Mathematical Statistics Seminar

Denis Chetverikov, University of California, Los Angeles, USA

On Cross-Validated Lasso

In this paper, we derive a rate of convergence of the Lasso estimator when the penalty parameter $lambda$ for the estimator is chosen using $K$-fold cross-validation; in particular, we show that in the model with the Gaussian noise and under fairly general assumptions on the candidate set of values of $lambda$, 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 $(slog p /n)^1/2cdot log^7/8(p n)$, 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 a small logarithmic factor $log^7/8(p n)$. 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 $slog^5(p n)$. 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 the assumption of the Gaussian noise fails; in particular, the prediction norm of the estimation error is with high-probability bounded from above up to a constant by $(slog^2(p n)/n)^1/4$ under mild regularity conditions.

(The authors are Denis Chetverikov, Zhipeng Liao, and Victor Chernozhukov.)

  • 21.06.2017 - 21.06.2017
  • 10:00-12:30
  • Weierstrass-Institute for Applied Analysis and Stochastics
  • Event homepage
Speakers

Denis Chetverikov, University of California, Los Angeles, USA