Ising Model Selection Using $\ell_{1}$-Regularized Linear Regression: A Statistical Mechanics Analysis
This work addresses model selection for Ising models, which is important for statistical physics and machine learning applications, but it is incremental as it extends existing ℓ₁-regularized methods with theoretical analysis.
The paper tackles the problem of Ising model selection using ℓ₁-regularized linear regression, showing that it achieves model selection consistency with sample complexity M = O(log N), matching ℓ₁-regularized logistic regression, and provides accurate predictions for non-asymptotic performance metrics like precision and recall.
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.