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We propose a randomized second-order method for optimization known as theNewton Sketch: it is based on performing an approximate Newton step using arandomly projected or sub-sampled Hessian. For self-concordant functions, weprove that the algorithm has super-linear convergence with exponentially highprobability, with convergence and complexity guarantees that are independent ofcondition numbers and related problem-dependent quantities. Given a suitableinitialization, similar guarantees also hold for strongly convex and smoothobjectives without self-concordance. When implemented using randomizedprojections based on a sub-sampled Hadamard basis, the algorithm typically hassubstantially lower complexity than Newton's method. We also describeextensions of our methods to programs involving convex constraints that areequipped with self-concordant barriers. We discuss and illustrate applicationsto linear programs, quadratic programs with convex constraints, logisticregression and other generalized linear models, as well as semidefiniteprograms.

Newton Sketch: A Linear-time Optimization Algorithm with Linear-Quadratic Convergence