Denis Chetverikov, University of California, Los Angeles, USA
On CrossValidated 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 crossvalidation; 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 crossvalidated 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 nonzero coefficients in the model. Thus, the crossvalidated 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 crossvalidated Lasso estimator; in particular, we show that under the same conditions as above, the number of nonzero 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 nontrivial bounds on the prediction norm of the estimation error of the crossvalidated Lasso estimator even if the assumption of the Gaussian noise fails; in particular, the prediction norm of the estimation error is with highprobability 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

10:0012:30

WeierstrassInstitute for Applied Analysis and Stochastics
Denis Chetverikov, University of California, Los Angeles, USA