提出一种在有限观测下求解随机优化问题的新方法，收敛速度达1/√N。
Convergence Rate of a Functional Learning Method for Contextual Stochastic Optimization
- 用参数化模型近似条件期望，联合学习与优化决策变量。
- 理论证明算法收敛速度为1/√N，依赖于观测样本数。
- 适合无法直接采样条件分布的实际优化场景，如在线决策。
我们研究一个涉及两个随机变量的随机优化问题：上下文变量X和依赖变量Y。目标是最小化非线性损失函数作用于条件期望𝔼[f(X,Y,β)∣X]的期望值，其中f是非线性函数，β为决策变量。重点关注实际中难以直接从条件分布Y∣X采样的情形，仅能获得独立同分布的观测对{(Xᵏ, Yᵏ)}_{k=0,1,2,…}。本文方法在预设的参数函数类中近似条件期望，分析一种同时学习与优化的算法，联合估计条件期望并优化外部目标。通过设计一种结合目标函数梯度平方范数与辅助参数模型均方误差的非最优性度量，证明该方法达到阶为𝒪(1/√N)的收敛速率，其中N为观测对的数量。
原文摘要 · Abstract (English)
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. 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 $\mathcal{O}\big(1/\sqrt{N}\big)$, where $N$ denotes the number of observed pairs.
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