  Equilibrium systems are a powerful way to express neural computations. As
special cases, they include models of great current interest in both
neuroscience and machine learning, such as deep neural networks, equilibrium
recurrent neural networks, deep equilibrium models, or meta-learning. Here, we
present a new principle for learning such systems with a temporally- and
spatially-local rule. Our principle casts learning as a least-control problem,
where we first introduce an optimal controller to lead the system towards a
solution state, and then define learning as reducing the amount of control
needed to reach such a state. We show that incorporating learning signals
within a dynamics as an optimal control enables transmitting activity-dependent
credit assignment information, avoids storing intermediate states in memory,
and does not rely on infinitesimal learning signals. In practice, our principle
leads to strong performance matching that of leading gradient-based learning
methods when applied to an array of problems involving recurrent neural
networks and meta-learning. Our results shed light on how the brain might learn
and offer new ways of approaching a broad class of machine learning problems.
