Conditional Risk Minimization with Side Information: A Tractable, Universal Optimal Transport Framework
This work addresses high-stakes decision-making problems in fields like finance by providing a tractable and interpretable method for distributionally robust conditional risk minimization, though it is incremental in building upon existing optimal transport approaches.
The paper tackles the challenge of constructing reliable conditional distributions for risk minimization with side information by introducing a universal optimal transport framework, which yields convex reformulations and scalable algorithms, demonstrated in portfolio optimization where conditional models outperform unconditional ones.
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.