  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.
