A cycle-based bound for subdominant eigenvalues of stochastic matrices

Kirkland, Steve

doi:10.1080/03081080701669309

Cited by 13 publications

(6 citation statements)

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“…Thus we find that the modulus of the subdominant eigenvalue of a stochastic matrix is critical in determining the long-term behaviour of the corresponding Markov chain. Because of that fact, there is a body of work on estimating the modulus of the subdominant eigenvalue for a stochastic matrix; see for instance [4,5,7,8].…”

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confidence: 99%

Subdominant eigenvalues for stochastic matrices with given column sums

Kirkland

2009

ELA

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Abstract. For any stochastic matrix A of order n, denote its eigenvalues as λ 1 (A), . . . , λn(A),Let c T be a row vector of order n whose entries are nonnegative numbers that sum to n. Define S(c), to be the set of n × n row-stochastic matrices with column sum vector c T . In this paper the quantity λ(c) = max{|λ 2 (A)||A ∈ S(c)} is considered. The vectors c T such that λ(c) < 1 are identified and in those cases, nontrivial upper bounds on λ(c) and weak ergodicity results for forward products are provided. The results are obtained via a mix of analytic and combinatorial techniques.

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confidence: 99%

Subdominant eigenvalues for stochastic matrices with given column sums

Kirkland

2009

ELA

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“…Let a be the minimum positive entry in Y P . Based on [27], the second largest eigenvalue λ 2 of Y P can be bounded by…”

Section: B Approximation Ratio Of the Feasible Communication Policymentioning

confidence: 99%

Communication-efficient Decentralized Machine Learning over Heterogeneous Networks

Zhou

Lin

Loghin

et al. 2020

Preprint

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In the last few years, distributed machine learning has been usually executed over heterogeneous networks such as a local area network within a multi-tenant cluster or a wide area network connecting data centers and edge clusters. In these heterogeneous networks, the link speeds among worker nodes vary significantly, making it challenging for state-of-theart machine learning approaches to perform efficient training. Both centralized and decentralized training approaches suffer from low-speed links. In this paper, we propose a decentralized approach, namely NetMax, that enables worker nodes to communicate via high-speed links and, thus, significantly speed up the training process. NetMax possesses the following novel features. First, it consists of a novel consensus algorithm that allows worker nodes to train model copies on their local dataset asynchronously and exchange information via peer-to-peer communication to synchronize their local copies, instead of a central master node (i.e., parameter server). Second, each worker node selects one peer randomly with a fine-tuned probability to exchange information per iteration. In particular, peers with high-speed links are selected with high probability. Third, the probabilities of selecting peers are designed to minimize the total convergence time. Moreover, we mathematically prove the convergence of NetMax. We evaluate NetMax on heterogeneous cluster networks and show that it achieves speedups of 3.7×, 3.4×, and 1.9× in comparison with the state-of-the-art decentralized training approaches Prague, Allreduce-SGD, and AD-PSGD, respectively.

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“…The eigenvalue localization problem for stochastic matrices is not new. Many researchers gave significant contribution to this context [6,9,10,12,13]. In this paper, we use Geršgorin disc theorem [7] to localize the non-Perron eigenvalues of S .…”

Section: Introductionmentioning

confidence: 99%