Imbedding Deep Neural Networks

Andrew Corbett, Dmitry Kangin

Keywords: Neural ODEs, Optimal Control, Deep Neural Networks, Invariant Imbedding

Abstract:

Continuous-depth neural networks, such as Neural ODEs, have refashioned the understanding of residual neural networks in terms of non-linear vector-valued optimal control problems. The common solution is to use the adjoint sensitivity method to replicate a forward-backward pass optimisation problem. We propose a new approach which explicates the network's `depth' as a fundamental variable, thus reducing the problem to a system of forward-facing initial value problems. This new method is based on the principal of `Invariant Imbedding' for which we prove a general solution, applicable to all non-linear, vector-valued optimal control problems with both running and terminal loss.
Our new architectures provide a tangible tool for inspecting the theoretical--and to a great extent unexplained--properties of network depth. They also constitute a resource of discrete implementations of Neural ODEs comparable to classes of imbedded residual neural networks. Through a series of experiments, we show the competitive performance of the proposed architectures for supervised learning and time series prediction.

One-sentence Summary: Invariant imbedding solution for (Bolza) optimal control problem derived and proved to yield new architectures of imbedded deep neural networks.

Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/arxiv:2202.00113/code)