Estimation with Norm Regularization

TL;DR

This paper provides a comprehensive analysis of estimation error bounds for norm regularized regression methods like Lasso, considering various aspects such as the norm, loss function, design matrix, and noise, with results applicable to a wide range of models.

Contribution

It generalizes estimation error analysis across all four aspects, characterizes the error set, and establishes bounds for diverse norms, design matrices, and noise models.

Findings

Estimation error decreases at rate 1/√n once sample size exceeds a threshold.

Sample complexity depends on the Gaussian width of the error set.

Bounds are applicable to isotropic/anisotropic subGaussian designs and convex loss functions.

Abstract

Analysis of non-asymptotic estimation error and structured statistical recovery based on norm regularized regression, such as Lasso, needs to consider four aspects: the norm, the loss function, the design matrix, and the noise model. This paper presents generalizations of such estimation error analysis on all four aspects compared to the existing literature. We characterize the restricted error set where the estimation error vector lies, establish relations between error sets for the constrained and regularized problems, and present an estimation error bound applicable to any norm. Precise characterizations of the bound is presented for isotropic as well as anisotropic subGaussian design matrices, subGaussian noise models, and convex loss functions, including least squares and generalized linear models. Generic chaining and associated results play an important role in the analysis. A key result from the analysis is that the sample complexity of all such estimators depends on the Gaussian width of a spherical cap corresponding to the restricted error set. Further, once the number of samples $n$ crosses the required sample complexity, the estimation error decreases as $\frac{c}{\sqrt{n}}$, where $c$ depends on the Gaussian width of the unit norm ball.