Equilibrium Propagation (EP) is a learning algorithm for training Energy-based Models (EBMs) on static inputs which leverages the variational description of their fixed points. Extending EP to time-varying inputs is a challenging problem, as the variational description must apply to the entire system trajectory rather than just fixed points, and careful consideration of boundary conditions becomes essential. In this work, we present Generalized Lagrangian Equilibrium Propagation (GLEP), which extends the variational formulation of EP to time-varying inputs. We demonstrate that GLEP yields different learning algorithms depending on the boundary conditions of the system, many of which are impractical for implementation. We then show that Hamiltonian Echo Learning (HEL) -- which includes the recently proposed Recurrent HEL (RHEL) and the earlier known Hamiltonian Echo Backpropagation (HEB) algorithms -- can be derived as a special case of GLEP. Notably, HEL is the only instance of GLEP we found that inherits the properties that make EP a desirable alternative to backpropagation for hardware implementations: it operates in a "forward-only" manner (i.e. using the same system for both inference and learning), it scales efficiently (requiring only two or more passes through the system regardless of model size), and enables local learning.

翻译：平衡传播（EP）是一种用于训练基于能量的模型（EBMs）处理静态输入的学习算法，它利用了模型不动点的变分描述。将EP推广至时变输入是一个具有挑战性的问题，因为变分描述必须适用于整个系统轨迹而不仅仅是固定点，并且对边界条件的仔细考量变得至关重要。在本工作中，我们提出了广义拉格朗日平衡传播（GLEP），它将EP的变分公式推广到时变输入。我们证明了GLEP会根据系统的边界条件产生不同的学习算法，其中许多算法在实现上并不实用。随后，我们证明了哈密顿回声学习（HEL）——包括最近提出的循环HEL（RHEL）和较早已知的哈密顿回声反向传播（HEB）算法——可以作为GLEP的一个特例推导出来。值得注意的是，HEL是我们发现的唯一一个继承了使EP成为硬件实现中反向传播的理想替代方案特性的GLEP实例：它以“仅前向”方式运行（即使用同一系统进行推理和学习），具有高效的可扩展性（无论模型规模大小，仅需两次或更多次系统遍历），并支持局部学习。