  We consider least-squares problems with quadratic regularization and propose
novel sketching-based iterative methods with an adaptive sketch size. The
sketch size can be as small as the effective dimension of the data matrix to
guarantee linear convergence. However, a major difficulty in choosing the
sketch size in terms of the effective dimension lies in the fact that the
latter is usually unknown in practice. Current sketching-based solvers for
regularized least-squares fall short on addressing this issue. Our main
contribution is to propose adaptive versions of standard sketching-based
iterative solvers, namely, the iterative Hessian sketch and the preconditioned
conjugate gradient method, that do not require a priori estimation of the
effective dimension. We propose an adaptive mechanism to control the sketch
size according to the progress made in each step of the iterative solver. If
enough progress is not made, the sketch size increases to improve the
convergence rate. We prove that the adaptive sketch size scales at most in
terms of the effective dimension, and that our adaptive methods are guaranteed
to converge linearly. Consequently, our adaptive methods improve the
state-of-the-art complexity for solving dense, ill-conditioned least-squares
problems. Importantly, we illustrate numerically on several synthetic and real
datasets that our method is extremely efficient and is often significantly
faster than standard least-squares solvers such as a direct factorization based
solver, the conjugate gradient method and its preconditioned variants.
