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Zeroth-Order Sampling Methods for Non-Log-Concave Distributions: Alleviating Metastability by Denoising Diffusion (2402.17886v4)

Published 27 Feb 2024 in stat.ML, cs.LG, math.PR, math.ST, stat.ME, and stat.TH

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.

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Citations (6)
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Summary

  • The paper introduces Zeroth-Order Diffusion Monte Carlo (ZOD-MC) to sample from unnormalized non-log-concave distributions by gradually denoising an initial distribution.
  • It provides a non-asymptotic convergence analysis using Kullback-Leibler divergence, showcasing improved efficiency over traditional sampling methods in low-dimensional settings.
  • Empirical results demonstrate that ZOD-MC effectively navigates mode separation and discontinuities, offering a promising tool for complex machine learning and statistical applications.

Enhanced Sampling from Non-Log-Concave Distributions through Zeroth-Order Diffusion Monte Carlo

Introduction to Sampling Challenges

Sampling from distributions defined by unnormalized densities is a classic and pervasive problem in computational statistics and machine learning. Traditional methods often struggle with distributions that are not log-concave or that exhibit challenging features such as high barriers between modes or discontinuities. This problem is exacerbated in high dimensions, a common setting in modern data-intensive applications.

Diffusion Monte Carlo Framework

The Zeroth-Order Diffusion Monte Carlo (ZOD-MC) method offers a promising approach to address these challenges. The central idea is to simulate a diffusion process that gradually denoises an initial distribution, morphing it into the target distribution. The cornerstone of ZOD-MC is an oracle-based meta-algorithm relying on Monte Carlo score estimators from samples approximating conditional distributions of a denoising process.

Theoretical Insights and Algorithmic Contributions

The ZOD-MC algorithm operationalizes this concept by using zeroth-order queries -- making no gradient information necessary. One of the key theoretical contributions is the non-asymptotic analysis of the convergence properties of this method, presenting guarantees in terms of Kullback-Leibler divergence. The results are particularly compelling for low-dimensional cases, establishing ZOD-MC as a significantly efficient sampler under such settings, surpassing the performance of alternative methods.

Experimental Validation

The paper extends its theoretical findings with an empirical evaluation, demonstrating the effectiveness of ZOD-MC across a range of non-log-concave distributions, including models with mode separation and discontinuities, as exemplified in the modified Gaussian mixtures and the Müller Brown potential. These experiments substantiate the method's insensitivity to mode separation and its ability to navigate around discontinuities, features that restrict the applicability of other sampling approaches.

Implications for Future Research

While the results are promising, especially in low-dimensional settings, one of the noted limitations is the potential exponential dependency on the dimensionality in terms of the oracle complexity for ZOD-MC. This aspect opens up avenues for further research, possibly towards developing strategies that mitigate the curse of dimensionality inherent in the rejection sampling component of ZOD-MC.

Comparative Advantage in Practical Scenarios

A noteworthy advantage of ZOD-MC, as highlighted by the experimental results, is its flexibility to cater to target distributions with non-smooth or discontinuous potential functions. This adaptability, coupled with its superior performance in challenging sampling scenarios, presents ZOD-MC as a valuable tool in the arsenal of modern sampling techniques, particularly for applications where gradient information is unavailable or unreliable.

Conclusion

In summary, the Zeroth-Order Diffusion Monte Carlo method introduces a viable and theoretically grounded approach to sample from complex distributions. Its development and analysis contribute to the broader endeavor of enhancing sampling methodologies, with implications that extend to various applications in machine learning, computational statistics, and beyond. The method stands out for its theoretical robustness, practical efficacy, and the potential it holds for further refinements and extensions to address the perennial challenges of sampling in high-dimensional spaces.