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How the brain performs credit assignment is a fundamental unsolved problem inneuroscience. 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 constraintsimposed by neural circuitry. Many such algorithms utilize the framework ofenergy-based models (EBMs), in which all free variables in the model areoptimized to minimize a global energy function. However, in the literature,these algorithms exist in isolation and no unified theory exists linking themtogether. Here, we provide a comprehensive theory of the conditions under whichEBMs can approximate BP, which lets us unify many of the BP approximationresults in the literature (namely, predictive coding, equilibrium propagation,and contrastive Hebbian learning) and demonstrate that their approximation toBP arises from a simple and general mathematical property of EBMs at free-phaseequilibrium. This property can then be exploited in different ways withdifferent energy functions, and these specific choices yield a family ofBP-approximating algorithms, which both includes the known results in theliterature and can be used to derive new ones.

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