  In this paper, we focus on the problem of stochastic optimization where the
objective function can be written as an expectation function over a closed
convex set. We also consider multiple expectation constraints which restrict
the domain of the problem. We extend the cooperative stochastic approximation
algorithm from Lan and Zhou [2016] to solve the particular problem. We close
the gaps in the previous analysis and provide a novel proof technique to show
that our algorithm attains the optimal rate of convergence for both optimality
gap and constraint violation when the functions are generally convex. We also
compare our algorithm empirically to the state-of-the-art and show improved
convergence in many situations.
