Unlocked Backpropagation using Wave Scattering

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.

💡 Research Summary

The paper tackles a fundamental limitation of modern deep learning: the forward‑backward lock inherent in back‑propagation, which forces a strict sequential dependency between the forward pass (computing activations) and the backward pass (propagating gradients). The authors observe that this lock mirrors the two‑point boundary value formulation of the Pontryagin Maximum Principle (PMP) in optimal control, where the state is fixed at the initial time and the costate (adjoint) is fixed at the final time.

To break this lock, they introduce an auxiliary “optimization time” τ, lifting the problem from a one‑dimensional trajectory on