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.
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