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.

翻译：我们从理论上分析了用于使用统计机械复制方法进行模型选择的模型的典型学习性学效为$ell=1美元常规线性回归($ell_1美元-LinR) 。 对于在抛磁阶段的典型随机常规图表,我们获得了对美元1美元-LinR的典型样本复杂性的准确估计。 值得注意的是,尽管模型特性不正确, $_ ell_ 1美元-LinR是符合与美元1美元1美元($ell_1美元-LogR)相同的样本复杂性的典型选择。 模拟显示理论预测与实验结果($1美元-LogRR)之间相当的一致, 即使是用于与许多典型循环的图表, 其中美元是Ising模型变量的数量。 此外,我们提供了一种有效的方法来准确预测1美元-1美元-LinRRRRR的不依赖行为, 例如精确和回顾。 模拟显示理论预测和实验结果之间的相当一致, 即使是用于与许多常规循环的图表, 也支持我们学习的典型的模型。