[Paper Review] Distributed Symmetry Breaking in Sampling (Optimal Distributed Randomly Coloring with Fewer Colors)

Weiming Feng, Thomas P. Hayes+1 more|arXiv (Cornell University)|Feb 20, 2018

Markov Chains and Monte Carlo Methods37 references12 citations

TL;DR

This paper introduces the Lazy Local Metropolis algorithm, a distributed MCMC method for almost-uniform sampling of proper q-colorings in graphs. By integrating distributed symmetry breaking into a local Metropolis filtering framework, it achieves optimal O(log n) mixing time under improved color bounds: q ≥ (2 + δ)∆ for general graphs and q ≥ (α* + δ)∆ with α* ≈ 1.763 for high-girth, high-degree graphs, outperforming prior parallelized Glauber dynamics by eliminating the Ω(∆) slowdown factor.

ABSTRACT

Markov Chain Monte Carlo (MCMC) algorithms are a widely-used algorithmic tool for sampling from high-dimensional distributions, a notable example is the equilibirum distribution of graphical models. The Glauber dynamics, also known as the Gibbs sampler, is the simplest example of an MCMC algorithm; the transitions of the chain update the configuration at a randomly chosen coordinate at each step. Several works have studied distributed versions of the Glauber dynamics and we extend these efforts to a more general family of Markov chains. An important combinatorial problem in the study of MCMC algorithms is random colorings. Given a graph G of maximum degree Δ and an integer k ≥ Δ+1, the goal is to generate a random proper vertex k-coloring of G. Jerrum (1995) proved that the Glauber dynamics has O(nlog{n}) mixing time when k > 2Δ. Fischer and Ghaffari (2018), and independently Feng, Hayes, and Yin (2018), presented a parallel and distributed version of the Glauber dynamics which converges in O(log{n}) rounds for k > (2+ε)Δ for any ε > 0. We improve this result to k > (11/6-δ)Δ for a fixed δ > 0. This matches the state of the art for randomly sampling colorings of general graphs in the sequential setting. Whereas previous works focused on distributed variants of the Glauber dynamics, our work presents a parallel and distributed version of the more general flip dynamics presented by Vigoda (2000) (and refined by Chen, Delcourt, Moitra, Perarnau, and Postle (2019)), which recolors local maximal two-colored components in each step.

Motivation & Objective

• To develop a distributed sampling algorithm that achieves optimal parallel speedup for almost-uniform proper q-coloring of graphs.
• To reduce the required number of colors q below the (2 + δ)∆ threshold used in prior parallel MCMC methods, especially for graphs with large girth and high degree.
• To overcome the inherent Ω(∆) time penalty in previous parallel Glauber dynamics by introducing a novel distributed symmetry-breaking step.
• To achieve optimal O(log n) mixing time under improved color bounds, matching the theoretical lower bound for message-passing distributed algorithms.

Proposed method

• Proposes the Lazy Local Metropolis algorithm, a distributed MCMC sampler that extends the Local Metropolis filtering paradigm with a new symmetry-breaking step.
• Each vertex independently proposes a random color and applies a local acceptance rule based on neighbor colorings, with a delayed update mechanism to reduce interference.
• Employs a path coupling argument with a carefully constructed coupling scheme to bound the Hamming distance between two coupled chains.
• Introduces a two-phase analysis: a burn-in phase [0, Tb] to reduce disagreement, followed by a contraction phase [Tb, Tm] where disagreements shrink by a constant factor per round.
• Uses a disagreement percolation argument to show that non-typical (bad) events occur with exponentially small probability, enabling concentration bounds.
• Applies a refined coupling to show that under good events, the expected number of differing vertices decreases by a constant factor per round, leading to O(log n) mixing time.

Experimental results

Research questions

• RQ1 Can the mixing time of distributed MCMC sampling for proper q-colorings be reduced to O(log n) while using fewer colors than the (2 + δ)∆ threshold?
• RQ2 Does the introduction of distributed symmetry breaking in the Local Metropolis framework enable faster mixing and improved color bounds?
• RQ3 Can the algorithm achieve optimal O(log n) mixing time for graphs with large girth and high maximum degree under the condition q ≥ (α* + δ)∆ with α* ≈ 1.763?
• RQ4 Is the O(log n) mixing time optimal for distributed message-passing algorithms when q = O(∆)?
• RQ5 How can path coupling be adapted in a distributed setting to analyze convergence under local, asynchronous updates?

Key findings

• The Lazy Local Metropolis algorithm achieves O(log n) mixing time for general graphs when q ≥ (2 + δ)∆, matching the optimal speedup of sequential Glauber dynamics.
• For graphs with girth ≥9 and sufficiently large maximum degree ∆ ≥ ∆₀(δ), the algorithm achieves O(log n) mixing time under the improved condition q ≥ (α* + δ)∆ with α* ≈ 1.763 satisfying α* = e^{1/α*}.
• The algorithm improves upon the Local Metropolis algorithm by reducing the required number of colors, achieving optimal parallelization without the Θ(∆) slowdown penalty.
• The mixing time is proven to be optimal for message-passing distributed algorithms, as O(log n) is the best possible under q = O(∆).
• The analysis shows that with high probability, the Hamming distance between two coupled chains decreases by a constant factor per round during the contraction phase, leading to logarithmic convergence.
• The probability of non-typical (bad) events—where disagreements percolate or color availability is low—is shown to be exponentially small, ensuring concentration of measure.