Sparse High-Dimensional Linear Regression. Algorithmic Barriers and a Local Search Algorithm

We consider a sparse high dimensional regression model where the goal is to recover a $k$-sparse unknown vector $\beta^*$ from $n$ noisy linear observations of the form $Y=X\beta^*+W \in \mathbb{R}^n$ where $X \in \mathbb{R}^{n \times p}$ has iid $N(0,1)$ entries and $W \in \mathbb{R}^n$ has iid $N(0,\sigma^2)$ entries. Under certain assumptions on the parameters, an intriguing assymptotic gap appears between the minimum value of $n$, call it $n^*$, for which the recovery is information theoretically possible, and the minimum value of $n$, call it $n_{\mathrm{alg}}$, for which an efficient algorithm is known to provably recover $\beta^*$. In \cite{gamarnikzadik} it was conjectured that the gap is not artificial, in the sense that for sample sizes $n \in [n^*,n_{\mathrm{alg}}]$ the problem is algorithmically hard. We support this conjecture in two ways. Firstly, we show that the optimal solution of the LASSO provably fails to $\ell_2$-stably recover the unknown vector $\beta^*$ when $n \in [n^*,c n_{\mathrm{alg}}]$, for some sufficiently small constant $c>0$. Secondly, we establish that $n_{\mathrm{alg}}$, up to a multiplicative constant factor, is a phase transition point for the appearance of a certain Overlap Gap Property (OGP) over the space of $k$-sparse vectors. The presence of such an Overlap Gap Property phase transition, which originates in statistical physics, is known to provide evidence of an algorithmic hardness. Finally we show that if $n>C n_{\mathrm{alg}}$ for some large enough constant $C>0$, a very simple algorithm based on a local search improvement rule is able both to $\ell_2$-stably recover the unknown vector $\beta^*$ and to infer correctly its support, adding it to the list of provably successful algorithms for the high dimensional linear regression problem.