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Calculus rules for proximal ε-subdifferentials and inexact proximity operators for weakly convex functions

Ewa Bednarczuk, Giovanni Bruccola, Gabriele Scrivanti, The Hung Tran

TL;DR

The work addresses optimization with $\rho$-weakly convex objectives by developing calculus rules for proximal $\varepsilon$-subdifferentials and analyzing inexact proximal operators. It derives a sum rule for the global proximal $\varepsilon$-subdifferentials of the sum of two $\rho$-weakly convex functions and demonstrates how the modulus of proximal subdifferentiability is tracked through the rule. The authors connect the $\varepsilon$-proximal operator to the $\varepsilon$-subdifferential and relate inexact proximal maps to Type-1 and Type-2 approximations, extending convex results to the weakly convex setting via duality arguments. This framework enables convergence analysis of inexact proximal algorithms in nonconvex data-analysis models and provides a principled method to manage inexactness in proximal steps.

Abstract

We investigate inexact proximity operators for weakly convex functions. To this aim, we derive sum rules for proximal ε-subdifferentials, by incorporating the moduli of weak convexity of the functions into the respective formulas. This allows us to investigate inexact proximity operators for weakly convex functions in terms of proximal ε-subdifferentials.

Calculus rules for proximal ε-subdifferentials and inexact proximity operators for weakly convex functions

TL;DR

The work addresses optimization with -weakly convex objectives by developing calculus rules for proximal -subdifferentials and analyzing inexact proximal operators. It derives a sum rule for the global proximal -subdifferentials of the sum of two -weakly convex functions and demonstrates how the modulus of proximal subdifferentiability is tracked through the rule. The authors connect the -proximal operator to the -subdifferential and relate inexact proximal maps to Type-1 and Type-2 approximations, extending convex results to the weakly convex setting via duality arguments. This framework enables convergence analysis of inexact proximal algorithms in nonconvex data-analysis models and provides a principled method to manage inexactness in proximal steps.

Abstract

We investigate inexact proximity operators for weakly convex functions. To this aim, we derive sum rules for proximal ε-subdifferentials, by incorporating the moduli of weak convexity of the functions into the respective formulas. This allows us to investigate inexact proximity operators for weakly convex functions in terms of proximal ε-subdifferentials.
Paper Structure (8 sections, 9 theorems, 74 equations)

This paper contains 8 sections, 9 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Weak convexity
  3. Proximal Subdifferentials
  4. Contribution
  5. Preliminaries
  6. Calculus rules
  7. inexact proximal maps
  8. Conclusions

Key Result

Proposition 1

Let $\mathcal{X}$ be a normed space. Let ${f: \mathcal{X} \rightarrow (-\infty,+\infty]}$ be $\gamma\text{-}$paraconvex with $\gamma>1$. Then there exists $C>0$ such that where

Theorems & Definitions (35)

  • Definition 1: $\gamma$-Paraconvexity
  • Definition 2: $(\gamma,C)-$Subdifferential jourani1996Subdifferentiability
  • Example 1
  • Proposition 1: jourani1996Subdifferentiability
  • Definition 3: Global Proximal Subdifferential
  • Definition 4: Global proximal $\varepsilon\text{-}$subdifferentials
  • Proposition 2
  • proof
  • Definition 5: $\varepsilon\text{-}$solution
  • Definition 6: $\varepsilon\text{-}C\text{-}$critical point
  • ...and 25 more