Diffusive Gibbs Sampling for Multi-Modal Distributions

Core Concepts

Diffusive Gibbs Sampling (DiGS) offers an innovative approach to sampling from multi-modal distributions by bridging disconnected modes using Gaussian convolution.

Abstract

The content introduces Diffusive Gibbs Sampling (DiGS) as a method to address the inadequate mixing of conventional Markov Chain Monte Carlo methods for multi-modal distributions. DiGS leverages diffusion models and Gaussian convolution to improve mode coverage in sampling tasks, demonstrating superior results compared to traditional methods like parallel tempering. The article discusses the application of DiGS in various domains, including mixtures of Gaussians, Bayesian neural networks, and molecular dynamics. It also provides detailed insights into score-based MCMC methods, convolution-based techniques, initialization strategies, hyperparameter selection, and multi-level noise scheduling. Introduction Generating samples from complex unnormalized probability distributions is crucial. Goal: Draw independent samples from the target distribution and estimate expectations. Score-Based MCMC Methods Unadjusted Langevin Algorithm (ULA) follows a transition rule based on a discrete-time Langevin SDE. Metropolis-adjusted Langevin Algorithm (MALA) corrects bias using the Metropolis-Hasting algorithm. Convolution-Based Method Gaussian convolution bridges disconnected modes in distributions effectively. Convolved distribution exhibits better connectivity between modes than the original distribution. Diffusive Gibbs Sampling DiGS uses a Gibbs sampler to sample from joint distribution p(x, ˜x). Alternately draws samples from conditional distributions p(˜x|x) and p(x|˜x). Comparison to Related Methods Contrasts with tempering-based sampling, score-based diffusion models, proximal MCMC methods like HMC. Empirical Evaluation Evaluates DiGS on synthetic problems like MoG-40, Bayesian neural networks, and real-world applications like molecular dynamics. Conclusion DiGS offers significant improvements in sampling multi-modal distributions efficiently and accurately.

Stats

Our approach exhibits a better mixing property for sampling multi-modal distributions than state-of-the-art methods such as parallel tempering.

Quotes

"Our approach exhibits a better mixing property for sampling multi-modal distributions than state-of-the-art methods such as parallel tempering."

Key Insights Distilled From

Diffusive Gibbs Sampling

by Wenl... at arxiv.org 03-21-2024

https://arxiv.org/pdf/2402.03008.pdf

Deeper Inquiries

How does DiGS compare to other advanced sampling techniques

Diffusive Gibbs Sampling (DiGS) offers significant improvements in sampling multi-modal distributions compared to other advanced sampling techniques. In the context provided, DiGS outperforms traditional methods like Metropolis-Adjusted Langevin Algorithm (MALA), Hamiltonian Monte Carlo (HMC), and parallel tempering (PT). While MALA and HMC struggle to explore distant modes efficiently, PT can cover all modes but may not distribute samples accurately across different modes. On the other hand, DiGS demonstrates superior mode coverage by effectively exploring all modes with correct weightings. This is evident in tasks such as generating samples from a mixture of 40 Gaussians or Bayesian neural networks where DiGS significantly outperformed the baselines in terms of accuracy and efficiency.

What are the implications of improved mode coverage in multi-modal sampling

Improved mode coverage in multi-modal sampling has several implications: Enhanced Representation: By capturing a broader range of modes within a distribution, improved mode coverage ensures that the generated samples are more representative of the entire distribution. Better Generalization: A sampler with enhanced mode coverage can generalize better to unseen data points or configurations by providing a more comprehensive understanding of the underlying distribution. Reduced Bias: Improved mode coverage helps mitigate bias towards specific regions within a distribution, leading to more balanced and accurate estimates. Increased Efficiency: Efficiently covering multiple modes reduces computational costs associated with exploring disconnected regions separately, making sampling processes more efficient.

How can the concept of bridging disconnected modes be applied in other fields beyond machine learning

The concept of bridging disconnected modes through techniques like Gaussian convolution used in Diffusive Gibbs Sampling can be applied beyond machine learning in various fields: Physics - In molecular dynamics simulations or statistical mechanics studies, bridging disconnected energy states could help model complex systems accurately and explore phase spaces efficiently. Finance - Understanding market behaviors involves dealing with multimodal distributions; bridging disconnected financial trends could lead to better risk assessment models or portfolio optimization strategies. Healthcare - Analyzing patient data often involves dealing with diverse health outcomes; connecting disparate medical conditions through multimodal analysis could improve diagnostic accuracy and treatment planning. Climate Science - Studying climate patterns requires analyzing complex datasets with multiple interacting variables; bridging disconnected climate phenomena could enhance predictive modeling for weather forecasting or climate change projections. By applying the concept of bridging disconnected modes across these diverse fields, researchers can gain deeper insights into complex systems and make more informed decisions based on comprehensive data representations.