Ising Model Selection Using ${\ell }_1$-Regularized Linear Regression: A Statistical Mechanics Analysis

Abstract

We theoretically analyze the typical learning performance of ${\ell }_1$-regularized linear regression (${\ell }_1$-LinR) for Ising model selection using the replica method from statistical mechanics. For typical random regular graphs in the paramagnetic phase, an accurate estimate of the typical sample complexity of ${\ell }_1$-LinR is obtained. Remarkably, despite the model misspecification, ${\ell }_1$-LinR is model selection consistent with the same order of sample complexity as ${\ell }_1$-regularized logistic regression (${\ell }_1$-LogR), i.e., $M=\mathcal{O}\left(\log N\right)$, where $N$ is the number of variables of the Ising model. Moreover, we provide an efficient method to accurately predict the non-asymptotic behavior of ${\ell }_1$-LinR for moderate $M,N$, such as precision and recall. Simulations show a fairly good agreement between theoretical predictions and experimental results, even for graphs with many loops, which supports our findings. Although this paper mainly focuses on ${\ell }_1$-LinR, our method is readily applicable for precisely characterizing the typical learning performances of a wide class of ${\ell }_1$-regularized $M$-estimators including ${\ell }_1$-LogR and interaction screening.

Accepted to NeurIPS 2021. Camera-ready version with supplementary materials