On Sampling with Approximate Transport Maps
This work addresses the problem of efficient sampling from complex distributions for researchers and practitioners in machine learning and statistics, providing insights into method selection based on target characteristics.
The paper clarifies the relative strengths and weaknesses of two approaches for using Normalizing Flows in sampling: flow-based proposals handle multimodal targets reliably up to moderately high dimensions, while reparametrization methods are more robust in high-dimensional settings and under poor training but struggle with multimodality.
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.