  Sketching techniques have become popular for scaling up machine learning
algorithms by reducing the sample size or dimensionality of massive data sets,
while still maintaining the statistical power of big data. In this paper, we
study sketching from an optimization point of view: we first show that the
iterative Hessian sketch is an optimization process with preconditioning, and
develop accelerated iterative Hessian sketch via the searching the conjugate
direction; we then establish primal-dual connections between the Hessian sketch
and dual random projection, and apply the preconditioned conjugate gradient
approach on the dual problem, which leads to the accelerated iterative dual
random projection methods. Finally to tackle the challenges from both large
sample size and high-dimensionality, we propose the primal-dual sketch, which
iteratively sketches the primal and dual formulations. We show that using a
logarithmic number of calls to solvers of small scale problem, primal-dual
sketch is able to recover the optimum of the original problem up to arbitrary
precision. The proposed algorithms are validated via extensive experiments on
synthetic and real data sets which complements our theoretical results.
