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

Abstract

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 energybased 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 * Corresponding author.

Preventing Deterioration of Classification Accuracy in Predictive Coding Networks

“…We have provided a detailed analysis of the dynamics of inference in predictive coding, and have shown that there is a tendency for that inference to slow down as training progresses. [20] separates the total energy of an Energy Based Model into the supervised loss (which depends on the errors at the top layer) and the internal energy (which corresponds to the energy of the hidden layers). We have shown that it is only the supervised loss which suffers from a slow-down in inference.…”

Understanding and Improving Optimization in Predictive Coding Networks

“…Previous work found that in the limit where ĥl → h l , IL approaches BP/SGD Bogacz 2017, 2019;Millidge et al 2022a;Rosenbaum 2022). This limit is equivalent, in our notation, to the limit where the β → 0 since in this case the output layer ĥL approaches its initial value h L , resulting in ĥl → h l .…”