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：本文聚焦于含协变量信息的随机规划问题，提出了一种嵌入非凸分段仿射决策规则的实证风险最小化方法，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。为解决该非凸且不可微的ERM问题，我们提出了一种增强型随机优化-最小化算法，并建立了其向（复合强）方向平稳点的渐近收敛性及复杂度分析。研究表明，所提出的基于PADR的ERM方法适用于一大类具有理论一致性保证与计算可处理性的非凸随机规划问题。数值实验表明，相较于现有先进方法，基于PADR的ERM方法在多种设定下均展现出优越性能，其具有显著更低的决策成本、更短的计算时间，并对特征维度及底层依赖关系的非线性具有鲁棒性。