Equilibrium Propagation Without Limits

TL;DR

This paper extends Equilibrium Propagation to finite nudges by modeling network states as Gibbs distributions, providing a rigorous gradient estimation method that does not rely on infinitesimal perturbations, thus broadening its applicability.

Contribution

It introduces a finite-nudge foundation for EP, proving the gradient correspondence with free energy differences and deriving a generalized algorithm based on loss-energy covariances.

Findings

Validates classic Contrastive Hebbian Learning as an exact gradient estimator for finite nudges.

Derives a generalized EP algorithm using path integrals of covariances.

Enables learning with strong error signals beyond infinitesimal approximations.

Abstract

We liberate Equilibrium Propagation (EP) from the limit of infinitesimal perturbations by establishing a finite-nudge foundation for local credit assignment. By modeling network states as Gibbs-Boltzmann distributions rather than deterministic points, we prove that the gradient of the difference in Helmholtz free energy between a nudged and free phase is exactly the difference in expected local energy derivatives. This validates the classic Contrastive Hebbian Learning update as an exact gradient estimator for arbitrary finite nudging, requiring neither infinitesimal approximations nor convexity. Furthermore, we derive a generalized EP algorithm based on the path integral of loss-energy covariances, enabling learning with strong error signals that standard infinitesimal approximations cannot support.