A Dynamical Mean-Field Theory for Learning in Restricted Boltzmann Machines
This work provides a theoretical foundation for learning in neural network models, but it appears incremental as it builds on existing dynamical mean-field theory approaches.
The authors tackled the problem of computing magnetizations in Restricted Boltzmann Machines by developing a message-passing algorithm and analyzing its dynamics using statistical mechanics methods, proving global convergence under a stability criterion and showing excellent agreement with numerical simulations.
We define a message-passing algorithm for computing magnetizations in Restricted Boltzmann machines, which are Ising models on bipartite graphs introduced as neural network models for probability distributions over spin configurations. To model nontrivial statistical dependencies between the spins' couplings, we assume that the rectangular coupling matrix is drawn from an arbitrary bi-rotation invariant random matrix ensemble. Using the dynamical functional method of statistical mechanics we exactly analyze the dynamics of the algorithm in the large system limit. We prove the global convergence of the algorithm under a stability criterion and compute asymptotic convergence rates showing excellent agreement with numerical simulations.