On Sampling Symmetric Gibbs Distributions on Sparse Random Graphs and Hypergraphs

Abstract

In this paper, we present a novel, polynomial time, algorithm for approximate sampling from symmetric Gibbs distributions on the sparse random graph and hypergraph. The examples of symmetric distributions we consider here include some important distributions on spin-systems and spin-glasses. These are: the q-state antiferromagnetic Potts model for q ≥ 2, including the (hyper)graph Ising model and random colourings. The uniform distribution over the Not-All-Equal solutions of a random k-SAT formula. Finally, we consider sampling from the spin-glass distribution called the k-spin model, i.e., this is the "diluted" version of the well-known Sherrington-Kirkpatrick model. Spin-glasses give rise to very intricate distributions which are also studied in mathematics, in neural computation, computational biology and many other areas. To our knowledge, this is the first rigorously analysed efficient sampling algorithm for spin-glasses which operates in a non trivial range of the parameters of the distribution. 
We present, what we believe to be, an elegant sampling algorithm. Our algorithm is unique in its approach and does not belong to any of the well-known families of sampling algorithms. We derive it by investigating the power and the limits of the approach that was introduced in [Efthymiou: SODA 2012] and combine it, in a novel way, with powerful notions from the Cavity method. 
Specifically, for a symmetric Gibbs distribution μ on the random (hyper)graph whose parameters are within an appropriate range, our sampling algorithm has the following properties: with probability 1-o(1) over the instances of the input (hyper)graph, it generates a configuration which is distributed within total variation distance n^{-Ω(1)} from μ. The time complexity is O((nlog n)²), where n is the size of the input (hyper)graph. 
We make a notable progress regarding impressive predictions of physicists relating phase-transitions of Gibbs distributions with the efficiency of the corresponding sampling algorithms. For most (if not all) the cases we consider here, our algorithm outperforms by far any other sampling algorithms in terms of the permitted range of the parameters of the Gibbs distributions. 
The use of notions and ideas from the Cavity method provides a new insight to the sampling problem. Our results imply that there is a lot of potential for further exploiting the Cavity method for algorithmic design.