On the diameter of subgradient sequences in o-minimal structures

TL;DR

This paper analyzes the behavior of subgradient sequences for definable functions in o-minimal structures, establishing convergence conditions based on step size and stratification techniques.

Contribution

It introduces a novel analysis linking subgradient sequence diameter to function variation within o-minimal structures, proving convergence under specific step size conditions.

Findings

Subgradient sequence diameter relates to function value variation.

Bounded subgradient sequences converge with step size of order 1/k.

Lipschitz L-regular stratifications are used to analyze sequences.

Abstract

We study subgradient sequences of locally Lipschitz functions definable in a polynomially bounded o-minimal structure. We show that the diameter of any subgradient sequence is related to the variation in function values, with error terms dominated by a double summation of step sizes. Consequently, we prove that bounded subgradient sequences converge if the step sizes are of order $1/k$. The proof uses Lipschitz $L$-regular stratifications in o-minimal structures to analyze subgradient sequences via their projections onto different strata.