Equilibrium Propagation for Non-Conservative Systems

TL;DR

This work addresses the limitation of Equilibrium Propagation (EP) to conservative, energy-based systems by introducing two non-conservative extensions that preserve the stationary-state learning paradigm. Asymmetric EP (AEP) adds a local correction based on the antisymmetric part of the Jacobian to recover the exact gradient, while Dyadic EP maps non-conservative dynamics to an extended energy landscape via a doubled state space, yielding equivalent gradient estimates. Empirically, AEP demonstrates faster and more accurate learning than prior approaches on MNIST, and enables training of feedforward architectures by enabling backward-like signals during augmented phases. The results suggest practical routes for credit assignment in dissipative hardware and asymmetric neural networks, with potential implications for neuromorphic and energy-based hardware implementations.

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 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.

Figures (7)

• Figure 1: Comparison of algorithm performance on MNIST using 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. \ref{['fig:one_epoch']}) and 20 epochs (Fig. \ref{['fig:twenty_epochs']}).
• Figure 2: Evolution of the asymmetry ratio $r_\text{str}$ (defined in Eq. \ref{['eq:anti-symmetric_ratio']}) during training on MNIST for AEP, EP, and VF, initialized from a symmetric configuration. The models use 50 hidden neurons.
• Figure 3: Impact of the structural asymmetry ratio $r_\text{str}$ on accuracy (top) and standard deviation over 10 runs (bottom). We compare VF (orange) and AEP (blue) under two training regimes: training only $\bm{J}^\text{in}$ (dashed) or all parameters (solid).
• 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.
• Figure 5: Cumulative loss as define by \ref{['eq:cumulative_loss']} over the first 5 epochs of training, for different asymmetry ratios $r_\text{str}$. We compare VF (orange) and AEP (blue), under two training regimes: training only $J^\text{in}$ (dashed) or all parameters (solid).