  We investigate the problem of testing the global null in the high-dimensional
regression models when the feature dimension $p$ grows proportionally to the
number of observations $n$. Despite a number of prior work studying this
problem, whether there exists a test that is model-agnostic, efficient to
compute and enjoys high power, still remains unsettled. In this paper, we
answer this question in the affirmative by leveraging the random projection
techniques, and propose a testing procedure that blends the classical $F$-test
with a random projection step. When combined with a systematic choice of the
projection dimension, the proposed procedure is proved to be minimax optimal
and, meanwhile, reduces the computation and data storage requirements. We
illustrate our results in various scenarios when the underlying feature matrix
exhibits an intrinsic lower dimensional structure (such as approximate block
structure or has exponential/polynomial eigen-decay), and it turns out that the
proposed test achieves sharp adaptive rates. Our theoretical findings are
further validated by comparisons to other state-of-the-art tests on the
synthetic data.
