  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.
