How the brain performs credit assignment is a fundamental unsolved problem in neuroscience. Many `biologically plausible' algorithms have been proposed, which compute gradients that approximate those computed by backpropagation (BP), and which operate in ways that more closely satisfy the constraints imposed by neural circuitry. Many such algorithms utilize the framework of energy-based models (EBMs), in which all free variables in the model are optimized to minimize a global energy function. However, in the literature, these algorithms exist in isolation and no unified theory exists linking them together. Here, we provide a comprehensive theory of the conditions under which EBMs can approximate BP, which lets us unify many of the BP approximation results in the literature (namely, predictive coding, equilibrium propagation, and contrastive Hebbian learning) and demonstrate that their approximation to BP arises from a simple and general mathematical property of EBMs at free-phase equilibrium. This property can then be exploited in different ways with different energy functions, and these specific choices yield a family of BP-approximating algorithms, which both includes the known results in the literature and can be used to derive new ones.

本文提供了能量基模型（EBMs）在逼近反向传播（BP）方面的综合理论，统一了预测编码、平衡传播和对比 Hebbian 学习等算法，从 EBMs 的自由相平衡这一简单而普遍的数学特性出发，在不同的能量函数下进行选择以得出一类逼近 BP 的算法。

Backpropagation at the Infinitesimal Inference Limit of Energy-Based
  Models: Unifying Predictive Coding, Equilibrium Propagation, and Contrastive
  Hebbian Learning

能量模型微观推断极限下的反向传播：预测编码、平衡传播和对比性赫布学习的统一