  In this paper, we revisit the large-scale constrained linear regression
problem and propose faster methods based on some recent developments in
sketching and optimization. Our algorithms combine (accelerated) mini-batch SGD
with a new method called two-step preconditioning to achieve an approximate
solution with a time complexity lower than that of the state-of-the-art
techniques for the low precision case. Our idea can also be extended to the
high precision case, which gives an alternative implementation to the Iterative
Hessian Sketch (IHS) method with significantly improved time complexity.
Experiments on benchmark and synthetic datasets suggest that our methods indeed
outperform existing ones considerably in both the low and high precision cases.
