  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
understanding of important differences among subpopulations of interest. In
this paper, we exploit multiple networks with Gaussian graphs to encode the
connectivity patterns of a large number of features on the subpopulations. To
uncover the heterogeneity of these structures across subpopulations, we suggest
a new framework of tuning-free heterogeneity pursuit (THP) via large-scale
inference, where the number of networks is allowed to diverge. In particular,
two new tests, the chi-based test and the linear functional-based test, 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 feature stem
from efficient multiple-network estimation by our newly suggested approach of
heterogeneous group square-root Lasso (HGSL) for high-dimensional
multi-response regression with heterogeneous noises. To solve this convex
program, we further introduce a tuning-free algorithm that is scalable and
enjoys provable convergence to the global optimum. Both computational and
theoretical advantages of our procedure are elucidated through simulation and
real data examples.
