An Ode to an ODE

TL;DR

This paper introduces ODEtoODE, a novel Neural ODE paradigm with time-dependent parameters evolving on a matrix manifold, improving training stability and solving gradient issues in deep networks.

Contribution

The paper proposes a new Neural ODE framework with parameters constrained on a manifold, providing stability, convergence guarantees, and improved performance over existing methods.

Findings

Enhanced training stability and convergence in deep neural networks.

Superior performance in reinforcement learning and supervised tasks.

Provable solutions to gradient vanishing and explosion problems.

Abstract

We present a new paradigm for Neural ODE algorithms, called ODEtoODE, where time-dependent parameters of the main flow evolve according to a matrix flow on the orthogonal group O(d). This nested system of two flows, where the parameter-flow is constrained to lie on the compact manifold, provides stability and effectiveness of training and provably solves the gradient vanishing-explosion problem which is intrinsically related to training deep neural network architectures such as Neural ODEs. Consequently, it leads to better downstream models, as we show on the example of training reinforcement learning policies with evolution strategies, and in the supervised learning setting, by comparing with previous SOTA baselines. We provide strong convergence results for our proposed mechanism that are independent of the depth of the network, supporting our empirical studies. Our results show an intriguing connection between the theory of deep neural networks and the field of matrix flows on compact manifolds.