  Message-passing algorithms based on the Belief Propagation (BP) equations
constitute a well-known distributed computational scheme. It is exact on
tree-like graphical models and has also proven to be effective in many problems
defined on graphs with loops (from inference to optimization, from signal
processing to clustering). The BP-based scheme is fundamentally different from
stochastic gradient descent (SGD), on which the current success of deep
networks is based. In this paper, we present and adapt to mini-batch training
on GPUs a family of BP-based message-passing algorithms with a reinforcement
field that biases distributions towards locally entropic solutions. These
algorithms are capable of training multi-layer neural networks with discrete
weights and activations with performance comparable to SGD-inspired heuristics
(BinaryNet) and are naturally well-adapted to continual learning. Furthermore,
using these algorithms to estimate the marginals of the weights allows us to
make approximate Bayesian predictions that have higher accuracy than point-wise
solutions.
