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：针对含有协变量信息的随机规划问题，本文提出了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。我们建立了基于PADR的ERM模型在无约束问题上的非渐近一致性结果，以及在约束问题上的渐近一致性结果。为求解非凸且不可微的ERM问题，我们开发了一种增强型随机主极小化算法，并建立了其渐近收敛到（复合强）方向驻点的性质及复杂度分析。研究表明，所提出的基于PADR的ERM方法适用于一类广泛的非凸SP问题，兼具理论一致性保障与计算可处理性。数值实验表明，相较于多种设置下的最优方法，基于PADR的ERM方法在显著降低成本和计算时间的同时，对特征维度及潜在依赖关系的非线性具有更强的鲁棒性。