  Non-asymptotic bounds for Gaussian and bootstrap approximation have recently
attracted significant interest in high-dimensional statistics. This paper
studies Berry-Esseen bounds for such approximations with respect to the
multivariate Kolmogorov distance, in the context of a sum of $n$ random vectors
that are $p$-dimensional and i.i.d. Up to now, a growing line of work has
established bounds with mild logarithmic dependence on $p$. However, the
problem of developing corresponding bounds with near $n^{-1/2}$ dependence on
$n$ has remained largely unresolved. Within the setting of random vectors that
have sub-Gaussian or sub-exponential entries, this paper establishes bounds
with near $n^{-1/2}$ dependence, for both Gaussian and bootstrap approximation.
In addition, the proofs are considerably distinct from other recent approaches
and make use of an "implicit smoothing" operation in the Lindeberg
interpolation.
