This paper proposes a distributionally robust approach to logistic
regression. We use the Wasserstein distance to construct a ball in the space of
probability distributions centered at the uniform distribution on the training
samples. If the radius of this ball is chosen judiciously, we can guarantee
that it contains the unknown data-generating distribution with high confidence.
We then formulate a distributionally robust logistic regression model that
minimizes a worst-case expected logloss function, where the worst case is taken
over all distributions in the Wasserstein ball. We prove that this optimization
problem admits a tractable reformulation and encapsulates the classical as well
as the popular regularized logistic regression problems as special cases. We
further propose a distributionally robust approach based on Wasserstein balls
to compute upper and lower confidence bounds on the misclassification
probability of the resulting classifier. These bounds are given by the optimal
values of two highly tractable linear programs. We validate our theoretical
out-of-sample guarantees through simulated and empirical experiments.

本文提出了一种使用 Wasserstein 距离构建概率分布空间球体，并在此基础上最小化包含了经典和常见正则化逻辑回归的最坏情况期望对数损失函数的鲁棒分布逻辑回归模型，同时使用基于 Wasserstein 球的分布鲁棒方法计算分类器误分类概率的置信度上下边界。