Higher-Order Hit-&-Run Samplers for Linearly Constrained Densities

TL;DR

This work proposes a novel constrained sampling algorithm, which combines strengths of higher-order information, like the target's log-density's gradients and curvature, with the Hit-&-Run proposal, a simple mechanism which guarantees the generation of feasible proposals, fulfilling the linear constraints.

Abstract

Markov chain Monte Carlo (MCMC) sampling of densities restricted to linearly constrained domains is an important task arising in Bayesian treatment of inverse problems in the natural sciences. While efficient algorithms for uniform polytope sampling exist, much less work has dealt with more complex constrained densities. In particular, gradient information as used in unconstrained MCMC is not necessarily helpful in the constrained case, where the gradient may push the proposal's density out of the polytope. In this work, we propose a novel constrained sampling algorithm, which combines strengths of higher-order information, like the target's log-density's gradients and curvature, with the Hit-&-Run proposal, a simple mechanism which guarantees the generation of feasible proposals, fulfilling the linear constraints. Our extensive experiments demonstrate improved sampling efficiency on complex constrained densities over various constrained and unconstrained samplers.

Figures (11)

• Figure 1: Our proposed smLHR sampler preconditions the direction using local curvature and clips the natural gradient step to prevent it from leaving the feasible region.
• Figure 2: Visualizations of (a) HR, (b) LHR and (c) smHR on a constrained 2-dimensional toy density. A visulization of smLHR is given in \ref{['fig:smlhr']}.
• Figure 3: Two-dimensional, exemplary visualizations of the (a) polytopes and (b) densities under consideration, as well as the effect of the respective scale $\sigma$ and angle $\theta$ parameters. The mode of the Gaussian densities is controlled by $\mu$ along the $(1, \ldots, 1)$ axis.
• Figure 4: Relative performance of the tested samplers on our benchmark problems as described in \ref{['sec:experiments']}. Individual dots show relative performance of individual sampling runs, the solid lines show average relative performance across all problems. Algorithms shown are RWMH , MALA , smMALA$_{\delta}$, Dikin , MAPLA , HR , LHR , smHR$_{\delta}$, and smLHR$_{\delta}$.
• Figure 5: Relative performance of smMALA$_{\varepsilon}$, smMALA$_{\delta}$, smHR$_{\varepsilon}$, smHR$_{\delta}$, smLHR$_{\varepsilon}$& smLHR$_{\delta}$using (a) different choices of metric for the bowtie and funnel, and (b) on the Gaussian targets. Colored bars show the 25%

Theorems & Definitions (6)

• Lemma 3.1
• Theorem 3.2
• proof : Proof of \ref{['lemma:ehr-pdf']}
• Lemma A.1
• proof
• proof : Proof of \ref{['thm:convergence']}