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

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