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arXiv · 2210.06907

No Dimension-Free Deterministic Algorithm Computes Approximate Stationarities of Lipschitzians

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.

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Lai Tian, Anthony Man-Cho So. 2022-10-13. No Dimension-Free Deterministic Algorithm Computes Approximate Stationarities of Lipschitzians. https://arxiv.org/abs/2210.06907

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