Log-concave polynomials IV: approximate exchange, tight mixing times, and near-optimal sampling of forests

Anari, Nima; Liu, Kuikui; Gharan, Shayan Oveis; Vinzant, Cynthia; Vuong, Thuy-Duong

doi:10.1145/3406325.3451091

Cited by 17 publications

(13 citation statements)

References 25 publications

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“…It is not hard to see, using the base-exchange property, that this procedure defines an ergodic, time-reversible Markov chain with stationary distribution π. Anari et al [3] show that this chain is rapidly mixing for any matroid N . In particular, in a recent follow-up work, they give a (tight) mixing time bound [5].…”

Section: Markov Chainsmentioning

confidence: 87%

“…steps, where n is the number of players, P the number of paths in the EP graph and Φ max the maximum value attained by Rosenthal's potential. 5 The notion of "ǫ-close" refers to the fact that the distribution seen after the indicated number of steps differs from the Gibbs distribution at most ǫ in total variation distance (see Section 2.4), a well-known distance measure for comparing probability distributions in Markov chain theory.…”

Section: Our Contributions and Techniquesmentioning

confidence: 99%

“…M-convexity is a property defined in the area of discrete convex analysis [41] (see Section 2.2). 6 The link between M-convexity and sampling has, roughly speaking, been established in a series of papers by Anari et al [2,4,5] through the theory of strongly log-concave polynomials, which in turn is also largely developed by Brändén and Huh [13] (under the name Lorentzian polynomials). In particular, Anari et al [4] give the first polynomial time algorithm for approximately sampling and counting the number of bases of a given matroid, resolving also an old conjecture by Mihail and Vazirani [40].…”

Section: Our Contributions and Techniquesmentioning

confidence: 99%

“…Theorem 2.7 (Anari et al [5]). Let N be matroid of rank r, and let π be an SLC probability distribution over the set of bases B given by π(α) ∝ w(α) for some weight function w : K → R ≥0 .…”

Section: Markov Chainsmentioning

confidence: 99%

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Sampling from the Gibbs Distribution in Congestion Games

Kleer¹

2021

Preprint

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Logit dynamics is a form of randomized game dynamics where players have a bias towards strategic deviations that give a higher improvement in cost. It is used extensively in practice. In congestion (or potential) games, the dynamics converges to the so-called Gibbs distribution over the set of all strategy profiles, when interpreted as a Markov chain. In general, logit dynamics might converge slowly to the Gibbs distribution, but beyond that, not much is known about their algorithmic aspects, nor that of the Gibbs distribution. In this work, we are interested in the following two questions for congestion games: i) Is there an efficient algorithm for sampling from the Gibbs distribution? ii) If yes, do there also exist natural randomized dynamics that converges quickly to the Gibbs distribution?We first study these questions in extension parallel congestion games, a well-studied special case of symmetric network congestion games. As our main result, we show that there is a simple variation on the logit dynamics (in which we in addition are also allowed to randomly interchange the strategies of two players) that converges quickly to the Gibbs distribution in such games. This answers both questions above affirmatively. We also address the first question for the class of so-called capacitated k-uniform congestion games.To prove our results, we rely on the recent breakthrough work of Anari, Liu, Oveis-Gharan and Vinzant (2019) concerning the approximate sampling of the base of a matroid according to strongly log-concave probability distribution.

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Section: Markov Chainsmentioning

confidence: 87%

Section: Our Contributions and Techniquesmentioning

confidence: 99%

Section: Our Contributions and Techniquesmentioning

confidence: 99%

Section: Markov Chainsmentioning

confidence: 99%

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Sampling from the Gibbs Distribution in Congestion Games

Kleer¹

2021

Preprint

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“…Entropy based techniques that could prove modified log-Sobolev inequalities (MLSI) were considered [CGM21,ALOV20]. Although modified log-Sobolev (MLS) constants can give tight bounds on mixing times, they are notoriously difficult to analyze.…”

mentioning

confidence: 99%

Optimal mixing for two-state anti-ferromagnetic spin systems

Chen¹,

Feng²,

Yin³

et al. 2022

Preprint

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A. We prove an optimal Ω −1 lower bound for modified log-Sobolev (MLS) constant of the Glauber dynamics for anti-ferromagnetic two-spin systems with vertices in the tree uniqueness regime. Specifically, this optimal MLS bound holds for the following classes of two-spin systems in the tree uniqueness regime:• all strictly anti-ferromagnetic two-spin systems (where both edge parameters , < 1), which cover the hardcore models and the anti-ferromagnetic Ising models; • general anti-ferromagnetic two-spin systems on regular graphs. Consequently, an optimal ( log ) mixing time holds for these anti-ferromagnetic two-spin systems when the uniqueness condition is satisfied. These MLS and mixing time bounds hold for any bounded or unbounded maximum degree, and the constant factors in the bounds depend only on the gap to the uniqueness threshold. We prove this by showing a boosting theorem for MLS constant for distributions satisfying certain spectral independence and marginal stability properties.

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A Queueing Network-Based Distributed Laplacian Solver

Gillani

Bagchi

2021

Algorithmica

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Log-concave polynomials IV: approximate exchange, tight mixing times, and near-optimal sampling of forests

Cited by 17 publications

References 25 publications

Sampling from the Gibbs Distribution in Congestion Games

Sampling from the Gibbs Distribution in Congestion Games

Optimal mixing for two-state anti-ferromagnetic spin systems

A Queueing Network-Based Distributed Laplacian Solver

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