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We consider the solution of a stochastic convex optimization problem$\mathbb{E}[f(x;\theta^*,\xi)]$ over a closed and convex set $X$ in a regimewhere $\theta^*$ is unavailable and $\xi$ is a suitably defined randomvariable. Instead, $\theta^*$ may be obtained through the solution of alearning problem that requires minimizing a metric $\mathbb{E}[g(\theta;\eta)]$in $\theta$ over a closed and convex set $\Theta$. Traditional approaches havebeen either sequential or direct variational approaches. In the case of theformer, this entails the following steps: (i) a solution to the learningproblem, namely $\theta^*$, is obtained; and (ii) a solution is obtained to theassociated computational problem which is parametrized by $\theta^*$. Suchavenues prove difficult to adopt particularly since the learning process has tobe terminated finitely and consequently, in large-scale instances, sequentialapproaches may often be corrupted by error. On the other hand, a variationalapproach requires that the problem may be recast as a possibly non-monotonestochastic variational inequality problem in the $(x,\theta)$ space; but thereare no known first-order stochastic approximation schemes are currentlyavailable for the solution of this problem. To resolve the absence ofconvergent efficient schemes, we present a coupled stochastic approximationscheme which simultaneously solves both the computational and the learningproblems. The obtained schemes are shown to be equipped with almost sureconvergence properties in regimes when the function $f$ is either stronglyconvex as well as merely convex.

On the solution of stochastic optimization and variational problems in imperfect information regimes