Physics

Nonlinear system

Iterative method

Mathematics

Biology

Algorithm

Mathematical optimization

Curse of dimensionality

Local convergence

Paleontology

Newton's method in optimization

Artificial intelligence

Set (abstract data type)

Programming language

Hessian matrix

Solver

Sketch

Context (archaeology)

Computer science

Quantum mechanics

Dimensionality reduction

Applied mathematics

Conjugate gradient method

Newton's method

Optimization methods & software

Sketching, a dimensionality reduction technique, has received much attention in the statistics community. In this paper, we study sketching in the context of Newton's method for solving finite-sum optimization problems in which the number of variables and data points are both large. We study two forms of sketching that perform dimensionality reduction in data space: Hessian subsampling and randomized Hadamard transformations. Each has its own advantages, and their relative tradeoffs have not been investigated in the optimization literature. Our study focuses on practical versions of the two methods in which the resulting linear systems of equations are solved approximately, at every iteration, using an iterative solver. The advantages of using the conjugate gradient method vs. a stochastic gradient iteration are revealed through a set of numerical experiments, and a complexity analysis of the Hessian subsampling method is presented.

An investigation of Newton-Sketch and subsampled Newton methods