  We propose a randomized second-order method for optimization known as the
Newton Sketch: it is based on performing an approximate Newton step using a
randomly projected or sub-sampled Hessian. For self-concordant functions, we
prove that the algorithm has super-linear convergence with exponentially high
probability, with convergence and complexity guarantees that are independent of
condition numbers and related problem-dependent quantities. Given a suitable
initialization, similar guarantees also hold for strongly convex and smooth
objectives without self-concordance. When implemented using randomized
projections based on a sub-sampled Hadamard basis, the algorithm typically has
substantially lower complexity than Newton's method. We also describe
extensions of our methods to programs involving convex constraints that are
equipped with self-concordant barriers. We discuss and illustrate applications
to linear programs, quadratic programs with convex constraints, logistic
regression and other generalized linear models, as well as semidefinite
programs.
