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

Machine learning algorithms minimizing average risk are susceptible to
distributional shifts. Distributionally Robust Optimization (DRO) addresses
this issue by optimizing the worst-case risk within an uncertainty set.
However, DRO suffers from over-pessimism, leading to low-confidence
predictions, poor parameter estimations as well as poor generalization. In this
work, we conduct a theoretical analysis of a probable root cause of
over-pessimism: excessive focus on noisy samples. To alleviate the impact of
noise, we incorporate data geometry into calibration terms in DRO, resulting in
our novel Geometry-Calibrated DRO (GCDRO) for regression. We establish the
connection between our risk objective and the Helmholtz free energy in
statistical physics, and this free-energy-based risk can extend to standard DRO
methods. Leveraging gradient flow in Wasserstein space, we develop an
approximate minimax optimization algorithm with a bounded error ratio and
elucidate how our approach mitigates noisy sample effects. Comprehensive
experiments confirm GCDRO's superiority over conventional DRO methods.

通过灵活利用数据的几何形态与最小风险优化算法相结合，本研究提出了一种新型的校准方法 —— 几何校准分布鲁棒优化 (Geometry-Calibrated DRO，GCDRO)，该方法减轻了噪声样本的影响，理论分析和广泛实验证明了其相较于常规分布鲁棒优化方法的优越性。