Derivatives and residual distribution of regularized M-estimators with application to adaptive tuning

TL;DR

This paper derives formulas for derivatives of regularized M-estimators in linear models, characterizes residual distributions in high dimensions, and proposes an adaptive tuning criterion that approximates out-of-sample error without prior noise or covariance knowledge.

Contribution

It provides a unified differentiability structure for convex regularized M-estimators, characterizes residual distributions in high-dimensional settings, and introduces a practical adaptive tuning method.

Findings

Derived general formulas for derivatives of regularized M-estimators.

Characterized residual distribution in high-dimensional regimes.

Proposed an adaptive criterion for tuning parameter selection.

Abstract

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 $y$ and $X$; this reveals a simple differentiability structure shared by all convex regularized M-estimators. (ii) Using these derivatives, we characterize the distribution of the residual $r_i = y_i-x_i^\top\hat\beta$ 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 $\hat\beta(y,X)$ and the effective degrees of freedom of the M-estimator, which are of independent interest.