Venue: ICLR (under review)
Year: 2021
Paper: https://openreview.net/forum?id=yKIAXjkJc2F
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.