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.

中文翻译：

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线性系统的随机重构：算法和收敛理论

我们将任意一致的线性系统的一系列重构发展为随机问题。重新制定由两个用户定义的参数控制：定义范数的正定矩阵，以及随机矩阵上的任意离散或连续分布。我们的重新表述有几个等效的解释，允许来自不同社区的研究人员利用他们特定领域的见解。特别是，我们的重新表述可以等效地视为随机优化问题、随机线性系统、随机不动点问题和概率交叉问题。我们证明了重新表述准确的充分、必要和充分条件。此外，我们提出并分析了三种用于解决重构问题的随机算法——基本的、并行和加速方法——具有全局线性收敛速度。速率可以解释为矩阵的条件数，它取决于系统矩阵和重新制定参数。这产生了一种我们称为随机预处理的新现象，它指的是找到导致足够小的条件数的参数（矩阵和分布）的问题。我们的基本方法可以等效地解释为随机梯度下降法、随机牛顿法、随机近点法、随机定点法和具有固定步长（松弛参数）的随机投影法，应用于重构。速率可以解释为矩阵的条件数，它取决于系统矩阵和重新制定参数。这产生了一种我们称为随机预处理的新现象，它指的是找到导致足够小的条件数的参数（矩阵和分布）的问题。我们的基本方法可以等效地解释为随机梯度下降法、随机牛顿法、随机近点法、随机不动点法和随机投影法，具有固定步长（松弛参数），应用于重构。速率可以解释为矩阵的条件数，它取决于系统矩阵和重新制定参数。这产生了一种我们称为随机预处理的新现象，它指的是找到导致足够小的条件数的参数（矩阵和分布）的问题。我们的基本方法可以等效地解释为随机梯度下降法、随机牛顿法、随机近点法、随机不动点法和随机投影法，具有固定步长（松弛参数），应用于重构。