Paper:
Hunting for Discriminatory Proxies in Linear Regression Models
A machine learning model may exhibit discrimination when used to make decisions involving people. One potential cause for such outcomes is that the model uses a statistical proxy for a protected demographic attribute. In this paper we formulate a definition of proxy use for the setting of linear regression and present algorithms for detecting proxies. Our definition follows recent work on proxies in classification models, and characterizes a model's constituent behavior that: 1) correlates closely with a protected random variable, and 2) is causally influential in the overall behavior of the model. We show that proxies in linear regression models can be efficiently identified by solving a second-order cone program, and further extend this result to account for situations where the use of a certain input variable is justified as a "business necessity". Finally, we present empirical results on two law enforcement datasets that exhibit varying degrees of racial disparity in prediction outcomes, demonstrating that proxies shed useful light on the causes of discriminatory behavior in models.

Review:
This paper describes an optimization based approach to identify variables that result in "discrimination by proxy" in linear regression models. The authors present a definition of proxy variables, proposes an optimization problem that can be solved to identify proxies, and present a surrogate problem that can be solved efficiently. They extend the results of this approach to settings where a "proxy" variable is exempt, and illustrate how an "exempt" variable can be strengthened. The paper includes experiments on two real-world datasets, which show that the ``defensible" proxies are generates.  PROS  - Well-motivated problem / novel approach. - Well-founded definition of a proxy for linear regression models.  CONS  - Reliability of detection is not considered - Experiments on real-world data do not validate the method (as there is no ground truth) - No guidance in how to set a key parameter for proxy detection (i.e., the association threshold \epsilon)?   Major Comments: -------------  This paper considers a well-motivated problem and proposes an interesting solution. However, I am concerned about several shortcomings in the current manuscript:   1. There is no discussion about how to control false positive / false negatives in proxy detection? If I were to implement their approach, and use it to detect proxies, then the first thing that I would like to have is some degree of certainty regarding how often I am detecting / not detecting proxies. I suspect that this is far more relevant and should be prioritized.  2. There is no empirical validation of the proposed method. The authors include experiments on real-world dataset, but these do not validate the method because we do not know the ground truth in these problems (i.e., the results are not falsifiable and just "make sense")  Both 1 and 2 affect the significance of the work. Both issues could be addressed by adding experiments on a toy problem where we know the ground truth. I'd advise designing a simple simulated regression problem where they know the ground truth. The proposed method should be able to correctly detect a proxy when it exists and avoid detecting a proxy when it does not exist. This would be a good way to discuss the potential to misreport proxy use.   Post Rebuttal -------------  I would like to thank the authors for the work that they put into their paper and the rebuttal. Many of the main concerns in my original review were addressed within the rebuttal. As a result, I am raising my score to a 7.  I expect that the authors will incorporate the FPR/FNR discussion into the text (as it is important for practitioners). The experiments should also be included as they provide some more convincing evidence.  In what follows, I provide some additional comments that can be quickly addressed to improve the paper. There are not intended as criticisms of the work, but simply intended to strengthen the paper in the long run:  1. Provide guidance in how to choose a key hyperarameter for proxy detection (i.e., the association threshold \epsilon)? Some key questions to answer include:  - Should practitioners pick a single value of epsilon? Or run the analysis for different values of epsilon? In the latter case, will there be conflicting results? How should be choose?   - Can epsilon be bounded after looking at a particular dataset? I think that bounds could be derived from Problem 1. If epsilon is too large, then \beta = 0 is the only feasible solution.   - What should a practitioner expect as they increase the value of \epsilon?   2. The paper is missing a discussion on how your work could be used. Say that we have identified a proxy, now what?  3. The related work is too broad. The entire discussion on fair classification methods (e.g., Dwork/Zafar et al) could be replaced with a statement that says: "Our approach aims to explain why a linear regression model may discriminate by proxy. Other works aim to produce classifiers that do not discriminate as defined by a specific definition of fairness [cite fair classification work]." Instead you may want to cite other papers that are focused on proxy detection.   4. The writing could be improved in later parts of the paper. Some tips.  - Text describing notation could be far shorter. For example, l. 103 - 106 could be replaced with "We consider a linear regression model \hat{Y} = \sum_j \alpha_j X_j" where "\alpha_j" represents the coefficient for variable j.   - There is some colloquial language that can be corrected. For example:  - l. 181 catch