  This paper deals with the convex feasibility problem, where the feasible set
is given as the intersection of a (possibly infinite) number of closed convex
sets. We assume that each set is specified algebraically as a convex
inequality, where the associated convex function is general (possibly
non-differentiable). For finding a point satisfying all the convex inequalities
we design and analyze random projection algorithms using special subgradient
iterations and extrapolated stepsizes. Moreover, the iterate updates are
performed based on parallel random observations of several constraint
components. For these minibatch stochastic subgradient-based projection methods
we prove sublinear convergence results and, under some linear regularity
condition for the functional constraints, we prove linear convergence rates. We
also derive conditions under which these rates depend explicitly on the
minibatch size. To the best of our knowledge, this work is the first deriving
conditions that show when minibatch stochastic subgradient-based projection
updates have a better complexity than their single-sample variants.
