We theoretically analyze the consistency of model selection for the least-absolute shrinkage and selection operator (Lasso) for high-dimensional Ising models. For a random regular (RR) graph of size $p$, node order $d$, and uniform coupling $\theta_0$, the postthreshold-free Lasso has the same order across the paramagnetic phase. Rigorously proven consistent model selection. Sample complexity as $n=\Omega{(d^3\log{p})}$, as that of $\ell_1$-regularized logistic regression ($\ell_1$-LogR). This result agrees with the conjecture of $\textit{Meng, Obuchi, and Kabashima 2021}$ using the inexact replication method of statistical physics, so we complement it with a rigorous proof. For general tree graphs, it has been demonstrated that under milder assumptions of dependent and inconsistent conditions, the same results as RR graphs can be obtained. Furthermore, we provide a rigorous proof of the consistency of Lasso’s model selection by thresholding the general tree-like graph of the paramagnetic phase without further assumptions about the conditions of dependence and inconsistency. The experimental results are in good agreement with the theoretical analysis.
