  In a physical neural system, where storage and processing are intimately
intertwined, the rules for adjusting the synaptic weights can only depend on
variables that are available locally, such as the activity of the pre- and
post-synaptic neurons, resulting in local learning rules. A systematic
framework for studying the space of local learning rules is obtained by first
specifying the nature of the local variables, and then the functional form that
ties them together into each learning rule. Such a framework enables also the
systematic discovery of new learning rules and exploration of relationships
between learning rules and group symmetries. We study polynomial local learning
rules stratified by their degree and analyze their behavior and capabilities in
both linear and non-linear units and networks. Stacking local learning rules in
deep feedforward networks leads to deep local learning. While deep local
learning can learn interesting representations, it cannot learn complex
input-output functions, even when targets are available for the top layer.
Learning complex input-output functions requires local deep learning where
target information is communicated to the deep layers through a backward
learning channel. The nature of the communicated information about the targets
and the structure of the learning channel partition the space of learning
algorithms. We estimate the learning channel capacity associated with several
algorithms and show that backpropagation outperforms them by simultaneously
maximizing the information rate and minimizing the computational cost, even in
recurrent networks. The theory clarifies the concept of Hebbian learning,
establishes the power and limitations of local learning rules, introduces the
learning channel which enables a formal analysis of the optimality of
backpropagation, and explains the sparsity of the space of learning rules
discovered so far.
