We present a data-driven framework for incorporating side information in dynamic optimization under uncertainty. Specifically, our approach uses predictive machine learning methods (such as k-nearest neighbors, kernel regression, and random forests) to weight the relative importance of various data-driven uncertainty sets in a robust optimization formulation. Through a novel measure concentration result for local machine learning methods, we prove that the proposed framework is asymptotically optimal for stochastic dynamic optimization with covariates. We also describe a general-purpose approximation for the proposed framework, based on overlapping linear decision rules, which is computationally tractable and produces high-quality solutions for dynamic problems with many stages. Across a variety of examples in shipment planning, inventory management, and finance, our method achieves improvements of up to 15 over alternatives and requires less than one minute of computation time on problems with twelve stages.
This paper follows a recent body of literature on data-driven optimization under uncertainty in operations research and management science. Much of this work has focused on the paradigm of distributionally robust optimization, in which the optimal solution is that which performs best in expectation over a worst-case probability distribution from an ambiguity set. Motivated by probabilistic guarantees, distributionally robust optimization has found particular applicability in data-driven settings in which the ambiguity set is constructed using historical data, such as @cite_7 @cite_9 @cite_5 @cite_14 . In particular, the final steps in our convergence result () draw heavily from similar techniques from @cite_5 and @cite_10 . In contrast to previous work, this paper develops a new measure concentration result for the weighted empirical distribution () which enables machine learning and covariates to be incorporated into sample robust optimization and Wasserstein-based distributionally robust optimization for the first time.
@cite_14 @cite_7 @cite_9 @cite_5 @cite_10
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We consider stochastic programs where the distribution of the uncertain parameters is only observable through a finite training dataset. Using the Wasserstein metric, we construct a ball in the space of (multivariate and non-discrete) probability distributions centered at the uniform distribution on the training samples, and we seek decisions that perform best in view of the worst-case distribution within this Wasserstein ball. The state-of-the-art methods for solving the resulting distributionally robust optimization problems rely on global optimization techniques, which quickly become computationally excruciating. In this paper we demonstrate that, under mild assumptions, the distributionally robust optimization problems over Wasserstein balls can in fact be reformulated as finite convex programs—in many interesting cases even as tractable linear programs. Leveraging recent measure concentration results, we also show that their solutions enjoy powerful finite-sample performance guarantees. Our theoretical results are exemplified in mean-risk portfolio optimization as well as uncertainty quantification. In this paper, we combine ideas from machine learning (ML) and operations research and management science (OR MS) in developing a framework, along with specific methods, for using data to prescribe decisions in OR MS problems. In a departure from other work on data-driven optimization and reflecting our practical experience with the data available in applications of OR MS, we consider data consisting, not only of observations of quantities with direct effect on costs revenues, such as demand or returns, but predominantly of observations of associated auxiliary quantities. The main problem of interest is a conditional stochastic optimization problem, given imperfect observations, where the joint probability distributions that specify the problem are unknown. We demonstrate that our proposed solution methods are generally applicable to a wide range of decision problems. We prove that they are computationally tractable and asymptotically optimal under mild conditions even when data is not independent and identically distributed (iid) and even for censored observations. As an analogue to the coefficient of determination @math , we develop a metric @math termed the coefficient of prescriptiveness to measure the prescriptive content of data and the efficacy of a policy from an operations perspective. To demonstrate the power of our approach in a real-world setting we study an inventory management problem faced by the distribution arm of an international media conglomerate, which ships an average of 1 billion units per year. We leverage both internal data and public online data harvested from IMDb, Rotten Tomatoes, and Google to prescribe operational decisions that outperform baseline measures. Specifically, the data we collect, leveraged by our methods, accounts for an 88 improvement as measured by our coefficient of prescriptiveness. Distributionally robust optimization is a paradigm for decision making under uncertainty where the uncertain problem data are governed by a probability distribution that is itself subject to uncertainty. The distribution is then assumed to belong to an ambiguity set comprising all distributions that are compatible with the decision maker's prior information. In this paper, we propose a unifying framework for modeling and solving distributionally robust optimization problems. We introduce standardized ambiguity sets that contain all distributions with prescribed conic representable confidence sets and with mean values residing on an affine manifold. These ambiguity sets are highly expressive and encompass many ambiguity sets from the recent literature as special cases. They also allow us to characterize distributional families in terms of several classical and or robust statistical indicators that have not yet been studied in the context of robust optimization. We determine conditions under which distributionally robust optimization problems based on our standardized ambiguity sets are computationally tractable. We also provide tractable conservative approximations for problems that violate these conditions. Motivated by data-driven decision making and sampling problems, we investigate probabilistic interpretations of robust optimization (RO). We establish a connection between RO and distributionally robust stochastic programming (DRSP), showing that the solution to any RO problem is also a solution to a DRSP problem. Specifically, we consider the case where multiple uncertain parameters belong to the same fixed dimensional space and find the set of distributions of the equivalent DRSP problem. The equivalence we derive enables us to construct RO formulations for sampled problems (as in stochastic programming and machine learning) that are statistically consistent, even when the original sampled problem is not. In the process, this provides a systematic approach for tuning the uncertainty set. The equivalence further provides a probabilistic explanation for the common shrinkage heuristic, where the uncertainty set used in an RO problem is a shrunken version of the original uncertainty set.