Paper:
When Worlds Collide: Integrating Different Counterfactual Assumptions in Fairness
Machine learning is now being used to make crucial decisions about people's lives. For nearly all of these decisions there is a risk that individuals of a certain race, gender, sexual orientation, or any other subpopulation are unfairly discriminated against. Our recent method has demonstrated how to use techniques from counterfactual inference to make predictions fair across different subpopulations. This method requires that one provides the causal model that generated the data at hand. In general, validating all causal implications of the model is not possible without further assumptions. Hence, it is desirable to integrate competing causal models to provide counterfactually fair decisions, regardless of which causal "world" is the correct one. In this paper, we show how it is possible to make predictions that are approximately fair with respect to multiple possible causal models at once, thus mitigating the problem of exact causal specification. We frame the goal of learning a fair classifier as an optimization problem with fairness constraints entailed by competing causal explanations. We show how this optimization problem can be efficiently solved using gradient-based methods. We demonstrate the flexibility of our model on two real-world fair classification problems. We show that our model can seamlessly balance fairness in multiple worlds with prediction accuracy.

Review:
The authors consider a novel supervised learning problem with fairness constraints, where the goal is to find an optimal predictor that is counterfactually fair from a list of candidate causal models. The parameterizations of each candidate causal model are known. The authors incorporate the fairness constraint as a regularization term in the loss function. Evaluations are performed on two real-world datasets, and results show that the proposed method balance fairness in multiple worlds with prediction accuracy.
While the idea of exploring a novel fairness measure in counterfactual semantics and enforcing it over multiple candidate models is interesting, there are a few issues I find confusing, which is listed next:
1. The Counterfactual Fairness definition (Def 1) is not clear. It is not immediate to see which counterfactual quantity the authors are trying to measure. Eq2, the probabilistic counterfactual fairness definition, measures the total causal effect (P(Y_x)) of the sensitive feature X on the predicted outcome Y. It is a relatively simple counterfactual quantity, which can be directly computed by physically setting X to a fixed value, without using Pearl’s algorithm of three steps.  
2. If the authors are referring to the total causal effect, it is thus unnecessary to use a rather complicated algorithm to compute counterfactuals (line87-94).
If the authors are indeed referring to the counterfactual fairness defined in [1], the motivation of using this novel counterfactual fairness has not been properly justified. A newly proposed fairness definition should often fall into one of following categories: i. It provides a stronger condition for existing fairness definitions; ii. It captures discriminations which are not covered by existing definitions; or iii. It provides a reasonable relaxation to improve prediction accuracy. I went back to the original counterfactual fairness paper [1] and found discussions regarding this problem. Since this a rather recent result, it would be better if the authors could explain it a bit further in the background section.
3. The contribution of this paper seems to be incremental unless I am missing something. The authors claim that the proposed technique “learns a fair predictor without knowing the true causal model”, but it still requires a finite list of known candidate causal models and then ranges over them. The natural question at this point is how to obtain a list of candidate models? In causal literature, “not knowing the true causal model” often means that only observational data is available, let alone a list of fully-parametrized possible models exists. The relaxation considered in this paper may be a good starting point, but it does not address the fundamental challenge of the unavailability of the underlying model.
Minor comments:
- All references to Figure2 in Sec2.2 should be Figure1.
[1] Matt J Kusner, Joshua R Loftus, Chris Russell, and Ricardo Silva. Counterfactual fairness. arXiv preprint arXiv:1703.06856, 2017.
Post-rebuttal: 
 Some of the main issues were addressed by the authors. One of the issues was around Definition 1, and I believe the authors can fix that in the camera-ready version. Connections with existing fairness measures can be added in the Supplement. 
Still, unless I am missing something, it seems the paper requires that each of the causal models is *fully* specified, which means that they know precisely the underlying structural functions and distributions over the exogenous variables. This is, generally, an overly strong requirement. 
I felt that the argument in the rebuttal saying that “These may come from expert knowledge or causal discovery algorithms like the popular PC or FCI algorithms [P. Spirtes, C. Glymour, and R. Scheines. “Causation, Prediction, and Search”, 2000]”  is somewhat misleading.  Even when the learning algorithms like FCI can pin down a unique causal structure (almost never the case), it’s still not the accurate to say that they provide the fully specified model with the structural functions and distributions over the exogenous. If one doesn’t have this type of knowledge and setting, one cannot run Pearl’s 3-step algorithm. I am, therefore, unable to find any reasonable justification or setting that would support the feasibility of the proposed approach.