Diffusive Gibbs Sampling (2402.03008v5)

Abstract: The inadequate mixing of conventional Markov Chain Monte Carlo (MCMC) methods for multi-modal distributions presents a significant challenge in practical applications such as Bayesian inference and molecular dynamics. Addressing this, we propose Diffusive Gibbs Sampling (DiGS), an innovative family of sampling methods designed for effective sampling from distributions characterized by distant and disconnected modes. DiGS integrates recent developments in diffusion models, leveraging Gaussian convolution to create an auxiliary noisy distribution that bridges isolated modes in the original space and applying Gibbs sampling to alternately draw samples from both spaces. A novel Metropolis-within-Gibbs scheme is proposed to enhance mixing in the denoising sampling step. DiGS exhibits a better mixing property for sampling multi-modal distributions than state-of-the-art methods such as parallel tempering, attaining substantially improved performance across various tasks, including mixtures of Gaussians, Bayesian neural networks and molecular dynamics.

• The paper introduces Diffusive Gibbs Sampling, which bridges isolated modes using Gaussian convolution to efficiently sample multimodal distributions.
• The method outperforms conventional techniques like parallel tempering in tasks ranging from Bayesian neural networks to molecular dynamics, achieving superior sample quality with fewer target density evaluations.
• Adaptive noise scheduling minimizes hyperparameter dependence, ensuring robust exploration and practical implementation across diverse complex inference problems.

Introduction to Diffusive Gibbs Sampling

Diffusive Gibbs Sampling (DiGS) is an innovative method addressing a critical issue in the Markov Chain Monte Carlo (MCMC) domain—inefficient sampling from multimodal distributions. Traditional techniques, such as Metropolis-Adjusted Langevin Algorithm (MALA) and Hamiltonian Monte Carlo (HMC), often struggle with distributions characterized by isolated modes due to difficulties in transitioning between these modes. DiGS not only offers a solution to this challenge but does so by integrating cutting-edge advancements in diffusion models with MCMC principles.

The Core Concept

The heart of DiGS lies in its unique fusion of Gaussian convolution and Gibbs sampling. Where conventional strategies fail, DiGS creates an auxiliary noisy distribution that effectively bridges the gaps between modes, enabling the sampler to move smoothly across the entire target distribution. It achieves this by constructing an intermediate space where isolated modes become connected through Gaussian convolution. Alternating Gibbs sampling then enables transitions between the original and auxiliary spaces, thereby efficiently capturing multimodal characteristics.

Numerical Performance

Empirical demonstrations reveal that DiGS substantially outperforms established MCMC methods on a variety of complex tasks. For instance, DiGS outstrips parallel tempering, a state-of-the-art method, on multimodal Gaussian mixture models, Bayesian neural networks, and molecular dynamics simulations, producing samples that more accurately represent the entire target distribution. It achieves this breakthrough with dramatically fewer evaluations of the target density, underscoring its computational efficiency.

Advantages over Related Methods

Relating to existing approaches, DiGS exhibits several advantages. Firstly, it obviates the intractability of score functions encountered in score-based diffusion models, making direct sampling from the model more feasible. Its Markov chain construction satisfies irreducibility and recurrence, implying thorough exploration and accurate capture of the distribution. Finally, DiGS's adaptive noise scheduling, inspired by noise schedules used in diffusion models, reduces dependency on precise hyperparameter selection, an often-laborious task.

Conclusion

Through its methods and results, DiGS offers a compelling avenue for researchers and practitioners dealing with multimodal distributions across disciplines. If sample quality and computational cost are of essence, DiGS provides an adept technique for challenging MCMC scenarios. Its innovations present a significant leap forward in the sampling capabilities of generative models, with the potential for wide-reaching impact in fields where understanding complex distributions is vital.