基于偏差原理的随机方差缩减梯度法在线性反问题中的早期停止
Early stopping of stochastic variance reduced gradient for linear inverse problems by the discrepancy principle
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中文总结 AI 辅助
本文为随机方差缩减梯度法配备偏差原理，在希尔伯特空间线性反问题中确立其正则化性质并推导收敛速率，为首个先验后停止规则下的随机迭代方法收敛速率结果，数值实验验证了理论。
中文摘要 AI 辅助
随机方差缩减梯度（SVRG）是随机梯度下降的一种变体，是求解大规模反问题的一种有前景的迭代方法。然而，为SVRG开发具有理论基础的先验后停止规则仍然是一个开放的挑战。在这项工作中，我们提供了配备偏差原理（最著名的先验后停止规则）的SVRG在希尔伯特空间中求解一类线性反问题的收敛性分析。我们确立了SVRG的正则化性质，此外，在适当的源条件下，我们推导了SVRG迭代的收敛速率。据我们所知，这些是在先验后停止规则下任何随机迭代方法用于反问题的首个收敛速率结果。理论发现得到了数值实验的支持。
英文摘要
Stochastic variance reduced gradient (SVRG) is a variant of stochastic gradient descent and is a promising iterative method for solving large-scale inverse problems. Nevertheless, the development of theoretically grounded a posteriori stopping rules for SVRG remains an open challenge. In this work, we provide a convergence analysis of SVRG equipped with the discrepancy principle, the most well-known a posteriori stopping rule, for solving a class of linear inverse problems in Hilbert spaces. We establish the regularizing property of SVRG, and moreover, under suitable source conditions, we derive convergence rates of SVRG iterates. To the best of our knowledge, these are the first convergence rate results of any stochastic iterative method for inverse problems under the a posteriori stopping rule. The theoretical findings are supported by numerical experiments.
发表机构
- The Chinese University of Hong Kong（香港中文大学）
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