← 研踪 PaperTrailarXiv:2611.05807 · v1
Near-Optimal Complexity of Finding Approximate Local Minima of Nonconvex Finite Sums
首次提交 2026-11-04 09:32
版本更新 2026-11-04 09:32
雷达观测 2026-11-05 10:08
摘要
We study the component-oracle complexity of finding approximate local minima of nonconvex finite sums under mean-squared averaged smoothness with parameter $L$ and a $ρ$-Lipschitz average Hessian. Each query returns the value, gradient, and one Hessian-vector product of a component. We show that the complexity in the high-accuracy regime $\varepsilon\le c_0L^2/(nρ)$ is $\tildeΘ(n^{3/4}L^{1/2}ρ^{1/4}Δ\varepsilon^{-7/4})$ for a sufficiently large initial gap $Δ$. We attain the upper bound by adapting the inverse solver and eigenvector routine of FastCubic to averaged smoothness. Our lower bound assigns the bridge edges of a nested chain to random components; a replicated spectral construction gives the same bound for unrestricted randomized algorithms. Under individual smoothness, a factor of $n^{1/4}$ remains for $\varepsilon\le cL^2/(n^2ρ)$, up to logarithmic factors. In the complementary low-accuracy regime, we propose PathVR, which uses $\tilde O(n+\sqrt{n}LΔ\varepsilon^{-2})$ queries and matches a lower bound from dense weak hiding.
作者备注：30 pages, 2 tables
观测到的版本
相邻记录可能跳过中间版本；首次发现 v2 及以上也会标为版本更新。
v1 · 2026-11-04 09:32 · 最新观测版本
Near-Optimal Complexity of Finding Approximate Local Minima of Nonconvex Finite Sums
We study the component-oracle complexity of finding approximate local minima of nonconvex finite sums under mean-squared averaged smoothness with parameter $L$ and a $ρ$-Lipschitz average Hessian. Each query returns the value, gradient, and one Hessian-vector product of a component. We show that the complexity in the high-accuracy regime $\varepsilon\le c_0L^2/(nρ)$ is $\tildeΘ(n^{3/4}L^{1/2}ρ^{1/4}Δ\varepsilon^{-7/4})$ for a sufficiently large initial gap $Δ$. We attain the upper bound by adapting the inverse solver and eigenvector routine of FastCubic to averaged smoothness. Our lower bound assigns the bridge edges of a nested chain to random components; a replicated spectral construction gives the same bound for unrestricted randomized algorithms. Under individual smoothness, a factor of $n^{1/4}$ remains for $\varepsilon\le cL^2/(n^2ρ)$, up to logarithmic factors. In the complementary low-accuracy regime, we propose PathVR, which uses $\tilde O(n+\sqrt{n}LΔ\varepsilon^{-2})$ queries and matches a lower bound from dense weak hiding.
查看此版本解读 →查看此版本原文 ↗