Convergence Rate of a Functional Learning Method for Contextual Stochastic Optimization

Abstract

We consider a stochastic optimization problem involving two random variables: a context variable $X$ and a dependent variable $Y$. The objective is to minimize the expected value of a nonlinear loss functional applied to the conditional expectation $\mathbb{E}[f(X,Y,{\hat{I}}^2)\mid X]$, where $f$ is a nonlinear function and ${\hat{I}}^2$ represents the decision variables. We focus on the practically important setting in which direct sampling from the conditional distribution of $Y\mid X$ is infeasible, and only a stream of i.i.d. observation pairs $\{(X^k,Y^k){\}}_{k=0,1,2,\dots }$ is available. In our approach, the conditional expectation is approximated within a prespecified parametric function class. We analyze a simultaneous learning-and-optimization algorithm that jointly estimates the conditional expectation and optimizes the outer objective. Using a specially designed measure of non-optimality, combining the squared norm of the objective function's gradient and the mean square error of the auxiliary parametric model, we establish that the method achieves a convergence rate of order $\mathcal{O}(1/\sqrt{N})$, where $N$ denotes the number of observed pairs.