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

arXiv: https://arxiv.org/abs/2305.03208 · HTML: https://mlsanity.com/paper/2305.03208/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 May 2023.

## Built on by (most cited first, 1 of 1)

- [Benign Nonconvex Landscape for Policy Optimization: Infinite-Horizon Discounted MDPs with General State and Action Spaces](https://mlsanity.com/paper/2609.21433/) (2026)

## Abstract (arXiv)

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.