  We propose a distributed bootstrap method for simultaneous inference on
high-dimensional massive data that are stored and processed with many machines.
The method produces a $\ell_\infty$-norm confidence region based on a
communication-efficient de-biased lasso, and we propose an efficient
cross-validation approach to tune the method at every iteration. We
theoretically prove a lower bound on the number of communication rounds
$\tau_{\min}$ that warrants the statistical accuracy and efficiency.
Furthermore, $\tau_{\min}$ only increases logarithmically with the number of
workers and intrinsic dimensionality, while nearly invariant to the nominal
dimensionality. We test our theory by extensive simulation studies, and a
variable screening task on a semi-synthetic dataset based on the US Airline
On-time Performance dataset. The code to reproduce the numerical results is
available at GitHub: https://github.com/skchao74/Distributed-bootstrap.
