This paper studies the effect of adding geometrically smoothed momentum to the randomized Kaczmarz algorithm, which is an instance of stochastic gradient descent on a linear least squares loss function. We prove a result about the expected error in the direction of singular vectors of the matrix defining the least squares loss. We present several numerical examples illustrating the utility of our result and pose several questions.

翻译：本文研究了在随机化Kaczmarz算法中引入几何平滑动量的效果，该算法是线性最小二乘损失函数上随机梯度下降的一个实例。我们证明了关于矩阵奇异向量方向上期望误差的一个结果，该矩阵定义了最小二乘损失。我们提供了若干数值算例来说明我们结果的实用性，并提出了几个待解决的问题。