Paper:
We propose a novel algorithm for learning fair representations that can simultaneously mitigate two notions of disparity among different demographic subgroups. Two key components underpinning the design of our algorithm are balanced error rate and conditional alignment of representations. We show how these two components contribute to ensuring accuracy parity and equalized false-positive and false-negative rates across groups without impacting demographic parity. Furthermore, we also demonstrate both in theory and on two real-world experiments that the proposed algorithm leads to a better utility-fairness trade-off on balanced datasets compared with existing algorithms on learning fair representations.

Review:
Update:
Thanks for providing additional results on the Adult dataset. I have increased my score. However I'd be nice to also see the balanced accuracy (i.e. sum of TPRs for each class divided by 2) results and compare to baseline trained with oversampling or re-weighted loss.
I would suggest authors to add more extensive comparisons to other methods using Adult dataset. There are quite a lot of papers in the fairness literature that experiment with the Adult dataset. They focus on different metrics and there is probably no method that is uniformly the best. Your paper demonstrated that your approach can succeed in achieving accuracy parity, but it would be good to also show tradeoffs with other metrics (in addition to DP). I was able to do some back-of-the-envelop calculations and your results seem fine, but a clear comparison would be good.
Below are some examples of the papers studying Adult dataset from the fairness angle:
[1] Mitigating unwanted biases with adversarial learning. Zhang et al., 2018. (already cited, but no comparison)
[2] What’s in a Name? Reducing Bias in Bios without Access to Protected Attributes. Romanov et al., 2019.
[3] Learning fair predictors with Sensitive Subspace Robustness. Yurochkin et al., 2019.
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This paper proposes an adversarial representation learning approach. The key difference with prior work is that the objective function is built around balanced error rates, one for classes, that is eventually used for classification, and two adversarial for predicting each of the protected attributes. Authors argue that proposed approach can simultaneously achieve accuracy parity and equalized odds.
The notion of accuracy parity does not seem to be very meaningful. For example, predicting uniformly at random seems like an intuitively fair classifier with EO gap 0 and DP gap 0. However it will not necessarily have error gap of 0 (i.e. satisfy accuracy parity), making me wonder if the notion of accuracy parity makes much sense.
I am not really sure what is the Err0 + Err1 metric used in Figures 1 and 2. Is it not normalized and can vary between 0 and 2? In which case it seems counterintuitive for performance quantification. If it is normalized, then results on COMPAS do not make sense. Err0 + Err1 of all methods is above 60%, which is worse than predicting uniformly at random.
Please report your TPRs for classes grouped by protected attribute when reporting the results in the context of group fairness. Further for Adult dataset, reporting balanced TPR as a measure of accuracy seems to make more sense given the class imbalance.