Iterative sketching and sketch-and-precondition are randomized algorithms used for solving overdetermined linear least-squares problems. When implemented in exact arithmetic, these algorithms produce high-accuracy solutions to least-squares problems faster than standard direct methods based on QR factorization. Recently, Meier, Nakatsukasa, Townsend, and Webb demonstrated numerical instabilities in a version of sketch-and-precondition in floating point arithmetic (arXiv:2302.07202). The work of Meier et al. raises the question: Is there a randomized least-squares solver that is both fast and stable? This paper resolves this question in the affirmative by proving that iterative sketching, appropriately implemented, is forward stable. Numerical experiments confirm the theoretical findings, demonstrating that iterative sketching is stable and faster than QR-based solvers for large problem instances.

翻译：迭代草图与草图预条件法是求解超定线性最小二乘问题的随机算法。在精确算术条件下，这些算法求解最小二乘问题的速度优于基于QR分解的标准直接方法，且能获得高精度解。近期，Meier、Nakatsukasa、Townsend及Webb在浮点算术实现中发现草图预条件法存在数值不稳定性（arXiv:2302.07202）。该研究引发疑问：是否存在兼具快速性与稳定性的随机最小二乘求解器？本文通过证明适当实现的迭代草图法具有前向稳定性，对此问题给出肯定答案。数值实验验证了理论结果，表明迭代草图法在大规模问题实例中既保持稳定性，又比基于QR的求解器运算速度更快。