  The problem of finding a solution to the linear system $Ax = b$ with certain
minimization properties arises in numerous scientific and engineering areas. In
the era of big data, the stochastic optimization algorithms become increasingly
significant due to their scalability for problems of unprecedented size. This
paper focuses on the problem of minimizing a strongly convex function subject
to linear constraints. We consider the dual formulation of this problem and
adopt the stochastic coordinate descent to solve it. The proposed algorithmic
framework, called fast stochastic dual coordinate descent, utilizes sampling
matrices sampled from user-defined distributions to extract gradient
information. Moreover, it employs Polyak's heavy ball momentum acceleration
with adaptive parameters learned through iterations, overcoming the limitation
of the heavy ball momentum method that it requires prior knowledge of certain
parameters, such as the singular values of a matrix. With these extensions, the
framework is able to recover many well-known methods in the context, including
the randomized sparse Kaczmarz method, the randomized regularized Kaczmarz
method, the linearized Bregman iteration, and a variant of the conjugate
gradient (CG) method. We prove that, with strongly admissible objective
function, the proposed method converges linearly in expectation. Numerical
experiments are provided to confirm our results.
