Equilibrium Propagation for Non-Conservative Systems

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 importance in applications, it is important to extend EP to nonconservative 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 nonconservative 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 using the MNIST database show that this algorithm achieves better performance and learns faster than previous proposals.

💡 Research Summary

Equilibrium Propagation (EP) is a biologically‑inspired learning scheme that derives weight updates from the stationary states of a dynamical system, thereby avoiding a separate backward pass. Its classical formulation, however, assumes that the dynamics are conservative – i.e., the force field derives from an energy function, which forces the Jacobian to be symmetric. This restriction excludes a large class of models that are inherently non‑conservative, such as feed‑forward neural networks, active optical systems, or any physical substrate driven far from thermodynamic equilibrium.

The paper first reviews the Vector Field (VF) algorithm, an earlier attempt to apply EP to non‑conservative forces. VF uses the same two‑phase nudging idea but replaces the exact gradient term with a product of the Jacobian and the state difference. The authors show analytically (Eq. 14) that this yields the true gradient only when the Jacobian is symmetric; otherwise the estimate is biased and can even point in the opposite direction when the antisymmetric part dominates.

To overcome this limitation the authors introduce Asymmetric EP (AEP). After the free phase the Jacobian at the equilibrium, (J_F(x_0,\theta)), is decomposed into its symmetric part (S_J) and antisymmetric part (A_J). During the nudged phase the dynamics are modified by a local corrective term (-2A_J(x-x_0)). This term exactly cancels the antisymmetric contribution, turning the effective Jacobian into the transpose of the original one, (J_F^{A}=J_F-2A_J=J_F^{\top}). Consequently, the post‑nudging state difference satisfies the same linear relation as in conservative EP, and the standard contrastive update (\Delta\theta\propto \partial_\theta F^\top (x_{+\beta}^A-x_{-\beta}^A)/(2\beta)) recovers the exact gradient (\mathrm{d}C/\mathrm{d}\theta). The correction is fully local: (A_J) is zero for unconnected units, and the factor ((x-x_0)) is already available at each synapse for the contrastive rule.

The second contribution is Dyadic EP (Dyadic EP), a variational construction that maps the original (n)-dimensional non‑conservative system onto a (2n)-dimensional conservative one. Two copies of the state, (z) and (z’), are introduced together with an energy functional (H(z,z’,\theta)) and a cost (D(z,z’)). By design, the dynamics of (z) follow the original force field while those of (z’) follow its transpose. Applying the standard EP energy‑based update to this augmented system yields, after infinitesimal nudging, the same gradient as AEP. Importantly, the positive and negative nudging phases can be executed in parallel, so the computational cost matches that of AEP while offering a clean variational interpretation. This construction echoes classic techniques for embedding dissipative systems into conservative ones (Bateman’s double‑oscillator) and relates to the Dual Propagation algorithm when the original Jacobian is symmetric.

Experimental validation uses continuous Hopfield networks trained on the MNIST digit classification task. In the symmetric‑weight regime, AEP outperforms the original EP and VF both in speed of convergence and final accuracy (≈0.5 % gain). When the network is forced to be highly asymmetric—rendering standard EP inapplicable—AEP and Dyadic EP still train all layers effectively, achieving >92 % test accuracy, whereas VF collapses to an extreme‑learning‑machine‑like behavior that only adapts the final read‑out layer. The authors also demonstrate that, for purely feed‑forward architectures, both methods reproduce the learning curves of conventional back‑propagation through time, confirming that the local correction fully compensates for the lack of symmetry.

In summary, the paper makes three major contributions: (1) a principled, locally implementable correction that extends EP to arbitrary differentiable non‑conservative dynamics (AEP); (2) a variational double‑state formulation (Dyadic EP) that yields the same exact gradient while allowing parallel computation; and (3) a thorough theoretical analysis together with empirical evidence showing that these methods close the performance gap left by previous attempts such as VF. The work opens the door to hardware‑friendly, energy‑based learning on neuromorphic chips, active photonic circuits, and other physical substrates where asymmetric interactions are the norm. Future directions include extending the framework to spiking neurons, stochastic dynamics, and quantum‑inspired platforms, as well as exploring meta‑learning strategies to automatically estimate the antisymmetric correction term.