What Decisions Need from a Distribution: Decision-Focused Learning via Optimality-Field Matching

Decision-Focused Learning (DFL) trains predictive models for downstream decision quality rather than predictive accuracy alone. For point predictions, recent work relates the regret gradient to prediction error and local solution sensitivity. In stochastic optimization, however, the optimal decision may depend on distributional information that a point prediction cannot represent. We show that a stochastic objective uses a predictive distribution through its *distributional optimality field*, a function over the decision space whose values are distributional functionals. Differentiating realized cost through the stochastic decision yields a sample gradient with the same structure: the field residual replaces prediction error and is transformed through the local solution sensitivity. When the field is available analytically, this gradient requires one solve and no predictive scenario sampling or differentiable optimization layer. Since the field being learned also determines the induced decision, this gradient constrains the field only where the decision currently lies. We therefore introduce **Functional-DFL**, which keeps the exact regret gradient and uses the same observation to match the field at additional decisions, so that regret remains controlled as the decision moves. Across two synthetic and two real-world problems, Functional-DFL achieves the best average decision quality among the compared baselines at low computational cost. Code is available at https://anonymous.4open.science/r/distributional_dfl-6B0B.