Dec 6, 2021
We theoretically investigate the typical learning performance of ℓ_1-regularized linear regression (ℓ_1-LinR) for Ising model selection using the replica method from statistical mechanics. For typical random regular (RR) graphs in the paramagnetic phase, we obtain an accurate estimate of the typical sample complexity of ℓ_1-LinR, which demonstrates that ℓ_1-LinR is model selection consistent with M=𝒪(log N) samples, where N is the number of variables of the Ising model. Moreover, we provide a computationally efficient method to accurately predict the non-asymptotic behavior of ℓ_1-LinR for moderate M and N, such as the precision and recall rates. Simulations show a fairly good agreement between the theoretical predictions and experimental results, even for graphs with many loops, which supports our findings. Although this paper focuses on ℓ_1-LinR, our method is readily applicable to precisely investigating the typical learning performance of a wide class of ℓ_1-regularized estimators for Ising model selection such as ℓ_1-regularized logistic regression and interaction screening.
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