The complexity of first-order optimization methods from a metric perspective

The complexity of first-order optimization methods from a metric perspective

A central tool for understanding first-order optimization algorithms is the Kurdyka-Lojasiewicz inequality. Standard approaches to such methods rely crucially on this inequality to leverage sufficient decrease conditions involving gradients or subgradients. However, the KL property fundamentally con...

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Veröffentlicht in:arXiv.org 2023-05
Hauptverfasser: Lewis, Adrian S, Tian, Tonghua
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Algorithms
Complexity
Metric space
Optimization
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description A central tool for understanding first-order optimization algorithms is the Kurdyka-Lojasiewicz inequality. Standard approaches to such methods rely crucially on this inequality to leverage sufficient decrease conditions involving gradients or subgradients. However, the KL property fundamentally concerns not subgradients but rather "slope", a purely metric notion. By highlighting this view, and avoiding any use of subgradients, we present a simple and concise complexity analysis for first-order optimization algorithms on metric spaces. This subgradient-free perspective also frames a short and focused proof of the KL property for nonsmooth semi-algebraic functions.
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Complexity
Metric space
Optimization
title The complexity of first-order optimization methods from a metric perspective
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