Wasserstein Introspective Neural Networks

We present Wasserstein introspective neural networks (WINN) that are both a generator and a discriminator within a single model. WINN provides a significant improvement over the recent introspective neural networks (INN) method by enhancing INN's generative modeling capability. WINN has three interesting properties: (1) A mathematical connection between the formulation of the INN algorithm and that of Wasserstein generative adversarial networks (WGAN) is made. (2) The explicit adoption of the Wasserstein distance into INN results in a large enhancement to INN, achieving compelling results even with a single classifier --- e.g., providing nearly a 20 times reduction in model size over INN for unsupervised generative modeling. (3) When applied to supervised classification, WINN also gives rise to improved robustness against adversarial examples in terms of the error reduction. In the experiments, we report encouraging results on unsupervised learning problems including texture, face, and object modeling, as well as a supervised classification task against adversarial attacks.Comment: Accepted to CVPR 2018 (Oral

Outlier detection using distributionally robust optimization under the Wasserstein metric

We present a Distributionally Robust Optimization (DRO) approach to outlier detection in a linear regression setting, where the closeness of probability distributions is measured using the Wasserstein metric. Training samples contaminated with outliers skew the regression plane computed by least squares and thus impede outlier detection. Classical approaches, such as robust regression, remedy this problem by downweighting the contribution of atypical data points. In contrast, our Wasserstein DRO approach hedges against a family of distributions that are close to the empirical distribution. We show that the resulting formulation encompasses a class of models, which include the regularized Least Absolute Deviation (LAD) as a special case. We provide new insights into the regularization term and give guidance on the selection of the regularization coefficient from the standpoint of a confidence region. We establish two types of performance guarantees for the solution to our formulation under mild conditions. One is related to its out-of-sample behavior, and the other concerns the discrepancy between the estimated and true regression planes. Extensive numerical results demonstrate the superiority of our approach to both robust regression and the regularized LAD in terms of estimation accuracy and outlier detection rates

Fair and Optimal Classification via Transports to Wasserstein-Barycenter

Fairness in automated decision-making systems has gained increasing attention as their applications expand to real-world high-stakes domains. To facilitate the design of fair ML systems, it is essential to understand the potential trade-offs between fairness and predictive power, and the construction of the optimal predictor under a given fairness constraint. In this paper, for general classification problems under the group fairness criterion of demographic parity (DP), we precisely characterize the trade-off between DP and classification accuracy, referred to as the minimum cost of fairness. Our insight comes from the key observation that finding the optimal fair classifier is equivalent to solving a Wasserstein-barycenter problem under $\ell_1$-norm restricted to the vertices of the probability simplex. Inspired by our characterization, we provide a construction of an optimal fair classifier achieving this minimum cost via the composition of the Bayes regressor and optimal transports from its output distributions to the barycenter. Our construction naturally leads to an algorithm for post-processing any pre-trained predictor to satisfy DP fairness, complemented with finite sample guarantees. Experiments on real-world datasets verify and demonstrate the effectiveness of our approaches.Comment: Code is at https://github.com/rxian/fair-classificatio

Matching Code and Law: Achieving Algorithmic Fairness with Optimal Transport

Increasingly, discrimination by algorithms is perceived as a societal and legal problem. As a response, a number of criteria for implementing algorithmic fairness in machine learning have been developed in the literature. This paper proposes the Continuous Fairness Algorithm (CFA$\theta$) which enables a continuous interpolation between different fairness definitions. More specifically, we make three main contributions to the existing literature. First, our approach allows the decision maker to continuously vary between specific concepts of individual and group fairness. As a consequence, the algorithm enables the decision maker to adopt intermediate ``worldviews'' on the degree of discrimination encoded in algorithmic processes, adding nuance to the extreme cases of ``we're all equal'' (WAE) and ``what you see is what you get'' (WYSIWYG) proposed so far in the literature. Second, we use optimal transport theory, and specifically the concept of the barycenter, to maximize decision maker utility under the chosen fairness constraints. Third, the algorithm is able to handle cases of intersectionality, i.e., of multi-dimensional discrimination of certain groups on grounds of several criteria. We discuss three main examples (credit applications; college admissions; insurance contracts) and map out the legal and policy implications of our approach. The explicit formalization of the trade-off between individual and group fairness allows this post-processing approach to be tailored to different situational contexts in which one or the other fairness criterion may take precedence. Finally, we evaluate our model experimentally.Comment: Vastly extended new version, now including computational experiment