Simultaneous support recovery in high dimensions: Benefits and perils of block $\ell_1/\ell_\infty$-regularization

TL;DR

This paper analyzes the high-dimensional behavior of $\, ext{l}_1/ ext{l}_ ext{infinity}$-regularized regression for joint estimation, revealing phase transitions and conditions where it outperforms or underperforms simpler methods depending on support overlap.

Contribution

It provides theoretical bounds, phase transition analysis, and insights into when block $ ext{l}_1/ ext{l}_ ext{infinity}$ regularization improves or worsens statistical efficiency.

Findings

Phase transition characterized by sample size and support overlap.

Improved efficiency when overlap exceeds 2/3.

Potential performance degradation for moderate to small overlap.

Abstract

Consider the use of $\ell_{1}/\ell_{\infty}$-regularized regression for joint estimation of a $\pdim \times \numreg$ matrix of regression coefficients. We analyze the high-dimensional scaling of $\ell_1/\ell_\infty$-regularized quadratic programming, considering both consistency in $\ell_\infty$-norm, and variable selection. We begin by establishing bounds on the $\ell_\infty$-error as well sufficient conditions for exact variable selection for fixed and random designs. Our second set of results applies to $\numreg = 2$ linear regression problems with standard Gaussian designs whose supports overlap in a fraction $\alpha \in [0,1]$ of their entries: for this problem class, we prove that the $\ell_{1}/\ell_{\infty}$-regularized method undergoes a phase transition--that is, a sharp change from failure to success--characterized by the rescaled sample size $\theta_{1,\infty}(n, p, s, \alpha) = n/\{(4 - 3 \alpha) s \log(p-(2- \alpha) s)\}$. An implication of this threshold is that use of $\ell_1 / \ell_{\infty}$-regularization yields improved statistical efficiency if the overlap parameter is large enough ($\alpha > 2/3$), but has \emph{worse} statistical efficiency than a naive Lasso-based approach for moderate to small overlap ($\alpha < 2/3$). These results indicate that some caution needs to be exercised in the application of $\ell_1/\ell_\infty$ block regularization: if the data does not match its structure closely enough, it can impair statistical performance relative to computationally less expensive schemes.