Sampling by Divergence Minimization

Abstract

We introduce a family of Markov Chain Monte Carlo (MCMC) methods designed to sample from target distributions with irregular geometry using an adaptive scheme. In cases where targets exhibit non-Gaussian behaviour, we propose that adaption should be regional in nature as opposed to global. Our algorithms minimize the information projection side of the Kullback-Leibler (KL) divergence between the proposal distribution class and the target to encourage proposals distributed similarly to the regional geometry of the target. Unlike traditional adaptive MCMC, this procedure rapidly adapts to the geometry of the current position as it explores the space without the need for a large batch of samples. We extend this approach to multimodal targets by introducing a heavily tempered chain to enable faster mixing between regions of interest. The divergence minimization algorithms are tested on target distributions with multiple irregularly shaped modes and we provide results demonstrating the effectiveness of our methods.

Entropy-based adaptive Hamiltonian Monte Carlo

“…Variations of the entropy objective. Recent work [18,11] have suggested to add the crossentropy term π(q) r(q |q) log π(q )dq dq to the entropy objective for optimizing the parameters of a Metropolis-Hastings kernel with proposal density r(q |q). Algorithm 1 can be adjusted to such variations, possibly by stopping gradients through ∇U as for optimizing the energy error term.…”