Zeroth-Order Sampling Methods for Non-Log-Concave Distributions: Alleviating Metastability by Denoising Diffusion

TL;DR

It is proved that ZOD-MC admits an inverse polynomial dependence on the desired sampling accuracy, albeit still suffering from the curse of dimensionality, and is a very efficient sampler, with performance exceeding latest samplers, including also-denoising-diffusion-based RDMC and RSDMC.

Abstract

This paper considers the problem of sampling from non-logconcave distribution, based on queries of its unnormalized density. It first describes a framework, Denoising Diffusion Monte Carlo (DDMC), based on the simulation of a denoising diffusion process with its score function approximated by a generic Monte Carlo estimator. DDMC is an oracle-based meta-algorithm, where its oracle is the assumed access to samples that generate a Monte Carlo score estimator. Then we provide an implementation of this oracle, based on rejection sampling, and this turns DDMC into a true algorithm, termed Zeroth-Order Diffusion Monte Carlo (ZOD-MC). We provide convergence analyses by first constructing a general framework, i.e. a performance guarantee for DDMC, without assuming the target distribution to be log-concave or satisfying any isoperimetric inequality. Then we prove that ZOD-MC admits an inverse polynomial dependence on the desired sampling accuracy, albeit still suffering from the curse of dimensionality. Consequently, for low dimensional distributions, ZOD-MC is a very efficient sampler, with performance exceeding latest samplers, including also-denoising-diffusion-based RDMC and RSDMC. Last, we experimentally demonstrate the insensitivity of ZOD-MC to increasingly higher barriers between modes or discontinuity in non-convex potential.

Figures (18)

• Figure 1: Accuracies of different methods for sampling Gaussian Mixture
• Figure 2: Sampling from asymmetric, unbalanced Gaussian Mixture.All diffusion-based methods (ZOD-MC, RDMC, RSDMC) use $2200$ oracles per score evaluation. Langevin and the proximal sampler are set to use the same total amount of oracles as diffusion based methods. While other methods suffer from metastability, ZOD-MC correctly samples all modes.
• Figure 3: Gaussian Mixture with further separated modes ($R=26$). ZOD-MC can overcome strengthened metastability and sample from every mode, while other methods are stuck at the mode at the origin, where every method is initialized.
• Figure 4: Accuracies of generated samples against dimension and Score Error. On the right, the result for SLIPS is not directly comparable as it has a different forward process.
• Figure 5: Generated samples for discontinuous Gaussian Mixture. Our method can recover the target distribution even under the presence of discontinuities. The same oracle complexity is again used in each method, $3200$ per score evaluation in diffusion-based approaches.

Theorems & Definitions (21)

• Lemma 1
• Remark 1
• Remark 2
• Proposition 1
• Theorem 1
• Remark 3
• Corollary 1
• Remark 4
• Proposition 2
• proof : Proof of Proposition \ref{['prop:OU decay']}