  We propose a stochastic gradient framework for solving stochastic composite
convex optimization problems with (possibly) infinite number of linear
inclusion constraints that need to be satisfied almost surely. We use smoothing
and homotopy techniques to handle constraints without the need for
matrix-valued projections. We show for our stochastic gradient algorithm
$\mathcal{O}(\log(k)/\sqrt{k})$ convergence rate for general convex objectives
and $\mathcal{O}(\log(k)/k)$ convergence rate for restricted strongly convex
objectives. These rates are known to be optimal up to logarithmic factors, even
without constraints. We demonstrate the performance of our algorithm with
numerical experiments on basis pursuit, a hard margin support vector machines
and a portfolio optimization and show that our algorithm achieves
state-of-the-art practical performance.
