Rapid Bayesian computation and estimation for neural networks via log-concave coupling

Abstract

This paper presents the study of a Bayesian estimation procedure for single-hidden-layer neural networks using ${\ell }_1$ controlled neuron weight vectors. We study the structure of the posterior density and provide a representation that makes it amenable to rapid sampling via Markov Chain Monte Carlo (MCMC). Let the neural network have $K$ neurons with internal weights of dimension $d$ and fix the outer weights. Thus, there are $Kd$ parameters overall. With $N$ data observations, use a gain parameter or inverse temperature of $\beta$ in the posterior density for the internal weights.
The posterior is intrinsically multi-modal and not naturally suited to rapid mixing of direct MCMC algorithms. For a continuous uniform prior on the ${\ell }_1$ ball, we demonstrate that the posterior density can be written as a mixture density with suitably defined auxiliary random variables, where the mixture components are log-concave. Furthermore, when the total number of model parameters $Kd$ is large enough that $Kd\ge C(\beta N{)}^2$, the mixing distribution of the auxiliary random variables is also log-concave. Thus, neuron parameters can be sampled from the posterior by only sampling log-concave densities. The authors refer to the pairing of weights with such auxiliary random variables as a log-concave coupling.