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. 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 paper introduces a learning-based robust control algorithm that provides robust stability and performance guarantees during learning. The approach uses Gaussian process (GP) regression based on data gathered during operation to update an initial model of the system and to gradually decrease the uncertainty related to this model. Embedding this data-based update scheme in a robust control framework guarantees stability during the learning process. Traditional robust control approaches have not considered online adaptation of the model and its uncertainty before. As a result, their controllers do not improve performance during operation. Typical machine learning algorithms that have achieved similar high-performance behavior by adapting the model and controller online do not provide the guarantees presented in this paper. In particular, this paper considers a stabilization task, linearizes the nonlinear, GP-based model around a desired operating point, and solves a convex optimization problem to obtain a linear robust controller. The resulting performance improvements due to the learning-based controller are demonstrated in experiments on a quadrotor vehicle. Self-paced learning and hard example mining re-weight training instances to improve learning accuracy. This paper presents two improved alternatives based on lightweight estimates of sample uncertainty in stochastic gradient descent (SGD): the variance in predicted probability of the correct class across iterations of mini-batch SGD, and the proximity of the correct class probability to the decision threshold. Extensive experimental results on six datasets show that our methods reliably improve accuracy in various network architectures, including additional gains on top of other popular training techniques, such as residual learning, momentum, ADAM, batch normalization, dropout, and distillation. Self-paced learning and hard example mining re-weight training instances to improve learning accuracy. This paper presents two improved alternatives based on lightweight estimates of sample uncertainty in stochastic gradient descent (SGD): the variance in predicted probability of the correct class across iterations of mini-batch SGD, and the proximity of the correct class probability to the decision threshold. Extensive experimental results on six datasets show that our methods reliably improve accuracy in various network architectures, including additional gains on top of other popular training techniques, such as residual learning, momentum, ADAM, batch normalization, dropout, and distillation. Topic models, such as Latent Dirichlet Allocation (LDA), posit that documents are drawn from admixtures of distributions over words, known as topics. The inference problem of recovering topics from admixtures, is NP-hard. Assuming separability, a strong assumption, [4] gave the first provable algorithm for inference. For LDA model, [6] gave a provable algorithm using tensor-methods. But [4,6] do not learn topic vectors with bounded @math error (a natural measure for probability vectors). Our aim is to develop a model which makes intuitive and empirically supported assumptions and to design an algorithm with natural, simple components such as SVD, which provably solves the inference problem for the model with bounded @math error. A topic in LDA and other models is essentially characterized by a group of co-occurring words. Motivated by this, we introduce topic specific Catchwords, group of words which occur with strictly greater frequency in a topic than any other topic individually and are required to have high frequency together rather than individually. A major contribution of the paper is to show that under this more realistic assumption, which is empirically verified on real corpora, a singular value decomposition (SVD) based algorithm with a crucial pre-processing step of thresholding, can provably recover the topics from a collection of documents drawn from Dominant admixtures. Dominant admixtures are convex combination of distributions in which one distribution has a significantly higher contribution than others. Apart from the simplicity of the algorithm, the sample complexity has near optimal dependence on @math , the lowest probability that a topic is dominant, and is better than [4]. Empirical evidence shows that on several real world corpora, both Catchwords and Dominant admixture assumptions hold and the proposed algorithm substantially outperforms the state of the art [5]. The vast majority of theoretical results in machine learning and statistics assume that the training data is a reliable reflection of the phenomena to be learned. Similarly, most learning techniques used in practice are brittle to the presence of large amounts of biased or malicious data. Motivated by this, we consider two frameworks for studying estimation, learning, and optimization in the presence of significant fractions of arbitrary data. The first framework, list-decodable learning, asks whether it is possible to return a list of answers such that at least one is accurate. For example, given a dataset of n points for which an unknown subset of I± n points are drawn from a distribution of interest, and no assumptions are made about the remaining (1âI±)n points, is it possible to return a list of poly(1 I±) answers? The second framework, which we term the semi-verified model, asks whether a small dataset of trusted data (drawn from the distribution in question) can be used to extract accurate information from a much larger but untrusted dataset (of which only an I±-fraction is drawn from the distribution). We show strong positive results in both settings, and provide an algorithm for robust learning in a very general stochastic optimization setting. This result has immediate implications for robustly estimating the mean of distributions with bounded second moments, robustly learning mixtures of such distributions, and robustly finding planted partitions in random graphs in which significant portions of the graph have been perturbed by an adversary. We study unsupervised generative modeling in terms of the optimal transport (OT) problem between true (but unknown) data distribution @math and the latent variable model distribution @math . We show that the OT problem can be equivalently written in terms of probabilistic encoders, which are constrained to match the posterior and prior distributions over the latent space. When relaxed, this constrained optimization problem leads to a penalized optimal transport (POT) objective, which can be efficiently minimized using stochastic gradient descent by sampling from @math and @math . We show that POT for the 2-Wasserstein distance coincides with the objective heuristically employed in adversarial auto-encoders (AAE) (, 2016), which provides the first theoretical justification for AAEs known to the authors. We also compare POT to other popular techniques like variational auto-encoders (VAE) (Kingma and Welling, 2014). Our theoretical results include (a) a better understanding of the commonly observed blurriness of images generated by VAEs, and (b) establishing duality between Wasserstein GAN (Arjovsky and Bottou, 2017) and POT for the 1-Wasserstein distance. We propose an approach to address data uncertainty for discrete optimization and network flow problems that allows controlling the degree of conservatism of the solution, and is computationally tractable both practically and theoretically. In particular, when both the cost coefficients and the data in the constraints of an integer programming problem are subject to uncertainty, we propose a robust integer programming problem of moderately larger size that allows controlling the degree of conservatism of the solution in terms of probabilistic bounds on constraint violation. When only the cost coefficients are subject to uncertainty and the problem is a 0−1 discrete optimization problem on n variables, then we solve the robust counterpart by solving at most n+1 instances of the original problem. Thus, the robust counterpart of a polynomially solvable 0−1 discrete optimization problem remains polynomially solvable. In particular, robust matching, spanning tree, shortest path, matroid intersection, etc. are polynomially solvable. We also show that the robust counterpart of an NP-hard α-approximable 0−1 discrete optimization problem, remains α-approximable. Finally, we propose an algorithm for robust network flows that solves the robust counterpart by solving a polynomial number of nominal minimum cost flow problems in a modified network. We present new algorithms to compute the mean of a set of empirical probability measures under the optimal transport metric. This mean, known as the Wasserstein barycenter, is the measure that minimizes the sum of its Wasserstein distances to each element in that set. We propose two original algorithms to compute Wasserstein barycenters that build upon the subgradient method. A direct implementation of these algorithms is, however, too costly because it would require the repeated resolution of large primal and dual optimal transport problems to compute subgradients. Extending the work of Cuturi (2013), we propose to smooth the Wasserstein distance used in the definition of Wasserstein barycenters with an entropic regularizer and recover in doing so a strictly convex objective whose gradients can be computed for a considerably cheaper computational cost using matrix scaling algorithms. We use these algorithms to visualize a large family of images and to solve a constrained clustering problem. We investigate a data-driven approach to two-stage stochastic linear optimization in which an uncertainty set is constructed around each data point. We propose an approximation algorithm for these sample robust optimization problems by optimizing a separate linear decision rule for each uncertainty set. We show that the proposed algorithm combines the asymptotic optimality and scalability of the sample average approximation while simultaneously offering improved out-of-sample performance guarantees. The practical value of our method is demonstrated in network inventory management and hospital scheduling. A common approach in coping with multiperiod optimization problems under uncertainty where statistical information is not really enough to support a stochastic programming model, has been to set up and analyze a number of scenarios. The aim then is to identify trends and essential features on which a robust decision policy can be based. This paper develops for the first time a rigorous algorithmic procedure for determining such a policy in response to any weighting of the scenarios. The scenarios are bundled at various levels to reflect the availability of information, and iterative adjustments are made to the decision policy to adapt to this structure and remove the dependence on hindsight. We consider sequential decision problems under uncertainty, where we seek to optimize an unknown function from noisy samples. This requires balancing exploration (learning about the objective) and exploitation (localizing the maximum), a problem well-studied in the multiarmed bandit literature. In many applications, however, we require that the sampled function values exceed some prespecified "safety" threshold, a requirement that existing algorithms fail to meet. Examples include medical applications where patient comfort must be guaranteed, recommender systems aiming to avoid user dissatisfaction, and robotic control, where one seeks to avoid controls causing physical harm to the platform. We tackle this novel, yet rich, set of problems under the assumption that the unknown function satisfies regularity conditions expressed via a Gaussian process prior. We develop an efficient algorithm called SAFEOPT, and theoretically guarantee its convergence to a natural notion of optimum reachable under safety constraints. We evaluate SAFEOPT on synthetic data, as well as two real applications: movie recommendation, and therapeutic spinal cord stimulation. We propose in this paper a new approach to train the Generative Adversarial Nets (GANs) with a mixture of generators to overcome the mode collapsing problem. The main intuition is to employ multiple generators, instead of using a single one as in the original GAN. The idea is simple, yet proven to be extremely effective at covering diverse data modes, easily overcoming the mode collapsing problem and delivering state-of-the-art results. A minimax formulation was able to establish among a classifier, a discriminator, and a set of generators in a similar spirit with GAN. Generators create samples that are intended to come from the same distribution as the training data, whilst the discriminator determines whether samples are true data or generated by generators, and the classifier specifies which generator a sample comes from. The distinguishing feature is that internal samples are created from multiple generators, and then one of them will be randomly selected as final output similar to the mechanism of a probabilistic mixture model. We term our method Mixture Generative Adversarial Nets (MGAN). We develop theoretical analysis to prove that, at the equilibrium, the Jensen-Shannon divergence (JSD) between the mixture of generators’ distributions and the empirical data distribution is minimal, whilst the JSD among generators’ distributions is maximal, hence effectively avoiding the mode collapsing problem. By utilizing parameter sharing, our proposed model adds minimal computational cost to the standard GAN, and thus can also efficiently scale to large-scale datasets. We conduct extensive experiments on synthetic 2D data and natural image databases (CIFAR-10, STL-10 and ImageNet) to demonstrate the superior performance of our MGAN in achieving state-of-the-art Inception scores over latest baselines, generating diverse and appealing recognizable objects at different resolutions, and specializing in capturing different types of objects by the generators. This work studies the design of reliable control laws of robotic systems operating in uncertain environments. We introduce a new approach to stochastic policy optimization based on probably approximately correct (PAC) bounds on the expected performance of control policies. An algorithm is constructed which directly minimizes an upper confidence bound on the expected cost of trajectories instead of employing a standard approach based on the expected cost itself. This algorithm thus has built-in robustness to uncertainty, since the bound can be regarded as a certificate for guaranteed future performance. The approach is evaluated on two challenging robot control scenarios in simulation: a car with side slip and a quadrotor navigating through obstacle-ridden environments. We show that the bound accurately predicts future performance and results in improved robustness measured by lower average cost and lower probability of collision. The performance of the technique is studied empirically and compared to several existing policy search algorithms. We consider Markov decision processes where the values of the parameters are uncertain. This uncertainty is described by a sequence of nested sets (that is, each set contains the previous one), each of which corresponds to a probabilistic guarantee for a different confidence level. Consequently, a set of admissible probability distributions of the unknown parameters is specified. This formulation models the case where the decision maker is aware of and wants to exploit some (yet imprecise) a priori information of the distribution of parameters, and it arises naturally in practice where methods for estimating the confidence region of parameters abound. We propose a decision criterion based on distributional robustness: the optimal strategy maximizes the expected total reward under the most adversarial admissible probability distributions. We show that finding the optimal distributionally robust strategy can be reduced to the standard robust MDP where parameters are known to belong to a single uncertainty set; hence, it can be computed in polynomial time under mild technical conditions. We consider the minimization of a convex objective function defined on a Hilbert space, which is only available through unbiased estimates of its gradients. This problem includes standard machine learning algorithms such as kernel logistic regression and least-squares regression, and is commonly referred to as a stochastic approximation problem in the operations research community. We provide a non-asymptotic analysis of the convergence of two well-known algorithms, stochastic gradient descent (a.k.a. Robbins-Monro algorithm) as well as a simple modification where iterates are averaged (a.k.a. Polyak-Ruppert averaging). Our analysis suggests that a learning rate proportional to the inverse of the number of iterations, while leading to the optimal convergence rate in the strongly convex case, is not robust to the lack of strong convexity or the setting of the proportionality constant. This situation is remedied when using slower decays together with averaging, robustly leading to the optimal rate of convergence. We illustrate our theoretical results with simulations on synthetic and standard datasets.