It's All in the Mix: Wasserstein Machine Learning with Mixed Features

Problem definition: The recent advent of data-driven and end-to-end decision-making across different areas of operations management has led to an ever closer integration of prediction models from machine learning and optimization models from operations research. A key challenge in this context is the presence of estimation errors in the prediction models, which tend to be amplified by the subsequent optimization model -- a phenomenon that is often referred to as the Optimizer's Curse or the Error-Maximization Effect of Optimization. Methodology/results: A contemporary approach to combat such estimation errors is offered by distributionally robust problem formulations that consider all data-generating distributions close to the empirical distribution derived from historical samples, where `closeness' is determined by the Wasserstein distance. While those techniques show significant promise in problems where all input features are continuous, they scale exponentially when binary and/or categorical features are present. This paper demonstrates that such mixed-feature problems can indeed be solved in polynomial time. We present a practically efficient algorithm to solve mixed-feature problems, and we compare our method against alternative techniques both theoretically and empirically on standard benchmark instances. Managerial implications: Data-driven operations management problems often involve prediction models with discrete features. We develop and analyze a methodology that faithfully accounts for the presence of discrete features, and we demonstrate that our approach can significantly outperform existing methods that are agnostic to the presence of discrete features, both theoretically and across standard benchmark instances.

Differential Privacy via Distributionally Robust Optimization

In recent years, differential privacy has emerged as the de facto standard for sharing statistics of datasets while limiting the disclosure of private information about the involved individuals. This is achieved by randomly perturbing the statistics to be published, which in turn leads to a privacy-accuracy trade-off: larger perturbations provide stronger privacy guarantees, but they result in less accurate statistics that offer lower utility to the recipients. Of particular interest are therefore optimal mechanisms that provide the highest accuracy for a pre-selected level of privacy. To date, work in this area has focused on specifying families of perturbations a priori and subsequently proving their asymptotic and/or best-in-class optimality. In this paper, we develop a class of mechanisms that enjoy non-asymptotic and unconditional optimality guarantees. To this end, we formulate the mechanism design problem as an infinite-dimensional distributionally robust optimization problem. We show that the problem affords a strong dual, and we exploit this duality to develop converging hierarchies of finite-dimensional upper and lower bounding problems. Our upper (primal) bounds correspond to implementable perturbations whose suboptimality can be bounded by our lower (dual) bounds. Both bounding problems can be solved within seconds via cutting plane techniques that exploit the inherent problem structure. Our numerical experiments demonstrate that our perturbations can outperform the previously best results from the literature on artificial as well as standard benchmark problems.