Convergence Rate of a Functional Learning Method for Contextual Stochastic Optimization
This addresses a practical bottleneck in contextual stochastic optimization for scenarios with limited data access, though it appears incremental as it builds on existing parametric approximation methods.
The paper tackles the problem of stochastic optimization with context-dependent variables where direct sampling from the conditional distribution is infeasible, achieving a convergence rate of O(1/√N) for a joint learning-and-optimization algorithm.
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 $\mathbb{E}[f(X, Y,β) \mid X]$, 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 $\{(X^k, Y^k)\}_{k=0,1,2,\ldots}$ 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, and establish that the method achieves a convergence rate of order $\mathcal{O}\big(1/\sqrt{N}\big)$, where $N$ denotes the number of observed pairs.