# Multi-task Learning for Gaussian Graphical Regressions with High Dimensional Covariates

arXiv: https://arxiv.org/abs/2205.10672 · HTML: https://mlsanity.com/paper/2205.10672/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 59 references that are in the map.
- First posted on arXiv 21 May 2022.

## Builds on (most cited first, 12 of 12)

- [A highly efficient semismooth Newton augmented Lagrangian method for solving Lasso problems](https://mlsanity.com/paper/1607.05428/) (2016)
- [An efficient Hessian based algorithm for solving large-scale sparse group Lasso problems](https://mlsanity.com/paper/1712.05910/) (2017)
- [Model Selection for High Dimensional Quadratic Regression via Regularization](https://mlsanity.com/paper/1501.00049/) (2014)
- [Two-Layer Feature Reduction for Sparse-Group Lasso via Decomposition of Convex Sets](https://mlsanity.com/paper/1410.4210/) (2014)
- [Hierarchical Sparse Modeling: A Choice of Two Group Lasso Formulations](https://mlsanity.com/paper/1512.01631/) (2015)
- [Learning Local Dependence In Ordered Data](https://mlsanity.com/paper/1604.07451/) (2016)
- [Asymptotically Normal and Efficient Estimation of Covariate-Adjusted Gaussian Graphical Model](https://mlsanity.com/paper/1309.5923/) (2013)
- [Mixed-Effect Time-Varying Network Model and Application in Brain Connectivity Analysis](https://mlsanity.com/paper/1806.03829/) (2018)
- [Sparse Group Lasso: Optimal Sample Complexity, Convergence Rate, and Statistical Inference](https://mlsanity.com/paper/1909.09851/) (2019)
- [On efficiently solving the subproblems of a level-set method for fused lasso problems](https://mlsanity.com/paper/1706.08732/) (2017)
- [High Dimensional Gaussian Graphical Regression Models with Covariates](https://mlsanity.com/paper/2011.05245/) (2020)
- [A Semi-smooth Newton Method for Solving Semidefinite Programs in Electronic Structure Calculations](https://mlsanity.com/paper/1708.08048/) (2017)

## Built on by (most cited first, 1 of 1)

- [Convex Estimation of Gaussian Graphical Regression Models with Covariates](https://mlsanity.com/paper/2410.06326/) (2024)

## Abstract (arXiv)

Gaussian graphical regression is a powerful means that regresses the precision matrix of a Gaussian graphical model on covariates, permitting the numbers of the response variables and covariates to far exceed the sample size. Model fitting is typically carried out via separate node-wise lasso regressions, ignoring the network-induced structure among these regressions. Consequently, the error rate is high, especially when the number of nodes is large. We propose a multi-task learning estimator for fitting Gaussian graphical regression models; we design a cross-task group sparsity penalty and a within task element-wise sparsity penalty, which govern the sparsity of active covariates and their effects on the graph, respectively. For computation, we consider an efficient augmented Lagrangian algorithm, which solves subproblems with a semi-smooth Newton method. For theory, we show that the error rate of the multi-task learning based estimates has much improvement over that of the separate node-wise lasso estimates, because the cross-task penalty borrows information across tasks. To address the main challenge that the tasks are entangled in a complicated correlation structure, we establish a new tail probability bound for correlated heavy-tailed (sub-exponential) variables with an arbitrary correlation structure, a useful theoretical result in its own right. Finally, the utility of our method is demonstrated through simulations as well as an application to a gene co-expression network study with brain cancer patients.