Automated data-driven decision-making systems are ubiquitous across a wide spread of online as well as offline services. These systems, depend on sophisticated learning algorithms and available data, to optimize the service function for decision support assistance. However, there is a growing concern about the accountability and fairness of the employed models by the fact that often the available historic data is intrinsically discriminatory, i.e., the proportion of members sharing one or more sensitive attributes is higher than the proportion in the population as a whole when receiving positive classification, which leads to a lack of fairness in decision support system. A number of fairness-aware learning methods have been proposed to handle this concern. However, these methods tackle fairness as a static problem and do not take the evolution of the underlying stream population into consideration. In this paper, we introduce a learning mechanism to design a fair classifier for online stream based decision-making. Our learning model, FAHT (Fairness-Aware Hoeffding Tree), is an extension of the well-known Hoeffding Tree algorithm for decision tree induction over streams, that also accounts for fairness. Our experiments show that our algorithm is able to deal with discrimination in streaming environments, while maintaining a moderate predictive performance over the stream.
In past work on fairness in machine learning, the focus has been on forcing the prediction of classifiers to have similar statistical properties for people of different demographics. To reduce the violation of these properties, fairness methods usually simply rescale the classifier scores, ignoring similarities and dissimilarities between members of different groups. Yet, we hypothesize that such information is relevant in quantifying the unfairness of a given classifier. To validate this hypothesis, we introduce Optimal Transport to Fairness (OTF), a method that quantifies the violation of fa…
Max Planck Institute for Software Systems (MPI-SWS)(马克斯·普朗克软件系统研究所 (MPI-SWS));University of Cambridge(剑桥大学);Alan Turing Institute(艾伦·图灵研究所);Leverhulme Centre for the Future of Intelligence(李弗休姆未来智能中心)
Consider a binary decision making process where a single machine learning classifier replaces a multitude of humans. We raise questions about the resulting loss of diversity in the decision making process. We study the potential benefits of using random classifier ensembles instead of a single classifier in the context of fairness-aware learning and demonstrate various attractive properties: (i) an ensemble of fair classifiers is guaranteed to be fair, for several different measures of fairness, (ii) an ensemble of unfair classifiers can still achieve fair outcomes, and (iii) an ensemble of cl…
PAC-Bayesian Generalization Guarantees for Fairness on Stochastic and Deterministic Classifiers
CNRS(法国国家科学研究中心);Institut d Optique Graduate School(高等光学学院);Inria(法国国家信息与自动化研究所);Institut Universitaire de France(法国大学研究院);Universite Claude Bernard Lyon 1(里昂第一大学);Univ Rennes(雷恩大学)
Machine learning algorithms are increasingly involved in sensitive decision-making process with adversarial implications on individuals. This paper presents mdfa, an approach that identifies the characteristics of the victims of a classifier's discrimination. We measure discrimination as a violation of multi-differential fairness. Multi-differential fairness is a guarantee that a black box classifier's outcomes do not leak information on the sensitive attributes of a small group of individuals. We reduce the problem of identifying worst-case violations to matching distributions and predicting …
University of South Australia(南澳大学);Simon Fraser University(西蒙菲莎大学)
The increasing application of machine learning techniques in everyday decision-making processes has brought concerns about the fairness of algorithmic decision-making. This paper concerns the problem of collider bias which produces spurious associations in fairness assessment and develops theorems to guide fairness assessment avoiding the collider bias. We consider a real-world application of auditing a trained classifier by an audit agency. We propose an unbiased assessment algorithm by utilising the developed theorems to reduce collider biases in the assessment. Experiments and simulations s…
Courant Institute of Mathematical Sciences, New York University(纽约大学柯朗数学科学研究所);Google AI(谷歌人工智能)
The potential for learned models to amplify existing societal biases has been broadly recognized. Fairness-aware classifier constraints, which apply equality metrics of performance across subgroups defined on sensitive attributes such as race and gender, seek to rectify inequity but can yield non-uniform degradation in performance for skewed datasets. In certain domains, imbalanced degradation of performance can yield another form of unintentional bias. In the spirit of constructing fairness-aware algorithms as societal imperative, we explore an alternative: Pareto-Efficient Fairness (PEF). Th…
Columbia University(哥伦比亚大学);Department of Computer Science and Data Science Institute(计算机科学与数据科学研究所)
We initiate the study of fair classifiers that are robust to perturbations in the training distribution. Despite recent progress, the literature on fairness has largely ignored the design of fair and robust classifiers. In this work, we develop classifiers that are fair not only with respect to the training distribution, but also for a class of distributions that are weighted perturbations of the training samples. We formulate a min-max objective function whose goal is to minimize a distributionally robust training loss, and at the same time, find a classifier that is fair with respect to a cl…
As AI-based systems increasingly impact many areas of our lives, auditing these systems for fairness is an increasingly high-stakes problem. Traditional group fairness metrics can miss discrimination against individuals and are difficult to apply after deployment. Counterfactual fairness describes an individualized notion of fairness but is even more challenging to evaluate after deployment. We present prediction sensitivity, an approach for continual audit of counterfactual fairness in deployed classifiers. Prediction sensitivity helps answer the question: would this prediction have been diff…
FADE: Towards Fairness-aware Generation for Domain Generalization via Classifier-Guided Score-based Diffusion Models
School of New Media and Communication, Tianjin University(天津大学新媒体与传播学院);School of Informatics, Xiamen University(厦门大学信息学院);Department of Computer Science, Baylor University(贝勒大学计算机科学系);Departments of Biostatistics and Epidemiology, Harvard University(哈佛大学生物统计学与流行病学系)
Artificial neural nets can represent and classify many types of data but are often tailored to particular applications -- e.g., for "fair" or "hierarchical" classification. Once an architecture has been selected, it is often difficult for humans to adjust models for a new task; for example, a hierarchical classifier cannot be easily transformed into a fair classifier that shields a protected field. Our contribution in this work is a new neural network architecture, the concept subspace network (CSN), which generalizes existing specialized classifiers to produce a unified model capable of learn…
In this paper, we study counterfactual fairness in text classification, which asks the question: How would the prediction change if the sensitive attribute referenced in the example were different? Toxicity classifiers demonstrate a counterfactual fairness issue by predicting that "Some people are gay" is toxic while "Some people are straight" is nontoxic. We offer a metric, counterfactual token fairness (CTF), for measuring this particular form of fairness in text classifiers, and describe its relationship with group fairness. Further, we offer three approaches, blindness, counterfactual augm…
Harvard John A. Paulson School of Engineering and Applied Sciences(哈佛大学约翰·A·保尔森工程与应用科学学院);Radcliffe Institute for Advanced Study(拉德克利夫高等研究院);Microsoft Research(微软研究院);Weizmann Institute of Science(魏茨曼科学研究所)
It is well understood that classification algorithms, for example, for deciding on loan applications, cannot be evaluated for fairness without taking context into account. We examine what can be learned from a fairness oracle equipped with an underlying understanding of ``true'' fairness. The oracle takes as input a (context, classifier) pair satisfying an arbitrary fairness definition, and accepts or rejects the pair according to whether the classifier satisfies the underlying fairness truth. Our principal conceptual result is an extraction procedure that learns the underlying truth; moreover…
We present a new approach to the global fairness verification of tree-based classifiers. Given a tree-based classifier and a set of sensitive features potentially leading to discrimination, our analysis synthesizes sufficient conditions for fairness, expressed as a set of traditional propositional logic formulas, which are readily understandable by human experts. The verified fairness guarantees are global, in that the formulas predicate over all the possible inputs of the classifier, rather than just a few specific test instances. Our analysis is formally proved both sound and complete. Exper…
Harvard University(哈佛大学);University of Pennsylvania(宾夕法尼亚大学);Rutgers University(罗格斯大学);University of Washington(华盛顿大学);Kaiser Permanente Washington Health Research Institute(凯撒医疗华盛顿健康研究所);Stanford University(斯坦福大学)
Algorithmic fairness plays an important role in machine learning and imposing fairness constraints during learning is a common approach. However, many datasets are imbalanced in certain label classes (e.g. "healthy") and sensitive subgroups (e.g. "older patients"). Empirically, this imbalance leads to a lack of generalizability not only of classification, but also of fairness properties, especially in over-parameterized models. For example, fairness-aware training may ensure equalized odds (EO) on the training data, but EO is far from being satisfied on new users. In this paper, we propose a t…
Google AI(谷歌人工智能);Toyota Technological Institute at Chicago(芝加哥丰田技术研究院);Cornell University(康奈尔大学);Kakao Mobility(Kakao出行)
Classifiers can be trained with data-dependent constraints to satisfy fairness goals, reduce churn, achieve a targeted false positive rate, or other policy goals. We study the generalization performance for such constrained optimization problems, in terms of how well the constraints are satisfied at evaluation time, given that they are satisfied at training time. To improve generalization performance, we frame the problem as a two-player game where one player optimizes the model parameters on a training dataset, and the other player enforces the constraints on an independent validation dataset…
National University of Singapore(新加坡国立大学);Univ. Lille(里尔大学);Inria(法国国家信息与自动化研究所);CNRS(法国国家科学研究中心);Centrale Lille(里尔中央理工学院)
In recent years, machine learning (ML) algorithms have been deployed in safety-critical and high-stake decision-making, where the fairness of algorithms is of paramount importance. Fairness in ML centers on detecting bias towards certain demographic populations induced by an ML classifier and proposes algorithmic solutions to mitigate the bias with respect to different fairness definitions. To this end, several fairness verifiers have been proposed that compute the bias in the prediction of an ML classifier--essentially beyond a finite dataset--given the probability distribution of input featu…
NTT(日本电信电话株式会社);The University of Tokyo(东京大学);Kyoto University(京都大学)
Machine learning is used to make decisions for individuals in various fields, which require us to achieve good prediction accuracy while ensuring fairness with respect to sensitive features (e.g., race and gender). This problem, however, remains difficult in complex real-world scenarios. To quantify unfairness under such situations, existing methods utilize {\it path-specific causal effects}. However, none of them can ensure fairness for each individual without making impractical functional assumptions on the data. In this paper, we propose a far more practical framework for learning an indivi…
Fairness-aware machine learning seeks to maximise utility in generating predictions while avoiding unfair discrimination based on sensitive attributes such as race, sex, religion, etc. An important line of work in this field is enforcing fairness during the training step of a classifier. A simple yet effective binary classification algorithm that follows this strategy is two-naive-Bayes (2NB), which enforces statistical parity - requiring that the groups comprising the dataset receive positive labels with the same likelihood. In this paper, we generalise this algorithm into N-naive-Bayes (NNB)…
University of California, Los Angeles(加利福尼亚大学洛杉矶分校);Mila(米拉研究所);Université de Montréal(蒙特利尔大学);Polytechnique Montréal(蒙特利尔理工学院)
As machine learning is increasingly used to make real-world decisions, recent research efforts aim to define and ensure fairness in algorithmic decision making. Existing methods often assume a fixed set of observable features to define individuals, but lack a discussion of certain features not being observed at test time. In this paper, we study fairness of naive Bayes classifiers, which allow partial observations. In particular, we introduce the notion of a discrimination pattern, which refers to an individual receiving different classifications depending on whether some sensitive attributes …
Google(谷歌);Rutgers University(罗格斯大学);Amazon(亚马逊);University of Washington(华盛顿大学)
Training and evaluation of fair classifiers is a challenging problem. This is partly due to the fact that most fairness metrics of interest depend on both the sensitive attribute information and label information of the data points. In many scenarios it is not possible to collect large datasets with such information. An alternate approach that is commonly used is to separately train an attribute classifier on data with sensitive attribute information, and then use it later in the ML pipeline to evaluate the bias of a given classifier. While such decoupling helps alleviate the problem of demogr…
Classifiers that achieve demographic balance by explicitly using protected attributes such as race or gender are often politically or culturally controversial due to their lack of individual fairness, i.e. individuals with similar qualifications will receive different outcomes. Individually and group fair decision criteria can produce counter-intuitive results, e.g. that the optimal constrained boundary may reject intuitively better candidates due to demographic imbalance in similar candidates. Both approaches can be seen as introducing individual resentment, where some individuals would have …
A plug-in algorithm to estimate Bayes Optimal Classifiers for fairness-aware binary classification has been proposed in (Menon & Williamson, 2018). However, the statistical efficacy of their approach has not been established. We prove that the plug-in algorithm is statistically consistent. We also derive finite sample guarantees associated with learning the Bayes Optimal Classifiers via the plug-in algorithm. Finally, we propose a protocol that modifies the plug-in approach, so as to simultaneously guarantee fairness and differential privacy with respect to a binary feature deemed sensitive.
Demystifying the Optimal Fair Classifier in Multi-Class Classification
College of Computer Science and Technology, Zhejiang University(浙江大学计算机科学与技术学院);College of Computer Science(计算机科学学院);Technology, Zhejiang University(技术,浙江大学);School of Communication Engineering, Hangzhou Dianzi University(杭州电子科技大学通信工程学院)
Fairness-aware classification is receiving increasing attention in the machine learning fields. Recently research proposes to formulate the fairness-aware classification as constrained optimization problems. However, several limitations exist in previous works due to the lack of a theoretical framework for guiding the formulation. In this paper, we propose a general framework for learning fair classifiers which addresses previous limitations. The framework formulates various commonly-used fairness metrics as convex constraints that can be directly incorporated into classic classification model…
Massachusetts Institute of Technology(麻省理工学院);University of Toronto(多伦多大学);University of Tübingen(蒂宾根大学);University of Adelaide(阿德莱德大学);Stanford University(斯坦福大学)
Deep learning models have reached or surpassed human-level performance in the field of medical imaging, especially in disease diagnosis using chest x-rays. However, prior work has found that such classifiers can exhibit biases in the form of gaps in predictive performance across protected groups. In this paper, we question whether striving to achieve zero disparities in predictive performance (i.e. group fairness) is the appropriate fairness definition in the clinical setting, over minimax fairness, which focuses on maximizing the performance of the worst-case group. We benchmark the performan…
Society has come to rely on algorithms like classifiers for important decision making, giving rise to the need for ethical guarantees such as fairness. Fairness is typically defined by asking that some statistic of a classifier be approximately equal over protected groups within a population. In this paper, current approaches to fairness are discussed and used to motivate algorithmic proposals that incorporate fairness into genetic programming for classification. We propose two ideas. The first is to incorporate a fairness objective into multi-objective optimization. The second is to adapt lex…
CSAIL, MIT(麻省理工学院计算机科学与人工智能实验室);IBM Research AI(IBM研究院人工智能部门)
Training classifiers under fairness constraints such as group fairness, regularizes the disparities of predictions between the groups. Nevertheless, even though the constraints are satisfied during training, they might not generalize at evaluation time. To improve the generalizability of fair classifiers, we propose fair mixup, a new data augmentation strategy for imposing the fairness constraint. In particular, we show that fairness can be achieved by regularizing the models on paths of interpolated samples between the groups. We use mixup, a powerful data augmentation strategy to generate th…
Université de Lorraine(洛林大学);CNRS(法国国家科学研究中心);Inria(法国国家信息与自动化研究所);LORIA(洛林研究与信息学实验室(此处按机构含义直译,若有更常见中文名需调整,但根据要求,LORIA作为机构名本身,保留英文并翻译))
Algorithmic decisions are now being used on a daily basis, and based on Machine Learning (ML) processes that may be complex and biased. This raises several concerns given the critical impact that biased decisions may have on individuals or on society as a whole. Not only unfair outcomes affect human rights, they also undermine public trust in ML and AI. In this paper we address fairness issues of ML models based on decision outcomes, and we show how the simple idea of "feature dropout" followed by an "ensemble approach" can improve model fairness. To illustrate, we will revisit the case of "Li…
Do Fairness Interventions Come at the Cost of Privacy: Evaluations for Binary Classifiers
Australian Artificial Intelligence Institute and the School of Computer Science, University of Technology Sydney(澳大利亚人工智能研究所和悉尼技术大学计算机学院);Data61, CSIRO(Data61和澳大利亚联邦科学与工业研究组织);City University of Macau(澳门城市大学)
Aalto University(阿尔托大学);Helsinki Institute for Information Technology HIIT(赫尔辛基信息技术研究所HIIT)
Our study revisits the problem of accuracy-fairness tradeoff in binary classification. We argue that comparison of non-discriminatory classifiers needs to account for different rates of positive predictions, otherwise conclusions about performance may be misleading, because accuracy and discrimination of naive baselines on the same dataset vary with different rates of positive predictions. We provide methodological recommendations for sound comparison of non-discriminatory classifiers, and present a brief theoretical and empirical analysis of tradeoffs between accuracy and non-discrimination.
As recent literature has demonstrated how classifiers often carry unintended biases toward some subgroups, deploying machine learned models to users demands careful consideration of the social consequences. How should we address this problem in a real-world system? How should we balance core performance and fairness metrics? In this paper, we introduce a MinDiff framework for regularizing classifiers toward different fairness metrics and analyze a technique with kernel-based statistical dependency tests. We run a thorough study on an academic dataset to compare the Pareto frontier achieved by …
With the rapid growth in language processing applications, fairness has emerged as an important consideration in data-driven solutions. Although various fairness definitions have been explored in the recent literature, there is lack of consensus on which metrics most accurately reflect the fairness of a system. In this work, we propose a new formulation : ACCUMULATED PREDICTION SENSITIVITY, which measures fairness in machine learning models based on the model's prediction sensitivity to perturbations in input features. The metric attempts to quantify the extent to which a single prediction dep…
Fairness of classification and regression has received much attention recently and various, partially non-compatible, criteria have been proposed. The fairness criteria can be enforced for a given classifier or, alternatively, the data can be adapated to ensure that every classifier trained on the data will adhere to desired fairness criteria. We present a practical data adaption method based on quantile preservation in causal structural equation models. The data adaptation is based on a presumed counterfactual model for the data. While the counterfactual model itself cannot be verified experi…
Singapore Univ. of Tech. and Design(新加坡科技设计大学);BITS Pilani(皮拉尼博拉理工学院)
Fairness is a critical trait in decision making. As machine-learning models are increasingly being used in sensitive application domains (e.g. education and employment) for decision making, it is crucial that the decisions computed by such models are free of unintended bias. But how can we automatically validate the fairness of arbitrary machine-learning models? For a given machine-learning model and a set of sensitive input parameters, our AEQUITAS approach automatically discovers discriminatory inputs that highlight fairness violation. At the core of AEQUITAS are three novel strategies to em…
Helmholtz AI(亥姆霍兹AI);Ecole Polytechnique Fédérale de Lausanne(洛桑联邦理工学院);Swisscom(瑞士电信)
The goal of fairness in classification is to learn a classifier that does not discriminate against groups of individuals based on sensitive attributes, such as race and gender. One approach to designing fair algorithms is to use relaxations of fairness notions as regularization terms or in a constrained optimization problem. We observe that the hyperbolic tangent function can approximate the indicator function. We leverage this property to define a differentiable relaxation that approximates fairness notions provably better than existing relaxations. In addition, we propose a model-agnostic mu…
Amazon Web Services(亚马逊云科技);MPI-IS(马克斯·普朗克智能系统研究所);University of Tübingen(图宾根大学);Rice University(莱斯大学)
Algorithmic fairness is frequently motivated in terms of a trade-off in which overall performance is decreased so as to improve performance on disadvantaged groups where the algorithm would otherwise be less accurate. Contrary to this, we find that applying existing fairness approaches to computer vision improve fairness by degrading the performance of classifiers across all groups (with increased degradation on the best performing groups). Extending the bias-variance decomposition for classification to fairness, we theoretically explain why the majority of fairness classifiers designed for lo…
Istituto Italiano di Tecnologia(意大利理工学院);University of Genova(热那亚大学);University of Waterloo(滑铁卢大学);University College London(伦敦大学学院)
We address the problem of algorithmic fairness: ensuring that sensitive variables do not unfairly influence the outcome of a classifier. We present an approach based on empirical risk minimization, which incorporates a fairness constraint into the learning problem. It encourages the conditional risk of the learned classifier to be approximately constant with respect to the sensitive variable. We derive both risk and fairness bounds that support the statistical consistency of our approach. We specify our approach to kernel methods and observe that the fairness requirement implies an orthogonali…
University of Maryland, Baltimore County(马里兰大学巴尔的摩县分校);University of Waikato(怀卡托大学);Télécom Paris, Institut Polytechnique de Paris(巴黎高等电信学院,巴黎综合理工学院);King Abdullah University of Science and Technology(阿卜杜拉国王科技大学);Carnegie Mellon University(卡内基梅隆大学);L3S Research Center(L3S研究中心);Leibniz University Hannover(莱布尼茨汉诺威大学)
As Artificial Intelligence (AI) is used in more applications, the need to consider and mitigate biases from the learned models has followed. Most works in developing fair learning algorithms focus on the offline setting. However, in many real-world applications data comes in an online fashion and needs to be processed on the fly. Moreover, in practical application, there is a trade-off between accuracy and fairness that needs to be accounted for, but current methods often have multiple hyperparameters with non-trivial interaction to achieve fairness. In this paper, we propose a flexible ensemb…
Université de Lorraine(洛林大学);CNRS(法国国家科学研究中心);Inria(法国国家信息与自动化研究所);LORIA(洛林计算机科学及其应用实验室)
Artificial Intelligence and Machine Learning are becoming increasingly present in several aspects of human life, especially, those dealing with decision making. Many of these algorithmic decisions are taken without human supervision and through decision making processes that are not transparent. This raises concerns regarding the potential bias of these processes towards certain groups of society, which may entail unfair results and, possibly, violations of human rights. Dealing with such biased models is one of the major concerns to maintain the public trust. In this paper, we address the que…
University of Illinois at Chicago(伊利诺伊大学芝加哥分校);Carnegie Mellon University(卡内基梅隆大学)
Developing classification methods with high accuracy that also avoid unfair treatment of different groups has become increasingly important for data-driven decision making in social applications. Many existing methods enforce fairness constraints on a selected classifier (e.g., logistic regression) by directly forming constrained optimizations. We instead re-derive a new classifier from the first principles of distributional robustness that incorporates fairness criteria into a worst-case logarithmic loss minimization. This construction takes the form of a minimax game and produces a parametri…
Lahore University of Management Science(拉合尔管理科学大学);Georgia State University(佐治亚州立大学);Massey University(梅西大学);Hazara University(哈扎拉大学);National University of Ireland Galway(爱尔兰国立高威大学)
Analysis of the fairness of machine learning (ML) algorithms recently attracted many researchers' interest. Most ML methods show bias toward protected groups, which limits the applicability of ML models in many applications like crime rate prediction etc. Since the data may have missing values which, if not appropriately handled, are known to further harmfully affect fairness. Many imputation methods are proposed to deal with missing data. However, the effect of missing data imputation on fairness is not studied well. In this paper, we analyze the effect on fairness in the context of graph dat…
Harvard John A. Paulson School of Engineering and Applied Science, Harvard University(哈佛大学哈佛约翰·A·保尔森工程与应用科学学院);Radcliffe Institute for Advanced Study(拉德克利夫高等研究院)
Algorithmic fairness, and in particular the fairness of scoring and classification algorithms, has become a topic of increasing social concern and has recently witnessed an explosion of research in theoretical computer science, machine learning, statistics, the social sciences, and law. Much of the literature considers the case of a single classifier (or scoring function) used once, in isolation. In this work, we initiate the study of the fairness properties of systems composed of algorithms that are fair in isolation; that is, we study fairness under composition. We identify pitfalls of naive…
National Research University Higher School of Economics(俄罗斯高等经济大学国立研究型大学(莫斯科));Moscow(莫斯科(注:此处“Moscow”为城市名,按要求不提取,但原机构名已包含,需保留));HSE(高等经济大学(注:HSE为原机构缩写,与“National Research University Higher School of Economics”为同一机构,按要求提取到大学级别即可))
The paper is devoted to the study of the model fairness and process fairness of the Russian demographic dataset by making predictions of divorce of the 1st marriage, religiosity, 1st employment and completion of education. Our goal was to make classifiers more equitable by reducing their reliance on sensitive features while increasing or at least maintaining their accuracy. We took inspiration from "dropout" techniques in neural-based approaches and suggested a model that uses "feature drop-out" to address process fairness. To evaluate a classifier's fairness and decide the sensitive features …
Fairness concerns about algorithmic decision-making systems have been mainly focused on the outputs (e.g., the accuracy of a classifier across individuals or groups). However, one may additionally be concerned with fairness in the inputs. In this paper, we propose and formulate two properties regarding the inputs of (features used by) a classifier. In particular, we claim that fair privacy (whether individuals are all asked to reveal the same information) and need-to-know (whether users are only asked for the minimal information required for the task at hand) are desirable properties of a deci…
University of Pennsylvania(宾夕法尼亚大学);Microsoft Research-New York City(微软研究院(纽约市))
The most prevalent notions of fairness in machine learning are statistical definitions: they fix a small collection of pre-defined groups, and then ask for parity of some statistic of the classifier across these groups. Constraints of this form are susceptible to intentional or inadvertent "fairness gerrymandering", in which a classifier appears to be fair on each individual group, but badly violates the fairness constraint on one or more structured subgroups defined over the protected attributes. We propose instead to demand statistical notions of fairness across exponentially (or infinitely)…