The least-control principle for local learning at equilibrium

A widely-accepted tenet of neuroscience is that learning relies on synaptic plasticity. Because plasticity is thought to be governed by local, activity-dependent rules, it is a longstanding mystery how synapses change in concert so that behavioral output improves. This problem is aggravated by the recurrent and multilayered architecture of the cortex, which makes it hard to determine the effects of a given synaptic modification. Here we present a novel principle for learning rooted in optimal control theory, which yields local plasticity rules for any neural dynamics which reaches an equilibrium, thus encompassing both multilayered and recurrent neural networks. Our principle casts learning as a least-control problem, where we first introduce an optimal controller to lead the dynamics towards a solution state, and then define learning as reducing the amount of control needed to reach such a state. We derive a gradient-following, activity-dependent plasticity rule for control minimization, and establish conditions under which it optimizes behavioral performance. Furthermore, we provide conditions for a network state to be optimally-controlled, and we design neural control circuits of varying complexity that meet our optimality conditions either approximately or exactly. Critically, unlike previous gradient-based cortical learning theories we do not require the control feedback to be vanishingly small, an assumption that is problematic in noisy circuits and at odds with experimental reports showing that feedback connections can influence cortical processing. We conduct a series of benchmarking experiments and find that our principle leads to strong performance matching that of backpropagation-of-error, currently the gold standard for deep learning. Finally, we establish a duality between control minimization and a probabilistic technique known as free-energy minimization. This connection allows interpreting an influential family of cortical models which fall under the umbrella of predictive processing as instantiations of our least-control principle.