标题： 关于随机方差减少梯度在线性反问题中的收敛性 显示英文标题

标题： On the convergence of stochastic variance reduced gradient for linear inverse problems

摘要： 随机方差减少梯度（SVRG）是一种基于方差减少的随机梯度下降的加速版本，对于解决大规模逆问题具有前景。 在本工作中，我们分析了SVRG及其一种结合了问题先验知识的正则化版本，用于解决希尔伯特空间中的线性逆问题。 我们证明，在适当的常数步长调度和正则性条件下，正则化的SVRG可以在无需任何早期停止规则的情况下，以噪声水平为基准达到最优收敛速率，而标准SVRG在先验停止规则下对于具有非光滑解的问题也是最优的。 分析基于显式的误差递归以及相对于锚点的内部循环更新的适当先验估计。 提供了数值实验以补充理论分析。 显示英文摘要

摘要： Stochastic variance reduced gradient (SVRG) is an accelerated version of stochastic gradient descent based on variance reduction, and is promising for solving large-scale inverse problems. In this work, we analyze SVRG and a regularized version that incorporates a priori knowledge of the problem, for solving linear inverse problems in Hilbert spaces. We prove that, with suitable constant step size schedules and regularity conditions, the regularized SVRG can achieve optimal convergence rates in terms of the noise level without any early stopping rules, and standard SVRG is also optimal for problems with nonsmooth solutions under a priori stopping rules. The analysis is based on an explicit error recursion and suitable prior estimates on the inner loop updates with respect to the anchor point. Numerical experiments are provided to complement the theoretical analysis.