  The Bonferroni adjustment, or the union bound, is commonly used to study rate
optimality properties of statistical methods in high-dimensional problems.
However, in practice, the Bonferroni adjustment is overly conservative. The
extreme value theory has been proven to provide more accurate multiplicity
adjustments in a number of settings, but only on ad hoc basis. Recently,
Gaussian approximation has been used to justify bootstrap adjustments in large
scale simultaneous inference in some general settings when $n \gg (\log p)^7$,
where $p$ is the multiplicity of the inference problem and $n$ is the sample
size. The thrust of this theory is the validity of the Gaussian approximation
for maxima of sums of independent random vectors in high-dimension. In this
paper, we reduce the sample size requirement to $n \gg (\log p)^5$ for the
consistency of the empirical bootstrap and the multiplier/wild bootstrap in the
Kolmogorov-Smirnov distance, possibly in the regime where the Gaussian
approximation is not available. New comparison and anti-concentration theorems,
which are of considerable interest in and of themselves, are developed as
existing ones interweaved with Gaussian approximation are no longer applicable.
