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On the diameter of subgradient sequences in o-minimal structures

Lexiao Lai, Mingzhi Song

TL;DR

The paper analyzes subgradient sequences of locally Lipschitz, definable functions in polynomially bounded o-minimal structures. By leveraging Lipschitz $L$-regular stratifications and a piecewise smooth decomposition, it relates the diameter of a subgradient sequence near a level set to the variation of function values, with error terms governed by a double sum of step sizes. A desingularizing function and strata-projected dynamics yield convergence when the step sizes decay as $O(1/k)$. This provides a rigorous convergence guarantee for nonconvex nonsmooth definable objectives via geometric stratifications, extending subgradient convergence theory beyond convex settings.

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.

On the diameter of subgradient sequences in o-minimal structures

TL;DR

The paper analyzes subgradient sequences of locally Lipschitz, definable functions in polynomially bounded o-minimal structures. By leveraging Lipschitz -regular stratifications and a piecewise smooth decomposition, it relates the diameter of a subgradient sequence near a level set to the variation of function values, with error terms governed by a double sum of step sizes. A desingularizing function and strata-projected dynamics yield convergence when the step sizes decay as . This provides a rigorous convergence guarantee for nonconvex nonsmooth definable objectives via geometric stratifications, extending subgradient convergence theory beyond convex settings.

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 . The proof uses Lipschitz -regular stratifications in o-minimal structures to analyze subgradient sequences via their projections onto different strata.

Paper Structure

This paper contains 5 sections, 15 theorems, 136 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. O-minimal structures and stratifications
  3. Piecewise smooth decomposition of definable functions
  4. Proof of Theorem \ref{['thm:diameter']}
  5. Lipschitz stratification

Key Result

Theorem 1.1

Let $f:\mathbb{R}^n\to \mathbb{R}$ be locally Lipschitz and definable in a polynomially bounded o-minimal structure. For any bounded $X\subset \mathbb{R}^n$, there exist $\bar{\alpha},\beta,\epsilon,\varsigma_1,\varsigma_2>0$ and $\theta\in (0,1)$ such that for any subgradient sequence $(x_k)_{k\in

Figures (1)

  • Figure 1: Constructed regions $\mathcal{N}_0(i,\alpha)$ for three adjacent strata.

Theorems & Definitions (31)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • Lemma 3.1
  • ...and 21 more