  We derive Berry-Esseen approximation bounds for general functionals of
independent random variables, based on chaos expansions methods. Our results
apply to $U$-statistics satisfying the weak assumption of decomposability in
the Hoeffding sense, and yield Kolmogorov distance bounds instead of the
Wasserstein bounds previously derived in the special case of degenerate
$U$-statistics. Linear and quadratic functionals of arbitrary sequences of
independent random variables are included as particular cases, with new fourth
moment bounds, and applications are given to Hoeffding decompositions, weighted
$U$-statistics, quadratic forms, and random subgraph weighing. In the case of
quadratic forms, our results recover and improve the bounds available in the
literature, and apply to matrices with non-empty diagonals.
