Quantum Algorithms for Non-smooth Non-convex Optimization

• URL: http://arxiv.org/abs/2410.16189v1
• Date: Mon, 21 Oct 2024 16:52:26 GMT
• Title: Quantum Algorithms for Non-smooth Non-convex Optimization
• Authors: Chengchang Liu, Chaowen Guan, Jianhao He, John C. S. Lui,
• Abstract summary: This paper considers the problem for finding the $(,epsilon)$-Goldstein stationary point of Lipschitz continuous objective. We construct a zeroth-order quantum estimator for the surrogate oracle function.
• Score: 30.576546266390714
• License:
• Abstract: This paper considers the problem for finding the $(\delta,\epsilon)$-Goldstein stationary point of Lipschitz continuous objective, which is a rich function class to cover a great number of important applications. We construct a zeroth-order quantum estimator for the gradient of the smoothed surrogate. Based on such estimator, we propose a novel quantum algorithm that achieves a query complexity of $\tilde{\mathcal{O}}(d^{3/2}\delta^{-1}\epsilon^{-3})$ on the stochastic function value oracle, where $d$ is the dimension of the problem. We also enhance the query complexity to $\tilde{\mathcal{O}}(d^{3/2}\delta^{-1}\epsilon^{-7/3})$ by introducing a variance reduction variant. Our findings demonstrate the clear advantages of utilizing quantum techniques for non-convex non-smooth optimization, as they outperform the optimal classical methods on the dependency of $\epsilon$ by a factor of $\epsilon^{-2/3}$.

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