In contextual optimization, a decision-maker observes historical samples of
uncertain variables and associated concurrent covariates, without knowing their
joint distribution. Given an additional covariate observation, the goal is to
choose a decision that minimizes some operational costs. A prevalent issue here
is covariate shift, where the marginal distribution of the new covariate
differs from historical samples, leading to decision performance variations
with nonparametric or parametric estimators. To address this, we propose a
distributionally robust approach that uses an ambiguity set by the intersection
of two Wasserstein balls, each centered on typical nonparametric or parametric
distribution estimators. Computationally, we establish the tractable
reformulation of this distributionally robust optimization problem.
Statistically, we provide guarantees for our Wasserstein ball intersection
approach under covariate shift by analyzing the measure concentration of the
estimators. Furthermore, to reduce computational complexity, we employ a
surrogate objective that maintains similar generalization guarantees. Through
synthetic and empirical case studies on income prediction and portfolio
optimization, we demonstrate the strong empirical performance of our proposed
models.

在上下文优化中，通过观察不确定变量的历史样本和相关联的并发协变量，不知道它们的联合分布。在给定附加协变量观测情况下，目标是选择最小化某些操作成本的决策。这里的一个普遍问题是协变量偏移，其中新协变量的边际分布与历史样本不同，导致具有非参数或参数估计器的决策性能变化。为了解决这个问题，我们提出了一个分布鲁棒方法，使用两个以典型的非参数或参数分布估计器为中心的 Wasserstein 球的交集作为模糊集合。在计算上，我们建立了这个分布鲁棒优化问题的易于计算的改写形式。在统计上，通过分析估计器的测度集中性，我们提供了我们的 Wasserstein 球交集方法在协变量偏移下的保证。此外，为了减少计算复杂性，我们采用了一个保持类似泛化保证的替代目标。通过对收入预测和投资组合优化的合成和实证案例研究，我们展示了我们提出的模型的强大实证性能。

上下文优化在协变量漂移下的鲁棒方法：通过相交的 Wasserstein 球

Contextual Optimization under Covariate Shift: A Robust Approach by  Intersecting Wasserstein Balls

Distributionally robust chance constrained programs minimize a deterministic
cost function subject to the satisfaction of one or more safety conditions with
high probability, given that the probability distribution of the uncertain
problem parameters affecting the safety condition(s) is only known to belong to
some ambiguity set. We study three popular approximation schemes for
distributionally robust chance constrained programs over Wasserstein balls,
where the ambiguity set contains all probability distributions within a certain
Wasserstein distance to a reference distribution. The first approximation
replaces the chance constraint with a bound on the conditional value-at-risk,
the second approximation decouples different safety conditions via Bonferroni's
inequality, and the third approximation restricts the expected violation of the
safety condition(s) so that the chance constraint is satisfied. We show that
the conditional value-at-risk approximation can be characterized as a tight
convex approximation, which complements earlier findings on classical
(non-robust) chance constraints, and we offer a novel interpretation in terms
of transportation savings. We also show that the three approximations can
perform arbitrarily poorly in data-driven settings, and that they are generally
incomparable with each other.

研究了三种分布鲁棒的机会限制规划的近似方案，并且在数据驱动的环境下表现得差别很大。其中一种方案采用条件风险价值的紧凑凸估计，并提供了一种新颖的解释方式。同时，他们的优劣无法相互比较。

关于 Wasserstein 球上数据驱动条件约束规划的逼近

On Approximations of Data-Driven Chance Constrained Programs over  Wasserstein Balls

Wasserstein balls, which contain all probability measures within a
pre-specified Wasserstein distance to a reference measure, have recently
enjoyed wide popularity in the distributionally robust optimization and machine
learning communities to formulate and solve data-driven optimization problems
with rigorous statistical guarantees. In this technical note we prove that the
Wasserstein ball is weakly compact under mild conditions, and we offer
necessary and sufficient conditions for the existence of optimal solutions. We
also characterize the sparsity of solutions if the Wasserstein ball is centred
at a discrete reference measure. In comparison with the existing literature,
which has proved similar results under different conditions, our proofs are
self-contained and shorter, yet mathematically rigorous, and our necessary and
sufficient conditions for the existence of optimal solutions are easily
verifiable in practice.

该技术笔记证明了在温和条件下 Wasserstein 球是弱紧的，并提供了存在最优解的必要和充分条件。同时，如果 Wasserstein 球以离散参考度量为中心，则表征解的稀疏性。相比于现有文献，在短而精炼的证明下，说明了存在最优解的必要和充分条件在实际中容易验证。