On efficient randomized algorithms for finding the PageRank vector

Gasnikov, A. V.; Dmitriev, Denis

doi:10.1134/s0965542515030069

Cited by 8 publications

(8 citation statements)

References 24 publications

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“…Also we want to point that the algorithm for solving equation (3.2) was chosen consciously from a set of modern methods for computing PageRank. We used review [5] of such methods. Since for our problem we need to estimate the error which is introduced to the function f (ϕ) value by approximate solution of the ranking problem (3.2), we considered only three methods: Markov Chain Monte Carlo (MCMC), Spillman's and Nemirovski-Nesterov's (NN).…”

Section: Algorithm 2 Methods For Model Learningmentioning

confidence: 99%

“…On the lower level, we use the linearly convergent method from [17] to calculate an approximation to the stationary distribution of the Markov process. We show in Section 5 that this method has the best among others [5] complexity bound for the two-level method as a whole. However, it is not enough to calculate the stationary distribution itself, since we need also to optimize the parameters of the random walk with respect to an objective function, which is based on the stationary distribution.…”

Section: Introductionmentioning

confidence: 99%

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Learning Supervised PageRank with Gradient-Free Optimization Methods

Bogolubsky,

Dvurechensky,

Gasnikov

et al. 2014

Preprint

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In this paper, we consider a problem of learning supervised PageRank models, which can account for some properties not considered by classical approaches such as the classical PageRank algorithm. Due to huge hidden dimension of the optimization problem we use random gradient-free methods to solve it. We prove 1 a convergence theorem and estimate the number of arithmetic operations needed to solve it with a given accuracy. We find the best settings of the gradient-free optimization method in terms of the number of arithmetic operations needed to achieve given accuracy of the objective. In the paper, we apply our algorithm to the web page ranking problem. We consider a parametric graph model of users' behavior and evaluate web pages' relevance to queries by our algorithm. The experiments show that our optimization method outperforms the untuned gradientfree method in the ranking quality.

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Section: Algorithm 2 Methods For Model Learningmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Learning Supervised PageRank with Gradient-Free Optimization Methods

Bogolubsky,

Dvurechensky,

Gasnikov

et al. 2014

Preprint

View full text Add to dashboard Cite

show abstract

“…Hence we need some numerical scheme which allows to calculate approximation for π q (ϕ) and dπq(ϕ) dϕ T for every q ∈ Q with a given accuracy. Motivated by the last requirement we have analysed state-of-the-art methods for finding the solution of Equation 3.2 in huge dimension summarized in the review Gasnikov and Dmitriev (2015) and power method, used in Page et al (1999); Backstrom and Leskovec (2011);Zhukovskii et al (2014). Only four methods allow to make the difference π q (ϕ) − πq , where πq is the approximation, small for some norm • which is crucial to estimate the error in the approximation of the function f (ϕ) value.…”

Section: Numerical Calculation Of the Value And The Gradient Of F (ϕ)mentioning

confidence: 99%

“…On the lower level of these methods, we use the linearly convergent method from Nes-terov and Nemirovski (2015) to calculate an approximation to the stationary distribution of Markov random walk. We analyze other methods from Gasnikov and Dmitriev (2015) and show that the chosen method is the most suitable since it allows to approximate the value of the loss function with any given accuracy and has lowest complexity estimation among others.…”

Section: Introductionmentioning

confidence: 99%

Learning Supervised PageRank with Gradient-Based and Gradient-Free Optimization Methods

Bogolubsky,

Dvurechensky,

Gasnikov

et al. 2016

Preprint

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In this paper, we consider a non-convex loss-minimization problem of learning Supervised PageRank models, which can account for some properties not considered by classical approaches such as the classical PageRank model. We propose gradient-based and random gradient-free methods to solve this problem. Our algorithms are based on the concept of an inexact oracle and unlike the state state-of-the-art gradient-based method we manage to provide theoretically the convergence rate guarantees for both of them. In particular, under the assumption of local convexity of the loss function, our random gradient-free algorithm guarantees decrease of the loss function value expectation. At the same time, we theoretically justify that without convexity assumption for the loss function our gradient-based algorithm allows to find a point where the stationary condition is fulfilled with a given accuracy. For both proposed optimization algorithms, we find the settings of hyperparameters which give the lowest complexity (i.e., the number of arithmetic operations needed to achieve the given accuracy of the solution of the loss-minimization problem). The resulting estimates of the complexity are also provided. Finally, we apply proposed optimization algorithms to the web page ranking problem and compare proposed and state-of-the-art algorithms in terms of the considered loss function.

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“…The problem becomes especially challenging in high dimensions since direct computations become unreliable due to non-linear time and memory efforts. Many studies have been devoted to the approximation of the PageRank vector based on random walks analysis and Markov Chain Monte Carlo methods [2,13,19,20,35,37]. Those methods are very attractive, both theoretically and practically, unless the spectral gap, i.e., the difference between the two largest eigenvalues of the transition matrix, is sufficiently large.…”

Section: Introductionmentioning

confidence: 99%

Efficient Numerical Methods to Solve Sparse Linear Equations with Application to PageRank

Anikin,

Gasnikov,

Gornov

et al. 2015

Preprint

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Over the last two decades, the PageRank problem received increased interest from the academic community as an efficient tool to estimate web-page importance in information retrieval. In spite of numerous developments, the design of efficient optimization algorithms for the PageRank problem is still problem is still ongoing.In this paper, we propose three new algorithms with linear-time convergence for solving the problem over boundeddegree graphs. The idea behind the algorithms is to set up the PageRank problem as a convex minimization problem over a unit simplex and then solve it using iterative methods with small iteration complexity. Our theoretical results are supported by extensive empirical evidence using real-world and simulated data.

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On efficient randomized algorithms for finding the PageRank vector

Cited by 8 publications

References 24 publications

Learning Supervised PageRank with Gradient-Free Optimization Methods

Learning Supervised PageRank with Gradient-Free Optimization Methods

Learning Supervised PageRank with Gradient-Based and Gradient-Free Optimization Methods

Efficient Numerical Methods to Solve Sparse Linear Equations with Application to PageRank

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