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On cross-validated Lasso

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  • Denis Chetverikov

    () (Institute for Fiscal Studies and UCLA)

  • . .

    (Institute for Fiscal Studies)

Abstract

In this paper, we derive a rate of convergence of the Lasso estimator when the penalty parameter ? for the estimator is chosen using K-fold cross-validation; in particular, we show that in the model with Gaussian noise and under fairly general assumptions on the candidate set of values of ?, the prediction norm of the estimation error of the cross-validated Lasso estimator is with high probability bounded from above up-to a constant by (s log p/n)1/2 (log7/8n) as long as p log n/n = o(1) and some other mild regularity conditions are satisfi ed where n is the sample size of available data, p is the number of covariates, and s is the number of non-zero coefficients in the model. Thus, the cross-validated Lasso estimator achieves the fastest possible rate of convergence up-to the logarithmic factor log7/8 n. In addition, we derive a sparsity bound for the cross-validated Lasso estimator; in particular, we show that under the same conditions as above, the number of non-zero coefficients of the estimator is with high probability bounded from above up-to a constant by s log5 n. Finally, we show that our proof technique generates non-trivial bounds on the prediction norm of the estimation error of the cross-validated Lasso estimator even if p is much larger than n and the assumption of Gaussian noise fails; in particular, the prediction norm of the estimation error is with high-probability bounded from above up-to a constant by (s log2(pn) / n)1/4 under mild regularity conditions.

Suggested Citation

  • Denis Chetverikov & . ., 2016. "On cross-validated Lasso," CeMMAP working papers CWP47/16, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  • Handle: RePEc:ifs:cemmap:47/16
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    File URL: https://www.ifs.org.uk/uploads/cemmap/wps/cwp471616.pdf
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    References listed on IDEAS

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    1. Albert Saiz & Uri Simonsohn, 2013. "Proxying For Unobservable Variables With Internet Document-Frequency," Journal of the European Economic Association, European Economic Association, vol. 11(1), pages 137-165, February.
    2. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2014. "Central limit theorems and bootstrap in high dimensions," CeMMAP working papers CWP49/14, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    3. Denis Chetverikov & Daniel Wilhelm, 2017. "Nonparametric Instrumental Variable Estimation Under Monotonicity," Econometrica, Econometric Society, vol. 85, pages 1303-1320, July.
    4. Belloni, Alexandre & Chernozhukov, Victor & Chetverikov, Denis & Kato, Kengo, 2015. "Some new asymptotic theory for least squares series: Pointwise and uniform results," Journal of Econometrics, Elsevier, vol. 186(2), pages 345-366.
    5. Alexandre Belloni & Victor Chernozhukov & Ivan Fernandez-Val & Christian Hansen, 2013. "Program evaluation with high-dimensional data," CeMMAP working papers CWP77/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    6. David Cesarini & Christopher T. Dawes & Magnus Johannesson & Paul Lichtenstein & Björn Wallace, 2009. "Genetic Variation in Preferences for Giving and Risk Taking," The Quarterly Journal of Economics, Oxford University Press, vol. 124(2), pages 809-842.
    7. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2012. "Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors," Papers 1212.6906, arXiv.org, revised Jan 2018.
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    Keywords

    Cross-Validated Lasso;

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