Quantum Algorithm for Online Convex Optimization
Jianhao He, Feidiao Yang, Jialin Zhang, Lvzhou Li

TL;DR
This paper introduces quantum algorithms for online convex optimization that outperform classical methods by achieving lower regret bounds with fewer queries, demonstrating potential quantum advantages in this domain.
Contribution
The paper presents the first quantum algorithms for online convex optimization, achieving lower regret bounds and fewer queries compared to classical algorithms.
Findings
Quantum algorithms achieve $O(\sqrt{T})$ regret with $O(1)$ queries, outperforming classical $O(\sqrt{nT})$ regret.
For strongly convex functions, quantum algorithms reach $O(\log T)$ regret with $O(1)$ queries, matching classical full-information bounds.
Quantum advantages are demonstrated in online convex optimization with minimal oracle queries.
Abstract
We explore whether quantum advantages can be found for the zeroth-order online convex optimization problem, which is also known as bandit convex optimization with multi-point feedback. In this setting, given access to zeroth-order oracles (that is, the loss function is accessed as a black box that returns the function value for any queried input), a player attempts to minimize a sequence of adversarially generated convex loss functions. This procedure can be described as a round iterative game between the player and the adversary. In this paper, we present quantum algorithms for the problem and show for the first time that potential quantum advantages are possible for problems of online convex optimization. Specifically, our contributions are as follows. (i) When the player is allowed to query zeroth-order oracles times in each round as feedback, we give a quantum algorithm…
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