1932

Abstract

Data sets that are terabytes in size are increasingly common, but computer bottlenecks often frustrate a complete analysis of the data, and diminishing returns suggest that we may not need terabytes of data to estimate a parameter or test a hypothesis. But which rows of data should we analyze, and might an arbitrary subset preserve the features of the original data? We review a line of work grounded in theoretical computer science and numerical linear algebra that finds that an algorithmically desirable sketch, which is a randomly chosen subset of the data, must preserve the eigenstructure of the data, a property known as subspace embedding. Building on this work, we study how prediction and inference can be affected by data sketching within a linear regression setup. We use statistical arguments to provide “inference-conscious” guides to the sketch size and show that an estimator that pools over different sketches can be nearly as efficient as the infeasible one using the full sample.

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/content/journals/10.1146/annurev-economics-022720-114138
2020-08-02
2024-04-24
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Literature Cited

  1. Achiloptas D. 2003. Database-friendly random projections: Johnson-Lindenstrauss with binary coins. J. Comput. Syst. Sci. 66:4671–87
    [Google Scholar]
  2. Agarwal PK, Har-Peled S, Varadarajan KR 2004. Approximating extent measures of points. J. Assoc. Comput. Mach. 51:4606–35
    [Google Scholar]
  3. Ahfock D, Astle W, Richardson S 2017. Statistical properties of sketching algorithms. arXiv:1706.03665 [stat.ME]
  4. Ailon N, Chazelle B. 2009. The fast Johnson–Lindenstrauss transform and approximate nearest neighbors. SIAM J. Comput. 39:1302–22
    [Google Scholar]
  5. Alon N, Matias Y, Onak K 1999. The space complexity of approximating the frequency moments. J. Comput. Syst. Sci. 58:1137–47
    [Google Scholar]
  6. Bai Z, Yin Y. 1993. Limit of the smallest eigenvalue of a large dimensional sample covariance matrix. Ann. Probab. 21:31275–94
    [Google Scholar]
  7. Belenzon S, Chatterji A, Dailey B 2017. Eponymous enterpreneurs. Am. Econ. Rev. 107:61638–55
    [Google Scholar]
  8. Boivin J, Ng S. 2006. Are more data always better for factor analysis. J. Econom. 132:169–94
    [Google Scholar]
  9. Boutidis C, Gittens A. 2013. Improved matrix algorithms via the Subsampled Randomized Hadamard Transform. SIAM J. Matrix Anal. 34:31301–40
    [Google Scholar]
  10. Breiman L. 1999. Pasting bites together for prediction in large data sets and on-line. Mach. Learn. 36:285–103
    [Google Scholar]
  11. Charikar M, Chen K, Farach-Colton M 2002. Finding frequent items in data streams. Proceedings of International Colloquium on Automata, Languages, and Programming693–703 Rome: EATCS
    [Google Scholar]
  12. Chawla N, Hall L, Bowyer K, Kegelmeyer P 2004. Learning ensembles from bites: a scalable and accurate approach. J. Mach. Learn. Res. 5:421–51
    [Google Scholar]
  13. Chen S, Varma R, Singh A, Kovacevic J 2016. A statistical perspective of sampling scores for linear regression Paper presented at the IEEE International Symposium on Information Theory Barcelona: July 10–15
  14. Chi J, Ipsen I. 2018. Randomized least squares regression: combining model and algorithm induced uncertainties. arXiv:1808.05924v1 [stat.ML]
  15. Christmann A, Steinwart I, Hubert M 2007. Robust learning from bites for data mining. Comput. Stat. Data Anal. 52:347–61
    [Google Scholar]
  16. Clarkson K, Woodruff D. 2013. Low rank approximation and regression in input sparsity time. Proceedings of the 45th ACM Symposium on the Theory of Computing81–90 New York: ACM
    [Google Scholar]
  17. Cohen M, Lee Y, Musco C, Musco C, Peng R, Sidford A 2015. Uniform sampling for matrix approximation. Proceedings of the 46th ACM Symposium on the Theory of Computing181–90 New York: ACM
    [Google Scholar]
  18. Cohen M, Nelson J, Woodruff D 2015. Optimal approximate matrix product in terms of stable rank. arXiv:1507.02268 [cs.DS]
  19. Cormode G, Garofalakis M, Haas PJ, Jermaine C 2011. Synopses for massive data: samples, histograms, wavelets, sketches. Found. Trends Datab. 4:1–294
    [Google Scholar]
  20. Cormode G, Muthukrishnan S. 2005. An improved data stream summary: the count-min sketch and applications. J. Algorithms 55:29–38
    [Google Scholar]
  21. Cramer JS. 1987. Mean and variance of in small and moderate samples. J. Econom. 35:253–66
    [Google Scholar]
  22. Dahiya Y, Konomis D, Woodruff D 2018. An empirical evaluation of sketching for numerical linear algebra Paper presented at the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining London: Aug 19–23
  23. Dasgupta A, Kumar R, Sarlos T 2010. A sparse Johnson–Lindenstrauss transform. arXiv:1004.4240 [cs.DS]
  24. Deaton A, Ng S. 1998. Parametric and nonparametric approaches to tax reform. J. Am. Stat. Assoc. 93:443900–9
    [Google Scholar]
  25. Dhillon P, Lu Y, Foster D, Ungar L 2013. New subsampling algorithms for faster least squares regression. Adv. Neural Inform. Proc. Syst. 26:360–68
    [Google Scholar]
  26. Drineas P, Kannan R, Mahoney M 2006. Fast Monte Carlo algorithms for matrices I: approximating matrix multiplications. SIAM J. Comput. 36:132–57
    [Google Scholar]
  27. Drineas P, Magdon-Ismail M, Mahoney M, Woodruff D 2012. Fast approximation of matrix coherence and statistical leverage. J. Mach. Learn. Res. 13:3441–72
    [Google Scholar]
  28. Drineas P, Mahoney M. 2005. On the Nyström method for approximating a Gram matrix for improved kernel-based learning. J. Mach. Learn. Res. 6:2152–25
    [Google Scholar]
  29. Drineas P, Mahoney M, Muthukrishnan S 2006. Sampling algorithms for L2 regression and applications. Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms1127–36 New York: ACM
    [Google Scholar]
  30. Drineas P, Mahoney M, Muthukrishnan S, Sarlos T 2011. Faster least squares approximation. Numer. Math. 117:219–49
    [Google Scholar]
  31. Du Mouchel W, Volinsky C, Johnson T, Cortes C, Pregibon D 1999. Squashing flat files flatter. Proceedings of the Fifth ACM Conference on Knowledge Discovery and Data Mining6–15 New York: ACM
    [Google Scholar]
  32. Eriksson-Bique S, Solberg M, Stefanelli M, Warkentin S, Abbey R, Ipsen I 2011. Importance sampling for a Monte Carlo matrix multiplication algorithm with application to information retrieval. SIAM J. Comput. 33:41689–706
    [Google Scholar]
  33. Geppert L, Ickstadt K, Munteanu A, Quedenfeld J, Sohler C 2017. Random projections for Bayesian regression. Stat. Comput. 27:79–101
    [Google Scholar]
  34. Ghashami M, Liberty E, Phillips M, Woodruff D 2016. Frequent directions: simple and deterministic matrix sketching. SIAM J. Comput. 45:51762–92
    [Google Scholar]
  35. Hansen BE. 2020. Econometrics Textb., Univ Wisconsin, Madison: https://www.ssc.wisc.edu/˜bhansen/econometrics/Econometrics.pdf
  36. Heince C, McWilliams B, Meinshausen N 2016. Dual loco: distributing statistical estimation using random projections. Proc. Mach. Learn. Res. 51:875–83
    [Google Scholar]
  37. Hogben L. 2007. Handbook of Linear Algebra London: Chapman & Hall
  38. Horvitz D, Thompson D. 1952. A generalization of sampling replacement from a finite universe. J. Am. Stat. Assoc. 47:663–85
    [Google Scholar]
  39. Ipsen I, Wentworth T. 2014. The effect of coherence on sampling from matrices with orthonormal columns and preconditioned least squares problems. SIAM J. Matrix Anal. Appl. 35:41490–520
    [Google Scholar]
  40. Johnson W, Lindenstrauss J. 1994. Extensions of Lipschitz maps into a Hilbert space. Contemp. Math. 26:189–206
    [Google Scholar]
  41. Jolliffe I. 1972. Discarding variables in a principal component analysis: artificial data. Appl. Stat. 21:2160–73
    [Google Scholar]
  42. Kane D, Nelson J. 2014. Sparser Johnson-Lindenstrauss transforms. J. Assoc. Comput. Mach. 61:14
    [Google Scholar]
  43. Li P, Hastie T, Church K 2006. Very sparse random projections. Proceedings of the 12th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining287–96 New York: ACM
    [Google Scholar]
  44. Ma P, Mahoney MW, Yu B 2014. A statistical perspective on algorithmic leveraging. Proc. Mach. Learn. Res. 32:191–99
    [Google Scholar]
  45. Madigan D, Raghavan N, Dumouchel W, Nason M, Posse C, Ridgeway G 1999. Likelihood-based data squashing: a modeling approach to instance construction Tech. Rep., AT&T Labs Res Florham Park, NJ:
  46. Mahoney MW. 2011. Randomized algorithms for matrices and data. Foundations and Trends in Machine Learning 32123–224 Delft, Neth.: Now Publ.
    [Google Scholar]
  47. McWilliams B, Krummenacher C, Lucic G, Buhmann J 2014. Fast and robust least squares estimation in corrupted linear models. Proceedings of the 27th International Conference on Neural Information Processing Systems 1415–23 New York: ACM
    [Google Scholar]
  48. Meng X, Mahoney M. 2013. Low-distortion subspace embeddings in input-sparsity time and applications to robust linear regression. Proceedings of the 45th ACM Symposium on the Theory of Computing91–100 New York: ACM
    [Google Scholar]
  49. Mitzenmacher M, Upfal E. 2006. Probability and Computing: Randomized Algorithms and Probabilistic Analysis Cambridge, UK: Cambridge Univ. Press
  50. Nelson J, Nguyen H. 2013a. OSNAP: faster numerical linear algebra algorithms via sparser subspace embeddings. Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science117–26 Piscataway, NJ: IEEE
    [Google Scholar]
  51. Nelson J, Nguyen H. 2013b. Sparsity lower bounds for dimensionality reducing maps. Proceedings of the 45th ACM Symposium on the Theory of Computing101–10 New York: ACM
    [Google Scholar]
  52. Nelson J, Nguyen H. 2014. Lower bounds for oblivious subspace embeddings. Proceedings of the 41st International Colloquium on Automata, Languages and Programming883–94 Rome: EATCS
    [Google Scholar]
  53. Ng S. 2017. Opportunities and challenges: lessons from analyzing terabytes of scanner data. Advances in Economics and Econometrics: Eleventh World Congress of the Econometric Society Vol II B Honore, A Pkes, M Piazzesi, L Samuelson 1–34 Cambridge, UK: Cambridge Univ. Press
    [Google Scholar]
  54. Owen A. 1990. Empirical likelihood ratio confidence region. Ann. Stat. 18:90–120
    [Google Scholar]
  55. Pilanci M, Wainwright M. 2015. Randomized sketches of convex programs with sharp guarantees. IEEE Trans. Inform. Theory 61:95096–115
    [Google Scholar]
  56. Pilanci M, Wainwright M. 2016. Iterative Hessian sketch: fast and accurate solution approximation for constrained least-squares. J. Mach. Learn. Res. 17:138
    [Google Scholar]
  57. Portnoy S, Koenker R. 1997. The Gaussian hare and the Laplacian tortoise: computability of squared-error versus absolute-error estimation. Stat. Sci. 12:4279–300
    [Google Scholar]
  58. Raskutti G, Mahoney M. 2016. A statistical perspective on randomized sketching for ordinary least squares. J. Mach. Learn. Res. 17:1–38
    [Google Scholar]
  59. Rudd P. 2000. An Introduction to Classical Econometric Theory Oxford, UK: Oxford Univ. Press
  60. Ruggles S, Flood S, Goeken R, Grover J, Meyer E et al. 2020. IPUMS USA: Version 10.0 [dataset]. IPUMS Minneapolis, MN: https://doi.org/10.18128/D010.V10.0
    [Crossref]
  61. Sarlos T. 2006. Improved approximation algorithms for large matrices via random projections. Proceedings of the 47 IEEE Symposium on Foundations of Computer Science143–52 Washington, DC: IEEE Comp. Soc.
    [Google Scholar]
  62. Wallace T. 1972. Weaker criteria and tests for linear restrictions. Econometrica 40:4689–98
    [Google Scholar]
  63. Wang H, Yang M, Stufken J 2019. Information-based optimal subdata selection for big data linear research. J. Am. Stat. Assoc. 114:525393–405
    [Google Scholar]
  64. Wang H, Zhu R, Ma P 2018. Optimal subsampling for large sample logistic regression. J. Am. Stat. Assoc. 113:522849–44
    [Google Scholar]
  65. Wang J, Lee J, Mahdav M, Kolar M, Srebo N 2017. Sketching meets random projection in the dual: a provable recovery algorithm for big and high dimensional data. Electron. J. Stat. 11:4896–944
    [Google Scholar]
  66. Wang S, Gittens A, Mahoney M 2018. Sketched ridge regression: optimization perspective, statistical perspective, and model averaging. J. Mach. Learn. Res. 18:1–50
    [Google Scholar]
  67. Woodruff D. 2014. Sketching as a tool for numerical linear algebra. Found. Trends Theor. Comput. Sci. 10:1–21–157
    [Google Scholar]
  68. Woolfe F, Liberty E, Vladmir R, Mark T 2008. A fast randomized algorithm for the approximation of matrices. Appl. Comput. Harmon. Anal. 25:3335–66
    [Google Scholar]
  69. Yin Y, Bai Z, Krishnaiah P 1988. On the limit of the largest eigenvalue of the largest dimensional sample covariance matrix. Probab. Theory Relat. Fields 78:4509–21
    [Google Scholar]
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