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

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.