Work

On the Complexity of Finding Stationary Points of p-th Order Hölder Smooth Functions in One Dimension

Year: 2026

Type: preprint

Abstract: We study the query complexity of finding an an $\varepsilon$-stationary points of a one-dimensional non-convex function whose $p$-th derivative is $β_p$-Hölder continuous. We determine the optimal complexity for deterministic and randomized algorithms and for oracles that return only derivatives, only function values, or full local information. Randomness provably helps when only derivatives are available, function values help in all cases, and higher-order derivatives never do. Together with known results in higher dimensions, this shows how these benefits depend on the dimension. Our deterministic lower bound also implies that the optimal accuracy exponents of objective-function-free adaptive regularization of order $p$ (OFFAR$p$) in every dimension $d\ge1$ and of deterministic gradient-only trapping in dimension $d=2$.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.16295

Authors: Zhou, Huanjian, Suzuki, Taiji

Institutions: The University of Tokyo, RIKEN Center for Advanced Intelligence Project

Language: English

Topic: Stochastic Gradient Optimization Techniques

Subfield: Artificial Intelligence

Field: Computer Science

Domain: Physical Sciences

Open Access status: green

Funders: Japan Society for the Promotion of Science, Core Research for Evolutional Science and Technology, Exploratory Research for Advanced Technology

Award: 大規模基盤モデルの革新的原理解明と方法論の新展開

OpenAlex ID: W7221756104

Cites: 3

Cited by: 1

Related to: 10