Equilibrium Propagation for Non-Conservative Systems
AsymEP and Dyadic EP recover exact gradients in non-conservative systems; AsymEP reaches 94.9% on highly asymmetric MNIST networks.
Key Findings
Methodology
The paper introduces Asymmetric EP (AsymEP) and Dyadic EP. A differentiable force field F(x,θ,u) relaxes to F(x0,θ,u)=0. AsymEP adds −2AJ(x0)(x−x0) during nudged learning, with AJ=(JF−JFᵀ)/2, so the modified Jacobian equals JFᵀ at equilibrium. Dyadic EP instead doubles the state to (z,z′) and defines H=−(z−z′)ᵀF((z+z′)/2,θ), yielding an equivalent variational construction.
Key Results
- On MNIST with a single hidden layer, AsymEP consistently outperformed standard EP and Vector Field (VF) after both one and 20 epochs, while inducing useful asymmetry. With 50 hidden neurons, EP preserved initial symmetry, whereas VF and AsymEP evolved non-zero structural asymmetry.
- Under controlled structural asymmetry, all-parameter AsymEP achieved 93.8±0.4% at rstr=0 and 94.9±0.2% at rstr=0.875, and remained trainable at rstr=1. VF fell to 5±3% or 8±4% at rstr=1, near chance.
- Theoretical analysis shows that VF uses JF⁻¹∂C/∂x, while the exact gradient requires (JFᵀ)⁻¹∂C/∂x. AsymEP supplies the missing transpose and is shown to agree with Backpropagation Through Time in the relevant limit.
Significance
The work removes the main restriction of standard Equilibrium Propagation: reliance on a conservative energy function and reciprocal connectivity. Exact credit assignment is extended to feedforward networks, non-reciprocal neural circuits, and physical systems maintained far from thermodynamic equilibrium. For neuroscience, it supports learning through intrinsic relaxation rather than a distinct backward circuit. For engineering, it creates a principled training route for optical, memristive, electrical, oscillator-based, and active-material substrates.
Technical Contribution
The central construction decomposes JF into symmetric SJ=(JF+JFᵀ)/2 and antisymmetric AJ=(JF−JFᵀ)/2 parts. Adding −2AJ(x−x0) makes the learning-phase Jacobian JFA(x0)=JFᵀ, exactly matching implicit differentiation. Dyadic EP embeds arbitrary force fields in a 2n-dimensional energy system and enables parallel positive and negative nudging. AsymEP is derived as its first-order projection and avoids physical time reversal.
Novelty
Unlike VF, which is exact only for conservative fields, and symmetry-penalty methods that force a system toward conservatism, this paper provides a general exact-gradient extension of EP for arbitrary differentiable non-conservative dynamics. Its fundamental novelty is a local antisymmetric-Jacobian correction, complemented by an independent doubled-state variational derivation.
Limitations
- The method assumes an invertible, sufficiently stable Jacobian near the free equilibrium. Strong nonlinearities, oscillations, multistability, or noise may prevent convergence; measuring and implementing Jacobian-based corrections can also impose substantial hardware overhead.
- Experiments focus mainly on one-hidden-layer continuous Hopfield networks. Although CIFAR-10 convolutional feedforward validation is reported, the supplied text gives no complete numerical table, so scalability to modern deep architectures remains uncertain.
Future Work
Future work should test deep convolutional and sparse architectures, stochastic and time-varying inputs, and online implementations that avoid storing x0 explicitly. System-level evaluations should compare energy, latency, robustness, and wiring costs on real memristive, optical, electrical, and oscillator hardware.
AI Executive Summary
Modern neural networks rely on backpropagation, which requires a distinct backward pathway, long-range error transmission, and gradient storage. These requirements are difficult to reproduce in biology and analog hardware. Equilibrium Propagation (EP) offers an alternative: it compares a free stationary state with a nearby target-nudged state, allowing local continuous dynamics to estimate gradients. Yet conventional EP assumes an energy function and therefore reciprocal interactions, excluding feedforward networks and many driven non-equilibrium systems.
Scurria and colleagues introduce Asymmetric EP (AsymEP) and Dyadic EP. AsymEP computes the free-equilibrium Jacobian JF, extracts its antisymmetric part AJ=(JF−JFᵀ)/2, and adds −2AJ(x−x0) during nudged relaxation. The resulting Jacobian becomes JFᵀ, exactly the operator required by implicit differentiation. Dyadic EP reaches the same goal by duplicating the state as (z,z′) and defining a variational energy H=−(z−z′)ᵀF((z+z′)/2).
On MNIST, AsymEP learned faster and performed better than EP and VF. With fixed structural asymmetry, it reached 94.9±0.2% at rstr=0.875 and remained trainable at rstr=1, whereas VF dropped to 5±3% or 8±4%. The framework therefore brings exact stationary-state credit assignment to non-reciprocal networks and potential physical substrates. Important challenges remain: stability, Jacobian implementation cost, large-scale deep learning, and full hardware validation.
Deep Analysis
Background
EP, introduced by Scellier and Bengio, estimates gradients from free and nudged equilibria. Later work addressed temporal locality, stochastic dynamics, clamping, and physical substrates. However, standard EP uses F=−∂E/∂x, which imposes reciprocal interactions. Feedforward networks, active materials, nonlinear optical systems, and other driven systems are intrinsically non-conservative and therefore fall outside the original theory.
Core Problem
The objective is to compute dC(x0,y)/dθ at a stationary point F(x0,θ,u)=0. VF replaces the exact post-synaptic factor (JFᵀ)⁻¹∂C/∂x with JF⁻¹∂C/∂x. The two coincide only when JF is symmetric. Increasing antisymmetry therefore corrupts learning, and in the purely antisymmetric limit the VF update can reverse the true gradient.
Innovation
- �� AsymEP: adds −2AJ(x−x0) during nudged relaxation so the effective Jacobian equals JFᵀ.
- �� Dyadic EP: doubles the state to (z,z′), defining an energy over the extended space; the mean follows inference and the difference carries error.
- �� Theory: AsymEP is the first-order projection of Dyadic EP and agrees with Backpropagation Through Time gradients in the appropriate limit.
Methodology
- �� Free phase: integrate dx/dt=F(x,θ) until x0 is reached.
- �� Jacobian decomposition: compute JF=∂F/∂x and AJ=(JF−JFᵀ)/2.
- �� Nudged phase: integrate dx/dt=F−β∂C/∂x−2AJ(x−x0) with ±β to obtain xA+β and xA−β.
- �� Update: use Δθ=ε(∂F/∂θ)ᵀ(xA+β−xA−β)/(2β).
- �� Dyadic route: define H=−(z−z′)ᵀF((z+z′)/2,θ), D=C((z+z′)/2), and evolve z and z′ under HT=H+βD.
Experiments
Datasets are MNIST, Fashion-MNIST, and CIFAR-10. Inputs are min–max normalized to [−1,1], with one-hot targets in {−1,1}. The main model is a one-hidden-layer continuous Hopfield network with tanh activation; Jin and Jdyn are trained. AsymEP is compared with standard EP, VF, and Backpropagation across hidden width, training duration, fixed structural asymmetry rstr, and input-only versus all-parameter training.
Results
With symmetric initialization, AsymEP outperformed EP and VF after one and 20 epochs on MNIST. Under fixed asymmetry, all-parameter AsymEP achieved 93.8±0.4% at rstr=0 and 94.9±0.2% at rstr=0.875, and still learned at rstr=1. VF reached only 5±3% or 8±4% at rstr=1, with collapse beginning around rstr≈0.5–0.8. Appendix experiments also report reduced oscillations and improved stability for AsymEP.
Applications
Potential applications include trainable feedforward networks, non-reciprocal optical and optoelectronic systems, memristor crossbars, coupled oscillators, and active materials. The substrate must provide differentiable local dynamics, stable relaxation, and learning-phase feedback or an equivalent antisymmetric correction. If the free state x0 can be retained, AsymEP requires two nudged relaxations for an update.
Limitations & Outlook
The principal limitations are stability and scale. The method requires an invertible Jacobian, convergent equilibria, and implementable antisymmetric corrections; noise and strong nonlinearities may violate these assumptions. Jacobian access can create memory, computation, or wiring costs, and feedback links may be needed during learning. The experiments remain modest in scale, while real-hardware energy, latency, and complete CIFAR-10 comparisons require further study.
Plain Language Accessible to non-experts
Imagine a factory that adjusts itself after receiving raw materials. It runs until every station and conveyor belt settles into a stable arrangement: this is the free state. Now place a label beside the output saying what the correct product should be. The factory receives a small hint and settles again. By comparing the two arrangements, we can infer which knobs and machines should change.
The difficulty is that some factories have one-way conveyors: station A affects B, but B does not affect A in the same way. Older methods pretend the factory is balanced in both directions, so their advice becomes unreliable as one-way behavior grows. AsymEP first measures the imbalance and adds a local correction during learning, making the comparison point toward the genuinely useful changes. Dyadic EP makes a second copy of the factory and lets both copies run together; their difference becomes the adjustment signal.
On MNIST, AsymEP retained about 94.9% accuracy under strong asymmetry, while VF fell to roughly 5–8% when the connections were completely one-sided. The lesson is that a physical system may learn from its own settling behavior without a separate reverse production line.
ELI14 Explained like you're 14
Picture a video game robot with lots of sliders. Its job is to recognize handwritten numbers. First, let it play by itself until its movements settle. Then give it a tiny clue—“the answer was 7”—and let it settle again. Comparing the two runs tells us which sliders should move and in what direction.
But imagine the robot’s roads are one-way: road A can affect road B, while B cannot push back equally. Ordinary EP acts as if every road works both ways, so its advice gets worse as the roads become more one-sided. AsymEP measures this imbalance and adds a correction while training. Dyadic EP takes a different approach: it makes a second robot, runs both versions together, and uses their disagreement as the learning signal.
In the MNIST tests, AsymEP stayed strong even when the connections were very asymmetric, reaching 94.9% accuracy at one difficult setting. VF dropped to about 5–8% when the connections were completely antisymmetric—basically random guessing. Pretty impressive, right? The robot can learn without a special reverse road, as long as its overall motion remains stable.
There is a catch: huge, noisy, or constantly wobbling robots may be harder to train. The researchers still need to test much larger networks and real hardware. But the idea is exciting: instead of sending a separate error message backward, the machine’s own settling process can help it discover what to fix!
Glossary
Equilibrium Propagation
A learning framework that uses stationary states for both inference and parameter updates. It estimates gradients from the difference between free and target-nudged equilibria.
The paper generalizes EP beyond conservative systems.
Non-conservative system
A dynamical system not generated by a single scalar energy, often containing non-reciprocal or antisymmetric interactions. Its stationary state need not be an energy minimum.
Feedforward networks and driven physical systems are target applications.
Asymmetric EP
An EP algorithm that adds an antisymmetric-Jacobian correction during the nudged phase. This makes the learning dynamics produce the transpose-Jacobian sensitivity required for the exact gradient.
The main algorithm uses Δθ=ε(∂F/∂θ)ᵀΔx/(2β).
Dyadic EP
A variational EP formulation that duplicates the state into two coupled variables and uses their difference as an error signal. It maps arbitrary dynamics into an augmented energy system.
It provides a parallel and principled counterpart to AsymEP.
Structural asymmetry ratio
rstr=||(Jdynᵀ−Jdyn)/2||F/||Jdyn||F, measuring antisymmetry in the recurrent connectivity matrix. It is distinct from state-dependent Jacobian asymmetry.
The experiments use rstr to control non-conservativity.
Open Questions Unanswered questions from this research
- 1 How to approximate or implement JF cheaply in million-parameter networks without adding prohibitive feedback wiring remains unresolved.
- 2 The behavior of AsymEP under strong noise, time-varying inputs, multistability, and unstable oscillations needs a fuller stochastic and dynamical-systems theory.
- 3 Energy, latency, robustness, and accuracy advantages on real chips are not yet fully quantified, especially relative to conventional backpropagation.
Applications
Immediate Applications
Analog neuromorphic training
Memristor arrays, adaptive circuits, or coupled oscillators could implement free and ±β nudged phases and update weights from local state differences. The hardware must retain x0 and support feedback or an equivalent correction during learning.
Non-reciprocal feedforward classifiers
AsymEP can train input, hidden, and output parameters in one-way or asymmetric networks, addressing the VF limitation that largely restricts learning to the final layer in purely feedforward structures.
Long-term Vision
Trainable physical intelligence
Optical, optoelectronic, active-material, and far-from-equilibrium systems could become adaptive computing media rather than fixed accelerators. Major obstacles include stability, device noise, feedback wiring, calibration, and large-scale control.
Abstract
Equilibrium Propagation (EP) is a physics-inspired learning algorithm that uses stationary states of a dynamical system both for inference and learning. In its original formulation it is limited to conservative systems, $\textit{i.e.}$ to dynamics which derive from an energy function. Given their applications, it is important to extend EP to non-conservative systems, $\textit{i.e.}$ systems with non-reciprocal interactions. Previous attempts to generalize EP to such systems failed to compute the exact gradient of the cost function. Here we propose a framework that extends EP to arbitrary non-conservative systems, including feedforward networks. We keep the key property of equilibrium propagation, namely the use of stationary states both for inference and learning. However, we modify the dynamics in the learning phase by a term proportional to the non-reciprocal part of the interaction so as to obtain the exact gradient of the cost function. This algorithm can also be derived using a variational formulation that generates the learning dynamics through an energy function defined over an augmented state space. Numerical experiments show that this algorithm achieves better performance and learns faster than previous proposals.