Adaptive Bayesian Sampling with Monte Carlo EM

TL;DR

This paper introduces an adaptive Bayesian sampling method that learns mass matrices using Monte Carlo EM, simplifying existing Riemannian approaches and achieving high accuracy with improved efficiency.

Contribution

It proposes a new online technique for learning mass matrices in energy-preserving dynamics within a Monte Carlo EM framework, reducing complexity and computational cost.

Findings

Achieves sampling accuracy comparable to Riemannian methods.

Significantly faster than existing adaptive samplers.

Effective on high-dimensional synthetic and real data.

Abstract

We present a novel technique for learning the mass matrices in samplers obtained from discretized dynamics that preserve some energy function. Existing adaptive samplers use Riemannian preconditioning techniques, where the mass matrices are functions of the parameters being sampled. This leads to significant complexities in the energy reformulations and resultant dynamics, often leading to implicit systems of equations and requiring inversion of high-dimensional matrices in the leapfrog steps. Our approach provides a simpler alternative, by using existing dynamics in the sampling step of a Monte Carlo EM framework, and learning the mass matrices in the M step with a novel online technique. We also propose a way to adaptively set the number of samples gathered in the E step, using sampling error estimates from the leapfrog dynamics. Along with a novel stochastic sampler based on Nos\'{e}-Poincar\'{e} dynamics, we use this framework with standard Hamiltonian Monte Carlo (HMC) as well as newer stochastic algorithms such as SGHMC and SGNHT, and show strong performance on synthetic and real high-dimensional sampling scenarios; we achieve sampling accuracies comparable to Riemannian samplers while being significantly faster.