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

Abstract: 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 regime where $\theta^*$ is unavailable and $\xi$ is a suitably defined random variable. Instead, $\theta^*$ may be obtained through the solution of a learning problem that requires minimizing a metric $\mathbb{E}[g(\theta;\eta)]$ in $\theta$ over a closed and convex set $\Theta$. Traditional approaches have been either sequential or direct variational approaches. In the case of the former, this entails the following steps: (i) a solution to the learning problem, namely $\theta^*$, is obtained; and (ii) a solution is obtained to the associated computational problem which is parametrized by $\theta^*$. Such avenues prove difficult to adopt particularly since the learning process has to be terminated finitely and consequently, in large-scale instances, sequential approaches may often be corrupted by error. On the other hand, a variational approach requires that the problem may be recast as a possibly non-monotone stochastic variational inequality problem in the $(x,\theta)$ space; but there are no known first-order stochastic approximation schemes are currently available for the solution of this problem. To resolve the absence of convergent efficient schemes, we present a coupled stochastic approximation scheme which simultaneously solves both the computational and the learning problems. The obtained schemes are shown to be equipped with almost sure convergence properties in regimes when the function $f$ is either strongly convex as well as merely convex.