Part of Advances in Neural Information Processing Systems 37 (NeurIPS 2024) Main Conference Track
Chengchang Liu, Chaowen Guan, Jianhao He, John C. S. Lui
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 novel 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 $\epsilon^{-2/3}$.