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Mathematics > Optimization and Control

arXiv:2210.06907 (math)
[Submitted on 13 Oct 2022]

Title:No Dimension-Free Deterministic Algorithm Computes Approximate Stationarities of Lipschitzians

Authors:Lai Tian, Anthony Man-Cho So
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Abstract:We consider the computation of an approximately stationary point for a Lipschitz and semialgebraic function $f$ with a local oracle. If $f$ is smooth, simple deterministic methods have dimension-free finite oracle complexities. For the general Lipschitz setting, only recently, Zhang et al. [47] introduced a randomized algorithm that computes Goldstein's approximate stationarity [25] to arbitrary precision with a dimension-free polynomial oracle complexity.
In this paper, we show that no deterministic algorithm can do the same. Even without the dimension-free requirement, we show that any finite time guaranteed deterministic method cannot be general zero-respecting, which rules out most of the oracle-based methods in smooth optimization and any trivial derandomization of Zhang et al. [47]. Our results reveal a fundamental hurdle of nonconvex nonsmooth problems in the modern large-scale setting and their infinite-dimensional extension.
Comments: 17 pages, submitted to SODA 2023
Subjects: Optimization and Control (math.OC)
Cite as: arXiv:2210.06907 [math.OC]
  (or arXiv:2210.06907v1 [math.OC] for this version)
  https://doi.org/10.48550/arXiv.2210.06907

Submission history

From: Lai Tian [view email]
[v1] Thu, 13 Oct 2022 11:20:05 UTC (927 KB)
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