Unlocked Backpropagation using Wave Scattering

TL;DR

This paper introduces a wave scattering-based reformulation of the maximum principle in optimal control, leading to new 'unlocked' algorithms for neural network training that avoid the traditional backpropagation lock.

Contribution

It presents a novel physical analogy using wave scattering to reformulate the maximum principle, enabling fully unlocked neural network training algorithms.

Findings

Derived a hyperbolic initial value problem from the maximum principle.

Introduced counter-propagating wave variables with finite speed.

Developed a family of algorithms suitable for neural network training.

Abstract

Both the backpropagation algorithm in machine learning and the maximum principle in optimal control theory are posed as a two-point boundary problem, resulting in a "forward-backward" lock. We derive a reformulation of the maximum principle in optimal control theory as a hyperbolic initial value problem by introducing an additional "optimization time" dimension. We introduce counter-propagating wave variables with finite propagation speed and recast the optimization problem in terms of scattering relationships between them. This relaxation of the original problem can be interpreted as a physical system that equilibrates and changes its physical properties in order to minimize reflections. We discretize this continuum theory to derive a family of fully unlocked algorithms suitable for training neural networks. Different parameter dynamics, including gradient descent, can be derived by demanding dissipation and minimization of reflections at parameter ports. These results also imply that any physical substrate that supports the scattering and dissipation of waves can be interpreted as solving an optimization problem.