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, i.e. to dynamics which derive from an energy function. Given their applications, it is important to extend EP to non-conservative systems, 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.

Today's artificial intelligence learns using an algorithm called backpropagation, which requires a dedicated, separate process to send error signals backward through the network. This is straightforward on a computer but very hard to reproduce in the brain or in energy-efficient physical hardware. Equilibrium Propagation is a promising alternative: it lets a physical system learn simply by relaxing into its natural resting state, with no separate backward machinery. Until now, however, it only worked for a special class of "balanced" systems, in which every connection between two units pushes back on its partner with equal strength. Many important systems lack this balance, including the feedforward networks used in modern AI, biological neural circuits, and devices driven by lasers — and earlier attempts to lift this restriction could no longer learn correctly. We introduce two methods that extend Equilibrium Propagation to these unbalanced systems while still computing the exact learning signal. The key idea is to slightly adjust the system's behavior during the learning phase, using only information already available locally at each connection. Tested on standard image-recognition tasks, our methods learn faster and more accurately than previous proposals, opening new paths toward efficient, brain-inspired learning hardware.