# An Investigation of Newton-Sketch and Subsampled Newton Methods

## TL;DR

This paper compares two sketching techniques, Hessian subsampling and randomized Hadamard transformations, for improving Newton's method in large-scale finite-sum optimization problems, analyzing their tradeoffs and practical performance.

## Contribution

It provides a comparative analysis of Hessian subsampling and randomized Hadamard sketching in Newton's method, including complexity and practical implementation insights.

## Key findings

- Conjugate gradient method outperforms stochastic gradient in this context.
- Complexity analysis favors Hessian subsampling under certain conditions.
- Numerical experiments demonstrate tradeoffs between the two sketching methods.

## Abstract

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.