Abstract
In order to solve robust PageRank problem a saddle-point Mirror Descent algorithm for solving convex-concave optimization problems is enhanced and studied. The algorithm is based on two proxy functions, which use specificities of value sets to be optimized on (min-max search). In robust PageRank case the ones are entropy-like function and square of Euclidean norm. The saddle-point Mirror Descent algorithm application to robust PageRank leads to concrete complexity results, which are being discussed alongside with illustrative numerical example.
Similar content being viewed by others
References
Juditsky, A. and Polyak, B., Robust Eigenvector of a Stochastic Matrix with Application to PageRank, 51 IEEE Conf. Decision Control, Maui, Hawaii, USA, 2012, pp. 3171–3176.
Brin, S. and Page, L., The Anatomy of a Large-Scale Hypertextual Web Search Engine, Computer Networks, 1998, vol. 30(1-7), pp. 107–117.
Langville, A.N. and Meyer, C.D., Google’s PageRank and Beyond: The Science of Search Engine Rankings, Princeton: Princeton Univ. Press, 2009.
Nazin, A.V., Estimating the Principal Eigenvector of a Stochastic Matrix: Mirror Descent Algorithms via Game Approach with Application to PageRank Problem, 49 IEEE Conf. Decision Control, Atlanta, Georgia, USA, 2010, pp. 792–797.
Polyak, B.T. and Tremba, A.A., Regularization-based Solution of the PageRank Problem for Large Matrices, Autom. Remote Control, 2012, vol. 73, no. 11, pp. 1877–1894.
Tremba, A. and Nazin, A., Extension of a Saddle Point Mirror Descent Algorithm with Application to Robust Page-Rank, 52 IEEE Conf. Decision Control, Florence, Italy, 2013, pp. 3691–3696.
Nemirovski, A., Juditsky, A., Lan, G., and Shapiro, A., Robust Stochastic Approximation Approach to Stochastic Programming, SIAM J. Optim., 2009, vol. 19, no. 4, pp. 1574–1609.
Juditsky, A.B., Nazin, A.V., Tsybakov, A.B., and Vayatis, N., Recursive Aggregation of Estimators by the Mirror Descent Algorithm with Averaging, Probl. Inform. Transmission, 2005, vol. 41, no. 4, pp. 368–384.
Rockafellar, R.T. and Wets, R.J.B., Variational Analysis, New York: Springer, 1998.
Beck, A. and Teboulle, M., Mirror Descent and Nonlinear Projected Subgradient Methods for Convex Optimization, Oper. Res. Lett., 2003, vol. 31, pp. 167–175.
Magaril-Il’yaev, G.G. and Tikhomirov, V.M., Convex Analysis: Theory and Applications, AMS, Translation of Mathematical Monographs, 2003, vol. 222.
Nemirovski, A. and Yudin, D., Problem Complexity and Method Efficiency in Optimization, New York: Wiley, 1983.
Chen, G. and Teboulle, M., A Proximal-Based Decomposition Method for Convex Minimization Problems, Math. Program., 1994, vol. 64, pp. 81–101.
Nesterov, Yu., Primal-Dual Subgradient Methods for Convex Problems, Math. Program., Ser. B, 2009, vol. 120, no. 1, pp. 221–259.
Juditsky, A. and Nemirovski, A., First Order Methods for Nonsmooth Convex Large-Scale Optimization. I. General Purpose Methods, in Optim. Machine Learning., Sra, S., Nowozin, S., and Wright, S., Eds., Boston: MIT Press, 2012, pp. 121–148.
Juditsky, A. and Nemirovski, A., First Order Methods for Nonsmooth Convex Large-Scale Optimization. II. Utilizing Problem’s Structure, in Optim. Machine Learning., Sra, S., Nowozin, S., and Wright, S., Eds., Boston: MIT Press, 2012, pp. 149–184.
Nazin, A.V. and Polyak, B.T., Randomized Algorithm to Determine the Eigenvector of a Stochastic Matrix with Application to the PageRank Problem, Autom. Remote Control, 2011, vol. 72, no. 2, pp. 342–352.
Nazin, A.V., The Adaptive Mirror Descent Algorithms in Problems of Stochastic Convex Optimization, Tr. Inst. Sist. Anal., 2014, vol. 64, no. 3, pp. 7–12.
Juditsky, A., Nemirovski, A., and Tauvel, C., Solving Variational Inequalities with Stochastic Mirror-Prox Algorithm, Stochast. Syst., 2011, vol. 1, no. 1, pp. 17–58.
Andersen, E.D., The MOSEK Optimization Tools Manual, Version 6.0, 2010, http://www.mosek.com.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.V. Nazin, A.A. Tremba, 2016, published in Avtomatika i Telemekhanika, 2016, No. 8, pp. 105–124.
Rights and permissions
About this article
Cite this article
Nazin, A.V., Tremba, A.A. Saddle point mirror descent algorithm for the robust PageRank problem. Autom Remote Control 77, 1403–1418 (2016). https://doi.org/10.1134/S0005117916080075
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117916080075