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.

翻译：平衡传播（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同时进行推断和学习。在其原始表述中，该算法仅限于保守系统，即动力学可由能量函数导出的系统。考虑到非保守系统（即具有非互易相互作用的系统）在应用中的重要性，将EP推广至此类系统至关重要。先前将EP推广至此类系统的尝试未能精确计算代价函数的梯度。本文提出一个框架，将EP扩展至任意非保守系统，包括前馈网络。我们保留了平衡传播的关键特性，即同时利用稳态进行推断和学习。然而，我们在学习阶段通过添加一个与相互作用非互易部分成正比的项来修改动力学，从而获得代价函数的精确梯度。该算法也可通过变分公式推导，该公式在扩展状态空间上定义的能量函数生成学习动力学。使用MNIST数据库的数值实验表明，该算法相比先前方案取得了更好的性能并实现了更快的学习速度。