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.

翻译：均衡传播（EP）是一种受物理学启发的学习算法，它利用动力系统的稳态进行推理和学习。在其原始表述中，该算法局限于保守系统，即其动力学源于能量函数的系统。鉴于其应用范围，将EP扩展到非保守系统（即具有非互易相互作用的系统）具有重要意义。此前将EP推广至此类系统的尝试未能精确计算损失函数的梯度。本文提出一种将EP扩展至任意非保守系统（包括前馈网络）的框架。我们保留了均衡传播的核心特性，即利用稳态进行推理和学习。然而，我们在学习阶段通过引入一个与相互作用非互易部分成正比的项来修正动力学过程，从而获得损失函数的精确梯度。该算法也可通过变分公式推导得出，该公式在增广状态空间上定义的能量函数生成学习动力学。数值实验表明，与先前方法相比，该算法实现了更优的性能和更快的学习速度。