Near-Optimal Quantum Algorithm for Minimizing the Maximal Loss

Abstract

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.

22 pages, 1 figure, To appear in The Twelfth International Conference on Learning Representations (ICLR 2024)