It is widely accepted that biased data leads to biased and thus potentially
unfair models. Therefore, several measures for bias in data and model
predictions have been proposed, as well as bias mitigation techniques whose aim
is to learn models that are fair by design. Despite the myriad of mitigation
techniques developed in the past decade, however, it is still poorly understood
under what circumstances which methods work. Recently, Wick et al. showed, with
experiments on synthetic data, that there exist situations in which bias
mitigation techniques lead to more accurate models when measured on unbiased
data. Nevertheless, in the absence of a thorough mathematical analysis, it
remains unclear which techniques are effective under what circumstances. We
propose to address this problem by establishing relationships between the type
of bias and the effectiveness of a mitigation technique, where we categorize
the mitigation techniques by the bias measure they optimize. In this paper we
illustrate this principle for label and selection bias on the one hand, and
demographic parity and ``We're All Equal'' on the other hand. Our theoretical
analysis allows to explain the results of Wick et al. and we also show that
there are situations where minimizing fairness measures does not result in the
fairest possible distribution.

存在情况下，偏见缓解技术能够更准确的模型，在无偏数据上测量时。我们通过对偏见类型和缓解技术的效果建立关系来解决这一问题，并用理论分析来解释实验结果，并展示最小化公平度量并不一定会产生最公平的分布。

如何做到公平？对标签和选择偏差的研究

How to be fair? A study of label and selection bias

We present an efficient stochastic algorithm (RSG+) for canonical correlation
analysis (CCA) using a reparametrization of the projection matrices. We show
how this reparametrization (into structured matrices), simple in hindsight,
directly presents an opportunity to repurpose/adjust mature techniques for
numerical optimization on Riemannian manifolds. Our developments nicely
complement existing methods for this problem which either require $O(d^3)$ time
complexity per iteration with $O(\frac{1}{\sqrt{t}})$ convergence rate (where
$d$ is the dimensionality) or only extract the top $1$ component with
$O(\frac{1}{t})$ convergence rate. In contrast, our algorithm offers a strict
improvement for this classical problem: it achieves $O(d^2k)$ runtime
complexity per iteration for extracting the top $k$ canonical components with
$O(\frac{1}{t})$ convergence rate. While the paper primarily focuses on the
formulation and technical analysis of its properties, our experiments show that
the empirical behavior on common datasets is quite promising. We also explore a
potential application in training fair models where the label of protected
attribute is missing or otherwise unavailable.

我们提出了一种有效的随机算法（RSG+），使用投影矩阵的重新参数化来进行规范相关分析（CCA），并展示了这种重新参数化如何直接提出机会来重新设计和调整用于 Riemann 流形上数值优化的成熟技术。