Mathematics > Optimization and Control
[Submitted on 13 Oct 2023 (v1), last revised 11 Sep 2024 (this version, v2)]
Title:The Computational Complexity of Finding Stationary Points in Non-Convex Optimization
View PDFAbstract:Finding approximate stationary points, i.e., points where the gradient is approximately zero, of non-convex but smooth objective functions $f$ over unrestricted $d$-dimensional domains is one of the most fundamental problems in classical non-convex optimization. Nevertheless, the computational and query complexity of this problem are still not well understood when the dimension $d$ of the problem is independent of the approximation error. In this paper, we show the following computational and query complexity results:
1. The problem of finding approximate stationary points over unrestricted domains is PLS-complete.
2. For $d = 2$, we provide a zero-order algorithm for finding $\varepsilon$-approximate stationary points that requires at most $O(1/\varepsilon)$ value queries to the objective function.
3. We show that any algorithm needs at least $\Omega(1/\varepsilon)$ queries to the objective function and/or its gradient to find $\varepsilon$-approximate stationary points when $d=2$. Combined with the above, this characterizes the query complexity of this problem to be $\Theta(1/\varepsilon)$.
4. For $d = 2$, we provide a zero-order algorithm for finding $\varepsilon$-KKT points in constrained optimization problems that requires at most $O(1/\sqrt{\varepsilon})$ value queries to the objective function. This closes the gap between the works of Bubeck and Mikulincer [2020] and Vavasis [1993] and characterizes the query complexity of this problem to be $\Theta(1/\sqrt{\varepsilon})$.
5. Combining our results with the recent result of Fearnley et al. [2022], we show that finding approximate KKT points in constrained optimization is reducible to finding approximate stationary points in unconstrained optimization but the converse is impossible.
Submission history
From: Alexandros Hollender [view email][v1] Fri, 13 Oct 2023 14:52:46 UTC (4,995 KB)
[v2] Wed, 11 Sep 2024 18:24:25 UTC (3,581 KB)
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