Polynomial-time sampling despite disorder chaos

TL;DR

This paper shows that even in models exhibiting disorder chaos, polynomial-time sampling algorithms like Glauber dynamics can still efficiently approximate distributions, challenging previous assumptions about the hardness implied by disorder chaos.

Contribution

It proves that disorder chaos does not necessarily prevent polynomial-time sampling, demonstrating this with the hardcore model on random graphs and Glauber dynamics.

Findings

Hardcore model on random graphs exhibits disorder chaos.

Glauber dynamics can sample efficiently in polynomial time.

Disorder chaos does not imply sampling hardness.

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 $\boldsymbol{G} \sim G(n,1/2)$: (1) the hardcore model (at fugacity $\lambda = 1$) on $\boldsymbol{G}$ exhibits disorder chaos, and (2) Glauber dynamics run for $O(n)$ time can approximately sample from the hardcore model on $\boldsymbol{G}$ (in Wasserstein distance).