Overcoming the limitations of phase transition by higher order analysis of regularization techniques

Abstract

We study the problem of estimating a sparse vector $\beta\in\mathbb{R}^{p}$ from the response variables $y=X\beta+w$, where $w\sim N(0,\sigma_{w}^{2}I_{n\times n})$, under the following high-dimensional asymptotic regime: given a fixed number $\delta$, $p\rightarrow\infty$, while $n/p\rightarrow\delta$. We consider the popular class of $\ell_{q}$-regularized least squares (LQLS), a.k.a. bridge estimators, given by the optimization problem \begin{equation*}\hat{\beta}(\lambda,q)\in\arg\min_{\beta}\frac{1}{2}\|y-X\beta\|_{2}^{2}+\lambda\|\beta\|_{q}^{q},\end{equation*} and characterize the almost sure limit of $\frac{1}{p}\|\hat{\beta}(\lambda,q)-\beta\|_{2}^{2}$, and call it asymptotic mean square error (AMSE). The expression we derive for this limit does not have explicit forms, and hence is not useful in comparing LQLS for different values of $q$, or providing information in evaluating the effect of $\delta$ or sparsity level of $\beta$. To simplify the expression, researchers have considered the ideal “error-free” regime, that is, $w=0$, and have characterized the values of $\delta$ for which AMSE is zero. This is known as the phase transition analysis.

In this paper, we first perform the phase transition analysis of LQLS. Our results reveal some of the limitations and misleading features of the phase transition analysis. To overcome these limitations, we propose the small error analysis of LQLS. Our new analysis framework not only sheds light on the results of the phase transition analysis, but also describes when phase transition analysis is reliable, and presents a more accurate comparison among different regularizers.