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We present pairwise fairness metrics for ranking models and regression models that form analogues of statistical fairness notions such as equal opportunity, equal accuracy, and statistical parity. Our pairwise formulation supports both discrete protected groups, and continuous protected attributes. We show that the resulting training problems can be efficiently and effectively solved using existing constrained optimization and robust optimization techniques developed for fair classification. Experiments illustrate the broad applicability and trade-offs of these methods.

A class of distortions termed functional Bregman divergences is defined, which includes squared error and relative entropy. A functional Bregman divergence acts on functions or distributions, and generalizes the standard Bregman divergence for vectors and a previous pointwise Bregman divergence that was defined for functions. A recently published result showed that the mean minimizes the expected Bregman divergence. The new functional definition enables the extension of this result to the continuous case to show that the mean minimizes the expected functional Bregman divergence over a set of functions or distributions. It is shown how this theorem applies to the Bayesian estimation of distributions. Estimation of the uniform distribution from independent and identically drawn samples is used as a case study. OverviewBregman divergences are a useful set of distortion functions that include squared error, relative entropy, logistic loss, Mahalanobis distance, and the Itakura-Saito function. Bregman divergences are popular in statistical estimation and information theory. Analysis using the concept of Bregman divergences has played a key role in recent advances in statistical learning [1][2][3][4][5][6][7][8][9], clustering [10,11], inverse problems [12], maximum entropy estimation [13], and the applicability of the data processing theorem [14]. Recently, it was discovered that the mean is the minimizer of the expected Bregman divergence for a set of d-dimensional points [10,15].In this paper we define a functional Bregman divergence that applies to functions and distributions, and we show that this new definition is equivalent to Bregman divergence applied to vectors. The functional definition generalizes a pointwise Bregman divergence that has been previously defined for measurable functions [7,16], and thus extends the class of distortion functions that are Bregman divergences; see Section 2.1.2 for an example. Most importantly, the functional definition enables one to solve functional minimization problems using standard methods from the calculus of variations; we extend the recent result on the expectation of vector Bregman divergence [10,15] to show that the mean minimizes the expected Bregman divergence for a set of functions or distributions. We show how this theorem links to Bayesian estimation of distributions. For distributions from the exponential family distributions, many popular divergences, such as relative entropy, can be expressed as a (different) Bregman divergence on the exponential distribution parameters. The functional Bregman definition enables stronger results and a more general application.In Section 1 we state a functional definition of the Bregman divergence and give examples for total squared difference, relative entropy, and squared bias. The relationship between the functional definition and previous Bregman definitions 1 2 is established. In Section 2 we present the main theorem: that the expectation of a set of functions minimizes the expected Bregman divergence. In Section ...