Sticky PDMP samplers for sparse and local inference problems

TL;DR

This paper introduces 'sticky' PDMP samplers, especially Sticky Zig-Zag, which improve inference efficiency in high-dimensional sparse models by enabling non-reversible jumps and spending time in sub-models, outperforming traditional methods.

Contribution

The paper develops a novel class of PDMP-based samplers with sticky features for better high-dimensional sparse inference, focusing on the Sticky Zig-Zag sampler.

Findings

Sticky PDMPs can be effectively applied to high-dimensional sparse models.

Sticky Zig-Zag shows favorable dependence on data size and dimension.

Numerical experiments demonstrate improved efficiency over traditional methods.

Abstract

We construct a new class of efficient Monte Carlo methods based on continuous-time piecewise deterministic Markov processes (PDMPs) suitable for inference in high dimensional sparse models, i.e. models for which there is prior knowledge that many coordinates are likely to be exactly $0$. This is achieved with the fairly simple idea of endowing existing PDMP samplers with 'sticky' coordinate axes, coordinate planes etc. Upon hitting those subspaces, an event is triggered during which the process sticks to the subspace, this way spending some time in a sub-model. This results in non-reversible jumps between different (sub-)models. While we show that PDMP samplers in general can be made sticky, we mainly focus on the Zig-Zag sampler. Compared to the Gibbs sampler for variable selection, we heuristically derive favourable dependence of the Sticky Zig-Zag sampler on dimension and data size. The computational efficiency of the Sticky Zig-Zag sampler is further established through numerical experiments where both the sample size and the dimension of the parameter space are large.