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 [, 2018, , 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 @math -mean and @math -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 @math exponential sampling algorithm and an @math 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 @math 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 @math or @math private constrained group-objective optimization.
Many commonly used learning algorithms work by iteratively updating an intermediate solution using one or a few data points in each iteration. Analysis of differential privacy for such algorithms often involves ensuring privacy of each step and then reasoning about the cumulative privacy cost of the algorithm. This is enabled by composition theorems for differential privacy that allow releasing of all the intermediate results. In this work, we demonstrate that for contractive iterations, not releasing the intermediate results strongly amplifies the privacy guarantees. We describe several applications of this new analysis technique to solving convex optimization problems via noisy stochastic gradient descent. For example, we demonstrate that a relatively small number of non-private data points from the same distribution can be used to close the gap between private and non-private convex optimization. In addition, we demonstrate that we can achieve guarantees similar to those obtainable using the privacy-amplification-by-sampling technique in several natural settings where that technique cannot be applied.