Bayesian Lasso Posterior Sampling via Parallelized Measure Transport

Abstract

It is well known that the Lasso can be interpreted as a Bayesian posterior mode estimate with a Laplacian prior. Obtaining samples from the full posterior distribution, the Bayesian Lasso, confers major advantages in performance as compared to having only the Lasso point estimate. Traditionally, the Bayesian Lasso is implemented via Gibbs sampling methods which suffer from lack of scalability, unknown convergence rates, and generation of samples that are necessarily correlated. We provide a measure transport approach to generate i.i.d samples from the posterior by constructing a transport map that transforms a sample from the Laplacian prior into a sample from the posterior. We show how the construction of this transport map can be parallelized into modules that iteratively solve Lasso problems and perform closed-form linear algebra updates. With this posterior sampling method, we perform maximum likelihood estimation of the Lasso regularization parameter via the EM algorithm. We provide comparisons to traditional Gibbs samplers using the diabetes dataset of Efron et al. Lastly, we give an example implementation on a computing system that leverages parallelization, a graphics processing unit, whose execution time has much less dependence on dimension as compared to a standard implementation.

Inference via low-dimensional couplings

“…where det ∇S := n i=1 ∂ k S k exists a.e., and where T η is the density of T ν η . There is a growing body of literature on the efficient numerical approximation of transport maps (see, e.g., [75,95,69,115,80,7]). Essentially all of these approaches employ numerical optimization to construct or realize the action of a map, and thus harness optimization to enhance integration.…”

Bayesian Lasso : Concentration and MCMC Diagnosis

“…A large number of theoretical results has been provided for LASSO. See [5], [6], [9], [12], [15] and the references herein. The most popular algorithms to find LASSO are LARS algorithm [8], ISTA and FISTA algorithms see e.g.…”