In this paper, we propose a novel adaptive stochastic extended iterative method, which can be viewed as an improved extension of the randomized extended Kaczmarz (REK) method, for finding the unique minimum Euclidean norm least-squares solution of a given linear system. In particular, we introduce three equivalent stochastic reformulations of the linear least-squares problem: stochastic unconstrained and constrained optimization problems, and the stochastic multiobjective optimization problem. We then alternately employ the adaptive variants of the stochastic heavy ball momentum (SHBM) method, which utilize iterative information to update the parameters, to solve the stochastic reformulations. We prove that our method converges linearly in expectation, addressing an open problem in the literature related to designing theoretically supported adaptive SHBM methods. Numerical experiments show that our adaptive stochastic extended iterative method has strong advantages over the non-adaptive one.

翻译：本文提出了一种新颖的自适应随机扩展迭代方法，该方法可视为随机扩展Kaczmarz（REK）方法的改进扩展，用于求解给定线性系统的唯一最小欧几里得范数最小二乘解。具体而言，我们引入了线性最小二乘问题的三种等效随机重构形式：随机无约束与约束优化问题，以及随机多目标优化问题。随后，我们交替采用自适应随机重球动量（SHBM）方法的变体——该方法利用迭代信息更新参数——来求解这些随机重构问题。我们证明了该方法在期望意义下线性收敛，从而解决了文献中关于设计具有理论支撑的自适应SHBM方法的一个开放性问题。数值实验表明，我们的自适应随机扩展迭代方法相较于非自适应方法具有显著优势。