  Despite their appeal as physics-inspired, energy-based and generative nature,
general Boltzmann Machines (BM) are considered intractable to train. This
belief led to simplified models of BMs with restricted intralayer connections
or layer-by-layer training of deep BMs. Recent developments in domain-specific
hardware -- specifically probabilistic computers (p-computer) with
probabilistic bits (p-bit) -- may change established wisdom on the tractability
of deep BMs. In this paper, we show that deep and unrestricted BMs can be
trained using p-computers generating hundreds of billions of Markov Chain Monte
Carlo (MCMC) samples per second, on sparse networks developed originally for
use in D-Wave's annealers. To maximize the efficiency of learning the
p-computer, we introduce two families of Mean-Field Theory assisted learning
algorithms, or xMFTs (x = Naive and Hierarchical). The xMFTs are used to
estimate the averages and correlations during the positive phase of the
contrastive divergence (CD) algorithm and our custom-designed p-computer is
used to estimate the averages and correlations in the negative phase. A custom
Field-Programmable-Gate Array (FPGA) emulation of the p-computer architecture
takes up to 45 billion flips per second, allowing the implementation of CD-$n$
where $n$ can be of the order of millions, unlike RBMs where $n$ is typically 1
or 2. Experiments on the full MNIST dataset with the combined algorithm show
that the positive phase can be efficiently computed by xMFTs without much
degradation when the negative phase is computed by the p-computer. Our
algorithm can be used in other scalable Ising machines and its variants can be
used to train BMs, previously thought to be intractable.
