On sampling symmetric Gibbs distributions on sparse random graphs and hypergraphs

TL;DR

A novel approach is built which utilises the notion of contiguity between Gibbs distributions and the so-called teacher-student model and is presented as the first, rigorously analysed, efficient algorithm for spin-glasses which operates in a non trivial range of the parameters.

Abstract

We introduce efficient algorithms for approximate sampling from symmetric Gibbs distributions on the sparse random (hyper)graph. The examples we consider include (but are not restricted to) important distributions on spin systems and spin-glasses such as the q state antiferromagnetic Potts model for $q\geq 2$, including the colourings, the uniform distributions over the Not-All-Equal solutions of random k-CNF formulas. Finally, we present an algorithm for sampling from the spin-glass distribution called the k-spin model. To our knowledge this is the first, rigorously analysed, efficient algorithm for spin-glasses which operates in a non trivial range of the parameters. Our approach builds on the one that was introduced in [Efthymiou: SODA 2012]. For a symmetric Gibbs distribution $μ$ on a random (hyper)graph whose parameters are within an certain range, our algorithm has the following properties: with probability $1-o(1)$ over the input instances, it generates a configuration which is distributed within total variation distance $n^{-Ω(1)}$ from $μ$. The time complexity is $O((n\log n)^2)$. The algorithm requires a range of the parameters which, for the graph case, coincide with the tree-uniqueness region, parametrised w.r.t. the expected degree d. For the hypergraph case, where uniqueness is less restrictive, we go beyond uniqueness. Our approach utilises in a novel way the notion of contiguity between Gibbs distributions and the so-called teacher-student model.

Figures (9)

• Figure 1: Structure of the Analysis
• Figure 2: Factor Graph
• Figure 3: Configuration ${\mathbold{\tau}}(\beta)$ when \ref{['eq:UpdateRuleB']} applies.
• Figure 4:
• Figure 5:

Theorems & Definitions (73)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Definition 4.1: Factor graph
• Definition 5.1: ${\tt SET}$
• Theorem 5.2
• Theorem 5.3