Near-ideal model selection by $\ell_1$ minimization

TL;DR

This paper demonstrates that the lasso method can nearly optimally select variables and estimate the mean in high-dimensional sparse models, achieving near-ideal error rates under broad, nonasymptotic conditions.

Contribution

It proves that solving a quadratic program with lasso yields near-optimal mean squared error in sparse high-dimensional settings, extending understanding of lasso's effectiveness.

Findings

Lasso nearly matches the ideal subset selection in mean estimation.

Performance is within a logarithmic factor of the oracle's error.

Results are nonasymptotic and depend on predictor collinearity.

Abstract

We consider the fundamental problem of estimating the mean of a vector $y=X\beta+z$, where $X$ is an $n\times p$ design matrix in which one can have far more variables than observations, and $z$ is a stochastic error term--the so-called "$p>n$" setup. When $\beta$ is sparse, or, more generally, when there is a sparse subset of covariates providing a close approximation to the unknown mean vector, we ask whether or not it is possible to accurately estimate $X\beta$ using a computationally tractable algorithm. We show that, in a surprisingly wide range of situations, the lasso happens to nearly select the best subset of variables. Quantitatively speaking, we prove that solving a simple quadratic program achieves a squared error within a logarithmic factor of the ideal mean squared error that one would achieve with an oracle supplying perfect information about which variables should and should not be included in the model. Interestingly, our results describe the average performance of the lasso; that is, the performance one can expect in an vast majority of cases where $X\beta$ is a sparse or nearly sparse superposition of variables, but not in all cases. Our results are nonasymptotic and widely applicable, since they simply require that pairs of predictor variables are not too collinear.