Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: The paper follows the axioms of cardinal welfare, in conjunction with an introduced notion of malfare which quantifies the amount of societal harm rather than welfare which quantifies overall benefit. This allows the authors to cast fair machine learning problems as risk minimization problems, where they aim to minimize the malfare function. The function, which follows the set of axioms, is a power mean function, defined over the set of groups of interest, weighing the loss on each, to output a combined, single, value. Casting the function as malfare allows them in turn to use a specific characteristic of the proposed formulation – contraction – which is crucial in establishing their generalization result, through bounding the values of the base loss function, and the use of Rademacher complexity. Finally, they show that for their notion of fair learning through the malfare function, in the agnostic case and under standard convex optimization assumptions, it is possible to efficiently learn through the subgradient method. The authors also perform an experimental evaluation of their suggested approach on the adult dataset.
main_review: I think that the paper does a nice job of formalizing and reasoning for the fairness objective they advocate for, and further – by showing that their formulation allows for generalization through the contraction property, which I thought was a nice observation. In terms of practicality, I think the proposed approach makes a good contribution for cases where we believe fairness should be measured in terms of accuracy in the different groups. In such cases, the algorithm they propose should be considered as one alternative in the fairness toolkit. In terms of overall novelty, my assessment is that the paper makes a moderate contribution. I think that the main theoretical contribution is through the observation that certain loss aggregation functions may allow for generalization using the contraction attribute, which leads to uniform sample complexity bounds. 

A main weakness that I see in the approach is that it weighs loss, in its general form, over the different groups. We have already seen that in many of the settings where fairness is a concern, false positives and false negatives can have entirely different effects on the individual (beneficial or detrimental), thus bundling them together and minimizing the overall loss may possibly result in an accumulation of mistakes of different types on different groups, which by itself may be problematic.

The presentation of the paper can be slightly improved, especially section 2, where the flow I thought was a bit confusing.
The submission appears to be technically sound, although I have only glanced at proofs in the appendix. The paper cites most of the relevant prior work, but could add a short discussion regarding existing group-based fairness techniques. 

Questions to authors:
1. Can the suggested approach be extended to also take into account the types of losses (false positives, false negatives)? How would your approach handle cases where the two types of losses have very different implications?
2. Assuming access to an oracle for standard 0-1 ERM in the agnostic case, would it be possible to harness this “building block” to optimize for your proposed malfare objective?
limitations_and_societal_impact: I believe that the authors have addressed the limitations of their work properly. I would appreciate further discussion regarding their suggested objective of fairness and in which cases the authors think it should be used.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: The paper is concerned with welfare-theoretic fair machine learning. Instead of enforcing statistical parity as a constraint, the paper instead casts fair learning as malfare minimization. Based on the axioms of cardinal welfarism, axioms for malfare are defined. Theoretical guarantees based on learning theory are established for malfare minimization. 
main_review: -I think the paper addresses an important topic and the way it casts fair machine learning is interesting and important. 

-I did not find the paper easy to read and it felt that too many results were packed in a limited number of pages. 

-It seems more natural to show that the axioms imply Th 2.5 then introduce definition 2.3 for the power mean. 

-Line 129: doesn't axiom 6 imply 5. 

-Axiom 4 (Def 2.2): the symbols should have some explanation. 

-Def 2.3: I think there is a typo for M_{\infty} , it should be S_i not S_g also brackets for the set. 

-The paper mentions that we cannot simply define utility as negative loss, but I did not find a strong argument in the paper against that. Why can't a naive transformation that shifts the utility to positive values work? 

-The  sentiment vector at the end takes only a fixed form which is the risk for each population. I assume that it not  possible to use learning theory guarantees without the sentiment taking on this form. But isn't there now a loss of generality to the formulation, couldn't different groups have different ways for evaluating sentiment just the way two individuals assign a different utility to the same item? 

-I think the representation would improve if at least in the main body, the brief (informal) statements of theorem are given instead. 

-I think the simplex notation is usually used for probability vectors, so w \in \Delta^{g-1}




limitations_and_societal_impact: I think the impacts were adequately discussed.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: This paper investigates an important challenge in welfare-centric fair ML, that is, how to connect the standard loss minimization view of ML with the utility maximization view of welfare-based fairness. The authors define malfare to measure societal harm: malfare is a complementary metric of welfare, and it is justified with axioms of cardinal welfare (and extensions to malfare). Utilizing the malfare metric, the authors cast fair ML that seeks equitable risk distribution to groups as malfare minimization over the expected losses of all groups. They further build up on the malfare minimization framework to extend PAC learning to fair-PAC (FPAC) learning, which is proposed as a formal notion of fair learnability. Their analysis of FPAC learning shows that standard PAC-learners may be converted to FPAC-learners under some conditions, and these connections further yield theoretical results on the statistical and computational learnability with FPAC learners.
main_review: This paper studies a new problem in fair machine learning. The adopted methodology is a novel combination of the cardinal welfare theory from economics, welfare-centric methods in fair ML and the well-established PAC learning theory. In my opinion, the results are significant as they fill important theoretical gaps in fair ML. The paper focuses on welfare-centric fair ML, which is a relatively new thread of fair ML that uses a social welfare function to represent fairness. Compared to the dominant parity based fairness notions used in ML, the welfare-centric view could provide a broader perspective on fairness driven by the well-beings of stakeholders and represent a wide range of principled fairness concepts. In existing literature, these potential advantages tend to be undermined by the lack of theoretical justification from the learning aspect, that is, there are no general learning performance guarantees on welfare-centric fair ML. This paper contributes FPAC learning as one possible framework to offer solid theoretical justifications, thus, in my opinion, greatly expands the potentials of welfare-metric fair ML.

The paper is well organized and clearly written. The authors provide sufficient introduction of the background and motivations of their task by citing related works. They also gave clear explanations of what distinguish the work from previous research. The presented technical results are solid, and the proofs and derivations are correct as far as I could verify. 

I find the paper well written for the most part, and I only have a few minor questions and suggestions.
1. From my understanding, the power mean formulation puts welfare and malfare on a spectrum from equality to inequality controlled by the power $p$. Do the authors think it is reasonable to interpret malfare as inequality measure? If so, I think it might be interesting to compare the FPAC framework or Empirical malfare minimization (EMM) with the results in [24]. On a related note, a potential reference to include is ‘An Axiomatic Theory of Fairness in Network Resource Allocation’ by Lan et al.: this paper adopts a similar axiomatic view on fairness and derive a mean-type format for fairness measures; if I recall correctly, the axioms they consider coincide with Axiom 1-5 used in the submission.
2. In the appendix of [11], the authors discussed a mapping between loss minimization to social welfare maximization. Although their welfare function definition is different from the one used in the submission, I think it is worth mentioning this result in the ‘Related work’.
3. The authors highlighted that EMM is more fitting in ML contexts since there often is no clearly neutral way to define utility based on loss. While I agree with this statement, I think in many cases the decision makers designing the learning task would have the expertise or authority to know what utility functions to consider. Certainly the utility definitions could be context dependent, but one may argue that it is desirably so to capture context-specific welfare considerations. Can the authors include brief examples of when utility definitions would be hard to define? 
4. Can the authors provide some intuition on which of the welfare/malfare axioms and welfare/malfare function properties appear to be indispensable for developing the theoretically sound FPAC learning framework? Along similar line, how hard will it be to extend the analysis to other welfare measures?

Minor comments:
1. Figure 1: the test hinge risk is shown as higher than the training hinge risk reversed pattern for the ‘Black’ group, while the other groups all have the reverse pattern. Is this figure correct?
2. Theorem 2.5 Condition 2: is the summation term $\sum_{i=1}^g f_p(\mathcal{S}_i)$ missing the weight vector?
3. Appendix proof of Theorem 2.4: line 558 and 559, convexity and concavity appear to be mixed up. 
limitations_and_societal_impact: The authors acknowledged the limitations of their work in several aspects, such as the condition needed to prove Theorem 3.9, the difficulty in extending the analysis to fair welfare maximization. I think the limitations are adequately discussed, and as these limitations are non-trivial to address, it is reasonable to leave as future work as indicated by the authors. 

As the paper studies fair ML, it naturally seeks to address potential negative societal impacts of ML. One aspect that may be useful to discuss is the welfare implication of EMM in practice. The authors state that EMM should be used when there is no explicit way to convert loss to utility, nevertheless, one may still wonder how the learner trained with EMM performs with respect to different utility definitions. For instance, if the predictions about whether income is above or below $50k per year are used to guide certain decisions, equitable risk distribution in the predictor does not necessarily guarantee desirable utility distribution in these decisions.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: The authors consider malfaire metrics,
measuring overall societal harm, with axiomatic justification via
standard axioms of welfare, and cast fair ML as malfare minimization
over the risk values (expected losses) of each group. this is not 
equivalent to simply defining utility as negative loss and maximizing welfare.

main_review: The paper tells us that malfare metrics have different properties to welfare 
metrics and there may be applications where minimizing malfare is possible 
but maximising welfare is not. I am not sure whether this is common 
or whether it just occurs for pathological situations. Nevertheless, 
it does seem important to know this. It would be useful (if possible) to give a 
brief mathematical explanation for the difference between welfare and malfare.
Is the only important difference possible Lipschitz-discontinuity or are there
others?

Results concerning PAC learning are generalised to the case of 
F(for fair)PAC learning (which takes into account malfare).
This is clearly solid work backed up by proofs in the supplementary material.

I just have one serious niggling doubt: axiom 6 is presented without any 
obvious justification. All other axioms have appropriate citations showing that 
they are standard (or at least have been previously proposed). 
Axiom 6 is essential to show that the malfare function must be a power means function. 
(One could also probably find applications where some of the other axioms 
maybe do not apply, but they do seem to be less contentious
than axiom 6). Axiom 6 is a strengthening of a traditional cardinal welfare axiom,
but is this strengthening really justified?

Figure 1: Is there a simple summary of what this figure is telling us? 
I think that results improve (in terms of fairness) as p increases, but the 
important question is why? Is there a one-line explanation for this?

line 320: This is a "generalisation" in the sense that the definition is slacker, 
which is not necessarily a good thing: the mean may converge when the frequency does not.
Is this a possibly important difference or not? In Theorem 3.9 you mean generalised UC
not UC, so don't put parentheses around "generalised".

Minor points:
- In certain figs (in the supplementary material), eg race-01, 
the y-axis has 1 several times which is clearly a typo.
- line 105: write pre-existing instead of preëxisting
- line 163: multiplicative linearity is axiom 6 in your list, not 5.
- line 164: typo?: "nats or of cross-entropy loss"
- line 167: "milquetoast" seems to be an inappropriate adjective. Perhaps 'weak' would be better.
- Figure 1 (legend): what happens if b_i is close to 0?
- Def. 3.3 use words instead of \forall, \exists
- line 279: acronym Agn not defined.
- line 291: careful with line numbers in middle of lines
- Def. 3.7 perhaps give a reference for Uniform Convergence.
- Fig. 2: what do dashed implications mean?
limitations_and_societal_impact: Yes.
review_title: null

Meta-review: The reviewers’ overall assessment of the paper was that (a) it proposes an interesting, relatively well-justified formulation of fairness through the concept of malware minimization. (2) The ensuing learnability analysis is non-trivial and the proposed algorithms can be practically informative.  Reviewers made several suggestions to the authors to improve the flow and exposition (in particular, it is important that the authors expand their discussion of the proposed axioms, in particular, axiom 6, in the main body of the paper). Assuming that the authors will reflect those changes in their next revision of the paper, I recommend acceptance.
Decision: Accept (Poster)