Fastest Mixing Markov Chain Research Articles

Geometric bounds on the fastest mixing Markov chain

In the Fastest Mixing Markov Chain problem, we are given a graph G=(V,E)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G = (V, E)$$\\end{document} and desire the discrete-time Markov chain with smallest mixing time τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document} subject to having equilibrium distribution uniform on V and non-zero transition probabilities only across edges of the graph. It is well-known that the mixing time τRW\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au _\ extsf {RW}$$\\end{document} of the lazy random walk on G is characterised by the edge conductance Φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi $$\\end{document} of G via Cheeger’s inequality: Φ-1≲τRW≲Φ-2log|V|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi ^{-1} \\lesssim \ au _\ extsf {RW} \\lesssim \\Phi ^{-2} \\log |V|$$\\end{document}. Analogously, we characterise the fastest mixing time τ⋆\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au ^\\star $$\\end{document} via a Cheeger-type inequality but for a different geometric quantity, namely the vertex conductance Ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Psi $$\\end{document} of G: Ψ-1≲τ⋆≲Ψ-2(log|V|)2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Psi ^{-1} \\lesssim \ au ^\\star \\lesssim \\Psi ^{-2} (\\log |V|)^2$$\\end{document}. This characterisation forbids fast mixing for graphs with small vertex conductance. To bypass this fundamental barrier, we consider Markov chains on G with equilibrium distribution which need not be uniform, but rather only ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}-close to uniform in total variation. We show that it is always possible to construct such a chain with mixing time τ≲ε-1(diamG)2log|V|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au \\lesssim \\varepsilon ^{-1} ({\ ext {diam}} G)^2 \\log |V|$$\\end{document}. Finally, we discuss analogous questions for continuous-time and time-inhomogeneous chains.

Dimension reduction for maximum matchings and the Fastest Mixing Markov Chain

Dimension reduction for maximum matchings and the Fastest Mixing Markov Chain

Decentralized Federated Learning With Markov Chain Based Consensus for Industrial IoT Networks

Federated learning (FL) provides a novel framework to collaboratively train a shared model in a distribution fashion by virtue of a central server. However, FL is inappropriate for a serverless scenario and also suffers from some major drawbacks in Industrial Internet of Things (IIoT) networks, such as unresilience to network failures and communication bottleneck effect. In this article, we propose a novel decentralized federated learning (DFL) approach for IIoT devices to achieve model consensus by exchanging model parameters only with their neighbors rather than a central server. We firstly formulate the problem of model consensus in DFL as a fastest mixing Markov chain problem and then optimize the consensus matrix to improve the convergence rate. Meanwhile, a practical medium access control protocol with time slotted channel hopping is taken into account to implement the proposed approach. Furthermore, we also propose an accumulated update compression method to alleviate communication cost. Finally, extensive simulation results demonstrate that the proposed approach improves accuracy and reduces communication cost especially under the nonindependent identically distribution data distribution.

Fastest mixing reversible Markov chain on friendship graph: Trade-off between transition probabilities among friends and convergence rate

A long-standing goal of social network research has been to alter the properties of network to achieve the desired outcome. In doing so, DeGroot’s consensus model has served as the popular choice for modeling the information diffusion and opinion formation in social networks. Achieving a trade-off between the cost associated with modifications made to the network and the speed of convergence to the desired state has shown to be a critical factor. This has been treated as the Fastest Mixing Markov Chain (FMMC) problem over a graph with given transition probabilities over a subset of edges. Addressing this multi-objective optimization problem over the friendship graph, this paper has provided the corresponding Pareto optimal points or the Pareto frontier. In the case of friendship graph with at least three blades, it is shown that the Pareto frontier is reduced to a global minimum point which is same as the optimal point corresponding to the minimum spanning tree of the friendship graph, i.e., the star topology. Furthermore, a lower limit for transition probabilities among friends has been provided, where values higher than this limit do not have any impact on the convergence rate.

Quadratic Growth Conditions for Convex Matrix Optimization Problems Associated with Spectral Functions

In this paper, we provide two types of sufficient conditions for ensuring the quadratic growth conditions of a class of constrained convex symmetric and non-symmetric matrix optimization problems regularized by nonsmooth spectral functions. These sufficient conditions are derived via the study of the $\mathcal{C}^2$-cone reducibility of spectral functions and the metric subregularity of their subdifferentials, respectively. As an application, we demonstrate how quadratic growth conditions are used to guarantee the desirable fast convergence rates of the augmented Lagrangian methods (ALM) for solving convex matrix optimization problems. Numerical experiments on an easy-to-implement ALM applied to the fastest mixing Markov chain problem are also presented to illustrate the significance of the obtained results.

Sparse Solutions to the Average Consensus Problem via Various Regularizations of the Fastest Mixing Markov-Chain Problem

In the consensus problem on multi-agent systems, in which the states of the agents represent opinions, the agents aim at reaching a common opinion (or consensus state) through local exchange of information. An important design problem is to choose the degree of interconnection of the subsystems to achieve a good trade-off between a small number of interconnections and a fast convergence to the consensus state, which is the average of the initial opinions under mild conditions. This paper addresses this problem through $l_1$ -norm and $l_0$ -“pseudo-norm” regularized versions of the well-known Fastest Mixing Markov-Chain (FMMC) problem. We show that such versions can be interpreted as robust forms of the FMMC problem and provide results to guide the choice of the regularization parameter.

Stochastic adaptive sampling for mobile sensor networks using kernel regression

In this paper, we provide a stochastic adaptive sampling strategy for mobile sensor networks to estimate scalar fields over surveillance regions using kernel regression, which does not require a priori statistical knowledge of the field. Our approach builds on a Markov Chain Monte Carlo (MCMC) algorithm, viz., the fastest mixing Markov chain under a quantized finite state space, for generating the optimal sampling probability distribution asymptotically. The proposed adaptive sampling algorithm for multiple mobile sensors is numerically evaluated under scalar fields. The comparison simulation study with a random walk benchmark strategy demonstrates the excellent performance of the proposed scheme.

Fastest mixing Markov chain problem for the union of two cliques

A symmetric, random walk on a graph G can be defined by prescribing weights to the edges in such a way that for each vertex the sum of the weights of the edges incident to the vertex is at most one. The fastest mixing, Markov chain (FMMC) problem for G is to determine the weighting that yields the fastest mixing random walk. We solve the FMMC problem in the case that G is the union of two complete graphs.

Distributed Averaging Via Lifted Markov Chains

Motivated by applications of distributed linear estimation, distributed control, and distributed optimization, we consider the question of designing linear iterative algorithms for computing the average of numbers in a network. Specifically, our interest is in designing such an algorithm with the fastest rate of convergence given the topological constraints of the network. As the main result of this paper, we design an algorithm with the fastest possible rate of convergence using a nonreversible Markov chain on the given network graph. We construct such a Markov chain by transforming the standard Markov chain, which is obtained using the Metropolis-Hastings method. We call this novel transformation <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pseudo-lifting</i> . We apply our method to graphs with geometry, or graphs with doubling dimension. Specifically, the convergence time of our algorithm (equivalently, the mixing time of our Markov chain) is proportional to the diameter of the network graph and hence optimal. As a byproduct, our result provides the fastest mixing Markov chain given the network topological constraints, and should naturally find their applications in the context of distributed optimization, estimation and control.

Fastest Mixing Markov Chain on Graphs with Symmetries

We show how to exploit symmetries of a graph to efficiently compute the fastest mixing Markov chain on the graph (i.e., find the transition probabilities on the edges to minimize the second-largest eigenvalue modulus of the transition probability matrix). Exploiting symmetry can lead to significant reduction in both the number of variables and the size of matrices in the corresponding semidefinite program, thus enable numerical solution of large-scale instances that are otherwise computationally infeasible. We obtain analytic or semi-analytic results for particular classes of graphs, such as edge-transitive and distance-transitive graphs. We describe two general approaches for symmetry exploitation, based on orbit theory and block-diagonalization, respectively. We also establish the connection between these two approaches.

Inequalities for the spectra of symmetric doubly stochastic matrices

We prove majorization relations for symmetric doubly stochastic matrices whose nonzero off-diagonal entries lie in an edge transitive graph. These results are then used to solve the fastest mixing Markov chain problem on edge transitive graphs.

Fastest Mixing Markov Chain on a Path

Fastest Mixing Markov Chain on a Path

Fastest Mixing Markov Chain on a Path

(2006). Fastest Mixing Markov Chain on a Path. The American Mathematical Monthly: Vol. 113, No. 1, pp. 70-74.

Bounding Fastest Mixing

In a recent work, Boyd, Diaconis and Xiao introduced a semidefinite programming approach for computing the fastest mixing Markov chain on a graph of allowed transitions, given a target stationary distribution. In this paper, we show that standard mixing time analysis techniques—variational characterizations, conductance, canonical paths—can be used to give simple, nontrivial lower and upper bounds on the fastest mixing time. To test the applicability of this idea, we consider several detailed examples including the Glauber dynamics of the Ising model.

Fastest Mixing Markov Chain on a Graph

We consider a symmetric random walk on a connected graph, where each edge is labeled with the probability of transition between the two adjacent vertices. The associated Markov chain has a uniform equilibrium distribution; the rate of convergence to this distribution, i.e., the mixing rate of the Markov chain, is determined by the second largest eigenvalue modulus (SLEM) of the transition probability matrix. In this paper we address the problem of assigning probabilities to the edges of the graph in such a way as to minimize the SLEM, i.e., the problem of finding the fastest mixing Markov chain on the graph. We show that this problem can be formulated as a convex optimization problem, which can in turn be expressed as a semidefinite program (SDP). This allows us to easily compute the (globally) fastest mixing Markov chain for any graph with a modest number of edges (say, $1000$) using standard numerical methods for SDPs. Larger problems can be solved by exploiting various types of symmetry and structure in the problem, and far larger problems (say, 100,000 edges) can be solved using a subgradient method we describe. We compare the fastest mixing Markov chain to those obtained using two commonly used heuristics: the maximum-degree method, and the Metropolis--Hastings algorithm. For many of the examples considered, the fastest mixing Markov chain is substantially faster than those obtained using these heuristic methods. We derive the Lagrange dual of the fastest mixing Markov chain problem, which gives a sophisticated method for obtaining (arbitrarily good) bounds on the optimal mixing rate, as well as the optimality conditions. Finally, we describe various extensions of the method, including a solution of the problem of finding the fastest mixing reversible Markov chain, on a fixed graph, with a given equilibrium distribution.