通过相交 Wasserstein 球实现分布鲁棒与对抗鲁棒的 Logistic 回归
Distributionally and Adversarially Robust Logistic Regression via Intersecting Wasserstein Balls
AI总结：
针对对抗鲁棒 Logistic 回归易过拟合的问题，提出利用辅助数据集相交 Wasserstein 模糊集的分布鲁棒优化方法，并给出凸重构与高效算法，实验表现优于基准。
摘要：
对抗鲁棒优化（ARO）已成为训练模型以防范测试阶段对抗攻击的事实标准。尽管这些模型对对抗攻击具有鲁棒性，但它们往往严重遭受过拟合。为解决这一问题，一些成功方法在训练阶段用替代方案替换经验分布，包括：(i) 位于模糊集中的最坏情况分布，从而得到 ARO 的分布鲁棒（DR）对应形式；(ii) 经验分布与由辅助（例如合成、外部、域外）数据集诱导的分布的混合。受前者启发，我们研究了用于 Logistic 回归的 ARO 的 Wasserstein DR 对应形式，并证明它具有可处理的凸优化重构。采用后一种设定，我们通过将其模糊集与使用辅助数据集构建的另一个模糊集相交，来修订该 DR 方法；只要能够估计数据生成分布与辅助分布之间的 Wasserstein 距离，这就会带来显著改进。我们研究了底层优化问题，开发了高效的求解算法，并证明所提方法在标准数据集上优于基准方法。
英文摘要：
Adversarially robust optimization (ARO) has emerged as the *de facto* standard for training models that hedge against adversarial attacks in the test stage. While these models are robust against adversarial attacks, they tend to suffer severely from overfitting. To address this issue, some successful methods replace the empirical distribution in the training stage with alternatives including *(i)* a worst-case distribution residing in an ambiguity set, resulting in a distributionally robust (DR) counterpart of ARO; *(ii)* a mixture of the empirical distribution with a distribution induced by an auxiliary (*e.g.*, synthetic, external, out-of-domain) dataset. Inspired by the former, we study the Wasserstein DR counterpart of ARO for logistic regression and show it admits a tractable convex optimization reformulation. Adopting the latter setting, we revise the DR approach by intersecting its ambiguity set with another ambiguity set built using the auxiliary dataset, which offers a significant improvement whenever the Wasserstein distance between the data generating and auxiliary distributions can be estimated. We study the underlying optimization problem, develop efficient solution algorithms, and demonstrate that the proposed method outperforms benchmark approaches on standard datasets.
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