Randomization has been shown to be beneficial to iterative methods for solving large-scale systems of linear equations, notably the Kaczmarz algorithm. We analyze the convergence of a broad class of pursuit algorithms that at each step pick $n$ members at random, from a system of linear equations, and update the iterate using the orthogonal projection to the intersection of the hyperplanes they represent. We identify, in this context, a specific degree-$n$ polynomial that non-linearly transforms the singular values of the system. This transformation to singular values and the corresponding condition number then characterizes the convergence rate, in expectation, of the pursuit. As a consequence, our results specify the convergence rate of the stochastic gradient descent algorithm, in terms of the mini-batch size $n$, when used for solving systems of linear equations.
翻译:随机化已被证明有利于解决大型线性方程式系统的迭代方法,特别是卡兹马兹算法。 我们分析了一大类追逐算法的趋同情况,该算法每步随机从线性方程式系统中挑选一美元成员,并使用正方形投影到它们所代表的超高平面交点的交叉点更新迭代法。 我们在此情况下确定了一个非线性地转换系统单值的特定度- 零美元多元值。 这种转换为单值和相应的条件号,然后成为追逐的趋同率的特征。 结果,我们的结果用微型批量大小来说明用于解决线性方程系统时的随机梯度梯度下降算法的趋同率 $ 。