In recent years, there has been a growing research interest in
decision-focused learning, which embeds optimization problems as a layer in
learning pipelines and demonstrates a superior performance than the
prediction-focused approach. However, for distributionally robust optimization
(DRO), a popular paradigm for decision-making under uncertainty, it is still
unknown how to embed it as a layer, i.e., how to differentiate decisions with
respect to an ambiguity set. In this paper, we develop such differentiable DRO
layers for generic mixed-integer DRO problems with parameterized second-order
conic ambiguity sets and discuss its extension to Wasserstein ambiguity sets.
To differentiate the mixed-integer decisions, we propose a novel dual-view
methodology by handling continuous and discrete parts of decisions via
different principles. Specifically, we construct a differentiable energy-based
surrogate to implement the dual-view methodology and use importance sampling to
estimate its gradient. We further prove that such a surrogate enjoys the
asymptotic convergency under regularization. As an application of the proposed
differentiable DRO layers, we develop a novel decision-focused learning
pipeline for contextual distributionally robust decision-making tasks and
compare it with the prediction-focused approach in experiments.

发展了可微分的分布鲁棒优化层，用于参数化二阶锥模糊集的混合整数分布鲁棒优化问题，并讨论其扩展到 Wasserstein 模糊集；通过处理决策的连续和离散部分，提出了新的双视图方法；以上下文分布鲁棒决策任务为应用，将可微分的分布鲁棒优化层应用于决策集中的学习，与预测集中方法进行了实验对比。

可微分分布鲁棒优化层

Differentiable Distributionally Robust Optimization Layers

We consider exact deterministic mixed-integer programming (MIP)
reformulations of distributionally robust chance-constrained programs (DR-CCP)
with random right-hand sides over Wasserstein ambiguity sets. The existing MIP
formulations are known to have weak continuous relaxation bounds, and,
consequently, for hard instances with small radius, or with large problem
sizes, the branch-and-bound based solution processes suffer from large
optimality gaps even after hours of computation time. This significantly
hinders the practical application of the DR-CCP paradigm. Motivated by these
challenges, we conduct a polyhedral study to strengthen these formulations. We
reveal several hidden connections between DR-CCP and its nominal counterpart
(the sample average approximation), mixing sets, and robust 0-1 programming. By
exploiting these connections in combination, we provide an improved formulation
and two classes of valid inequalities for DR-CCP. We test the impact of our
results on a stochastic transportation problem numerically. Our experiments
demonstrate the effectiveness of our approach; in particular our improved
formulation and proposed valid inequalities reduce the overall solution times
remarkably. Moreover, this allows us to significantly scale up the problem
sizes that can be handled in such DR-CCP formulations by reducing the solution
times from hours to seconds.

本研究提出了一种基于混合整数编程的改进模型和两类合法不等式，用于加强 Wasserstein 模糊集下分布鲁棒机会约束规划（DR-CCP）的精确确定性混合整数规划（MIP）重构。相关实验结果表明，该改进模型和提出的合法不等式不仅可明显降低求解时间，还可扩大此类 DR-CCP 问题的处理规模。