Tuning-Free Heterogeneous Inference in Massive Networks

Abstract

Heterogeneity is often natural in many contemporary applications involving massive data. While posing new challenges to effective learning, it can play a crucial role in powering meaningful scientific discoveries through the integration of information among subpopulations of interest. In this article, we exploit multiple networks with Gaussian graphs to encode the connectivity patterns of a large number of features on the subpopulations. To uncover the underlying sparsity structures across subpopulations, we suggest a framework of large-scale tuning-free heterogeneous inference, where the number of networks is allowed to diverge. In particular, two new tests, the chi-based and the linear functional-based tests, are introduced and their asymptotic null distributions are established. Under mild regularity conditions, we establish that both tests are optimal in achieving the testable region boundary and the sample size requirement for the latter test is minimal. Both theoretical guarantees and the tuning-free property stem from efficient multiple-network estimation by our newly suggested heterogeneous group square-root Lasso for high-dimensional multi-response regression with heterogeneous noises. To solve this convex program, we further introduce a scalable algorithm that enjoys provable convergence to the global optimum. Both computational and theoretical advantages are elucidated through simulation and real data examples. Supplementary materials for this article are available online.

Keywords:

• Efficiency
• Heterogeneous group square-root Lasso
• Heterogeneous learning
• High dimensionality
• Large-scale inference
• Multiple networks
• Scalability
• Sparsity

Supplementary Material

The online supplementary materials contain a scalable HGSL algorithm with provable convergence, the proofs of Theorems 2.1-3.1 and Propositions 2.1-2.3, as well as the proofs of key lemmas and additional technical details. Additional computational cost comparison with existing methods is also provided.

Acknowledgments

Part of this work was completed while the last two authors visited the Departments of Statistics at University of California, Berkeley and Stanford University. These authors sincerely thank both departments for their hospitality.

Additional information

Funding

This work was supported by NSF Grant DMS-1812030, NIH funding: NIH Grant 1R01GM131407-01, NSF CAREER Awards DMS-0955316, and DMS-1150318, a grant from the Simons Foundation, and Adobe Data Science Research Award.