The problem of minimizing the maximum of $N$ convex, Lipschitz functions plays significant roles in optimization and machine learning. It has a series of results, with the most recent one requiring $O(N\epsilon^{-2/3} + \epsilon^{-8/3})$ queries to a first-order oracle to compute an $\epsilon$-suboptimal point. On the other hand, quantum algorithms for optimization are rapidly advancing with speedups shown on many important optimization problems. In this paper, we conduct a systematic study for quantum algorithms and lower bounds for minimizing the maximum of $N$ convex, Lipschitz functions. On one hand, we develop quantum algorithms with an improved complexity bound of $\tilde{O}(\sqrt{N}\epsilon^{-5/3} + \epsilon^{-8/3})$. On the other hand, we prove that quantum algorithms must take $\tilde{\Omega}(\sqrt{N}\epsilon^{-2/3})$ queries to a first order quantum oracle, showing that our dependence on $N$ is optimal up to poly-logarithmic factors.

翻译：最小化$N$个凸Lipschitz函数的最大值问题在优化和机器学习中具有重要作用。已有系列研究成果，最新结果需调用一阶预言机$O(N\epsilon^{-2/3} + \epsilon^{-8/3})$次以计算$\epsilon$-次优解。另一方面，量子优化算法正快速发展，已在诸多重要优化问题上展现出加速效果。本文系统研究了最小化$N$个凸Lipschitz函数最大值的量子算法及其下界。一方面，我们提出复杂度为$\tilde{O}(\sqrt{N}\epsilon^{-5/3} + \epsilon^{-8/3})$的改进量子算法；另一方面，我们证明量子算法对一阶量子预言机至少需要$\tilde{\Omega}(\sqrt{N}\epsilon^{-2/3})$次查询，从而表明算法对$N$的依赖在多项式对数因子意义上达到最优。