# Stochastic Reformulations of Linear Systems: Algorithms and Convergence Theory

## TL;DR

This paper introduces a flexible stochastic reformulation framework for linear systems, enabling new algorithms with proven linear convergence and a novel concept of stochastic preconditioning to optimize problem conditioning.

## Contribution

It develops a family of stochastic reformulations of linear systems with convergence analysis and introduces stochastic preconditioning for improved efficiency.

## Key findings

- Reformulations are governed by user-defined parameters and have multiple interpretations.
- Proposed algorithms achieve global linear convergence rates.
- Introduction of stochastic preconditioning to optimize condition numbers.

## Abstract

We develop a family of reformulations of an arbitrary consistent linear system into a stochastic problem. The reformulations are governed by two user-defined parameters: a positive definite matrix defining a norm, and an arbitrary discrete or continuous distribution over random matrices. Our reformulation has several equivalent interpretations, allowing for researchers from various communities to leverage their domain specific insights. In particular, our reformulation can be equivalently seen as a stochastic optimization problem, stochastic linear system, stochastic fixed point problem and a probabilistic intersection problem. We prove sufficient, and necessary and sufficient conditions for the reformulation to be exact. Further, we propose and analyze three stochastic algorithms for solving the reformulated problem---basic, parallel and accelerated methods---with global linear convergence rates. The rates can be interpreted as condition numbers of a matrix which depends on the system matrix and on the reformulation parameters. This gives rise to a new phenomenon which we call stochastic preconditioning, and which refers to the problem of finding parameters (matrix and distribution) leading to a sufficiently small condition number. Our basic method can be equivalently interpreted as stochastic gradient descent, stochastic Newton method, stochastic proximal point method, stochastic fixed point method, and stochastic projection method, with fixed stepsize (relaxation parameter), applied to the reformulations.