Quantum Algorithm for Online Convex Optimization

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 $T$ 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 $O(1)$ times in each round as feedback, we give a quantum algorithm that achieves $O(\sqrt{T})$ regret without additional dependence of the dimension $n$, which outperforms the already known optimal classical algorithm only achieving $O(\sqrt{nT})$ regret. Note that the regret of our quantum algorithm has achieved the lower bound of classical first-order methods. (ii) We show that for strongly convex loss functions, the quantum algorithm can achieve $O(\log T)$ regret with $O(1)$ queries as well, which means that the quantum algorithm can achieve the same regret bound as the classical algorithms in the full information setting.