Advertisement

Statistical Leveraging Methods in Big Data

  • Xinlian Zhang
  • Rui Xie
  • Ping Ma
Chapter
Part of the Springer Handbooks of Computational Statistics book series (SHCS)

Abstract

With the advance in science and technologies in the past decade, big data becomes ubiquitous in all fields. The exponential growth of big data significantly outpaces the increase of storage and computational capacity of high performance computers. The challenge in analyzing big data calls for innovative analytical and computational methods that make better use of currently available computing power. An emerging powerful family of methods for effectively analyzing big data is called statistical leveraging. In these methods, one first takes a random subsample from the original full sample, then uses the subsample as a surrogate for any computation and estimation of interest. The key to success of statistical leveraging methods is to construct a data-adaptive sampling probability distribution, which gives preference to those data points that are influential to model fitting and statistical inference. In this chapter, we review the recent development of statistical leveraging methods. In particular, we focus on various algorithms for constructing subsampling probability distribution, and a coherent theoretical framework for investigating their estimation property and computing complexity. Simulation studies and real data examples are presented to demonstrate applications of the methodology.

Keywords

Randomized algorithm Leverage scores Subsampling Least squares Linear regression 

Notes

Acknowledgements

This work was funded in part by NSF DMS-1440037(1222718), NSF DMS-1438957(1055815), NSF DMS-1440038(1228288), NIH R01GM122080, NIH R01GM113242.

References

  1. Agarwal A, Duchi JC (2011) Distributed delayed stochastic optimization. In: Advances in neural information processing systems, pp 873–881Google Scholar
  2. Avron H, Maymounkov P, Toledo S (2010) Blendenpik: supercharging LAPACK’s least-squares solver. SIAM J Sci Comput 32:1217–1236MathSciNetCrossRefGoogle Scholar
  3. Bhlmann P, van de Geer S (2011) Statistics for high-dimensional data: methods, theory and applications, 1st edn. Springer, BerlinGoogle Scholar
  4. Chatterjee S, Hadi AS (1986) Influential observations, high leverage points, and outliers in linear regression. Stat Sci 1(3):379–393MathSciNetCrossRefGoogle Scholar
  5. Chen X, Xie M (2014) A split-and-conquer approach for analysis of extraordinarily large data. Stat Sin 24:1655–1684Google Scholar
  6. Clarkson KL, Woodruff DP (2013) Low rank approximation and regression in input sparsity time. In: Proceedings of the forty-fifth annual ACM symposium on theory of computing. ACM, New York, pp 81–90Google Scholar
  7. Clarkson KL, Drineas P, Magdon-Ismail M, Mahoney MW, Meng X, Woodruff DP (2013) The Fast Cauchy Transform and faster robust linear regression. In: Proceedings of the twenty-fourth annual ACM-SIAM symposium on discrete algorithms. Society for Industrial and Applied Mathematics, Philadelphia, pp 466–477CrossRefGoogle Scholar
  8. Coles S, Bawa J, Trenner L, Dorazio P (2001) An introduction to statistical modeling of extreme values, vol 208. Springer, BerlinCrossRefGoogle Scholar
  9. Drineas P, Mahoney MW, Muthukrishnan S (2006) Sampling algorithms for 2 regression and applications. In: Proceedings of the 17th annual ACM-SIAM symposium on discrete algorithms, pp 1127–1136Google Scholar
  10. Drineas P, Mahoney MW, Muthukrishnan S, Sarlós T (2010) Faster least squares approximation. Numer Math 117(2):219–249MathSciNetCrossRefGoogle Scholar
  11. Drineas P, Magdon-Ismail M, Mahoney MW, Woodruff DP (2012) Fast approximation of matrix coherence and statistical leverage. J Mach Learn Res 13:3475–3506Google Scholar
  12. Duchi JC, Agarwal A, Wainwright MJ (2012) Dual averaging for distributed optimization: convergence analysis and network scaling. IEEE Trans Autom Control 57(3):592–606MathSciNetCrossRefGoogle Scholar
  13. Friedman J, Hastie T, Tibshirani R (2001) The elements of statistical learning, vol 1. Springer series in statistics. Springer, BerlinGoogle Scholar
  14. Golub GH, Van Loan CF (1996) Matrix computations. Johns Hopkins University Press, BaltimorezbMATHGoogle Scholar
  15. Hesterberg T (1995) Weighted average importance sampling and defensive mixture distributions. Technometrics 37(2):185–194MathSciNetCrossRefGoogle Scholar
  16. Hoaglin DC, Welsch RE (1978) The hat matrix in regression and ANOVA. Am Stat 32(1):17–22zbMATHGoogle Scholar
  17. Lichman M (2013) UCI machine learning repositoryGoogle Scholar
  18. Ma P, Sun X (2015) Leveraging for big data regression. Wiley Interdiscip Rev Comput Stat 7(1):70–76MathSciNetCrossRefGoogle Scholar
  19. Ma P, Mahoney MW, Yu B (2014) A statistical perspective on algorithmic leveraging. In: Proceedings of the 31st international conference on machine learning (ICML-14), pp 91–99Google Scholar
  20. Ma P, Mahoney MW, Yu B (2015) A statistical perspective on algorithmic leveraging. J Mach Learn Res 16:861–911MathSciNetzbMATHGoogle Scholar
  21. Ma P, Zhang X, Ma J, Mahoney MW, Yu B, Xing X (2016) Optimal subsampling methods for large sample linear regression. Technical report, Department of Statistics, University of GeorgiaGoogle Scholar
  22. Mahoney MW (2011) Randomized algorithms for matrices and data. Foundations and trends in machine learning. NOW Publishers, Boston. Also available at: arXiv:1104.5557Google Scholar
  23. Mahoney MW, Drineas P (2009) CUR matrix decompositions for improved data analysis. Proc Natl Acad Sci 106(3):697–702MathSciNetCrossRefGoogle Scholar
  24. McCullagh P, Nelder JA (1989) Generalized linear models, vol 37. CRC, Boca RatonCrossRefGoogle Scholar
  25. Meng X, Mahoney MW (2013) Low-distortion subspace embeddings in input-sparsity time and applications to robust linear regression. In: Proceedings of the forty-fifth annual ACM symposium on theory of computing. ACM, New York, pp 91–100Google Scholar
  26. Meng X, Saunders MA, Mahoney MW (2014) LSRN: a parallel iterative solver for strongly over-or underdetermined systems. SIAM J Sci Comput 36(2):C95–C118MathSciNetCrossRefGoogle Scholar
  27. Raskutti G, Mahoney MW (2016) A statistical perspective on randomized sketching for ordinary least-squares. J Mach Learn Res 17(214):1–31MathSciNetzbMATHGoogle Scholar
  28. Velleman PF, Welsch ER (1981) Efficient computing of regression diagnostics. Am Stat 35(4): 234–242zbMATHGoogle Scholar
  29. Wang H, Zhu R, Ma P (2017) Optimal subsampling for large sample logistic regression. J Am Stat Assoc (in press)Google Scholar
  30. Xie R, Sriram TN, Ma P (2017) Sequential leveraging sampling method for streaming time series data. Technical report, Department of Statistics University of GeorgiaGoogle Scholar
  31. Zhang Y, Duchi JC, Wainwright MJ (2013) Divide and conquer kernel ridge regression: a distributed algorithm with minimax optimal rates. CoRR. abs/1305.5029Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of GeorgiaAthensUSA

Personalised recommendations