Spurious Stationary Points and Hardness Results for Bregman Proximal-Type Algorithms

Core Concepts

The existence of spurious stationary points, which satisfy the zero extended Bregman stationarity measure but are not true stationary points, poses fundamental challenges to the analysis and design of Bregman proximal-type algorithms. These spurious points can trap the algorithms within a small neighborhood, even for convex problems, highlighting the inherent distinction between Euclidean and Bregman geometries.

Abstract

This paper investigates the reliability of existing stationarity measures based on Bregman divergence in distinguishing between stationary and non-stationary points. The key findings are: The authors introduce an extended Bregman stationarity measure that is well-defined and continuous over the entire domain, including the boundary. This measure unifies and extends previous stationarity measures. The authors prove that the extended stationarity measure being zero is necessary for a point to be stationary. However, they show that this condition is not sufficient, as there exist "spurious stationary points" that satisfy the zero stationarity measure but are not true stationary points. The authors establish a hardness result: Bregman proximal-type algorithms are unable to escape from a spurious stationary point in finite steps when the initial point is unfavorable, even for convex problems. This highlights the inherent challenges in the design and analysis of Bregman divergence-based methods. The authors provide simple convex and non-convex counter-examples to illustrate the existence of spurious stationary points. They also show that spurious points generally exist for a broad class of convex problems with polytopal constraints. The paper's findings introduce both fundamental theoretical and numerical challenges to the machine learning and optimization communities, calling for new principles in algorithm design to address the issue of spurious stationary points.

Stats

None.

Quotes

None.

Key Insights Distilled From

Spurious Stationarity and Hardness Results for Mirror Descent

by He Chen,Jiaj... at arxiv.org 04-15-2024

https://arxiv.org/pdf/2404.08073.pdf

Deeper Inquiries

How can we design Bregman proximal-type algorithms that are able to reliably escape from spurious stationary points in finite steps

To design Bregman proximal-type algorithms that can reliably escape from spurious stationary points in finite steps, several strategies can be considered. Adaptive Step Sizes: Implementing adaptive step size strategies can help the algorithm navigate away from spurious stationary points. By adjusting the step size based on the local geometry and progress of the optimization, the algorithm can avoid getting trapped. Restart Mechanisms: Introducing restart mechanisms can be beneficial. If the algorithm detects that it is not making progress or is stuck near a spurious stationary point, it can restart from a different point or adjust its search direction to break free from the trap. Incorporating Curvature Information: Utilizing curvature information from the Bregman divergence can guide the algorithm towards regions of faster convergence and away from spurious stationary points. By leveraging the curvature of the objective function, the algorithm can make more informed decisions about the direction of optimization. Exploration-Exploitation Balance: Balancing exploration and exploitation in the optimization process can help the algorithm explore different regions of the search space while also exploiting promising areas. This balance can prevent the algorithm from getting stuck in local optima or spurious stationary points.

What are the potential implications of the existence of spurious stationary points on the practical performance of Bregman divergence-based methods in real-world applications

The existence of spurious stationary points can have significant implications on the practical performance of Bregman divergence-based methods in real-world applications. Convergence Issues: Spurious stationary points can hinder the convergence of optimization algorithms, leading to suboptimal solutions or premature convergence. This can impact the efficiency and effectiveness of the optimization process. Algorithm Robustness: The presence of spurious stationary points challenges the robustness of optimization algorithms. Without mechanisms to escape from these points, algorithms may struggle to find the true optimal solution, affecting the reliability of the optimization process. Computational Resources: Dealing with spurious stationary points may require additional computational resources and time. Algorithms may need to perform extra iterations or computations to overcome these points, increasing the overall computational cost of optimization. Algorithm Selection: The presence of spurious stationary points may influence the choice of optimization algorithms. Practitioners may need to consider the algorithm's ability to handle such points and its robustness in real-world scenarios when selecting an optimization approach.

Are there any connections between the challenges posed by spurious stationary points and the design of stationarity measures for non-convex optimization problems in general

The challenges posed by spurious stationary points in Bregman divergence-based methods can shed light on the design of stationarity measures for non-convex optimization problems in general. Measure Effectiveness: The existence of spurious stationary points highlights the importance of designing stationarity measures that can accurately distinguish between true stationary points and spurious ones. This calls for the development of more robust and reliable measures that can capture the true optimization landscape. Algorithm Evaluation: The challenges posed by spurious stationary points emphasize the need to evaluate optimization algorithms based on their ability to handle such points. Stationarity measures play a crucial role in assessing algorithm performance and convergence properties in non-convex optimization scenarios. Theoretical Understanding: Exploring the implications of spurious stationary points can deepen our theoretical understanding of optimization landscapes and the behavior of algorithms in non-convex settings. This can lead to the development of more effective optimization strategies and theoretical frameworks for non-convex optimization problems.