Conditional risk minimization arises in high-stakes decisions where risk must be assessed in light of side information, such as stressed economic conditions, specific customer profiles, or other contextual covariates. Constructing reliable conditional distributions from limited data is notoriously difficult, motivating a series of optimal-transport-based proposals that address this uncertainty in a distributionally robust manner. Yet these approaches remain fragmented, each constrained by its own limitations: some rely on point estimates or restrictive structural assumptions, others apply only to narrow classes of risk measures, and their structural connections are unclear. We introduce a universal framework for distributionally robust conditional risk minimization, built on a novel union-ball formulation in optimal transport. This framework offers three key advantages: interpretability, by subsuming existing methods as special cases and revealing their deep structural links; tractability, by yielding convex reformulations for virtually all major risk functionals studied in the literature; and scalability, by supporting cutting-plane algorithms for large-scale conditional risk problems. Applications to portfolio optimization with rank-dependent expected utility highlight the practical effectiveness of the framework, with conditional models converging to optimal solutions where unconditional ones clearly do not.

翻译：条件风险最小化出现在高风险决策中，其中风险评估必须结合侧信息进行，例如紧张的经济状况、特定客户画像或其他情境协变量。从有限数据构建可靠的条件分布是众所周知的难题，这推动了一系列基于最优传输的解决方案，以分布鲁棒的方式处理这种不确定性。然而，这些方法仍然较为零散，各自受限于自身的缺陷：一些方法依赖于点估计或限制性的结构假设，另一些仅适用于狭窄的风险度量类别，且它们之间的结构关联尚不明确。我们提出了一个用于分布鲁棒条件风险最小化的通用框架，该框架建立在最优传输中一种新颖的并集球表述之上。该框架具有三个关键优势：可解释性——通过将现有方法归纳为特例并揭示其深层结构联系；可处理性——通过为文献中几乎所有主要风险泛函提供凸重构形式；以及可扩展性——通过支持大规模条件风险问题的割平面算法。在基于秩依赖期望效用的投资组合优化中的应用，突显了该框架的实际有效性：条件模型收敛到最优解，而无条件模型则明显无法做到。