  Much of modern learning theory has been split between two regimes: the
classical offline setting, where data arrive independently, and the online
setting, where data arrive adversarially. While the former model is often both
computationally and statistically tractable, the latter requires no
distributional assumptions. In an attempt to achieve the best of both worlds,
previous work proposed the smooth online setting where each sample is drawn
from an adversarially chosen distribution, which is smooth, i.e., it has a
bounded density with respect to a fixed dominating measure. We provide tight
bounds on the minimax regret of learning a nonparametric function class, with
nearly optimal dependence on both the horizon and smoothness parameters.
Furthermore, we provide the first oracle-efficient, no-regret algorithms in
this setting. In particular, we propose an oracle-efficient improper algorithm
whose regret achieves optimal dependence on the horizon and a proper algorithm
requiring only a single oracle call per round whose regret has the optimal
horizon dependence in the classification setting and is sublinear in general.
Both algorithms have exponentially worse dependence on the smoothness parameter
of the adversary than the minimax rate. We then prove a lower bound on the
oracle complexity of any proper learning algorithm, which matches the
oracle-efficient upper bounds up to a polynomial factor, thus demonstrating the
existence of a statistical-computational gap in smooth online learning.
Finally, we apply our results to the contextual bandit setting to show that if
a function class is learnable in the classical setting, then there is an
oracle-efficient, no-regret algorithm for contextual bandits in the case that
contexts arrive in a smooth manner.
