(Bandit) Convex Optimization with Biased Noisy Gradient Oracles
This work addresses optimization challenges in online learning for researchers, but it is incremental as it builds on prior methods without introducing a new algorithm.
The paper tackles the problem of bandit convex optimization by proposing a novel framework that abstracts gradient estimation as an oracle, unifying previous works and revealing limitations in achieving optimal rates with existing methods.
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.