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On the equivalence of a Hessian-free inequality and Lipschitz continuous Hessian

Radu I. Boţ, Minh N. Dao, Tianxiang Liu, Bruno F. Lourenço, Naoki Marumo

TL;DR

This work resolves the question of whether a Hessian-free Jensen-type inequality implies Frechet differentiability with a Lipschitz derivative in a Hilbert-to-reflexive Banach setting. It employs a slice-based approach, using φ_{y^*} = y* ∘ F and the Baillon–Haddad theorem to obtain L-Lipschitz gradients of the slices, then assembles these into a global derivative F' that is L-Lipschitz. The result generalizes the known finite-dimensional converse to the Hilbert space framework and clarifies how gradient-free regularity enforces higher-order smoothness. It also outlines avenues for extension and notes limitations related to norm differentiability and cocoercivity arguments.

Abstract

It is known that if a twice differentiable function has a Lipschitz continuous Hessian, then its gradients satisfy a Jensen-type inequality. In particular, this inequality is Hessian-free in the sense that the Hessian does not actually appear in the inequality. In this paper, we show that the converse holds in a generalized setting: if a continuos function from a Hilbert space to a reflexive Banach space satisfies such an inequality, then it is Fréchet differentiable and its derivative is Lipschitz continuous. Our proof relies on the Baillon-Haddad theorem.

On the equivalence of a Hessian-free inequality and Lipschitz continuous Hessian

TL;DR

This work resolves the question of whether a Hessian-free Jensen-type inequality implies Frechet differentiability with a Lipschitz derivative in a Hilbert-to-reflexive Banach setting. It employs a slice-based approach, using φ_{y^*} = y* ∘ F and the Baillon–Haddad theorem to obtain L-Lipschitz gradients of the slices, then assembles these into a global derivative F' that is L-Lipschitz. The result generalizes the known finite-dimensional converse to the Hilbert space framework and clarifies how gradient-free regularity enforces higher-order smoothness. It also outlines avenues for extension and notes limitations related to norm differentiability and cocoercivity arguments.

Abstract

It is known that if a twice differentiable function has a Lipschitz continuous Hessian, then its gradients satisfy a Jensen-type inequality. In particular, this inequality is Hessian-free in the sense that the Hessian does not actually appear in the inequality. In this paper, we show that the converse holds in a generalized setting: if a continuos function from a Hilbert space to a reflexive Banach space satisfies such an inequality, then it is Fréchet differentiable and its derivative is Lipschitz continuous. Our proof relies on the Baillon-Haddad theorem.

Paper Structure

This paper contains 6 sections, 6 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Proof of the Theorem
  3. \ref{['bounded']} Linearity and boundedness of $y^* \mapsto \phi_{y^*}'(x)$ for every $x \in X$.
  4. \ref{['derivative']} Constructing the Fréchet derivative of $F$.
  5. \ref{['Lipschitz']} Lipschitz continuity of the derivative.
  6. Final remarks

Key Result

Lemma 1

Let $f \colon \mathbb{R}^d \to \mathbb{R}$ be a twice differentiable function with $L$-Lipschitz continuous Hessian. Then, for any $x_1,\ldots, x_n \in \mathbb{R}^d$ and $\lambda_1,\ldots, \lambda_n \geq 0$ such that $\sum _{i=1}^n \lambda_i = 1$, we have

Theorems & Definitions (9)

  • Lemma 1: Hessian-free inequality marumo2024parameter
  • Theorem 1
  • Corollary 1
  • Theorem 2: A piece of the Baillon--Haddad Theorem BC10
  • Lemma 2
  • proof
  • Lemma 3
  • proof : Proof of \ref{['lem:F_differentiability']}
  • proof : Proof of $\ref{['item:grad_free']} \implies \ref{['item:lip_F_derivative']}$ in \ref{['thm:hess_free_converse']}