  We prove novel algorithmic guarantees for several online problems in the
smoothed analysis model. In this model, at each time an adversary chooses an
input distribution with density function bounded above by $\tfrac{1}{\sigma}$
times that of the uniform distribution; nature then samples an input from this
distribution. Crucially, our results hold for {\em adaptive} adversaries that
can choose an input distribution based on the decisions of the algorithm and
the realizations of the inputs in the previous time steps.
  This paper presents a general technique for proving smoothed algorithmic
guarantees against adaptive adversaries, in effect reducing the setting of
adaptive adversaries to the simpler case of oblivious adversaries. We apply
this technique to prove strong smoothed guarantees for three problems:
  -Online learning: We consider the online prediction problem, where instances
are generated from an adaptive sequence of $\sigma$-smooth distributions and
the hypothesis class has VC dimension $d$. We bound the regret by
$\tilde{O}\big(\sqrt{T d\ln(1/\sigma)} + d\sqrt{\ln(T/\sigma)}\big)$. This
answers open questions of [RST11,Hag18].
  -Online discrepancy minimization: We consider the online Koml\'os problem,
where the input is generated from an adaptive sequence of $\sigma$-smooth and
isotropic distributions on the $\ell_2$ unit ball. We bound the $\ell_\infty$
norm of the discrepancy vector by $\tilde{O}\big(\ln^2\!\big(
\frac{nT}{\sigma}\big) \big)$.
  -Dispersion in online optimization: We consider online optimization of
piecewise Lipschitz functions where functions with $\ell$ discontinuities are
chosen by a smoothed adaptive adversary and show that the resulting sequence is
$\big( {\sigma}/{\sqrt{T\ell}}, \tilde O\big(\sqrt{T\ell}
\big)\big)$-dispersed. This matches the parameters of [BDV18] for oblivious
adversaries, up to log factors.
