Improved Regret Bounds for Projection-free Bandit Convex Optimization

TL;DR

This paper introduces a projection-free algorithm for bandit convex optimization that achieves improved regret bounds of O(T^{3/4}) with linear oracle calls, matching the best known bounds in full information settings.

Contribution

It presents the first projection-free bandit convex optimization algorithm with improved regret bounds of O(T^{3/4}) using only O(T) oracle calls, advancing scalable high-dimensional online learning.

Findings

Achieves O(T^{3/4}) expected regret bound.

Uses only O(T) calls to the linear optimization oracle.

Improves over previous O(T^{4/5}) regret bounds.

Abstract

We revisit the challenge of designing online algorithms for the bandit convex optimization problem (BCO) which are also scalable to high dimensional problems. Hence, we consider algorithms that are \textit{projection-free}, i.e., based on the conditional gradient method whose only access to the feasible decision set, is through a linear optimization oracle (as opposed to other methods which require potentially much more computationally-expensive subprocedures, such as computing Euclidean projections). We present the first such algorithm that attains $O(T^{3/4})$ expected regret using only $O(T)$ overall calls to the linear optimization oracle, in expectation, where $T$ is the number of prediction rounds. This improves over the $O(T^{4/5})$ expected regret bound recently obtained by \cite{Karbasi19}, and actually matches the current best regret bound for projection-free online learning in the \textit{full information} setting.