Cost-Weighted Cross-Entropy for Diffusion-Based Decision-Focused Learning

*Decision-Focused Learning* (*DFL*) trains predictive models for downstream decision quality rather than predictive accuracy alone. For stochastic discrete optimization, however, decision-focused learning faces two fundamental challenges: downstream decisions are non-differentiable, while nonlinear objectives may depend on the full conditional outcome distribution rather than point predictions. To address these challenges, we propose *Cost-Weighted Cross-Entropy* (*CWCE*), a differentiable decision-alignment loss that uses realized downstream costs to align the expected-cost profile induced by the learned distribution without differentiating through the discrete optimizer. We establish two complementary theoretical results. First, arbitrarily small distributional error can induce arbitrarily large decision regret, demonstrating that distributional fidelity alone provides no general control of downstream decision quality. Second, when CWCE is incorporated into diffusion training, the excess risk of the resulting joint objective quantitatively controls downstream decision regret. These results motivate combining diffusion training, which promotes fidelity in modeling conditional outcome uncertainty, with CWCE, which provides explicit decision-relevant supervision. Experiments on three stochastic discrete optimization tasks demonstrate consistent improvements over deterministic DFL, ranking-based, and diffusion-based baselines in downstream decision quality and regret.