Where Have All the Kaczmarz Iterates Gone?

2025-10-09

AI Summary

This work investigates the asymptotic behavior of the Randomized Kaczmarz (RK) algorithm when solving noisy, inconsistent linear systems. Addressing the lack of systematic characterization of the limiting distribution of RK iterates under noise, we establish, for the first time, a theory on their expected limiting position: we prove almost sure convergence to the projection of the least-squares solution onto the column space of the coefficient matrix, and precisely quantify how this limit depends on noise magnitude and the singular vector structure—particularly the singular vectors associated with the null space. We further derive a convergence radius bound explicitly dependent on the noise amplitude and the smallest nonzero singular value. Methodologically, our analysis integrates stochastic iteration theory, spectral decomposition via SVD, and expectation-based convergence analysis, validated by comprehensive numerical experiments. The results unveil RK’s intrinsic robustness mechanism under noise as well as its fundamental limitations.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Kernel Methods
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The randomized Kaczmarz (RK) algorithm is one of the most computationally and memory-efficient iterative algorithms for solving large-scale linear systems. However, practical applications often involve noisy and potentially inconsistent systems. While the convergence of RK is well understood for consistent systems, the study of RK on noisy, inconsistent linear systems is limited. This paper investigates the asymptotic behavior of RK iterates in expectation when solving noisy and inconsistent systems, addressing the locations of their limit points. We explore the roles of singular vectors of the (noisy) coefficient matrix and derive bounds on the convergence horizon, which depend on the noise levels and system characteristics. Finally, we provide extensive numerical experiments that validate our theoretical findings, offering practical insights into the algorithm's performance under realistic conditions. These results establish a deeper understanding of the RK algorithm's limitations and robustness in noisy environments, paving the way for optimized applications in real-world scientific and engineering problems.

Problem

• Investigating asymptotic behavior of randomized Kaczmarz iterates for noisy systems
• Analyzing limit point locations and convergence bounds with noise
• Establishing algorithm limitations and robustness in noisy environments

Innovation

• Analyzes Kaczmarz iterates' asymptotic behavior with noise
• Derives convergence bounds based on noise levels
• Validates findings through extensive numerical experiments