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.

翻译：理解一阶优化算法的核心工具是Kurdyka-Lojasiewicz不等式。这类方法的标准分析方案关键依赖于该不等式，通过利用包含梯度或次梯度的充分下降条件。然而，KL性质本质上关注的并非次梯度，而是“斜率”这一纯度量概念。通过凸显这一视角并完全避免使用次梯度，我们对度量空间上的一阶优化算法提出了简洁的复杂度分析。这种无次梯度视角还为非光滑半代数函数的KL性质提供了一个简短而聚焦的证明。