On the solution of stochastic optimization problems in imperfect information regimes

We consider the solution of a stochastic convex optimization problem E[f(x;θ*,ξ)] in x over a closed and convex set X in a regime where θ* is unavailable. Instead, θ* may be learnt by minimizing a suitable metric E[g(θη)] in θ over a closed and convex set Θ. We present a coupled stochastic approximation scheme for the associated stochastic optimization problem with imperfect information. The schemes are shown to be equipped with almost sure convergence properties in regimes where the function f is both strongly convex as well as merely convex. Rate estimates are provided in both a strongly convex as well as a merely convex regime, where the use of averaging facilitates the development of a bound.