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.

Physical systems that naturally perform energy descent offer a direct route to accelerating machine learning. Oscillator Ising Machines (OIMs) exemplify this idea: their GHz-frequency dynamics mirror both the optimization of energy-based models (EBMs) and gradient descent on loss landscapes, while intrinsic noise corresponds to Langevin dynamics - supporting sampling as well as optimization. Equilibrium Propagation (EP) unifies these processes into descent on a single total energy landscape, enabling local learning rules without global backpropagation. We show that EP on OIMs achieves competitive accuracy ($\sim 97.2 \pm 0.1 \%$ on MNIST, $\sim 88.0 \pm 0.1 \%$ on Fashion-MNIST), while maintaining robustness under realistic hardware constraints such as parameter quantization and phase noise. These results establish OIMs as a fast, energy-efficient substrate for neuromorphic learning, and suggest that EBMs - often bottlenecked by conventional processors - may find practical realization on physical hardware whose dynamics directly perform their optimization.

Spiking Neural Networks (SNNs) promise energy-efficient, sparse, biologically inspired computation. Training them with Backpropagation Through Time (BPTT) and surrogate gradients achieves strong performance but remains biologically implausible. Equilibrium Propagation (EP) provides a more local and biologically grounded alternative. However, existing EP frameworks, primarily based on deterministic neurons, either require complex mechanisms to handle discontinuities in spiking dynamics or fail to scale beyond simple visual tasks. Inspired by the stochastic nature of biological spiking mechanism and recent hardware trends, we propose a stochastic EP framework that integrates probabilistic spiking neurons into the EP paradigm. This formulation smoothens the optimization landscape, stabilizes training, and enables scalable learning in deep convolutional spiking convergent recurrent neural networks (CRNNs). W e provide theoretical guarantees showing that the proposed stochastic EP dynamics approximate deterministic EP under mean-field theory, thereby inheriting its underlying theoretical guarantees. The proposed framework narrows the gap to both BPTT-trained SNNs and EP-trained non-spiking CRNNs in vision benchmarks while preserving locality, highlighting stochastic EP as a promising direction for neuromorphic and on-chip learning.

Abstract--Equilibrium propagation has been proposed as a biologically plausible alternative to the backpropagation algorithm. The local nature of gradient computations, combined with the use of convergent RNNs to reach equilibrium states, make this approach well-suited for implementation on neuro-morphic hardware. However, previous studies on equilibrium propagation have been restricted to networks containing only dense layers or relatively small architectures with a few convo-lutional layers followed by a final dense layer . These networks have a significant gap in accuracy compared to similarly sized feedforward networks trained with backpropagation. In this work, we introduce the Hopfield-Resnet architecture, which incorporates residual (or skip) connections in Hopfield networks with clipped ReLU as the activation function. The proposed architectural enhancements enable the training of networks with nearly twice the number of layers reported in prior works.

In its original formulation EP was described in the case of real-time neuronal dynamics, which is computationally costly.

It thus provides a compelling framework for training neuromorphic systems and understanding learning in neurobiology.

We propose a method for training dynamical systems governed by Lagrangian mechanics using Equilibrium Propagation. Our approach extends Equilibrium Propagation - initially developed for energy-based models - to dynamical trajectories by leveraging the principle of action extremization. Training is achieved by gently nudging trajectories toward desired targets and measuring how the variables conjugate to the parameters to be trained respond. This method is particularly suited to systems with periodic boundary conditions or fixed initial and final states, enabling efficient parameter updates without requiring explicit backpropagation through time. In the case of periodic boundary conditions, this approach yields the semiclassical limit of Quantum Equilibrium Propagation. Applications to systems with dissipation are also discussed.