Convergence of MCMC and Loopy BP in the Tree Uniqueness Region for the Hard-Core Model

Efthymiou, Charilaos; Hayes, Thomas P.; Štefankovič, Daniel; Vigoda, Eric; Yin, Yitong

doi:10.1137/17m1127144

Cited by 18 publications

(10 citation statements)

References 47 publications

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“…Here we want to utilize that when a vertex z has large internal branching factor (i.e., most of z's neighbors are internal to the block) then these neighbors are not worst-case but are from the stationary distribution of the block (conditional on a fixed coloring on the block's outer boundary). Then we want to exploit the so-called "local uniformity results" first utilized by Dyer and Frieze [8] (and then expanded upon in [19,17,9,12]). The relevant property in this context is that if a set of ∆ vertices receive independently at random colors (uniformly distributed over all k colors) then the expected number of available colors (i.e., colors that do not appear in this set) is ≈ k exp(−∆/k).…”

Section: 2mentioning

confidence: 99%

Sampling Random Colorings of Sparse Random Graphs

Efthymiou

Hayes²,

Štefankovič

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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We study the mixing properties of the single-site Markov chain known as the Glauber dynamics for sampling k-colorings of a sparse random graph G(n, d/n) for constant d. The best known rapid mixing results for general graphs are in terms of the maximum degree ∆ of the input graph G and hold when k > 11∆/6 for all G. Improved results hold when k > α∆ for graphs with girth ≥ 5 and ∆ sufficiently large where α ≈ 1.7632 . . . is the root of α = exp(1/α); further improvements on the constant α hold with stronger girth and maximum degree assumptions.For sparse random graphs the maximum degree is a function of n and the goal is to obtain results in terms of the expected degree d. The following rapid mixing results for G(n, d/n) hold with high probability over the choice of the random graph for sufficiently large constant d. Mossel and Sly (2009) proved rapid mixing for constant k, and Efthymiou (2014) improved this to k linear in d. The condition was improved to k > 3d by Yin and Zhang (2016) using non-MCMC methods.Here we prove rapid mixing when k > αd where α ≈ 1.7632 . . . is the same constant as above. Moreover we obtain O(n 3 ) mixing time of the Glauber dynamics, while in previous rapid mixing results the exponent was an increasing function in d. Our proof analyzes an appropriately defined block dynamics to "hide" high-degree vertices. One new aspect in our improved approach is utilizing so-called local uniformity properties for the analysis of block dynamics. To analyze the "burn-in" phase we prove a concentration inequality for the number of disagreements propagating in large blocks. * A full version of this work is available online at https: //arxiv.org/abs/1707.03796.

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Section: 2mentioning

confidence: 99%

Sampling Random Colorings of Sparse Random Graphs

Efthymiou

Hayes²,

Štefankovič

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…In the low fugacity regime with λ < λ c (∆), Weitz's algorithm applies to G bip n,∆ , and Efthymiou, Hayes,Štefankovic, Vigoda, and Yin [14] have also shown that the Glauber dynamics have mixing time O(n log n). For λ > λ c (∆), the Glauber dynamics for the hard-core model on G bip n,∆ are known to mix slowly [29], and perhaps Theorem 2 can be improved to work for all λ > λ c (∆).…”

Section: Discussionmentioning

confidence: 99%

Algorithms for #BIS-hard problems on expander graphs

Jenssen

Keevash

Perkins

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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We give an FPTAS and an efficient sampling algorithm for the high-fugacity hard-core model on bounded-degree bipartite expander graphs and the low-temperature ferromagnetic Potts model on bounded-degree expander graphs. The results apply, for example, to random (bipartite) ∆-regular graphs, for which no efficient algorithms were known for these problems (with the exception of the Ising model) in the non-uniqueness regime of the infinite ∆-regular tree. We also find efficient counting and sampling algorithms for proper q-colorings of random ∆-regular bipartite graphs when q is sufficiently small as a function of ∆. * University of Oxford, jenssen@maths.ox.ac.uk. † University of Oxford, keevash@maths.ox.ac.uk. Research supported in part by ERC Consolidator Grant 647678. ‡ University of Illinois at Chicago, math@willperkins.org. Research supported in part by EPSRC grant EP/P009913/1.1 is the hard-core partition function (also known as the independence polynomial in graph theory).A fully polynomial-time approximation scheme (FPTAS) is an algorithm that for every ε > 0 outputs an ε-relative approximation to Z G (λ) (that is, a numberẐ so that e −εẐ ≤ Z G (λ) ≤ e εẐ ) and runs in time polynomial in |V (G)| and 1/ε. An efficient sampling scheme is a randomized algorithm that for every ε > 0 runs in time polynomial in |V (G)| and 1/ε and outputs an independent set I with distribution µ alg so that µ G,λ − µ alg T V < ε.The computational complexity of the approximate counting and sampling problems for the hard-core model is well understood for bounded-degree graphs. For graphs of maximum degree at most ∆, when λ < λ c (∆) = (∆−1) ∆−1 (∆−2) ∆ , there is an FPTAS and an efficient sampling scheme due to Weitz [37]; whereas when λ > λ c (∆) both computational problems are hard: there is no polynomial-time algorithm unless NP=RP [34,35,15]. The value λ c (∆) is the uniqueness threshold of the hard-core model on the infinite ∆-regular tree [24].On the other hand, if we restrict ourselves to bipartite graphs, then the computational complexity of these tasks are open problems. The class #BIS is the class of problems polynomial-time equivalent to approximating the number of independent sets of a bipartite graph [13], and many interesting approximate counting and sampling problems have been shown to be #BIS-hard [18,10,16]. In particular, Cai, Galanis, Goldberg, Guo, Jerrum,Štefankovič, and Vigoda [9] showed that for all ∆ ≥ 3 and all λ > λ c (∆), it is #BIS-hard to approximate the hard-core partition function at fugacity λ on a bipartite graph of maximum degree ∆. Resolving the complexity of #BIS is a major open problem in the field of approximate counting.One direction for partial progress on any intermediate complexity class is to find subclasses of instances for which the problem is tractable (e.g. results showing that the Unique Games problem is tractable on expander graphs [1,28]). For #BIS, we would like to find subclasses of bipartite graphs on which we can efficiently approximate the number of independent sets or the hard-...

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“…For example, for the hardcore model, such result would give an efficient algorithm for sampling from the hardcore model in the uniqueness regime, on all graphs including those with unbounded maximum degree, which remains to be open even for static and approximate sampling. So far we only have efficient static and approximate sampling algorithms for graphs with bounded maximum degree [47] or graphs with large girth and sufficiently large degree [6].…”

Section: Discussionmentioning

confidence: 99%

“…Previously, the Glauber dynamics is known to be rapidly mixing for the hardcore model with λ < λ c (∆) on amenable graphs [15,47] as well as graphs with large girth and degree [6], and also with λ ≤ 2 ∆−2 [44,5] on general graphs. And the perfect sampling methods of [9,26] giveÕ(n)-time perfect samplers also when λ ≤ 2 ∆−2 .…”

Section: Dynamic Sampling From the Spin Systemsmentioning

confidence: 99%

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Dynamic sampling from graphical models

Feng

Vishnoi

Yin

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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In this paper, we study the problem of sampling from a graphical model when the model itself is changing dynamically with time. This problem derives its interest from a variety of inference, learning, and sampling settings in machine learning, computer vision, statistical physics, and theoretical computer science. While the problem of sampling from a static graphical model has received considerable attention, theoretical works for its dynamic variants have been largely lacking. The main contribution of this paper is an algorithm that can sample dynamically from a broad class of graphical models over discrete random variables. Our algorithm is parallel and Las Vegas: it knows when to stop and it outputs samples from the exact distribution. We also provide sufficient conditions under which this algorithm runs in time proportional to the size of the update, on general graphical models as well as well-studied specific spin systems. In particular we obtain, for the Ising model (ferromagnetic or anti-ferromagnetic) and for the hardcore model the first dynamic sampling algorithms that can handle both edge and vertex updates (addition, deletion, change of functions), both efficient within regimes that are close to the respective uniqueness regimes, beyond which, even for the static and approximate sampling, no local algorithms were known or the problem itself is intractable. Our dynamic sampling algorithm relies on a local resampling algorithm and a new "equilibrium" property that is shown to be satisfied by our algorithm at each step, and enables us to prove its correctness. This equilibrium property is robust enough to guarantee the correctness of our algorithm, helps us improve bounds on fast convergence on specific models, and should be of independent interest.

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Convergence of MCMC and Loopy BP in the Tree Uniqueness Region for the Hard-Core Model

Cited by 18 publications

References 47 publications

Sampling Random Colorings of Sparse Random Graphs

Sampling Random Colorings of Sparse Random Graphs

Algorithms for #BIS-hard problems on expander graphs

Dynamic sampling from graphical models

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