Convergence Rate of a Functional Learning Method for Contextual Stochastic Optimization

2026-03-13

AI Summary

This work addresses nonlinear stochastic optimization problems involving conditional expectations when direct sampling from the conditional distribution \(Y|X\) is infeasible. Leveraging an i.i.d. stream of observational data \(\{(X^k, Y^k)\}\), the authors propose a joint learning-and-optimization algorithm that simultaneously approximates the conditional expectation within a parameterized function class and optimizes the outer objective. By integrating techniques from function approximation, stochastic approximation, and nonlinear functional optimization—all under the minimal assumption of access only to observed data pairs—the method establishes, for the first time, a theoretical convergence rate of \(O(1/\sqrt{N})\). This provides a theoretically grounded and practically viable solution to real-world problems where conditional sampling is unavailable.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Social networks and social learning
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 $\mathbb{E}[f(X, Y,β) \mid X]$, 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.

Problem

• stochastic optimization
• conditional expectation
• contextual optimization
• nonlinear loss functional
• i.i.d. observations

Innovation

• contextual stochastic optimization
• conditional expectation approximation
• simultaneous learning-and-optimization
• convergence rate
• functional learning