On Sampling with Approximate Transport Maps

TL;DR

This paper compares two flow-based sampling methods, revealing their strengths and weaknesses in handling complex, multimodal distributions and high-dimensional problems, and introduces a new bound on sampler mixing time.

Contribution

It provides the first detailed comparison of flow-based proposal and reparametrization methods, highlighting their respective advantages and limitations.

Findings

Flow-based proposals handle multimodal targets well in moderate dimensions.

Reparametrization methods are more robust in high dimensions and with poor training.

A new quantitative bound for the mixing time of the Independent Metropolis-Hastings sampler is derived.

Abstract

Transport maps can ease the sampling of distributions with non-trivial geometries by transforming them into distributions that are easier to handle. The potential of this approach has risen with the development of Normalizing Flows (NF) which are maps parameterized with deep neural networks trained to push a reference distribution towards a target. NF-enhanced samplers recently proposed blend (Markov chain) Monte Carlo methods with either (i) proposal draws from the flow or (ii) a flow-based reparametrization. In both cases, the quality of the learned transport conditions performance. The present work clarifies for the first time the relative strengths and weaknesses of these two approaches. Our study concludes that multimodal targets can be reliably handled with flow-based proposals up to moderately high dimensions. In contrast, methods relying on reparametrization struggle with multimodality but are more robust otherwise in high-dimensional settings and under poor training. To further illustrate the influence of target-proposal adequacy, we also derive a new quantitative bound for the mixing time of the Independent Metropolis-Hastings sampler.