Convergence Rate of a Functional Learning Method for Contextual Stochastic Optimization

TL;DR

This paper analyzes a stochastic optimization method that jointly learns and optimizes a conditional expectation functional, achieving a convergence rate of 1/√N using streaming data without direct sampling from the conditional distribution.

Contribution

It introduces and analyzes a joint learning-and-optimization algorithm for conditional expectations in stochastic optimization, establishing its convergence rate with streaming data.

Findings

Achieves a convergence rate of O(1/√N)

Handles infeasible direct sampling from conditional distributions

Jointly estimates conditional expectation and optimizes the objective

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 $\mathbb{E}[f(X, Y,\beta) \mid X]$, where $f$ is a nonlinear function and $\beta$ 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.