  Projection-based iterative methods for solving large over-determined linear
systems are well-known for their simplicity and computational efficiency. It is
also known that the correct choice of a sketching procedure (i.e.,
preprocessing steps that reduce the dimension of each iteration) can improve
the performance of iterative methods in multiple ways, such as, to speed up the
convergence of the method by fighting inner correlations of the system, or to
reduce the variance incurred by the presence of noise. In the current work, we
show that sketching can also help us to get better theoretical guarantees for
the projection-based methods. Specifically, we use good properties of Gaussian
sketching to prove an accelerated convergence rate of the sketched relaxation
(also known as Motzkin's) method. The new estimates hold for linear systems of
arbitrary structure. We also provide numerical experiments in support of our
theoretical analysis of the sketched relaxation method.
