Overview
The research investigates Quantile Randomized Kaczmarz (QRK) as a solver for streaming linear systems that are subject to Massart noise. QRK has previously been recognized as an efficient method for addressing corrupted linear systems. This work specifically focuses on clarifying the maximum tolerable corruption level and determining the explicit subsample size required for QRK's effective operation. It establishes linear convergence for corruption levels reaching approximately 7% and identifies an order-optimal batch size of $D=O(\log T)$ for each update operation within the streaming context.
Research Context
Quantile Randomized Kaczmarz (QRK) has garnered significant attention due to its demonstrated efficiency in solving linear systems corrupted by noise. Prior work by Cai et al. (SIAM J. Matrix Anal. Appl. 47(2):802-823, 2026) established that a specific number of samples, $O(\log T/\log(1/\beta))$, is both necessary and sufficient for computing the quantile to ensure QRK's linear convergence over $T$ iterations. This convergence was demonstrated for linear systems possessing a $\beta$-fraction of arbitrary corruptions, contingent on $\beta$ being sufficiently small. However, critical aspects remained unaddressed by this prior research: the precise upper limit for the corruption level $\beta$ and the explicit calculation of the required subsample size $D$, free from hidden constants.
Approach
This paper addresses the identified gaps by studying streaming linear systems specifically under Massart noise conditions. The methodology employs QRK with an order-optimal batch size, $D=O(\log T)$, for each update iteration. A key aspect of this approach is leveraging the independence of samples across different iterations, a characteristic inherent to the streaming setting. This independence facilitates a sharper analytical framework compared to non-streaming scenarios. The analysis aims to yield explicit and computable bounds for two primary parameters: the tolerable corruption level and the necessary subsample size.
Findings
- The study establishes linear convergence for QRK when applied to streaming linear systems affected by Massart noise.
- The analysis indicates that QRK can maintain linear convergence for corruption levels of up to approximately 7%.
- An order-optimal batch size of $D=O(\log T)$ is determined for each update in the streaming context.
- The independence of samples from previous iterations, characteristic of the streaming setting, enabled a sharper analysis.
- This sharper analysis resulted in explicit, computable bounds for both the tolerable corruption level and the required subsample size $D$.
Why This Matters
The findings provide explicit, computable bounds for the tolerable corruption level and the necessary subsample size for QRK operating on streaming linear systems with Massart noise. Understanding these parameters, particularly the maximum tolerable corruption, offers practical guidance for implementing QRK in scenarios where data streams are inherently noisy.
Potential Applications
The paper discusses that constants associated with the established bounds can improve under conditions of oblivious noise, indicating a potential for enhanced performance or broader applicability in specific noise models. No further specific applications are detailed in the source.