AI 中文总结
本研究证明确定性算法在非光滑非凸优化中寻找Goldstein稳定点的复杂度至少为(1/ε)^Ω(d)，而随机算法可达到维度无关复杂度，从而揭示随机化带来的指数级计算优势。
AI 中文摘要
我们研究了寻找非光滑非凸Lipschitz函数的$(\delta,\epsilon)$-Goldstein稳定点的复杂性。迄今为止，已知随机一阶算法可以以与维度无关的预言机复杂度解决此任务[Zhang et al., 2020]，而确定性算法则不能，因为其复杂度必须至少随维度$d$线性增长[Jordan et al., 2023, Tian and So, 2024]。这留下了一个开放问题：确定性算法是否仍能以关于$d$的多项式预言机复杂度解决该问题。我们通过证明确定性算法的下界为$(1/\epsilon)^{\Omega(d)}$来否定地回答这个问题，从而弥合了先前已知下界和上界之间的指数差距，并解决了Jordan et al. [2023]提出的一个开放问题。我们进一步讨论了该结果对较弱平稳性概念、寻找下降方向和确定性平滑的若干扩展和影响。总体而言，我们的结果确立了随机化在非光滑非凸优化中提供的指数计算优势。
英文摘要
We study the complexity of finding $(δ,ε)$-Goldstein stationary points of nonsmooth nonconvex Lipschitz functions. By now, it is known that randomized first-order algorithms can solve this task with a dimension-free oracle complexity [Zhang et al., 2020], whereas deterministic algorithms cannot, as their complexity must scale at least linearly with the dimension $d$ [Jordan et al., 2023, Tian and So, 2024]. This leaves open whether deterministic algorithms can nevertheless solve the problem with oracle complexity polynomial in $d$. We answer this question negatively by proving a lower bound of order $(1/ε)^{Ω(d)}$ for deterministic algorithm, closing the exponential gap between the previously known lower and upper bounds and resolving an open problem posed by Jordan et al. [2023]. We further discuss several extensions and implications of this result to weaker stationarity notions, finding a descent direction and deterministic smoothing. Overall, our results establish an exponential computational advantage in nonsmooth nonconvex optimization offered by randomization.
Comments13 pages
