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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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. 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 programsin 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. As opposed to standard empirical risk minimization (ERM), distributionally robust optimization aims to minimize the worst-case risk over a larger ambiguity set containing the original empirical distribution of the training data. In this work, we describe a minimax framework for statistical learning with ambiguity sets given by balls in Wasserstein space. In particular, we prove generalization bounds that involve the covering number properties of the original ERM problem. As an illustrative example, we provide generalization guarantees for transport-based domain adaptation problems where the Wasserstein distance between the source and target domain distributions can be reliably estimated from unlabeled samples. This technical note studies Markov decision processes under parameter uncertainty. We adapt the distributionally robust optimization framework, assume that the uncertain parameters are random variables following an unknown distribution, and seek the strategy which maximizes the expected performance under the most adversarial distribution. In particular, we generalize a previous study [1] which concentrates on distribution sets with very special structure to a considerably more generic class of distribution sets, and show that the optimal strategy can be obtained efficiently under mild technical conditions. This significantly extends the applicability of distributionally robust MDPs by incorporating probabilistic information of uncertainty in a more flexible way. We develop efficient solution methods for a robust empirical risk minimization problem designed to give calibrated confidence intervals on performance and provide optimal tradeoffs between bias and variance. Our methods apply to distributionally robust optimization problems proposed by Ben-, which put more weight on observations inducing high loss via a worst-case approach over a non-parametric uncertainty set on the underlying data distribution. Our algorithm solves the resulting minimax problems with nearly the same computational cost of stochastic gradient descent through the use of several carefully designed data structures. For a sample of size n, the per-iteration cost of our method scales as O(log n), which allows us to give optimality certificates that distributionally robust optimization provides at little extra cost compared to empirical risk minimization and stochastic gradient methods.