Local Gibbs sampling beyond local uniformity

2025-02-15

AI Summary

This work addresses local sampling from high-dimensional distributions—such as Ising models—overcoming the fundamental limitation of existing algorithms that require a “local uniformity” assumption. We propose the first local Gibbs sampler that operates without this assumption, built upon a local query framework, local induced distribution analysis for high-dimensional models, and adaptive Markov chain construction. Our method enables efficient approximate sampling from near-critical spin systems on unbounded-degree graphs. Theoretically, it achieves exponential speedup in local time complexity compared to prior state-of-the-art methods, and natively supports dynamic graph updates. Crucially, this is the first algorithm to achieve truly local, efficient sampling in the near-critical regime—where correlations become long-range—thereby significantly extending the applicability boundary of local sampling techniques to previously intractable parameter regimes.

Technology Category

• Search and Optimization: Local Search
• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

Local samplers are algorithms that generate random samples based on local queries to high-dimensional distributions, ensuring the samples follow the correct induced distributions while maintaining time complexity that scales locally with the query size. These samplers have broad applications, including deterministic approximate counting [He, Wang, Yin, SODA '23; Feng et al., FOCS '23], sampling from infinite or high-dimensional Gibbs distributions [Anand, Jerrum, SICOMP '22; He, Wang, Yin, FOCS '22], and providing local access to large random objects [Biswas, Rubinfield, Yodpinyanee, ITCS '20]. In this work, we present a local sampler for Gibbs distributions of spin systems whose efficiency does not rely on the"local uniformity"property, which imposes unconditional marginal lower bounds -- a key assumption required by all prior local samplers. For fundamental models such as the Ising model, our algorithm achieves local efficiency in near-critical regimes, providing an exponential improvement over existing methods. Additionally, our approach is applicable to spin systems on graphs with unbounded degrees and supports dynamic sampling within the same near-critical regime.

Problem

• Develops local sampler for Gibbs distributions
• Eliminates reliance on local uniformity
• Improves efficiency in near-critical regimes

Innovation

• Local Gibbs sampling algorithm
• Exponential efficiency improvement
• Dynamic sampling in near-critical regimes