Optimistic Robust Optimization With Applications To Machine Learning

Robust Optimization has traditionally taken a pessimistic, or worst-case viewpoint of uncertainty which is motivated by a desire to find sets of optimal policies that maintain feasibility under a variety of operating conditions. In this paper, we explore an optimistic, or best-case view of uncertainty and show that it can be a fruitful approach. We show that these techniques can be used to address a wide variety of problems. First, we apply our methods in the context of robust linear programming, providing a method for reducing conservatism in intuitive ways that encode economically realistic modeling assumptions. Second, we look at problems in machine learning and find that this approach is strongly connected to the existing literature. Specifically, we provide a new interpretation for popular sparsity inducing non-convex regularization schemes. Additionally, we show that successful approaches for dealing with outliers and noise can be interpreted as optimistic robust optimization problems. Although many of the problems resulting from our approach are non-convex, we find that DCA or DCA-like optimization approaches can be intuitive and efficient

Bootstrap Robust Prescriptive Analytics

We address the problem of prescribing an optimal decision in a framework where its cost depends on uncertain problem parameters $Y$ that need to be learned from data. Earlier work by Bertsimas and Kallus (2014) transforms classical machine learning methods that merely predict $Y$ from supervised training data $[(x_1, y_1), \dots, (x_n, y_n)]$ into prescriptive methods taking optimal decisions specific to a particular covariate context $X=\bar x$. Their prescriptive methods factor in additional observed contextual information on a potentially large number of covariates $X=\bar x$ to take context specific actions $z(\bar x)$ which are superior to any static decision $z$. Any naive use of limited training data may, however, lead to gullible decisions over-calibrated to one particular data set. In this paper, we borrow ideas from distributionally robust optimization and the statistical bootstrap of Efron (1982) to propose two novel prescriptive methods based on (nw) Nadaraya-Watson and (nn) nearest-neighbors learning which safeguard against overfitting and lead to improved out-of-sample performance. Both resulting robust prescriptive methods reduce to tractable convex optimization problems and enjoy a limited disappointment on bootstrap data. We illustrate the data-driven decision-making framework and our novel robustness notion on a small news vendor problem as well as a small portfolio allocation problem

Sinkhorn Distributionally Robust Optimization

We study distributionally robust optimization (DRO) with Sinkhorn distance -- a variant of Wasserstein distance based on entropic regularization. We derive convex programming dual reformulation for a general nominal distribution. Compared with Wasserstein DRO, it is computationally tractable for a larger class of loss functions, and its worst-case distribution is more reasonable for practical applications. To solve the dual reformulation, we develop a stochastic mirror descent algorithm using biased gradient oracles and analyze its convergence rate. Finally, we provide numerical examples using synthetic and real data to demonstrate its superior performance.Comment: 56 pages, 8 figure