  The Iterative Hard Thresholding (IHT) algorithm has been considered
extensively as an effective deterministic algorithm for solving sparse
optimizations. The IHT algorithm benefits from the information of the batch
(full) gradient at each point and this information is a crucial key for the
convergence analysis of the generated sequence. However, this strength becomes
a weakness when it comes to machine learning and high dimensional statistical
applications because calculating the batch gradient at each iteration is
computationally expensive or impractical. Fortunately, in these applications
the objective function has a summation structure that can be taken advantage of
to approximate the batch gradient by the stochastic mini-batch gradient. In
this paper, we study the mini-batch Stochastic IHT (SIHT) algorithm for solving
the sparse optimizations. As opposed to previous works where increasing and
variable mini-batch size is necessary for derivation, we fix the mini-batch
size according to a lower bound that we derive and show our work. To prove
stochastic convergence of the objective value function we first establish a
critical sparse stochastic gradient descent property. Using this stochastic
gradient descent property we show that the sequence generated by the stochastic
mini-batch SIHT is a supermartingale sequence and converges with probability
one. Unlike previous work we do not assume the function to be a restricted
strongly convex. To the best of our knowledge, in the regime of sparse
optimization, this is the first time in the literature that it is shown that
the sequence of the stochastic function values converges with probability one
by fixing the mini-batch size for all steps.
