Near-Optimal Quantum Algorithm for Minimizing the Maximal Loss
<|reference_start|>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 $O(N\epsilon^{-2/3} + \epsilon^{-8/3})$ 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.<|reference_end|>
@article{wang2024near-optimal,
    title={Near-Optimal Quantum Algorithm for Minimizing the Maximal Loss},
    author={Hao Wang, Chenyi Zhang, Tongyang Li},
    journal={arXiv preprint arXiv:2402.12745},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.12745},
    primaryClass={quant-ph cs.DS math.OC}
}