The Exponential Price of Determinism in Nonsmooth Nonconvex Optimization

2026-09-20

AI Summary

研究证明了在非光滑非凸优化中，确定性算法找到近似平稳点的复杂度至少为(1/ε)^Ω(d)，表明随机化方法具有指数级计算优势。

Abstract

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.

Problem

• nonsmooth nonconvex optimization
• deterministic algorithms
• complexity
• dimension
• Goldstein stationary points

Innovation

• nonsmooth nonconvex optimization
• deterministic algorithms
• randomized algorithms
• complexity lower bound