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The gradient's limit of a definable family of functions admits a variational stratification

Sholom Schechtman

TL;DR

The paper addresses how gradient behavior behaves in the limit of a definable family of functions $(f_a)$ converging to $F$ as $a\to 0$. It introduces the limit set $D_F$ of Clarke-like Jacobians and proves that, under definability in an o-minimal structure, the pair $(F,D_F)$ admits a definable $C^p$ variational stratification, and in many cases yields a conservative field. A key contribution is the stability of variational stratifications under graph-closure and a parametric closure result, which together enable gradient-limit analyses for smoothing methods. These results provide theoretical guarantees for convergence and gradient-consistency of smoothing approaches to nonsmooth or non-Lipschitz objectives, with broad implications for optimization and numerical methods in definable settings.

Abstract

It is well-known that the convergence of a family of smooth functions does not imply the convergence of its gradients. In this work, we show that if the family is definable in an o-minimal structure (for instance semialgebraic, subanalytic, or any composition of the previous with exp, log), then the gradient's limit admits a variational stratification and, under mild assumptions, is a conservative set-valued field in the sense introduced by Bolte and Pauwels. Immediate implications of this result on convergence guarantees of smoothing methods are discussed. The result is established in a general form, where the functions in the original family might be non Lipschitz continuous, be vector-valued and the gradients are replaced by their Clarke Jacobians or an arbitrary mapping satisfying a definable variational stratification. In passing, we investigate various stability properties of definable variational stratifications which might be of independent interest.

The gradient's limit of a definable family of functions admits a variational stratification

TL;DR

The paper addresses how gradient behavior behaves in the limit of a definable family of functions converging to as . It introduces the limit set of Clarke-like Jacobians and proves that, under definability in an o-minimal structure, the pair admits a definable variational stratification, and in many cases yields a conservative field. A key contribution is the stability of variational stratifications under graph-closure and a parametric closure result, which together enable gradient-limit analyses for smoothing methods. These results provide theoretical guarantees for convergence and gradient-consistency of smoothing approaches to nonsmooth or non-Lipschitz objectives, with broad implications for optimization and numerical methods in definable settings.

Abstract

It is well-known that the convergence of a family of smooth functions does not imply the convergence of its gradients. In this work, we show that if the family is definable in an o-minimal structure (for instance semialgebraic, subanalytic, or any composition of the previous with exp, log), then the gradient's limit admits a variational stratification and, under mild assumptions, is a conservative set-valued field in the sense introduced by Bolte and Pauwels. Immediate implications of this result on convergence guarantees of smoothing methods are discussed. The result is established in a general form, where the functions in the original family might be non Lipschitz continuous, be vector-valued and the gradients are replaced by their Clarke Jacobians or an arbitrary mapping satisfying a definable variational stratification. In passing, we investigate various stability properties of definable variational stratifications which might be of independent interest.
Paper Structure (25 sections, 34 theorems, 72 equations)

This paper contains 25 sections, 34 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Paper organization.
  3. Preliminaries
  4. Notations.
  5. Submanifolds
  6. Stratifications
  7. Functions definable in an o-minimal structure
  8. Conservative set-valued fields
  9. Variational stratifications
  10. Examples.
  11. Computational properties of definable variational stratifications
  12. Closing the graph of a conservative mapping
  13. Closure preserves conservativity
  14. Parametric closure
  15. Definable limits of conservative fields
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let the function $f: x,a \mapsto f_a(x)$ be definable and, for each $a>0$, $f_a$ be continuous. Let $\partial f_a$ in eq:def_cons_intro denote the Clarke subgradient of $f_a$ and assume that for all $x$, $\lim_{(x',a) \rightarrow (x,0)}f_a(x') = F(x)$. Then, the pair $(F,D_F)$ admits a variational

Theorems & Definitions (81)

  • Theorem 1.1: Particular case of Theorem \ref{['thm:main']}
  • Lemma 2.1: lee2022manifolds
  • Definition 2.2
  • Definition 2.3: Whitney stratification of a set
  • Remark 2.4
  • Definition 2.5: Whitney stratification of a function van96
  • Definition 2.6: Thom ($a_f$) stratification
  • Definition 2.7
  • Proposition 2.8: van96
  • Proposition 2.9: van96
  • ...and 71 more