  Stochastic and adversarial data are two widely studied settings in online
learning. But many optimization tasks are neither i.i.d. nor fully adversarial,
which makes it of fundamental interest to get a better theoretical
understanding of the world between these extremes. In this work we establish
novel regret bounds for online convex optimization in a setting that
interpolates between stochastic i.i.d. and fully adversarial losses. By
exploiting smoothness of the expected losses, these bounds replace a dependence
on the maximum gradient length by the variance of the gradients, which was
previously known only for linear losses. In addition, they weaken the i.i.d.
assumption by allowing, for example, adversarially poisoned rounds, which were
previously considered in the expert and bandit setting. Our results extend this
to the online convex optimization framework. In the fully i.i.d. case, our
bounds match the rates one would expect from results in stochastic
acceleration, and in the fully adversarial case they gracefully deteriorate to
match the minimax regret. We further provide lower bounds showing that our
regret upper bounds are tight for all intermediate regimes in terms of the
stochastic variance and the adversarial variation of the loss gradients.
