indsigt - Algorithms and Data Structures - # Proximal Subdifferentials and Inexact Proximity Operators for Weakly Convex Functions

Calculus Rules for Proximal ε-Subdifferentials and Inexact Proximity Operators of Weakly Convex Functions

Q: How can the results on proximal ε-subdifferentials and inexact proximity operators be extended to more general classes of non-convex functions beyond weakly convex functions

The results on proximal ε-subdifferentials and inexact proximity operators for weakly convex functions can be extended to more general classes of non-convex functions by considering broader notions of convexity and subdifferentials. One approach is to explore the concept of paraconvexity, which encompasses weak convexity as a special case. By generalizing the calculus rules and properties derived for weakly convex functions to the broader class of paraconvex functions, we can extend the results to a wider range of non-convex functions. This extension would involve adapting the sum rules, criticality conditions, and approximation techniques to suit the characteristics of paraconvex functions. Additionally, incorporating more sophisticated subdifferential concepts, such as proximal subdifferentials for paraconvex functions, can provide a framework for analyzing inexact proximity operators for a more diverse set of non-convex functions.

Q: What are the potential applications of the developed theory in areas such as optimization, machine learning, or signal processing

The developed theory on proximal ε-subdifferentials and inexact proximity operators for weakly convex functions has various potential applications in optimization, machine learning, and signal processing. In optimization, the theory can be utilized to enhance algorithms for solving large-scale non-convex optimization problems. By incorporating inexact proximity operators and proximal ε-subdifferentials, optimization algorithms can handle weakly convex objectives more efficiently, leading to improved convergence rates and solution accuracy. The ability to analyze critical points and approximate proximal points for weakly convex functions can guide the development of optimization strategies that are robust and effective in dealing with non-convexity. In machine learning, the theory can be applied to optimization problems arising in training deep learning models, where non-convex loss functions are common. By leveraging the insights from inexact proximal operators, machine learning algorithms can be designed to handle weakly convex objectives more effectively, leading to better model performance and faster convergence. The theory can also aid in developing regularization techniques that exploit the properties of weakly convex functions to improve generalization and model interpretability. In signal processing, the theory can be used to optimize signal reconstruction, denoising, and compression algorithms that involve non-convex optimization problems. By incorporating inexact proximity operators and proximal ε-subdifferentials, signal processing techniques can be tailored to handle weakly convex functions more efficiently, leading to improved signal recovery and processing performance. The theory can also be applied to design adaptive signal processing algorithms that can adapt to the underlying structure of weakly convex functions in signal data.

Q: Can the insights gained from the analysis of inexact proximal operators be leveraged to design efficient algorithms for solving optimization problems with weakly convex objectives

The insights gained from the analysis of inexact proximal operators can be leveraged to design efficient algorithms for solving optimization problems with weakly convex objectives. By incorporating the concept of proximal ε-subdifferentials and inexact proximity operators, algorithms can be developed to handle the non-convexity of weakly convex functions more effectively. One approach is to design optimization algorithms that utilize inexact proximity operators to approximate the proximal points of weakly convex functions. By incorporating the criticality conditions and sum rules derived for proximal ε-subdifferentials, these algorithms can efficiently navigate the non-convex landscape of weakly convex objectives, leading to faster convergence and improved solution quality. Furthermore, the analysis of inexact proximal operators can guide the development of adaptive optimization strategies that dynamically adjust the level of inexactness in the proximity operators based on the characteristics of the weakly convex functions. This adaptive approach can enhance the robustness and convergence properties of optimization algorithms for weakly convex objectives, making them more suitable for real-world applications in various domains.

Kernekoncepter

This work investigates proximal ε-subdifferentials and derives sum rules for weakly convex functions, incorporating the corresponding moduli of weak convexity. It also analyzes inexact proximity operators for weakly convex functions in terms of proximal ε-subdifferentials and the related notion of criticality.

Resumé

The content starts by introducing the concept of weakly convex functions, which are a generalization of convex functions. It then discusses proximal subdifferentials, which are a suitable tool for defining criticality for weakly convex functions.

The main contributions of the work are:

Providing sufficient and necessary conditions for the sum rule of the global proximal ε-subdifferentials for the sum of two ρ-weakly convex functions. The modulus of proximal subdifferentiability and the modulus of weak convexity ρ are incorporated into the calculus rules.
Investigating the relationship between the ε-proximal operator of a ρ-weakly convex function f and the ε-proximal subdifferential of f, using the derived calculus rules.
Relating the notion of inexact (approximate) proximal point to Type-1 and Type-2 approximations proposed in the convex setting.

The work also provides several auxiliary results, such as the globalization property of proximal subdifferentials for paraconvex functions, and the characterization of weakly convex functions as the difference between a convex and a quadratic function.

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

by Ewa Bednarcz... kl. arxiv.org 04-24-2024

https://arxiv.org/pdf/2211.14525.pdf

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

Dybere Forespørgsler

How can the results on proximal ε-subdifferentials and inexact proximity operators be extended to more general classes of non-convex functions beyond weakly convex functions

The results on proximal ε-subdifferentials and inexact proximity operators for weakly convex functions can be extended to more general classes of non-convex functions by considering broader notions of convexity and subdifferentials. One approach is to explore the concept of paraconvexity, which encompasses weak convexity as a special case. By generalizing the calculus rules and properties derived for weakly convex functions to the broader class of paraconvex functions, we can extend the results to a wider range of non-convex functions. This extension would involve adapting the sum rules, criticality conditions, and approximation techniques to suit the characteristics of paraconvex functions. Additionally, incorporating more sophisticated subdifferential concepts, such as proximal subdifferentials for paraconvex functions, can provide a framework for analyzing inexact proximity operators for a more diverse set of non-convex functions.

What are the potential applications of the developed theory in areas such as optimization, machine learning, or signal processing

The developed theory on proximal ε-subdifferentials and inexact proximity operators for weakly convex functions has various potential applications in optimization, machine learning, and signal processing.
In optimization, the theory can be utilized to enhance algorithms for solving large-scale non-convex optimization problems. By incorporating inexact proximity operators and proximal ε-subdifferentials, optimization algorithms can handle weakly convex objectives more efficiently, leading to improved convergence rates and solution accuracy. The ability to analyze critical points and approximate proximal points for weakly convex functions can guide the development of optimization strategies that are robust and effective in dealing with non-convexity.
In machine learning, the theory can be applied to optimization problems arising in training deep learning models, where non-convex loss functions are common. By leveraging the insights from inexact proximal operators, machine learning algorithms can be designed to handle weakly convex objectives more effectively, leading to better model performance and faster convergence. The theory can also aid in developing regularization techniques that exploit the properties of weakly convex functions to improve generalization and model interpretability.
In signal processing, the theory can be used to optimize signal reconstruction, denoising, and compression algorithms that involve non-convex optimization problems. By incorporating inexact proximity operators and proximal ε-subdifferentials, signal processing techniques can be tailored to handle weakly convex functions more efficiently, leading to improved signal recovery and processing performance. The theory can also be applied to design adaptive signal processing algorithms that can adapt to the underlying structure of weakly convex functions in signal data.

Can the insights gained from the analysis of inexact proximal operators be leveraged to design efficient algorithms for solving optimization problems with weakly convex objectives

The insights gained from the analysis of inexact proximal operators can be leveraged to design efficient algorithms for solving optimization problems with weakly convex objectives. By incorporating the concept of proximal ε-subdifferentials and inexact proximity operators, algorithms can be developed to handle the non-convexity of weakly convex functions more effectively.
One approach is to design optimization algorithms that utilize inexact proximity operators to approximate the proximal points of weakly convex functions. By incorporating the criticality conditions and sum rules derived for proximal ε-subdifferentials, these algorithms can efficiently navigate the non-convex landscape of weakly convex objectives, leading to faster convergence and improved solution quality.
Furthermore, the analysis of inexact proximal operators can guide the development of adaptive optimization strategies that dynamically adjust the level of inexactness in the proximity operators based on the characteristics of the weakly convex functions. This adaptive approach can enhance the robustness and convergence properties of optimization algorithms for weakly convex objectives, making them more suitable for real-world applications in various domains.