Near-Optimal Quantum Algorithm for Minimizing the Maximal Loss

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(Nε-2/3 + ε-8/3) queries to a first-order oracle to compute an ε-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 Õ(√Nε-5/3 + ε-8/3). On the other hand, we prove that quantum algorithms must take Ω̃(√Nε-2/3) queries to a first order quantum oracle, showing that our dependence on N is optimal up to poly-logarithmic factors.