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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…

Chaotic Dynamics · Physics 2025-07-21 Serge Massar

Computing gradients of a cost function is central to design-based optimization and machine learning algorithms. Equilibrium propagation provides an exact method to compute gradients in hardware by exploiting the inherent physical laws. The…

Disordered Systems and Neural Networks · Physics 2025-08-11 Marc Berneman , Daniel Hexner

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…

Machine Learning · Computer Science 2025-12-01 Elon Litman

Recurrent neural networks (RNNs) trained using Equilibrium Propagation (EP), a biologically plausible training algorithm, have demonstrated strong performance in various tasks such as image classification and reinforcement learning.…

Machine Learning · Computer Science 2025-08-21 Yoshimasa Kubo , Jean Erik Delanois , Maxim Bazhenov

Artificial neural networks, one of the most successful approaches to supervised learning, were originally inspired by their biological counterparts. However, the most successful learning algorithm for artificial neural networks,…

Machine Learning · Computer Science 2020-06-18 Matilde Tristany Farinha , Sérgio Pequito , Pedro A. Santos , Mário A. T. Figueiredo

Finding spike-based learning algorithms that can be implemented within the local constraints of neuromorphic systems, while achieving high accuracy, remains a formidable challenge. Equilibrium Propagation is a promising alternative to…

Neural and Evolutionary Computing · Computer Science 2021-02-18 Erwann Martin , Maxence Ernoult , Jérémie Laydevant , Shuai Li , Damien Querlioz , Teodora Petrisor , Julie Grollier

Expectation propagation (EP) is a deterministic approximation algorithm that is often used to perform approximate Bayesian parameter learning. EP approximates the full intractable posterior distribution through a set of local approximations…

Machine Learning · Statistics 2015-11-19 Yingzhen Li , Jose Miguel Hernandez-Lobato , Richard E. Turner

We introduce Gradient Equilibrium Propagation (GradEP), a mechanism that extends Equilibrium Propagation (EP) to train energy gradients rather than energy minima, enabling EP to be applied to tasks where the learning objective depends on…

Disordered Systems and Neural Networks · Physics 2026-05-26 Alex Gower

Efficient feature selection from high-dimensional datasets is a very important challenge in many data-driven fields of science and engineering. We introduce a statistical mechanics inspired strategy that addresses the problem of sparse…

Machine Learning · Statistics 2021-04-07 Alfredo Braunstein , Thomas Gueudré , Andrea Pagnani , Mirko Pieropan

We introduce a principled method to train end-to-end analog neural networks by stochastic gradient descent. In these analog neural networks, the weights to be adjusted are implemented by the conductances of programmable resistive devices…

Neural and Evolutionary Computing · Computer Science 2020-06-11 Jack Kendall , Ross Pantone , Kalpana Manickavasagam , Yoshua Bengio , Benjamin Scellier

Expectation Propagation (Minka, 2001) is a widely successful algorithm for variational inference. EP is an iterative algorithm used to approximate complicated distributions, typically to find a Gaussian approximation of posterior…

Computation · Statistics 2016-04-01 Guillaume Dehaene , Simon Barthelmé

Exact inference in the linear regression model with spike and slab priors is often intractable. Expectation propagation (EP) can be used for approximate inference. However, the regular sequential form of EP (R-EP) may fail to converge in…

Machine Learning · Statistics 2011-12-13 José Miguel Hernández-Lobato , Daniel Hernández-Lobato

Recurrent Backpropagation and Equilibrium Propagation are supervised learning algorithms for fixed point recurrent neural networks which differ in their second phase. In the first phase, both algorithms converge to a fixed point which…

Machine Learning · Computer Science 2018-05-24 Benjamin Scellier , Yoshua Bengio

Bayesian inference is a popular method to build learning algorithms but it is hampered by the fact that its key object, the posterior probability distribution, is often uncomputable. Expectation Propagation (EP) (Minka (2001)) is a popular…

Machine Learning · Statistics 2016-12-16 Guillaume P. Dehaene

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…

Neural and Evolutionary Computing · Computer Science 2026-03-19 Sankar Vinayak Elayedam , Gopalakrishnan Srinivasan

As deep learning applications continue to deploy increasingly large artificial neural networks, the associated high energy demands are creating a need for alternative neuromorphic approaches. Optics and photonics are particularly compelling…

Optics · Physics 2025-08-19 Nicola Dal Cin , Florian Marquardt , Clara C. Wanjura

Large multilayer neural networks trained with backpropagation have recently achieved state-of-the-art results in a wide range of problems. However, using backprop for neural net learning still has some disadvantages, e.g., having to tune a…

Machine Learning · Statistics 2015-07-16 José Miguel Hernández-Lobato , Ryan P. Adams

This tutorial paper surveys provably optimal alternatives to end-to-end backpropagation (E2EBP) -- the de facto standard for training deep architectures. Modular training refers to strictly local training without both the forward and the…

Machine Learning · Computer Science 2022-08-10 Shiyu Duan , Jose C. Principe

Binary Neural Networks (BNNs), which constrain both weights and activations to binary values, offer substantial reductions in computational complexity, memory footprint, and energy consumption. These advantages make them particularly well…

Machine Learning · Computer Science 2026-02-18 Luca Colombo , Fabrizio Pittorino , Daniele Zambon , Carlo Baldassi , Manuel Roveri , Cesare Alippi

Reinforcement learning (RL) has enabled robust quadruped locomotion over complex terrain, but most learned controllers are trained offline with backpropagation in massively parallel simulation and deployed as fixed policies, limiting…

Neural and Evolutionary Computing · Computer Science 2026-05-12 Zhuangyu Han , Abhronil Sengupta