Near-Optimal Quantum Algorithm for Minimizing the Maximal Loss
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Keywords: Quantum Algorithms, Quantum Query Complexity, Convex Optimization, Minimizing Loss
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TL;DR: We conduct a systematic study of quantum algorithms and lower bounds for minimizing the maximum of $N$ convex, Lipschitz functions.
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 of 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.
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Primary Area: optimization
Submission Number: 5095
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