## STA 290 Seminar: Denis Chetverikov

### STA 290 Seminar Series

DATE: Thursday, March 2^{nd} 2017, 4:10pm

LOCATION: MSB 1147, Colloquium Room. Refreshments at 3:30pm in MSB 4110

SPEAKER: **Denis Chetverikov****,** UCLA

TITLE: **“****On Cross-Validated Lasso”**

ABSTRACT: In this paper, we derive a rate of convergence of the Lasso estimator when the penalty parameter λ 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 λ, 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} . (log^{7/8} (*pn*)), 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} (*pn*). 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 nonzero coefficients of the estimator is with high probability bounded from above up to a constant by *s* log^{5}(*pn*). 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 (*s* log2(*pn*)=n)^{1=4} under mild regularity conditions.