Equilibrium Propagation in Energy-Based Models

• Equilibrium Propagation is a learning framework for energy-based models that uses free and nudged phases to locally compute gradients without a separate backward pass.
• It employs a free phase where predictions emerge from system relaxation and a nudged phase where weak supervision shifts state dynamics to reveal gradient information.
• Extensions of EP include bias reduction techniques and adaptations for spiking, quantum, and deep residual architectures, enhancing both biological plausibility and computational performance.

Equilibrium Propagation (EP) is a learning framework for energy-based models in which inference and learning are both implemented by the relaxation dynamics of the same recurrent system. In its original formulation, the network first settles to a free fixed point that encodes the prediction, then to a nearby nudged fixed point obtained by adding a small supervised perturbation, and the contrast between these two equilibria yields the gradient of a well-defined objective. The framework was introduced as a way to bridge energy-based models and backpropagation while avoiding a separate backward circuit, with error derivatives emerging implicitly from the dynamics of the nudged phase (Scellier et al., 2016).

1. Conceptual foundations

EP starts from a convergent recurrent network whose state is driven toward a stationary point by an internal energy. The defining idea is that supervised learning can be implemented by a local perturbation of this equilibrium rather than by an explicit backward pass. In the free phase, the input is clamped and the network relaxes until it produces a prediction. In the nudged phase, the output units are slightly attracted toward the target, or equivalently toward a configuration of lower cost, and this perturbation propagates backward through the hidden units as the network relaxes again. The resulting change in local activities encodes the parameter gradient (Scellier et al., 2016).

This formulation was proposed partly to resolve a long-standing tension between gradient-based learning and biological plausibility. Standard backpropagation requires distinct computations in the forward and backward phases and typically assumes a dedicated backward circuit. EP instead uses the same leaky-integrator neural computation in both phases. The only local difference between the two phases can be whether synaptic changes are enabled. In the original account, this makes it plausible that a mechanism similar to backpropagation could be implemented by physical or biological substrates that naturally relax to equilibria (Scellier et al., 2016).

EP also positions itself relative to earlier contrastive algorithms. It shares similarities with Contrastive Hebbian Learning and Contrastive Divergence because it learns from differences between two nearby states, but it differs in that the contrast is tied to the gradient of a well-defined supervised objective. This distinction is central to the method’s theoretical identity.

2. Mathematical formulation

Let $s$ denote the network state, typically partitioned into hidden variables $h$ and outputs $y$, with inputs $x$ clamped. Let $\theta$ be the trainable parameters and $\hat y$ the target. EP defines an internal energy $E(s,\theta,x)$ and a supervised cost $C(s,\hat y)$, then combines them into the augmented energy

$$F(s,\theta,x,\hat y,\beta)=E(s,\theta,x)+\beta\,C(s,\hat y),$$

where $\beta$ is the nudging strength (Scellier et al., 2016).

The free phase corresponds to $h$0 and follows

$h$1

The nudged phase uses a small $h$2 and follows

$h$3

The free equilibrium $h$4 satisfies

$h$5

and the nudged equilibrium $h$6 satisfies

$h$7

For a Hopfield-style model with symmetric interactions, the original paper uses

$h$8

with symmetric weights $h$9 and supervised quadratic cost

$y$0

In that setting, the output nudging force is

$y$1

so the output layer is only weakly attracted toward the target, and the hidden-layer perturbations arise from the recurrent dynamics rather than from an explicitly imposed backward computation (Scellier et al., 2016).

The training objective is the free-phase cost

$y$2

and EP computes its gradient through the two nearby equilibria:

$y$3

The derivation uses the fixed-point dependence of $y$4 on $y$5 together with symmetry of cross-derivatives. In the deterministic formulation, exact fixed points and symmetric couplings are the main structural assumptions.

3. Local credit assignment and relation to backpropagation

For pairwise energies, EP yields explicitly local synaptic rules. In the symmetric Hopfield form,

$y$6

so the parameter update becomes

$y$7

and similarly

$y$8

The update is therefore a difference of pre-post products measured at two nearby equilibria, which gives EP its standard characterization as a local contrastive Hebbian rule (Scellier et al., 2016).

The original derivation also identifies a temporal interpretation related to spike-timing dependent plasticity. Along the nudged trajectory from $y$9 to $x$0, the tied-weights rule

$x$1

integrates to the same contrastive update. This is one of the main reasons EP is repeatedly presented as biologically plausible.

Its relation to backpropagation becomes explicit after linearization around the free fixed point. In a recurrent multilayer network,

$x$2

so the small activity changes during the nudged phase correspond to backpropagated error derivatives in the linear regime (Scellier et al., 2016). A recurrent interpretation via discrete-time dynamics later made this connection operational: EP’s fixed-point estimator matches the gradient computed by Backpropagation Through Time in the limit of infinitesimal nudging for convergent recurrent networks (Ernoult et al., 2020).

A common misconception is that EP is simply another name for Contrastive Hebbian Learning. The original formulation explicitly distinguishes itself by computing the gradient of a supervised objective defined via local perturbations, rather than contrasting arbitrary free and clamped states. A second misconception is that finite nudging is harmless. In practice, the one-sided estimator has a systematic first-order bias $x$3, whereas a symmetric three-phase estimator using $x$4 and $x$5 reduces the bias to $x$6 (Laborieux et al., 2021). This bias reduction became a decisive step in scaling EP beyond small benchmarks.

4. Implementations and architectural adaptations

The standard training loop clamps the input, relaxes the free dynamics to $x$7, runs a nudged relaxation with small $x$8 to obtain $x$9, then updates parameters from the contrast between the two states. In digital simulations, the state is often updated by discrete-time Euler descent,

$\theta$0

sometimes with clipping for bounded activations. The original experiments on permutation-invariant MNIST used recurrent symmetric multilayer networks with 1, 2, and 3 hidden layers of 500 units each, discrete-time leaky integrator dynamics, hard-sigmoid activations, minibatches of size 20, Glorot-Bengio initialization, persistent states to speed inference, and brief nudged phases; training error reached $\theta$1 for all tested architectures, with test error between $\theta$2 and $\theta$3 (Scellier et al., 2016).

A major practical issue is the long relaxation time required by deep recurrent networks. The discrete-time reformulation of EP introduced a primitive function $\theta$4 whose gradient defines the state update, simplifying implementation and extending the method to convolutional neural networks. The same work proposed Continual Equilibrium Propagation, in which weights are adjusted continually in the second phase using only local information in space and time; under slow synaptic changes and small nudging, the per-step neuron and synapse updates follow the gradients of Backpropagation Through Time (Ernoult et al., 2020).

Bias reduction was the next major scaling step. The three-phase symmetric estimator made it possible to train deep ConvNets on CIFAR-10, including architectures with distinct forward and backward connections. With bidirectional weights and a softmax readout trained by cross-entropy, symmetric EP reached $\theta$5 test error and $\theta$6 train error on CIFAR-10, close to a Backpropagation Through Time baseline at $\theta$7 test error and $\theta$8 train error (Laborieux et al., 2021).

EP has also been extended to sequence classification by combining convergent recurrent networks with modern Hopfield attention. In that construction, the full token sequence is stored as a static memory, preserving EP’s requirement of static input across phases while allowing sequence-aware retrieval. Reported accuracies were $\theta$9 on IMDB and $\hat y$0 on SNLI (Bal et al., 2022).

In spiking settings, EP has been scaled to convolutional convergent recurrent networks by combining predictive coding with spiking communication. A key architectural result is that max pooling and its inverse introduce a mismatch in spiking EP, whereas average pooling with nearest-neighbor upsampling preserves the adjoint-like relationship needed for accurate gradient estimation. On MNIST and FashionMNIST, this yielded test errors of $\hat y$1 and $\hat y$2, respectively, comparable to convergent RNNs and SNNs trained by Backpropagation Through Time (Lin et al., 2024).

5. Generalizations of the EP principle

Several later developments relax the assumptions of the original fixed-point, infinitesimal-nudge, conservative formulation. One line of work extends EP to holomorphic networks with complex-valued states and parameters. Under holomorphicity and adiabatic time-scale separation, the gradient can be computed exactly for finite-amplitude teaching signals as the first Fourier coefficient of a local observable measured during a continuous oscillation $\hat y$3, eliminating the need for separate free and nudged phases (Laborieux et al., 2022).

Another line reframes EP statistically rather than deterministically. In stochastic EP, Langevin dynamics,

$\hat y$4

induces a Boltzmann distribution $\hat y$5, and the gradient becomes a contrast between expectations under the nudged and free distributions. A closely related thermal formulation shows that finite temperature permits a covariance estimator in the free phase,

$\hat y$6

so thermal fluctuations allow training without output clamping (Massar et al., 2024).

Finite-nudge theory has also been strengthened. A Gibbs–Boltzmann treatment at nonzero temperature proves that the gradient of the difference in Helmholtz free energy between nudged and free phases is exactly the difference in expected local energy derivatives, validating a finite-nudge contrastive update for arbitrary nudging strength and leading to a generalized path-integral EP algorithm based on loss–energy covariances (Litman, 27 Nov 2025). This directly addresses the small-$\hat y$7 limitation of classical EP.

Quantum generalizations replace the classical energy with a Hamiltonian and the free equilibrium with an eigenstate, typically the ground state. In this setting, the mean energy functional plays the role of the minimized quantity, and the gradient is estimated from differences in expectation values of $\hat y$8 between free and nudged eigenstates. This was worked out both in a general quantum and thermal treatment and in an explicit framework for quantum systems such as the transverse-field Ising model and quantum harmonic oscillator networks (Massar et al., 2024, Scellier, 2024).

Perhaps the sharpest departure from the original framework concerns non-conservative systems. Classical EP assumes reciprocity, since the dynamics must derive from an energy. A later extension introduced Asymmetric EP and Dyadic EP for arbitrary differentiable vector fields, including feedforward networks, by modifying the learning-phase dynamics with a term proportional to the antisymmetric part of the Jacobian. In the $\hat y$9 limit this recovers the exact gradient while preserving the core EP principle that both inference and learning rely on stationary states (Scurria et al., 3 Feb 2026).

The same conceptual move has been pushed into trajectory optimization. For Lagrangian systems, EP can be formulated over action-extremizing trajectories rather than static fixed points; with periodic boundary conditions or fixed endpoints, two nudged trajectories suffice to estimate gradients from time integrals of conjugate variables (Massar, 12 May 2025). For linear damped periodic systems such as RLC networks or damped Newtonian mechanics, an effective action over periodic steady states restores an exact EP-style gradient rule based on free and nudged responses to periodic forcing (Berneman et al., 25 Jun 2025).

6. Empirical record, hardware implications, and limitations

EP’s empirical record is now broader than the original Hopfield-like setting. Beyond the early MNIST results, architectural work on deeper energy-based networks introduced the Hopfield-Resnet family, where residual connections and clipped $E(s,\theta,x)$0 improve the conditioning of the energy landscape and permit substantially deeper models. Hopfield-Resnet13 achieved $E(s,\theta,x)$1 accuracy on CIFAR-10, approximately $E(s,\theta,x)$2 higher than the previous best EP result and comparable to ResNet13 trained using backpropagation at $E(s,\theta,x)$3 (P et al., 30 Sep 2025). A different brain-inspired route, the Feedback-regulated REsidual recurrent neural network, uses feedback regulation to reduce the spectral radius and residual connections to alleviate vanishing gradients; on MNIST, a convolutional FRE-RNN trained with EP reached $E(s,\theta,x)$4 test accuracy, compared with $E(s,\theta,x)$5 for the cited P-EP baseline, while substantially reducing the required iteration counts (Liu et al., 5 Aug 2025).

Hardware-oriented studies emphasize locality and reduced data movement. In resistive networks trained in situ with EP, measurement uncertainty was modeled as additive post-activation noise at equilibrium readout. Deep multilayer architectures remained trainable up to a critical uncertainty $E(s,\theta,x)$6 for a $E(s,\theta,x)$7–$E(s,\theta,x)$8–$E(s,\theta,x)$9 resistive network, independent of whether the dataset was MNIST, KMNIST, or FashionMNIST. Moderate uncertainty below this limit improved both convergence and performance, with optimal accuracies of $C(s,\hat y)$0 on MNIST, $C(s,\hat y)$1 on KMNIST, and $C(s,\hat y)$2 on FashionMNIST; averaging over $C(s,\hat y)$3 independent samples reduced effective uncertainty as $C(s,\hat y)$4 (Peters et al., 28 Mar 2025).

Oscillator hardware provides another concrete substrate. On oscillator Ising machines, EP is implemented directly in the device dynamics, with local updates expressed in terms of cosine phase observables at free and nudged equilibria. Reported accuracies were $C(s,\hat y)$5 on MNIST and $C(s,\hat y)$6 on Fashion-MNIST, alongside robustness to Gaussian phase noise and to modest quantization, including $C(s,\hat y)$7-bit parameters and $C(s,\hat y)$8-bit phase detection (Gower, 14 Oct 2025). Closely related work on phase-only and amplitude–phase oscillator networks showed that EP can train couplings while simultaneously promoting synchronization under realistic frequency dispersion; a synchronized Kuramoto network reached $C(s,\hat y)$9 test accuracy on MNIST, and an amplitude–phase model reached $$F(s,\theta,x,\hat y,\beta)=E(s,\theta,x)+\beta\,C(s,\hat y),$$0 (Rageau et al., 16 Apr 2025).

Stability of the relaxation itself has become a technical topic. Introducing neuron-specific time steps, or heterogeneous time constants, leaves the fixed points unchanged but improves numerical stability by acting as a diagonal preconditioner. In a single-hidden-layer EP network with $$F(s,\theta,x,\hat y,\beta)=E(s,\theta,x)+\beta\,C(s,\hat y),$$1 hidden units trained on MNIST, KMNIST, and Fashion-MNIST, heterogeneous time constants preserved MNIST performance around $$F(s,\theta,x,\hat y,\beta)=E(s,\theta,x)+\beta\,C(s,\hat y),$$2–$$F(s,\theta,x,\hat y,\beta)=E(s,\theta,x)+\beta\,C(s,\hat y),$$3 while producing modest gains on KMNIST and Fashion-MNIST relative to a scalar time-step baseline (Kubo et al., 3 Mar 2026).

The central limitations remain recognizable from the original proposal. Classical EP requires symmetric weights, convergence to fixed points, and sufficiently small perturbations that remain in the same energy basin. On digital hardware, the free phase can be slow, especially as depth increases. Finite $$F(s,\theta,x,\hat y,\beta)=E(s,\theta,x)+\beta\,C(s,\hat y),$$4 introduces estimator bias unless symmetric or more elaborate variants are used. Some architectures, especially deep or asymmetric ones, need auxiliary design choices such as residual links, pooling constraints, feedback regulation, or modified learning dynamics to remain trainable. Later work suggests that these are not merely engineering details: they delimit the extent to which EP retains exact gradient status, local plasticity, and physical realizability across conservative, stochastic, quantum, non-conservative, and dynamical regimes.

Taken together, these developments define EP less as a single algorithm than as a family of learning principles centered on one claim: parameter gradients can be extracted from the response of a dynamical system to weak supervision, using only local contrasts between nearby operating states. In the original energy-based setting this claim is exact in the infinitesimal-nudge limit (Scellier et al., 2016); subsequent work has expanded the same principle to finite nudges, noise, oscillations, spiking communication, deep residual architectures, periodic dynamics, non-conservative flows, and quantum systems (Laborieux et al., 2022, Litman, 27 Nov 2025, Scurria et al., 3 Feb 2026, Scellier, 2024).