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

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 $d=2$.