Quantum Algorithm for Online Exp-concave Optimization

TL;DR

This paper introduces a quantum algorithm for online exp-concave optimization that achieves lower regret than classical methods by leveraging quantum techniques to estimate the Hessian, demonstrating quantum advantage in this setting.

Contribution

The paper develops quantum online quasi-Newton methods for bandit exp-concave optimization, achieving improved regret bounds over classical algorithms.

Findings

Achieves $O(n\log T)$ regret with $O(1)$ queries per round.

Demonstrates quantum advantage by surpassing classical regret bounds.

Introduces quantum Hessian estimation for online optimization.

Abstract

We explore whether quantum advantages can be found for the zeroth-order feedback online exp-concave optimization problem, which is also known as bandit exp-concave optimization with multi-point feedback. We present quantum online quasi-Newton methods to tackle the problem and show that there exists quantum advantages for such problems. Our method approximates the Hessian by quantum estimated inexact gradient and can achieve $O(n\log T)$ regret with $O(1)$ queries at each round, where $n$ is the dimension of the decision set and $T$ is the total decision rounds. Such regret improves the optimal classical algorithm by a factor of $T^{2/3}$.