Differentially Private Simple Linear Regression

Economics and social science research often require analyzing datasets of sensitive personal information at fine granularity, with models fit to small subsets of the data. Unfortunately, such fine-grained analysis can easily reveal sensitive individual information. We study algorithms for simple linear regression that satisfy differential privacy, a constraint which guarantees that an algorithms output reveals little about any individual input data record, even to an attacker with arbitrary side information about the dataset. We consider the design of differentially private algorithms for simple linear regression for small datasets, with tens to hundreds of datapoints, which is a particularly challenging regime for differential privacy. Focusing on a particular application to small-area analysis in economics research, we study the performance of a spectrum of algorithms we adapt to the setting. We identify key factors that affect their performance, showing through a range of experiments that algorithms based on robust estimators (in particular, the Theil-Sen estimator) perform well on the smallest datasets, but that other more standard algorithms do better as the dataset size increases.

The Cost of a Reductions Approach to Private Fair Optimization

Through the lens of information-theoretic reductions, 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. Quantifying the complexity, both statistically and computationally, of making such models satisfy the rigorous definition of differential privacy is our end goal. We resolve a few open questions and show applicability to fair machine learning, hypothesis testing, and to optimizing non-standard measures of classification loss. Furthermore, our sample complexity bounds are tight amongst all strategies that jointly minimize a composition of functions. 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 give the first polynomial-time algorithms to solve the problem with $(epsilon, 0)$ or $(epsilon, delta)$ differential privacy guarantees when defined on a convex decision set (for example, the $ell_P$ unit ball) with convex constraints and losses. Accompanying information-theoretic lower bounds for the problem are presented. In addition, 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, resulting in an exponential improvement in certain parameter regimes. We introduce a class of bounded divergence linear optimizers, which could be of independent interest, and specialize to pure and approximate differential privacy.

Learning to Prune: Speeding up Repeated Computations

It is common to encounter situations where one must solve a sequence of similar computational problems. Running a standard algorithm with worst-case runtime guarantees on each instance will fail to take advantage of valuable structure shared across the problem instances. For example, when a commuter drives from work to home, there are typically only a handful of routes that will ever be the shortest path. A naive algorithm that does not exploit this common structure may spend most of its time checking roads that will never be in the shortest path. More generally, we can often ignore large swaths of the search space that will likely never contain an optimal solution. We present an algorithm that learns to maximally prune the search space on repeated computations, thereby reducing runtime while provably outputting the correct solution each period with high probability. Our algorithm employs a simple explore-exploit technique resembling those used in online algorithms, though our setting is quite different. We prove that, with respect to our model of pruning search spaces, our approach is optimal up to constant factors. Finally, we illustrate the applicability of our model and algorithm to three classic problems: shortest-path routing, string search, and linear programming. We present experiments confirming that our simple algorithm is effective at significantly reducing the runtime of solving repeated computations.

Unleashing Linear Optimizers for Group-Fair Learning and Optimization

Most systems and learning algorithms optimize average performance or average loss -- one reason being computational complexity. However, many objectives of practical interest are more complex than simply average loss. This arises, for example, when balancing performance or loss with fairness across people. We prove that, from a computational perspective, optimizing arbitrary objectives that take into account performance over a small number of groups is not significantly harder to optimize than average performance. Our main result is a polynomial-time reduction that uses a linear optimizer to optimize an arbitrary (Lipschitz continuous) function of performance over a (constant) number of possibly-overlapping groups. This includes fairness objectives over small numbers of groups, and we further point out that other existing notions of fairness such as individual fairness can be cast as convex optimization and hence more standard convex techniques can be used. Beyond learning, our approach applies to multi-objective optimization, more generally.

Learning Certifiably Optimal Rule Lists for Categorical Data

We present the design and implementation of a custom discrete optimization technique for building rule lists over a categorical feature space. Our algorithm produces rule lists with optimal training performance, according to the regularized empirical risk, with a certificate of optimality. By leveraging algorithmic bounds, efficient data structures, and computational reuse, we achieve several orders of magnitude speedup in time and a massive reduction of memory consumption. We demonstrate that our approach produces optimal rule lists on practical problems in seconds. Our results indicate that it is possible to construct optimal sparse rule lists that are approximately as accurate as the COMPAS proprietary risk prediction tool on data from Broward County, Florida, but that are completely interpretable. This framework is a novel alternative to CART and other decision tree methods for interpretable modeling.