Abstract We explore whether quantum advantages can be found for the zeroth-order online convex optimization (OCO) 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 OCO. 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 ( T ) regret without additional dependence of the dimension n, which outperforms the already known optimal classical algorithm only achieving O ( n T ) 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.

中文翻译：

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在线凸优化的量子算法

摘要 我们探讨了零阶在线凸优化（OCO）问题是否可以找到量子优势，该问题也称为多点反馈强盗凸优化。在这种情况下，给定对零阶预言的访问（即，损失函数作为一个黑盒访问，返回任何查询输入的函数值），玩家试图最小化一系列对抗性生成的凸损失函数。这个过程可以描述为玩家和对手之间的 T 轮迭代博弈。在本文中，我们针对该问题提出了量子算法，并首次展示了 OCO 问题可能具有潜在的量子优势。具体来说，我们的贡献如下。它优于已知的仅实现 O ( n T ) 遗憾的最优经典算法。请注意，我们的量子算法的遗憾已经达到了经典一阶方法的下限。(ii) 我们表明，对于强凸损失函数，量子算法也可以通过 O(1) 查询实现 O(log T) 后悔，