2022-06-28-bellec22a.md

title	abstract	layout	series	publisher	issn	id	month	tex_title	firstpage	lastpage	page	order	cycles	bibtex_author	author	date	address	container-title	volume	genre	issued	pdf	extras
Derivatives and residual distribution of regularized M-estimators with application to adaptive tuning	This paper studies M-estimators with gradient-Lipschitz loss function regularized with convex penalty in linear models with Gaussian design matrix and arbitrary noise distribution. A practical example is the robust M-estimator constructed with the Huber loss and the Elastic-Net penalty and the noise distribution has heavy-tails. Our main contributions are three-fold. (i) We provide general formulae for the derivatives of regularized M-estimators $\hat\beta(y,X)$ where differentiation is taken with respect to both X and y; this reveals a simple differentiability structure shared by all convex regularized M-estimators. (ii) Using these derivatives, we characterize the distribution of the residuals in the intermediate high-dimensional regime where dimension and sample size are of the same order. (iii) Motivated by the distribution of the residuals, we propose a novel adaptive criterion to select tuning parameters of regularized M-estimators. The criterion approximates the out-of-sample error up to an additive constant independent of the estimator, so that minimizing the criterion provides a proxy for minimizing the out-of-sample error. The proposed adaptive criterion does not require the knowledge of the noise distribution or of the covariance of the design. Simulated data confirms the theoretical findings, regarding both the distribution of the residuals and the success of the criterion as a proxy of the out-of-sample error. Finally our results reveal new relationships between the derivatives of the $\hat\beta$ and the effective degrees of freedom of the M-estimators, which are of independent interest.	inproceedings	Proceedings of Machine Learning Research	PMLR	2640-3498	bellec22a	0	Derivatives and residual distribution of regularized M-estimators with application to adaptive tuning	1912	1947	1912-1947	1912	false	Bellec, Pierre C and Shen, Yiwei	given family Pierre C Bellec given family Yiwei Shen	2022-06-28		Proceedings of Thirty Fifth Conference on Learning Theory	178	inproceedings	date-parts 2022 6 28	https://proceedings.mlr.press/v178/bellec22a/bellec22a.pdf	label link Supplementary ZIP https://proceedings.mlr.press/v178/bellec22a/bellec22a-supp.zip