Equilibrium Propagation Without Limits

2025-11-26

AI Summary

Equilibrium Propagation (EP) suffers from a fundamental limitation—its reliance on infinitesimal perturbations, which hinders effective learning under strong error signals. Method: We propose a finite-nudging framework that dispenses with infinitesimal approximations by modeling network states as Gibbs–Boltzmann distributions. Leveraging statistical physics tools—including Helmholtz free energy differences, path integrals, and loss-energy covariance—we derive an exact gradient estimator valid for arbitrary finite perturbations, and rigorously prove that contrastive Hebbian updates constitute an unbiased estimator of this gradient. Contribution/Results: This work establishes, for the first time, a local credit assignment theory for EP without requiring infinitesimal approximations or convexity assumptions. It yields a generalized EP algorithm that significantly improves learning efficiency and robustness under strong supervised signals, thereby resolving long-standing theoretical and practical bottlenecks in EP-based learning.

Technology Category

• Machine Learning: Learning Theory
• Natural Language Processing: Learning & Optimization for NLP
• Search and Optimization: Learning to Search

Application Category

• Economics, Online Markets and Human Computation: Social networks and social learning
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• Establishes finite-nudge local credit assignment for Equilibrium Propagation
• Proves gradient of Helmholtz free energy difference equals expected energy derivatives
• Derives generalized EP algorithm using loss-energy covariance path integrals

Innovation

• Gibbs-Boltzmann distributions model network states
• Contrastive Hebbian Learning as exact gradient estimator
• Generalized EP algorithm uses loss-energy covariance path integral