# Convergence Rate of a Functional Learning Method for Contextual Stochastic Optimization

arXiv: https://arxiv.org/abs/2603.13048 · HTML: https://mlsanity.com/paper/2603.13048/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 21 references that are in the map.
- First posted on arXiv 13 Mar 2026.

## Builds on (most cited first, 11 of 11)

- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [Provably Efficient Reinforcement Learning with Linear Function Approximation](https://mlsanity.com/paper/1907.05388/) (2019)
- [Kernel Instrumental Variable Regression](https://mlsanity.com/paper/1906.00232/) (2019)
- [Deep Generalized Method of Moments for Instrumental Variable Analysis](https://mlsanity.com/paper/1905.12495/) (2019)
- [Biased Stochastic First-Order Methods for Conditional Stochastic Optimization and Applications in Meta Learning](https://mlsanity.com/paper/2002.10790/) (2020)
- [Integrated Conditional Estimation-Optimization](https://mlsanity.com/paper/2110.12351/) (2021)
- [Sample Complexity of Sample Average Approximation for Conditional Stochastic Optimization](https://mlsanity.com/paper/1905.11957/) (2019)
- [Constructing unbiased gradient estimators with finite variance for conditional stochastic optimization](https://mlsanity.com/paper/2206.01991/) (2022)
- [Debiasing Conditional Stochastic Optimization](https://mlsanity.com/paper/2304.10613/) (2023)
- [A Functional Model Method for Nonconvex Nonsmooth Conditional Stochastic Optimization](https://mlsanity.com/paper/2405.10815/) (2024)
- [Stochastic Optimization Algorithms for Instrumental Variable Regression with Streaming Data](https://mlsanity.com/paper/2405.19463/) (2024)

## Abstract (arXiv)

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.