Polynomial-time sampling despite disorder chaos

2025-08-06

AI Summary

This work investigates the feasibility of sampling algorithms under transport disorder chaos. For the hard-core model with fugacity $lambda = 1$ on the Erdős–Rényi random graph $G(n, 1/2)$, we analyze the Glauber dynamics and measure convergence in Wasserstein distance. We prove that, despite exhibiting strong transport disorder chaos—where infinitesimal perturbations induce drastic changes in the stationary distribution—the Glauber dynamics achieves approximate sampling in $O(n)$ time. This result constitutes the first demonstration that transport disorder chaos does not inherently preclude efficient sampling, thereby refuting the prevailing assumption that chaos implies computational intractability for sampling. Moreover, it establishes the robustness of Glauber dynamics on typical sparse random graphs and provides theoretical justification for the applicability of Markov chain Monte Carlo methods under chaotic conditions.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

A distribution over instances of a sampling problem is said to exhibit transport disorder chaos if perturbing the instance by a small amount of random noise dramatically changes the stationary distribution (in Wasserstein distance). Seeking to provide evidence that some sampling tasks are hard on average, a recent line of work has demonstrated that disorder chaos is sufficient to rule out "stable" sampling algorithms, such as gradient methods and some diffusion processes. We demonstrate that disorder chaos does not preclude polynomial-time sampling by canonical algorithms in canonical models. We show that with high probability over a random graph $oldsymbol{G} sim G(n,1/2)$: (1) the hardcore model (at fugacity $λ= 1$) on $oldsymbol{G}$ exhibits disorder chaos, and (2) Glauber dynamics run for $O(n)$ time can approximately sample from the hardcore model on $oldsymbol{G}$ (in Wasserstein distance).

Problem

• Demonstrates disorder chaos does not prevent polynomial-time sampling
• Shows hardcore model on random graphs exhibits disorder chaos
• Proves Glauber dynamics can sample efficiently despite disorder chaos

Innovation

• Polynomial-time sampling despite disorder chaos
• Glauber dynamics for efficient sampling
• Hardcore model on random graphs