The Cost of a Reductions Approach to Private Fair Optimization

We examine a reductions approach to fair optimization and learning where a black-box optimizer is used to learn a fair model for classification or regression [Alabi et al., 2018, Agarwal et al., 2018] and explore the creation of such fair models that adhere to data privacy guarantees (specifically differential privacy). For this approach, we consider two suites of use cases: the first is for optimizing convex performance measures of the confusion matrix (such as $G$-mean and $H$-mean); the second is for satisfying statistical definitions of algorithmic fairness (such as equalized odds, demographic parity, and the gini index of inequality). The reductions approach to fair optimization can be abstracted as the constrained group-objective optimization problem where we aim to optimize an objective that is a function of losses of individual groups, subject to some constraints. We present two differentially private algorithms: an $(\epsilon, 0)$ exponential sampling algorithm and an $(\epsilon, \delta)$ algorithm that uses a linear optimizer to incrementally move toward the best decision. We analyze the privacy and utility guarantees of these empirical risk minimization algorithms. Compared to a previous method for ensuring differential privacy subject to a relaxed form of the equalized odds fairness constraint, the $(\epsilon, \delta)$ differentially private algorithm we present provides asymptotically better sample complexity guarantees. The technique of using an approximate linear optimizer oracle to achieve privacy might be applicable to other problems not considered in this paper. Finally, we show an algorithm-agnostic lower bound on the accuracy of any solution to the problem of $(\epsilon, 0)$ or $(\epsilon, \delta)$ private constrained group-objective optimization.