  We study the bootstrap for the maxima of the sums of independent random
variables, a problem of high relevance to many applications in modern
statistics. Since the consistency of bootstrap was justified by Gaussian
approximation in Chernozhukov et al. (2013), quite a few attempts have been
made to sharpen the error bound for bootstrap and reduce the sample size
requirement for bootstrap consistency. In this paper, we show that the sample
size requirement can be dramatically improved when we make the inference
slightly conservative, that is, to inflate the bootstrap quantile
$t_{\alpha}^*$ by a small fraction, e.g. by $1\%$ to $1.01\,t^*_\alpha$. This
simple procedure yields error bounds for the coverage probability of
conservative bootstrap at as fast a rate as $\sqrt{(\log p)/n}$ under suitable
conditions, so that not only the sample size requirement can be reduced to
$\log p \ll n$ but also the overall convergence rate is nearly parametric.
Furthermore, we improve the error bound for the coverage probability of the
standard non-conservative bootstrap to $[(\log (np))^3 (\log p)^2/n]^{1/4}$
under general assumptions on data. These results are established for the
empirical bootstrap and the multiplier bootstrap with third moment match. An
improved coherent Lindeberg interpolation method, originally proposed in Deng
and Zhang (2017), is developed to derive sharper comparison bounds, especially
for the maxima.
