On Model Selection Consistency of Lasso for High-Dimensional Ising Models

We theoretically analyze the model selection consistency of least absolute shrinkage and selection operator (Lasso), both with and without post-thresholding, for high-dimensional Ising models. For random regular (RR) graphs of size p with regular node degree d and uniform couplings θ0, it is rigorously proved that Lasso textitwithout post-thresholding is model selection consistent in the whole paramagnetic phase with the same order of sample complexity n=Omega(d3logp) as that of ℓ1-regularized logistic regression (ℓ1-LogR). This result is consistent with the conjecture in Meng, Obuchi, and Kabashima 2021 using the non-rigorous replica method from statistical physics and thus complements it with a rigorous proof. For general tree-like graphs, it is demonstrated that the same result as RR graphs can be obtained under mild assumptions of the dependency condition and incoherence condition. Moreover, we provide a rigorous proof of the model selection consistency of Lasso with post-thresholding for general tree-like graphs in the paramagnetic phase without further assumptions on the dependency and incoherence conditions. Experimental results agree well with our theoretical analysis.