Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: This paper proposes to formulate the unfairness as the smallest OT cost between the score function of the classifier and the closest fair score function over the same data. The author proposes to compute the OT cost as differentiable fairness regularizer, and validates the effectiveness of their method on two datasets.
strengths_and_weaknesses: Strengths:

The overall formulation of unfairness and OT cost is well-established. The idea of formulating unfairness as smallest OT cost is novel, and the relaxation of OT regularization is also sound. The author also analyzes the computational complexty and compare it with baseline method of L1 linear constraints.

Weaskness:

My main concern is the high computational complexty. On large datasets, applying the proposed method seems to have much longer training time than Norm baseline. Also, computing the OT regularization requires more computational resources than Norm baseline. Can the author present a case on the running time of the proposed algorithm on large-scale applications?

Details of the proposed method are not clear. For example, it is not clear how the cost matrix in Eq. 7 is calculated; it is also not clear which part (training loss, training distribution, or predicted score) is adjusted during training to achieve fairness.
questions: In line 267-268, 'For the cost function c, we use the Euclidean distance between non-protected features', does it mean that the optimal transport is between training distribution and some weighted training distribution, i.e. during training, the training distribution is adjusted, instead of predictions of training samples?
limitations: The limitations and potential negative societal impact are adequately addressed.
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: Traditionally, fairness in machine learning force the prediction of classifiers to have similar statistical properties for different demographic groups. An example is rescaling the score function of the classifier. However, such rescaling ignores the similarities and dissimilarities between members of different groups. To this end, the authors introduce OTF (Optimal Transport to Fairness) to quantify the unfairness of a probabilistic classifiers, which is defined as the smallest optimal transport cost between the score function of the classifier to a fair score function over the same data. In addition, OTF can be computed as a differentiable fairness regularizer for a class of linear fairness definitions. In experiments, OTF shows better fairness-prediction trade-off.
strengths_and_weaknesses: Strength: 

- The paper is well structured. The motivation is clear.

Weakness: 

- A motivating example showing the computation of OTF for a classifier will highly improve the readability of the paper. In the final edits, such an example should be added.
- Experiments are limited as the authors consider only two datasets.



questions: - For different OT methods, it is unclear whether for a fixed ROC AUC score, violation of fairness metric is different. Or is it the case that both ROC AUC score and fairness violation score are different for different OT methods?


limitations: Limitations are addressed in the paper.
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: This paper proposed to quantify the unfairness of a classifier in machine learning by the Optimal Transport (OT) theory. By introducing the definition of *linear fairness notion* under the probabilistic perspective, the author proposed to quantify the unfairness with OT cost between a classifier and a set of linear fair score functions. With the above definition of linear fairness notion, the fairness regularizations are reformulated as the constraints of this OT problem. Based on this theoretical result, the author applied entropic smoothing and debiased methods to solve the above OT problem. The empirical results on two datasets (i.e., *Adult* and *Bank*) indicate that the proposed methods achieve a better AUC-fairness trade-off under the demographic disparity (DP) and equalized odds (EO) metrics, compared to several selected baselines.
strengths_and_weaknesses: Strengths:
1. This paper proposed new concepts, i.e., *Linear Fariness Notion* and *Optimal Transport to Fairness (OTF)*, which provide a new perspective to introduce the fair constraints during the learning process. The derivation seems correct.
2. Compared to the previous OT-based related work, this paper proposed to implement the OT-based fairness regularizer for DP and EO with a differentiable manner. And this paper is also applicable for the scenario of multiple sensitive attributes.

Weakness:
1. The applicability to the regression problem was not discussed in this paper, while other OT-based related work can also be applied to regression problems [1].
2. The author only conducted the experiments on two datasets, which seems not enough. More empirical results on other datasets (e.g., Toxicity Classification [3]) are expected.
3. The paper only provides the experimental results on train sets in the main body, and provides limited analysis of the test results in the appendix.
4. The applicability to high-dimensional real-world data is not discussed.
questions: 1. In Line 83, it is unclear why *"multiple linear fairness notions can be combined into a new linear fairness notion by stacking their respective g functions"*. The expression *"stacking"* is not mathematically clear. What does it mean? The element-wise multiplication of g functions or addition? Could you explain more about this claim?
2. The authors only provide the train set scores in the main body of this paper and leave the test set results in the appendix. The author claimed that test results are with large noise due to the test set size. But I still think it is crucial to provide all test results in the main body of the paper. Although these results may be noisy and "not so beautiful", but the authors still need to analyze these results and provide discussions (e.g., w.r.t. the stability and the variance of these results).
3. About the adjusted $OTF_{\epsilon}$. A common solution to debias the entropy-regularized OT problem is the unbiased Sinkhorn divergence [2], which is  defined as $SOT_{\epsilon}(\mu, \nu) \triangleq OT_{\epsilon}(\mu,\nu) - \frac{1}{2}( OT_{\epsilon}(\mu,\mu) +  OT_{\epsilon}(\nu,\nu))$. Could the author discuss the difference between the proposed $OTF_{\epsilon}$ and the debiased Sinkhorn divergence, and emphasize why the adjusted $OTF_{\epsilon}$ is preferred in this work?
4. How about the experimental results on other datasets (e.g., Toxicity Classification [3])? It will be better to provide more empirical results on different datasets to support the consistently better performance of the proposed method.
5. About the applicability on high dimensional data. In this paper, only logistic regression model is used as the probabilistic classifier. The reviewer understands that it is because the feature dimension in Adult and Bank datasets is very low ($d_{\mathcal{X}}\leq 20$ if I remember correctly). In the scenario of high dimensional data (e.g, images data like CelebA [4], text data like [3]), the one-time computation of the transport cost matrix $C$ (e.g., like the Euclidean distance used in this paper) on original space seems no longer feasible. However, if we compute the cost matrix with the embedded representation from a deep neural network, will the proposed method still be applicable? Will it introduce more computational overhead? Or what kind of modifications do we need?

minors:
- Could the authors discuss the feasibility or limitation of their proposed method for the *sufficiency rule* (which encourages $Y\bot A|\hat{Y}$)?
- It would be better if the author could carefully check the typos for the future version. For example in Eq.(2), it seems it should be "$\forall l \in \{0,1\}$", but the notation "$\in$" was missed; missing space between words in Line 100.

References:
- [1] Chiappa, Silvia, et al. "A General Approach to Fairness with Optimal Transport." AAAI. 2020.
- [2] Genevay, Aude, Gabriel Peyré, and Marco Cuturi. "Learning generative models with sinkhorn divergences." International Conference on Artificial Intelligence and Statistics. PMLR, 2018.
- [3] Jigsaw. Jigsaw unintended bias in toxicity classification, 2018.
- [4] Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Large-scale celebfaces attributes (celeba)
dataset.
limitations: - The main body of this paper only shows the results on the train set. More analysis and discussion on test results are needed.
- The feasibility or limitation for the regression problem is not discussed in this paper.
- The applicability to real-world high-dimensional data (e.g., images, text) has not been discussed in the current version.
See the Questions and Weakness part for details.
review_title: null

Meta-review: metareview: Reviewers are fairly positive about this paper. This paper takes an optimal transport approach to projecting an unfair score function to a fairness constrained set by minimizing transport cost. This addresses the issue of making similar changes to individuals with similar features instead of just post processing score thresholds to match the necessary fairness criteria. This work also provides a nice regularization term reflecting the OT cost to be optimized during training. They handle fairness measures that are linear in a certain sense.

 Main concerns that were brought was:

a) Computational complexity : Due to pairwise transport and cost matrices being involved - reviewers were worried about time complexity. Answer from authors is to do batch computation which is reasonable.

b) More experimental validation was suggested. Authors responded to adding one additional evaluation. However, it would be great if authors could add additional evaluations before camera ready as there is plenty of time.

c) There was question about using unbiased Sinkhorn divergence . Authors clarified that this would lead to a multi level optimization problem while their approach is simpler.

I believe the concerns were adequately addressed and not major. 

In any case, I reiterate to the authors to consider adding another evaluation on additional datasets as suggested by one of the reviewers before camera ready.
confidence: Certain
Decision: Accept