Swendsen‐Wang dynamics for general graphs in the tree uniqueness region

Blanca, Antonio; Chen, Zongchen; Vigoda, Eric

doi:10.1002/rsa.20858

“…Previously, our understanding of the speed of convergence of the SW dynamics on random Δ-regular graphs was very limited. For the special case of q = 2, which corresponds to the Ising model, it was established in [4] that the spectral gap of the SW dynamics is Ω(1) for all p < p u (2, Δ); this implies an O(n) mixing time bound. In addition, Guo and Jerrum [33] established an O(n 10 ) mixing time bound for the SW dynamics that applies to any graph and any p ∈ (0, 1).…”

Section: Log(1/δ))mentioning

confidence: 99%

Random-Cluster Dynamics on Random Regular Graphs in Tree Uniqueness

Blanca

¹

,

Gheissari

²

2021

Commun. Math. Phys.

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We establish rapid mixing of the random-cluster Glauber dynamics on random $$\varDelta $$ Δ -regular graphs for all $$q\ge 1$$ q ≥ 1 and $$p<p_u(q,\varDelta )$$ p < p u ( q , Δ ) , where the threshold $$p_u(q,\varDelta )$$ p u ( q , Δ ) corresponds to a uniqueness/non-uniqueness phase transition for the random-cluster model on the (infinite) $$\varDelta $$ Δ -regular tree. It is expected that this threshold is sharp, and for $$q>2$$ q > 2 the Glauber dynamics on random $$\varDelta $$ Δ -regular graphs undergoes an exponential slowdown at $$p_u(q,\varDelta )$$ p u ( q , Δ ) . More precisely, we show that for every $$q\ge 1$$ q ≥ 1 , $$\varDelta \ge 3$$ Δ ≥ 3 , and $$p<p_u(q,\varDelta )$$ p < p u ( q , Δ ) , with probability $$1-o(1)$$ 1 - o ( 1 ) over the choice of a random $$\varDelta $$ Δ -regular graph on n vertices, the Glauber dynamics for the random-cluster model has $$\varTheta (n \log n)$$ Θ ( n log n ) mixing time. As a corollary, we deduce fast mixing of the Swendsen–Wang dynamics for the Potts model on random $$\varDelta $$ Δ -regular graphs for every $$q\ge 2$$ q ≥ 2 , in the tree uniqueness region. Our proof relies on a sharp bound on the “shattering time”, i.e., the number of steps required to break up any configuration into $$O(\log n)$$ O ( log n ) sized clusters. This is established by analyzing a delicate and novel iterative scheme to simultaneously reveal the underlying random graph with clusters of the Glauber dynamics configuration on it, at a given time.

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“…This is compatible with the analogous results in Markov chain literature. See the introduction of [ 32 ] for a well written summary of Markov chain mixing in the Ising model.…”

Section: Why the Ising Model: A Summary Of Our Contributionsmentioning

confidence: 99%

Dynamics of Coordinate Ascent Variational Inference: A Case Study in 2D Ising Models

Plummer

¹

,

Pati

²

,

Bhattacharya

³

2020

Entropy

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Variational algorithms have gained prominence over the past two decades as a scalable computational environment for Bayesian inference. In this article, we explore tools from the dynamical systems literature to study the convergence of coordinate ascent algorithms for mean field variational inference. Focusing on the Ising model defined on two nodes, we fully characterize the dynamics of the sequential coordinate ascent algorithm and its parallel version. We observe that in the regime where the objective function is convex, both the algorithms are stable and exhibit convergence to the unique fixed point. Our analyses reveal interesting discordances between these two versions of the algorithm in the region when the objective function is non-convex. In fact, the parallel version exhibits a periodic oscillatory behavior which is absent in the sequential version. Drawing intuition from the Markov chain Monte Carlo literature, we empirically show that a parameter expansion of the Ising model, popularly called the Edward–Sokal coupling, leads to an enlargement of the regime of convergence to the global optima.

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“…There is a long line of work studying the connection between spatial mixing (i.e., decay of correlations) properties of Gibbs distributions and the speed of convergence of reversible Markov chains (see, e.g., [30,1,50,47,38,39,15,20,44]). These results focus on local Markov chains, such as the Glauber dynamics, but there has also been some recent progress in understanding this connection for non-local Markov chains such as the SW dynamics [6,5,13]. In particular, it was established in [5] that the strong spatial mixing (SSM) property implies that the mixing time T mix (SW ) of the SW dynamics is O(n), where n := |V | is the number of vertices.…”

Section: Introductionmentioning

confidence: 99%

Entropy decay in the Swendsen-Wang dynamics on ${\mathbb Z}^d$

Blanca,

Caputo,

Parisi

et al. 2020

Preprint

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We study the mixing time of the Swendsen-Wang dynamics for the ferromagnetic Ising and Potts models on the integer lattice Z d . This dynamics is a widely used Markov chain that has largely resisted sharp analysis because it is non-local, i.e., it changes the entire configuration in one step. We prove that, whenever strong spatial mixing (SSM) holds, the mixing time on any n-vertex cube of Z d is O(log n), improving on the previous best known bound of O(n). SSM is a standard condition corresponding to exponential decay of correlations with distance between spins on the lattice and is known to hold in d = 2 dimensions throughout the high-temperature (single phase) region. Our result follows from a modified log-Sobolev inequality, which expresses the fact that the dynamics contracts relative entropy at a constant rate at each step. The proof of this fact utilizes a new factorization of the entropy in the joint probability space over spins and edges that underlies the Swendsen-Wang dynamics. This factorization leads to several additional results, including mixing time bounds for a number of natural local and non-local Markov chains on the joint space, as well as for the standard random-cluster dynamics.

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Swendsen‐Wang dynamics for general graphs in the tree uniqueness region

Cited by 14 publications

References 58 publications

Random-Cluster Dynamics on Random Regular Graphs in Tree Uniqueness

Random-Cluster Dynamics on Random Regular Graphs in Tree Uniqueness

Dynamics of Coordinate Ascent Variational Inference: A Case Study in 2D Ising Models

Entropy decay in the Swendsen-Wang dynamics on ${\mathbb Z}^d$

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