Convergence Rate of a Functional Learning Method for Contextual Stochastic Optimization

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 mathbbE[f(X, Y,β) mid X], where f is a nonlinear function and β 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 (Xk, Yk)k=0,1,2,ldots 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 Obig(1/√Nbig), where N denotes the number of observed pairs.