Focusing on stochastic programming (SP) with covariate information, this paper proposes an empirical risk minimization (ERM) method embedded within a nonconvex piecewise affine decision rule (PADR), which aims to learn the direct mapping from features to optimal decisions. We establish the nonasymptotic consistency result of our PADR-based ERM model for unconstrained problems and asymptotic consistency result for constrained ones. To solve the nonconvex and nondifferentiable ERM problem, we develop an enhanced stochastic majorization-minimization algorithm and establish the asymptotic convergence to (composite strong) directional stationarity along with complexity analysis. We show that the proposed PADR-based ERM method applies to a broad class of nonconvex SP problems with theoretical consistency guarantees and computational tractability. Our numerical study demonstrates the superior performance of PADR-based ERM methods compared to state-of-the-art approaches under various settings, with significantly lower costs, less computation time, and robustness to feature dimensions and nonlinearity of the underlying dependency.

翻译：聚焦于含协变量信息的随机规划问题，本文提出一种嵌入非凸分段仿射决策规则（PADR）的经验风险最小化（ERM）方法，旨在学习从特征到最优决策的直接映射。针对无约束问题，我们建立了基于PADR的ERM模型的非渐近一致性结果；针对约束问题，则建立了渐近一致性结果。为解决非凸且不可微的ERM问题，我们开发了一种增强型随机极大化-极小化算法，并在复杂度分析的基础上建立了其到（复合强）方向稳定点的渐近收敛性。研究表明，所提出的基于PADR的ERM方法可适用于一类广泛的非凸随机规划问题，兼具理论一致性保证与计算可处理性。数值实验表明，在各种设置下，基于PADR的ERM方法相较于现有最优方法表现出更优性能：显著更低的成本、更少的计算时间，以及对特征维度和潜在依赖关系非线性性的鲁棒性。