or hybrid models that balance precision and computation could be promising avenues for continued research. Furthermore, transferring these methodologies into real-world applications could spur innovations in areas like automated control systems, financial modeling, and more complex biological simulations.

In conclusion, the paper marks a pivotal step towards more efficient and practical neural ODEs, reducing the gap between theoretical potential and practical implementation, and thereby expanding the horizon for future research and deployment of sophisticated AI technologies.