面向含Massart噪声的流式线性系统的分位数随机Kaczmarz方法
Quantile Randomized Kaczmarz for Streaming Linear Systems with Massart Noise
AI总结：
本文针对含Massart噪声的流式线性系统，提出分位数随机Kaczmarz方法，明确了可容忍约7%污染水平，给出显式子样本量边界，实现线性收敛。
AI中文摘要：
分位数随机Kaczmarz（Quantile Randomized Kaczmarz, QRK）已被证明是求解受污染线性系统的高效算法，受到广泛关注。Cai等人（《SIAM矩阵分析与应用杂志》47卷2期：802-823，2026）近期研究表明，当求解含β比例任意污染的线性系统时，只要β足够小，QRK计算分位数所需的样本量为O(log T/log(1/β))，这是实现T次迭代线性收敛的必要且充分条件。然而，目前仍不清楚污染水平β最大可达多少，以及如何计算所需子样本量D的显式值（不含隐藏常数）。本文研究采用QRK求解含Massart噪声的流式线性系统，每次更新使用阶最优批大小D=O(log T)。流式设置中各迭代的样本相互独立，可实现更精细的分析，从而得到可容忍污染水平和所需子样本量的显式、可计算边界。特别地，我们证明当污染水平最高约7%时，算法可实现线性收敛。此外，我们还讨论了在无感知噪声情况下常数的改进情况。
英文摘要：
Quantile randomized Kaczmarz (QRK) has proven to be an efficient solver for corrupted linear systems and has received much attention. It was recently shown by Cai et al. (SIAM J. Matrix Anal. Appl. 47(2):802-823, 2026) that using $O(\log T/\log(1/β))$ samples for computing the quantile is necessary and sufficient for QRK to converge linearly over $T$ iterations when solving linear systems with a $β$-fraction of arbitrary corruptions, as long as $β$ is small enough. However, it remains unclear how large the corruption level $β$ can be, and how to compute the required subsample size $D$ explicitly, without hidden constants. This paper studies streaming linear systems with Massart noise via QRK using an order-optimal batch size $D=O(\log T)$ in each update. The independence of samples from previous iterations in the streaming setting enables a sharper analysis, yielding explicit, computable bounds on both the tolerable corruption level and the required subsample size. In particular, we establish linear convergence for corruption levels of up to approximately 7%. We also discuss how the constants improve under oblivious noise.
