Lagrangian-based Equilibrium Propagation: generalisation to arbitrary boundary conditions & equivalence with Hamiltonian Echo Learning

TL;DR

This paper introduces GLEP, a generalized variational framework for equilibrium propagation with time-varying inputs, and shows that Hamiltonian Echo Learning is a special case inheriting desirable hardware-friendly properties.

Contribution

The work extends equilibrium propagation to dynamic inputs via GLEP and establishes its connection to Hamiltonian Echo Learning, highlighting practical algorithms for energy-based models.

Findings

GLEP generalizes EP to time-varying inputs with boundary condition considerations.

Hamiltonian Echo Learning is derived as a special case of GLEP.

HEL inherits properties like forward-only operation and local learning.

Abstract

Equilibrium Propagation (EP) is a learning algorithm for training Energy-based Models (EBMs) on static inputs which leverages the variational description of their fixed points. Extending EP to time-varying inputs is a challenging problem, as the variational description must apply to the entire system trajectory rather than just fixed points, and careful consideration of boundary conditions becomes essential. In this work, we present Generalized Lagrangian Equilibrium Propagation (GLEP), which extends the variational formulation of EP to time-varying inputs. We demonstrate that GLEP yields different learning algorithms depending on the boundary conditions of the system, many of which are impractical for implementation. We then show that Hamiltonian Echo Learning (HEL) -- which includes the recently proposed Recurrent HEL (RHEL) and the earlier known Hamiltonian Echo Backpropagation (HEB) algorithms -- can be derived as a special case of GLEP. Notably, HEL is the only instance of GLEP we found that inherits the properties that make EP a desirable alternative to backpropagation for hardware implementations: it operates in a "forward-only" manner (i.e. using the same system for both inference and learning), it scales efficiently (requiring only two or more passes through the system regardless of model size), and enables local learning.