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 $\theta_0$, it is rigorously proved that Lasso \textit{without post-thresholding} is model selection consistent in the whole paramagnetic phase with the same order of sample complexity $n=\Omega{(d^3\log{p})}$ as that of $\ell_1$-regularized logistic regression ($\ell_1$-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.
翻译:我们从理论上分析了高维Ising 模型中最不绝对缩缩和甄选操作员(Lasso)的模型选择一致性,无论是否持有后保存,对于高维Ising 模型(Lasso) 。对于以常规节点为单位的随机常规(RR)图($P$,美元美元)和统一的组合($@theta_0美元),我们严格地证明,Lasso \ textit{不带后保存} 是整个地磁阶段的模型选择一致性,其样本复杂性的顺序相同($ ⁇ Omega{(d}3\log{p}}),与美元/ell_1美元($1美元)的常规后勤回归($1美元-LogR)相同。这一结果与Meng、Obuchi和Kabashima 2021 的配置一致,使用了统计物理的非硬性复制法方法,从而用严格的证据来补充它。对于树类图一样,也证明了RRRgetgress的相同的结果,可以在对依赖性条件和一致性的假设下获得。此外,我们以精确的模型模型的理论模型模型分析结果,我们同意。