Machine learning is a powerful method of extracting meaning from data; unfortunately, current digital hardware is extremely energy-intensive. There is interest in an alternative analog computing implementation that could match the performance of traditional machine learning while being significantly more energy-efficient. However, it remains unclear how to train such analog computing systems while adhering to locality constraints imposed by the physical (as opposed to digital) nature of these systems. Local learning algorithms such as Equilibrium Propagation and Coupled Learning have been proposed to address this issue. In this paper, we develop an algorithm to exactly calculate gradients using a graph theoretic and analytical framework for Kirchhoff's laws. We also introduce Generalized Equilibrium Propagation, a framework encompassing a broad class of Hebbian learning algorithms, including Coupled Learning and Equilibrium Propagation, and show how our algorithm compares. We demonstrate our algorithm using numerical simulations and show that we can train resistor networks without the need for a replica or readout over all resistors, only at the output layer. We also show that under the analytical gradient approach, it is possible to update only a subset of the resistance values without a strong degradation in performance.

翻译：机器学习是从数据中提取意义的强大方法；然而，当前的数字硬件能耗极高。人们关注一种替代性的模拟计算实现方式，它有望在保持传统机器学习性能的同时显著提升能效。但如何在这种模拟计算系统中进行训练，同时遵守其物理（而非数字）特性所施加的局部性约束，仍是一个未解决的问题。为应对此挑战，已有研究提出了如平衡传播和耦合学习等局部学习算法。本文中，我们基于基尔霍夫定律的图论与解析框架，开发了一种精确计算梯度的算法。我们还提出了广义平衡传播框架，该框架涵盖了一类广泛的赫布学习算法，包括耦合学习与平衡传播，并展示了我们算法的比较优势。通过数值模拟，我们验证了该算法能够训练电阻网络，且无需在所有电阻上复制或读取数据，仅需在输出层进行操作。此外，我们证明在解析梯度方法下，仅更新部分电阻值并不会导致性能显著下降。