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Asymmetric Equilibrium Propagation claims to compute the exact cost gradient for arbitrary non-conservative dynamics by adding a local Jacobian-based correction during the nudged phase.

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2026-08-03 04:53 UTC pith:5TWQUUTV

load-bearing objection AEP's antisymmetric-Jacobian correction is a clean, correct fix to the Vector Field estimator; the convergence assumption on the augmented dynamics is the main thing a referee should push on. the 2 major comments →

arxiv 2602.03670 v2 pith:5TWQUUTV submitted 2026-02-03 cs.LG cs.AIcs.NEmath.DSphysics.class-ph

Equilibrium Propagation for Non-Conservative Systems

Antonino Emanuele Scurria , Dimitri Vanden Abeele , Bortolo Matteo Mognetti , Serge Massar This is my paper
classification cs.LG cs.AIcs.NEmath.DSphysics.class-ph
keywords equilibrium propagationnon-conservative systemsnon-reciprocal interactionslocal learning rulesgradient estimationenergy-based learningcontinuous Hopfield networksfeedforward networks
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper tries to extend Equilibrium Propagation (EP), a physics-inspired learning rule that compares two stationary states of a dynamical system, to systems that do not derive from an energy function. Earlier attempts, such as the Vector Field algorithm, produced biased gradients whenever the Jacobian of the dynamics is asymmetric. The paper proposes Asymmetric EP (AEP), which adds a local corrective force built from the antisymmetric part of the Jacobian at the free equilibrium, and proves that in the infinitesimal-nudging limit the resulting update matches the exact gradient of the cost. It also introduces Dyadic EP, a variational formulation that doubles the state space and obtains the same exact gradients with a saddle-point energy. If correct, this gives the first EP-style local learning rule that is exact beyond conservative systems, including feedforward networks.

Core claim

The paper's central claim is that AEP computes the exact gradient of the cost function in the limit of infinitesimal nudging for arbitrary differentiable non-conservative dynamics. In AEP, the nudged equilibrium xβ_A satisfies F(xβ_A, θ) − β ∂C/∂x(xβ_A) − 2 A_J(x0, θ)(xβ_A − x0) = 0, where A_J is the antisymmetric part of the Jacobian at the free equilibrium. Because the augmented dynamics have Jacobian J_F^T(x0) at equilibrium, the derivative of the nudged state with respect to β equals (J_F^T(x0))^{-1} ∂C/∂x(x0), which is exactly the post-synaptic factor in the true gradient formula. Therefore the contrastive update Δθ = ε (∂F/∂θ)^T (xβ_A − x−β_A)/(2β) converges to gradient descent on the

What carries the argument

The central object is the antisymmetric part of the Jacobian, A_J = (J_F − J_F^T)/2, evaluated at the free equilibrium. Adding the correction −2 A_J(x0)(x − x0) to the nudged dynamics flips the effective Jacobian at x0 from J_F to J_F^T, since J_F = S_J + A_J and J_F^T = S_J − A_J. This transpose is exactly what the gradient formula requires. The paper's second mechanism, Dyadic EP, constructs an energy H(z, z') = −(z − z')^T F((z + z')/2, θ) over a doubled state space; the difference d = z − z' acts as a physical error signal that relaxes to β (J_F^T)^{-1} ∇C, while the midpoint m follows the original inference dynamics.

Load-bearing premise

The load-bearing premise is that the augmented AEP dynamics in Eq. (20) actually converge to the two stationary states x±β_A used in the update; the paper assumes this relaxation without a proof, and also notes that the exact synaptic mechanism for realizing the corrective force remains future work.

What would settle it

Take a linear non-conservative system dx/dt = J x with J = [[a, −b], [b, a]], a quadratic cost C, and compute the AEP contrast (xβ_A − x−β_A)/(2β) as β → 0; if it does not converge to (J^T)^{-1} ∂C/∂x, then Eq. (25) is false. A second check is to simulate the augmented flow from several initial conditions: if for some bounded F the state never settles to a fixed point within the integration time, the algorithm's update is undefined.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Any differentiable non-conservative system with an invertible Jacobian at a stationary state can be trained by local, contrastive learning rules that require no explicit backward circuit.
  • Feedforward networks, where earlier EP-style rules cannot train hidden layers, become trainable because the AEP correction creates effective backward pathways during the learning phase.
  • The Dyadic EP formulation shows that the variational energy-based principle behind EP can be applied universally to stationary-state dynamics, and its doubled state space allows the positive and negative nudging phases to run in parallel.
  • On MNIST, AEP learns faster and reaches higher accuracy than the Vector Field baseline, and in a purely feedforward architecture it improves test accuracy from about 64% to about 93%.
  • The paper identifies the precise failure mode of the earlier Vector Field update: its gradient error grows with the spectral radius of S_J^{-1} A_J, so the correction is needed exactly when the antisymmetric part of the Jacobian is large.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The exact-gradient property depends only on the linearization at the free equilibrium, so a testable extension is to reuse a single estimate of A_J across an entire mini-batch, amortizing the cost of the Jacobian decomposition.
  • The feedforward result suggests a concrete hardware recipe: physical feedback connections that are inactive during inference and activated only during the nudged phase would realize the AEP correction in analog or neuromorphic substrates.
  • Dyadic EP's saddle-point dynamics interpret the difference variable as a physical error signal; one could probe this directly by recording z − z' during the nudged phase in a physical implementation, providing an experimental check of the learning signal.
  • The same doubling trick may extend to stochastic dynamics if the stationary distribution replaces the fixed point and the Jacobian is replaced by the appropriate linear response, although the paper does not establish this.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper proposes two algorithms, Asymmetric EP (AEP) and Dyadic EP, that extend Equilibrium Propagation to non-conservative dynamical systems. AEP modifies the nudged-phase dynamics by adding the local term −2A_J(x0,θ)(x−x0), where A_J is the antisymmetric part of the Jacobian at the free equilibrium, so that the Jacobian of the augmented system at x0 becomes J_F^T. The paper proves that the β-derivative of the resulting nudged state is (J_F^T)^{-1}∂C/∂x, which is exactly the post-synaptic term in the gradient formula, and therefore the contrastive update in Eq. (21) recovers dC/dθ in the β→0 limit. Dyadic EP doubles the state space and defines an energy H(z,z′,θ)=−(z−z′)^T F((z+z′)/2,θ), reproducing the same gradient through a saddle-point flow. The theoretical core is supported by appendices, and MNIST experiments compare AEP with EP, VF, and feedforward-only training.

Significance. If the algorithmic assumptions are justified, this is a meaningful advance: it would be the first EP-style algorithm that computes exact cost gradients for arbitrary non-conservative dynamics, with possible applications to neuromorphic hardware and physical learning. The algebraic core is clean and self-contained: the decomposition J_F=S_J+A_J and the identity J_F−2A_J=J_F^T are exactly the right mechanism for correcting the VF gradient error. The Dyadic EP construction is also conceptually interesting and provides a variational interpretation. The paper is transparent about experimental details and includes explicit learning rules in the appendices. However, the central claim as stated depends on a convergence assumption that is neither proved nor verified, and the feedforward illustration contains an incorrect equation. This prevents the paper from currently supporting its broad advertised claim.

major comments (2)
  1. [§3, Algorithm 1 step 3, Eq. (20)] The proof shows that the augmented Jacobian at the free equilibrium x0 is J_F^T, and hence the fixed point equation has a locally unique branch xβ_A with derivative (J_F^T)^{-1}∂C/∂x. However, Algorithm 1 assumes that the non-gradient flow (20), started at x0, actually relaxes to x±β_A for the finite ±β used in training. Only local stability is demonstrated: because J_FA(x0)=J_F^T(x0) has the same spectrum as J_F, local stability at x0 follows, but it does not imply global convergence or even convergence for the finite β=0.5 and nnudge=10 used in the experiments. Appendix F.2.2 explicitly states that with fixed integration steps "incomplete convergence degrades the accuracy of the gradient estimates." If the trajectory has not reached a stationary state, Eq. (21) is not the β-derivative computed in Eq. (25); the implemented update is a different, biased estimator. Since the paper adverti
  2. [§5.3, Eq. (41)] The displayed augmented forces in the feedforward example do not follow from Eq. (20). For the dynamics in Eq. (35), the off-diagonal block of J_F(x0) is diag(ρ′(o0)) W_{h→o} diag(ρ′(h0)). Therefore the AEP correction adds +diag(ρ′(h0)) W_{h→o}^T diag(ρ′(o0)) (o−o0) to F_h and −diag(ρ′(o0)) W_{h→o} diag(ρ′(h0)) (h−h0) to F_o. Equation (41) instead writes ρ′(h)⊙(W^T(o−o0)) and ρ′(o)⊙W(h−h0), dropping the ρ′ factors inside the W terms. As written, Eq. (41) is not the dynamics of Algorithm 1, and the explicit demonstration that AEP creates effective backward connections is incorrect. These formulas should be corrected, and the reported feedforward experiments should be checked against the corrected augmented dynamics.
minor comments (5)
  1. [§5.3] The sentence "As demonstrated in (Scurria, 2026)" cites an in-preparation work that is not available for verification. This is not load-bearing for the main result, but it should be removed or replaced with a verifiable reference.
  2. [§4, Algorithm 2] In Algorithm 2, the free phase says "starting from ... z0" before z0 is introduced. Specify that z0 is the common initial condition (not necessarily the free fixed point), or define it before use.
  3. [Appendix A, Eq. (46)] The Neumann expansion requires ∥S_J^{-1}A_J∥<1; this condition should be stated explicitly before Eq. (44)–(46).
  4. [§3, Eqs. (21)–(25)] The exact-gradient result is stated only in the β→0 limit, but the implemented update uses a finite difference at finite β. The paper does not quantify the finite-β bias of Eq. (21) for AEP. This is not fatal—standard EP also uses finite β—but a bias estimate or at least a remark would strengthen the relation between theory and experiments.
  5. [Appendix B] The BPTT equivalence is only sketched and the notation s, t is not fully defined. Since the main proof is already in §3, this appendix can be shortened or made more precise.

Circularity Check

0 steps flagged

No significant circularity: the AEP and Dyadic EP gradient derivations are self-contained; the only self-reference is a non-load-bearing citation.

full rationale

The central identity, Eq. (25), is obtained by differentiating the augmented fixed-point equation (22) and using the fact that the augmented Jacobian at x0 is J_F^T. This is a direct linear-algebra derivation from the stated dynamics, not an input fitted to the cost or to experimental data. The correction term -2A_J(x0)(x-x0) is constructed from the antisymmetric part of the vector-field Jacobian, which is independent of the cost gradient, so the result is not self-definitional. The exact gradient formula Eq. (14) is independently derived by differentiating F(x0,theta)=0. Dyadic EP similarly defines H and D from F and C (Eq. 26) and then derives the adjoint relation (Eq. 65) at stationarity; this is a variational construction with proof, not a renaming of a known result or an ansatz smuggled in via citation. The only self-referential element is the side remark in Section 5.3 citing Scurria (2026, in preparation) for exact backpropagation dynamics; it is not used to justify the paper's load-bearing claims, which are established by the equations in the same section. Appendix F.2.2's admission that 'incomplete convergence degrades the accuracy of the gradient estimates' is a finite-step convergence caveat for the experiments, not a circular reduction of the beta -> 0 identity. Thus there are no circular steps in the derivation chain.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 2 invented entities

The theoretical central claim rests on standard implicit-differentiation assumptions: existence, uniqueness, invertibility, and convergence of the augmented flows. No numbers are fitted to make the derivation work. The doubled variables and augmented energy are invented mathematical structures without independent empirical evidence, but they are not claimed to be physical entities.

free parameters (5)
  • Nudging parameter β = 0.5 in experiments; β→0 in theory
    Controls the contrastive update's bias-variance trade-off; exactness only holds in the β→0 limit. Not fitted to data, but central to the estimator.
  • Learning rate ε = 0.05 (input-hidden), 0.01 (hidden-output)
    Standard SGD hyperparameter chosen for the MNIST experiments; not fitted to validate the theory.
  • Time step for dynamics integration = 0.5 (symmetric init / feedforward), 0.3 (fixed r_str)
    Integration step used to approximate continuous relaxation to stationarity; chosen for stability.
  • Free/nudged integration steps = n_free = 20/30, n_nudge = 10
    Assumed sufficient for convergence to stationary states; not proven, and convergence is load-bearing for the algorithm.
  • Structural asymmetry ratio r_str = swept in {0, 0.25, ..., 1}
    Experimental design parameter controlling the degree of non-reciprocity; not fitted to maximize accuracy.
axioms (5)
  • domain assumption The dynamical system relapses to a unique stationary state x0 with F(x0)=0 and invertible Jacobian J_F(x0).
    Stated in §2.2 and used in Eq. (13)-(16) for implicit differentiation; without it the gradient formula is undefined.
  • domain assumption The augmented AEP dynamics in Eq. (20) converge to stationary states x±β_A for ±β.
    Assumed in Algorithm 1 step 3 and Eq. (22); global convergence is not proven, only local stability via J_F^T.
  • domain assumption The Dyadic EP saddle-point flow in Eq. (28) converges to a stationary point (zβ,z′β).
    Assumed in Algorithm 2; required for the gradient estimate to be well-defined.
  • domain assumption The cost function C does not depend explicitly on the parameters θ.
    Used in Appendix C.1, Eq. (57), when simplifying dET/dθ; standard for the supervised-learning setting.
  • ad hoc to paper The separation ∥d∥ = ∥z−z′∥ is O(β), allowing truncation of O(∥d∥^3) terms.
    Used in Appendix D to derive the illustrative Hopfield-like energy; not needed for the exact proof in Appendix C.2.
invented entities (2)
  • Doubled state space (z, z′) in Dyadic EP no independent evidence
    purpose: Construct a scalar generator H(z,z′,θ) for non-conservative dynamics, enabling a variational formulation of learning.
    Purely mathematical construction with no observable physical prediction; it is a theoretical tool.
  • Augmented energy H(z,z′,θ) = −(z−z′)ᵀF((z+z′)/2) no independent evidence
    purpose: Acts as a variational generator whose saddle-point flow reproduces the original dynamics on the diagonal z=z′ and yields exact gradients when nudged.
    Mathematical device inspired by Bateman/Galley doubling; no independent falsifiable handle is proposed.

pith-pipeline@v1.3.0-alltime-deepseek · 21558 in / 19819 out tokens · 200284 ms · 2026-08-03T04:53:06.597280+00:00 · methodology

0 comments
read the original 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.

Figures

Figures reproduced from arXiv: 2602.03670 by Antonino Emanuele Scurria, Bortolo Matteo Mognetti, Dimitri Vanden Abeele, Serge Massar.

Figure 2
Figure 2. Figure 2: studies the evolution of the asymmetry ratio rstr. The results are reported for 50 hidden neurons. As expected, EP preserves the initial weight symmetry. In contrast, VF and AEP induce non-trivial evolution of rstr following two distinct patterns, resulting in three distinct network con￾figurations. A complementary figure is available in Ap￾pendix F.1. 5.2. Fixed Asymmetry Ratio While the previous section … view at source ↗
Figure 1
Figure 1. Figure 1: Comparison of algorithm performance on MNIST us￾ing a layered architecture with one hidden layer and symmetric initialization. Squares denote AEP, circles EP, and triangles VF. Test accuracy (averaged over 10 runs) is shown after one epoch (Fig. 1a) and 20 epochs (Fig. 1b) [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Impact of the structural asymmetry ratio rstr on accuracy (top) and standard deviation over 10 runs (bottom). We compare VF (orange) and AEP (blue) under two training regimes: training only J in (dashed) or all parameters (solid). structural asymmetry ratio. In practice, the training appears to adjust the neuronal state such that neurons connected by strongly asymmetric weights have low activation. Finally… view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of the mean accuracy and standard deviation (over 10 runs) during training on MNIST for AEP, EP, and VF. Models use 50 hidden neurons. F.2.1. LEARNING RULES Parametrization and notation. To enforce a fixed asym￾metry ratio, we explicitly parameterize the independent ele￾ments of Eq. (38). We introduce two parameter vectors θ S and θ A of size M = Ndyn(Ndyn − 1)/2, which encode the off-diagonal el… view at source ↗
Figure 5
Figure 5. Figure 5: Cumulative loss as define by (119) over the first 5 epochs of training, for different asymmetry ratios rstr. We compare VF (orange) and AEP (blue), under two training regimes: training only J in (dashed) or all parameters (solid). In [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Asymmetry ratio of the Jacobian rjac defined in equation (121) after training for different asymmetry ratios rstr. We compare VF (orange) and AEP (blue), under two training regimes: training only J in (dashed) or all parameters (solid). For the input weights connecting to the hidden layer, we get the usual formula: ∆J in ik ∝ 1 2β h (h +β i − h −β i )ρ ′ (h 0 i )uk i , (123) while for the feedforward weigh… view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of AEP and VF on a feedforward network. Test accuracy on MNIST is shown as a function of training epochs for a single-hidden-layer network with 20 neurons. Curves report the mean and standard deviation over 10 runs. Best accuracies are 92.7% ± 0.5% (AEP) and 64.3% ± 2.0% (VF). 19 [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗

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