Estimation with Norm Regularization
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 analy...
Saved in:
Main Authors | , , , |
---|---|
Format | Journal Article |
Language | English |
Published |
09.05.2015
|
Subjects | |
Online Access | Get full text |
Cover
Loading…
Summary: | 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. |
---|---|
DOI: | 10.48550/arxiv.1505.02294 |