  Stochastic iterative algorithms have gained recent interest in machine
learning and signal processing for solving large-scale systems of equations,
$Ax=b$. One such example is the Randomized Kaczmarz (RK) algorithm, which acts
only on single rows of the matrix $A$ at a time. While RK randomly selects a
row of $A$ to work with, Motzkin's Method (MM) employs a greedy row selection.
Connections between the two algorithms resulted in the Sampling
Kaczmarz-Motzkin (SKM) algorithm which samples a random subset of $\beta$ rows
of $A$ and then greedily selects the best row of the subset. Despite their
variable computational costs, all three algorithms have been proven to have the
same theoretical upper bound on the convergence rate. In this work, an improved
analysis of the range of random (RK) to greedy (MM) methods is presented. This
analysis improves upon previous known convergence bounds for SKM, capturing the
benefit of partially greedy selection schemes. This work also further
generalizes previous known results, removing the theoretical assumptions that
$\beta$ must be fixed at every iteration and that $A$ must have normalized
rows.
