  Algorithms for bandit convex optimization and online learning often rely on
constructing noisy gradient estimates, which are then used in appropriately
adjusted first-order algorithms, replacing actual gradients. Depending on the
properties of the function to be optimized and the nature of ``noise'' in the
bandit feedback, the bias and variance of gradient estimates exhibit various
tradeoffs. In this paper we propose a novel framework that replaces the
specific gradient estimation methods with an abstract oracle. With the help of
the new framework we unify previous works, reproducing their results in a clean
and concise fashion, while, perhaps more importantly, the framework also allows
us to formally show that to achieve the optimal root-$n$ rate either the
algorithms that use existing gradient estimators, or the proof techniques used
to analyze them have to go beyond what exists today.
