Learning Joint Graphical Models With Computational Efficiency, Dynamic Regularization, and Adaptation

Multi-sourced datasets are common in studies of variable interactions, for example, heterogeneous gene networks, cross-domain recommendation, etc., where each source induces a related but distinct dependency structure. Joint learning of multiple graphical models (i.e., multiple precision matrices) has emerged as an important tool in analyzing such data. Unlike separate learning, joint learning can leverage shared structural patterns across graphs to yield more accurate results. In this paper, we present an adaptive method named MIGHT (Multi-task Iterative Graphical Hard Thresholding) to estimate multiple graphs jointly. We introduce an $\boldsymbol{\ell}_\mathbf{0}$-based similarity measure, reformulate the joint model as a series of multi-task learning problems, and solve them using a dynamically regularized algorithm based on iterative hard thresholding. By leveraging the efficiency of the iterative procedure and the inherent parallelizability of the nodewise framework, the proposed method is computationally efficient and amenable to parallel implementation. Theoretically, we derive the non-asymptotic error bound for the resulting estimator under joint Gaussian graphical models. Furthermore, the proposed algorithm is adaptive to heterogeneous column-wise signal strengths: for nodes with strong signals, our estimator achieves improved error bounds and selection consistency adaptively, and also exhibits asymptotic normality without an additional debiasing step. The performance of our method is illustrated through numerical simulations and real-data analysis on a cancer gene-expression RNA-seq dataset.