Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: The work proposes a method to guarantee fair prediction intervals in regression tasks. It takes the approach of quantile regressions to obtain the prediction intervals and uses a previous result on the form of the fair quantile regressor to develop an approach for fair prediction intervals. Additionally, it uses conformal inference approach to provide intervals that have valid coverage in finite samples. The experiments on 4 real datasets show that the method maintains competitive coverage, length of prediction intervals, and prediction error compared to baselines while reducing unfairness.
strengths_and_weaknesses: Strengths
- addresses a new problem.
- the work seems to be important given the importance of providing valid prediction intervals in safety-critical settings while ensuring some form of fairness.
- skilfully combines advances from fair quantile regression and conformal inference.

Weaknesses
- both the problem definition and motivation is not clear from the Introduction.
- there seems to be a discrepancy between the fairness goal and the method. The aim is to define fairness of prediction intervals but the fairness metric is defined for the quantile regression functions which are one of the ways of constructing the intervals but not the only way.
- presentation is not easily understandable to readers not familiar with quantile regression-based prediction intervals and conformal inference.

The main concern is the presentation of the method. The theoretical analysis seems sound but I am not familiar with the techniques and literature so I cannot comment on its validity and significance. In my opinion, improving the presentation would take substantial changes to the writing than is possible to verify from the author response.

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## After the response

The response adequately addresses my concerns. The presentation of the problem statement and overview of the method has been greatly improved. I appreciate the honest response to comparison with related work [1] and meticulous comparison in running time. 

The work addresses an important problem via a principled approach. Thus, moving the fair learning field forward beyond notions based on averages to quantiles. Hence I have increased my score to 7, Accept.
questions: Questions:

Regarding presentation:

1. The relation between DP fairness defined for the interval prediction function (Def. 1) and DP fairness defined for quantile functions (between \nu’s) is not clear. How does one ensure another?

The importance or challenge in building a method that gives both fair quantile regression and conformal prediction intervals is not presented convincingly.

Regarding related work:

2. How does the proposed method for fair quantile regression relate to the method in Agarwal et al. [1]? Their method also covers the case of quantile regression with check loss and DP fairness. Does the addition of the split-conformal step applicable to their method as well to get miscoverage guarantees?

3. How does the work differ from Romano et al. [42]? Is the difference due to the fairness objective?

Regarding experiments:

4. The abstract claims that the experiments uncover the fairness vs coverage frontiers. Table 1 reports the coverage and fairness metrics but does not show the frontier. Does the claim refer to Figure 2 which reports the frontier of fairness vs error.

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Suggestions (not expecting a response):

I would suggest tightening up the motivations for fairness (between group-specific quantile regression functions) and uncertainty estimates in the Introduction (first page). It does not clearly answer why these are required. For example what is the issue with heteroscedastic data or in the case of uncertainty estimates, do the high-stakes areas require prediction intervals or confidence intervals.

Please describe why it is challenging to combine fair regression functions with conformal inference such that the significance of the proposed method and analysis can be conveyed.
 
The example of salary prediction in Introduction is unclear. What is the fairness objective?

The \alpha_lo and \alpha_hi needs to be explained at page 4, first equation. Do these refer to upper and lower quantiles, \alpha/2 and 1-\alpha/2 for a miscoverage rate of \alpha?

The description of the method can be improved. Consider giving a sketch of the method before hand before describing each of the steps.

Consider discussing other potential fairness metrics in case of real-valued predictions like equalizing calibration metrics across groups.

Discuss related work Williamson and Menon 2019, Fairness risk measures. The fairness metric defined for conditional variance-at-risk is another possibility to go beyond mean predictions.
https://proceedings.mlr.press/v97/williamson19a.html

Please define DP fairness briefly to give context to Introduction and Related work.

The left hand side of Proposition 1 that is the squared loss term is not clear.

A tip on writing style is to limit the use of superlatives like tremendous as they are not precise and redundant sometimes.
limitations: Limitations are adequately discussed. Consider discussing whether the method allows for relaxing the fairness constraint that is the KS distance can be less than some non-zero value.
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 work investigates the incorporation of group-fairness into the established learning problem of conformalized quantile regression. In particular, the authors extend previously developed quantile regression techniques to handle demographic parity constraints between groups. To do this,  the authors develop a postprocessing technique for modifying a given quantile regressor such that it becomes fair with respect to demographic parity. Within this framework the authors provide theoretical gurentess with respect to the efficacy of their method. In particular Theorems 1, 2 demonstrate that the proposed technique produces a model which has both reasonable predictive power as well as fairness. Further, the authors compare their technique to two other fair regression methods and show that their method iid frequently superior in terms of fairness and error.


strengths_and_weaknesses: ## Strengths
- The paper is well written (barring a few minor issues mentioned later).
- The authors make non-trivial contributions to a well established problem, namely introducing a notion of group fairness to the quantile regression problem.
- A mix of theoretical experimental results are provided.
- The proposed method is shown to outperform two state-of-the-art methods in experiments.
- The authors make a point to address the practicality of their method (e.g. Remark 1).
 
## Weaknesses
- The experiments performed are somewhat limited. 
- Running time and computability of certain quantities appears to be missing. What is the worst case running time of Algorithm 1. How feasible is it to compute equation 8 (does this have a closed form solution or must we rely on numerical approximations)?
- The proposed model only works with one fairness metric (namely DP). It would be nice to see similar results for a fairness metric which is dependent on the underlying $Y$.
- Allowed unfairness does not appear to be a tunable parameter.
- See Questions section for more details comments.
questions: ## Questions and Comments 


- **Slack in fairness**: In practice achieving perfect fairness (i.e. equal DP across all groups) is infeasible or at the very least undesirable. As such it is useful to have a so called "tunable slackness variable" $\varepsilon$, such that fairness can be framed as $|M_i - M_j| \leq \varepsilon$ between any two groups $i, j$, where $M$ is a statistical metric applied to both groups. The current model requires $\varepsilon=0$. Can we say anything theoretically about when $\varepsilon > 0$. Similarly, is a model designer able to control the desired unfairness (i.e. $\varepsilon$) in the practice? If so, are there any experimental results on error-fairness tradeoffs as we have vary $\varepsilon$?

- **Equation 2**: Equation 2 seems to be misspecified compared to [33]. Also the suppression of the dependence that $q$ has on $x$ and $\alpha$ in this definition may lead to confusion. It may be more clear to use $q_{\alpha}(x)$ instead of simply $q$, especially since $q$ is defined to be a function two lines earlier.

- **Citations for legal justifications**: Legal justification is given for both demographic parity as well as choice of sensitive feature S. In both instances the authors cite computer science papers (not legal works). Although the claims are mentioned in these citations, it would be better to directly cite the actual source, thus avoiding any whisper-down-the-lane issues that may arise.

- **Distribution notation**: The variable $\mathbb{P}$ is used to denote distributions. Perhaps $P_{X}$, $D$, or $\mathcal{D}$ would be more standard. Note that  [33, 41] do not use $\mathbb{P}$ to denote distributions.

- **Interpretation of DP**: Prior to the definition of demographic parity, the authors mention that demographic parity can be thought of as a stronger version of the so called "four-fifths rule". While this is a reasonable interpretation for DP in the context of classification, this example does not necessarily translate to regression. For example in a regression problem what does it mean for a candidate to "receive a job"?. In [1] (the paper from which this interpretation is pulled) the authors are quite careful to point out that they are referring to DP in the context of classification during this example. This distinction should either be made more precise, or an alternative interpretation should be given. 

     
- **Comparison to other fair models**: In the experiments related to Figure 2, the authors mention that the models they compare two are not quantile regressors so they adapt these models to the quantile regression paradigm. Can the authors give a more percicse description of this adaptation procedure? Is this as simple as altering the labels in the training set, then running the original learning scheme, or is there something else being done? Additionally, the comparisons in this figure are for linear models only do we see similar results for non-linear models?


- **Minor comments/issues**:
        
    - NeurIPS latex format appears to be partially incorrect (no space for authorship and line numbers are missing).
	
    - The first sentence of the abstract “Fairness and impartiality have been among the predominant pursuits in human society” comes across as a slight exaggeration. One could make the argument that fairness is more often than not, treated as an afterthought rather than primary goal, i.e. ensuring systems are just fair enough to be acceptable for the time. As such I think some readers are likely to take issue with this claim as-is. 
        
    - Page 1, second to last line: "categories ... would" should have a common instead of \dots
	
    - Page 2, last paragraph: The first sentence "While current approaches ... " is incomplete.
       
    - Page 3, second paragraph: "$\mathbb{E}$ and $\mathbb{P}$ represent for the generic ..." should be "$\mathbb{E}$ and $\mathbb{P}$ represent the generic ...". Also this sentence starts with math notation. 
    - Page 3, Section 3, first paragraph: "... since then it becomes more and more pervasive...", should be "... since then it has become more pervasive ..."
    - Page 3, Section 3, first paragraph: "As a brief introduction to the quantile regression", this sentence is incomplete and also should not have "the" before "quantile regression".

    - Definition 1: It is more standard to give these definitions as the difference between a sententive group and the population, rather than between two group, e.g. $|M_g - M|$ for all groups $g$, where $M_g$ is the statistical metric applied to $g$ and $M$ is the statical metric applied to the whole population.
         
    - Page 4, last paragraph: "Proposition 1 pronounces", pronounces is out of place, perhaps "implies" would be better.

    - Page 7, Remark 2: "...tails as Corollary 1 in Appendix", missing "the" before "Appendix".

    - Page 6, last paragraph: "...presented in Lemma 3in the ...", missing a space between "3" and "in".

   - Page 8, Government Salary: "The yearly salary for over 200 hundred examples...", I believe this should be "The yearly salary for over 200,000 examples...", based on the size of gov_census.rda in the supplement.

    - Figure 2: It may be better to set the minimum of the $x$-axis to $0$.

     - Inconsistent use of "supplement" and "appendix" throughout the paper.


limitations: The authors have adequately addressed the limitations of their work.
review_title: null

final_score: rating: 4: Borderline reject: Technically solid paper where reasons to reject, e.g., limited evaluation, outweigh reasons to accept, e.g., good evaluation. Please use sparingly.
review_content: summary: In this paper, the authors study the fairness problem in quantile regression settings. Based on the theories from the Wasserstein barycenter problem and two smoothing strategies, the authors propose the CFQP algorithm. They also theoretically analyze the results of the algorithm. Finally, they conduct experiments on four real-world datasets to prove the effectiveness of the algorithm.
strengths_and_weaknesses: **Originality**

\+ The authors propose to study the fairness problem in the quantile regression setting, which is a new problem.

\- The methods proposed by the authors seem a combination of the Wasserstein barycenter problem studied in [10] and two smoothing strategies in [6][7], which makes the contribution marginal.

**Quality**

\- The advantages of the two smoothing strategies are not clear. On the one hand, more discussions about the two smoothing strategies are needed. For example, why the jittering strategy works when the sample sizes of subgroups are small? And what are the second-order efficiency and relative deficiency brought by the kernel smoothing strategy? On the other hand, the authors are encouraged to conduct the ablation study by testing the experimental results if removing either or both of the smoothing strategies.

**Clarity**

\- The paper is not well-written. Several unclear parts need further clarification. There are also several typos to fix.
  1. The three conditions in Section 5 are confusing. What do these conditions mean and are they practical? More discussions are needed.
  2. As mentioned above, the advantages of the two smoothing strategies should be clarified.
  3. In Section 2, it is not clear what the subscript $n+1$ means since there are $n$ training samples. I understand this may refer to the unknown test sample, but please clarify it.
  4. Figure 1 caption: "affacted" --> "affected", "quartile" --> "quantile".

**Significance**

\+ The method is generally sound with both theoretical and experimental results.
questions: See details in the Strengths And Weaknesses section, especially the points in the Originality, Quality, and Clarity parts.
limitations: More limitations are demonstrated in the Strengths and Weaknesses section.
review_title: null

final_score: rating: 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
review_content: summary: This paper studies Demographic Parity in Quantile Regression. The authors extend from previous work in Conformalized Quantile Regression and use optimal transport and functional synchronization techniques to incorporate DP into Quantile Regression. Experiments show that the fair version of Quantile Regression significantly reduces the gap in DP while keeping a good coverage rate and length of the interval.
strengths_and_weaknesses: Pros:
1. Consider the fairness issue in quantile regression is novel. It seems that there is no other paper considering this specific problem.
2. The proposed techniques look sound and make sense to me.
3. Good experimental results. Compared to the non-fair version, the proposed method gets a small gap in terms of KS at a minimal cost of coverage rate and interval length.

Cons:
1. There is limited context characterizing the fairness issue in quantile regression. Probably the authors can describe without a careful design how unfairness can be propagated in quantile regression through a vivid example, and how such unfairness is different from other problems, e.g., ordinary regression.
2. Some key information in experiments is missing. The authors mention in Fig. 2 they compare to 'some state-of-the-art fairness algorithms' but there is no reference and discussion.
questions: 1. It seems that the template is not in submission mode.
limitations: Limitations and negative societal impacts are adequately addressed.
review_title: null

Meta-review: metareview: This work extends the group-level fairness definitions (that were primarily established for supervised learning tasks) to the problem of conformalized quantile regression. A conceptual contribution is to redefine the group-level fairness using the average prediction to quantiles. Based on this adaptation, the authors further developed a postprocessing technique to revise a trained quantile regressor to satisfy the modified fairness definition for quantiles. 

All reviewers acknowledged that the paper is reasonably written and the main idea delivers smoothly. A mixture of theoretical and experimental results are provided

There were some questions raised prior to the rebuttal and were successfully addressed, including clarifying the running time of computing quantiles, comparing them to other fair approaches, and partially misinterpreting DP. The authors are strongly encouraged to incorporate these comments into the final version. 

Some reviewers had remaining questions about the novelty of the paper in light of prior results but the meta reviewer feels the introduced concept of fairness on quantile can be a good addition to the literature and might inspire follow-up works. 
confidence: Less certain
Decision: Accept