  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.
