(Bandit) Convex Optimization with Biased Noisy Gradient Oracles

TL;DR

This paper introduces a new abstract framework for bandit convex optimization that unifies previous methods and reveals fundamental limits on achieving optimal convergence rates with current gradient estimation techniques.

Contribution

It proposes a novel, abstract oracle-based framework that unifies existing approaches and demonstrates the necessity of new techniques for optimal convergence in bandit convex optimization.

Findings

Unified analysis of existing gradient estimation methods.

Showed current techniques cannot achieve optimal rates without new approaches.

Provided a formal framework for future research in bandit convex optimization.

Abstract

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.