Construction of ambiguity set in robust optimization relies on the choice of divergences between probability distributions. In distribution learning, choosing appropriate probability distributions based on observed data is critical for approximating the true distribution. To improve the performance of machine learning models, there has recently been interest in designing objective functions based on Lp-Wasserstein distance rather than the classical Kullback-Leibler (KL) divergence. In this paper, we derive concentration and asymptotic results using Bregman divergence. We propose a novel asymmetric statistical divergence called Wasserstein-Bregman divergence as a generalization of L2-Wasserstein distance. We discuss how these results can be applied to the construction of ambiguity set in robust optimization.
In @cite_22 , they formulate a robust optimization problem in terms of a KL divergence constraint and show that the problem can be converted into a convex optimization problem which can be solved analytically. In @cite_35 , they show that chance constraints with KL divergence ambiguity sets can be reformulated into a traditional chance constraint problem with different risk levels.
@cite_35 @cite_22
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In this paper we study distributionally robust optimization (DRO) problems where the ambiguity set of the probability distribution is defined by the Kullback-Leibler (KL) divergence. We consider DRO problems where the ambiguity is in the objective function, which takes a form of an expectation, and show that the resulted minimax DRO problems can be formulated as a one-layer convex minimization problem. We also consider DRO problems where the ambiguity is in the constraint. We show that ambiguous expectation-constrained programs may be reformulated as a one-layer convex optimization problem that takes the form of the Benstein approximation of Nemirovski and Shapiro (2006). We further consider distributionally robust probabilistic programs. We show that the optimal solution of a probability minimization problem is also optimal for the distributionally robust version of the same problem, and also show that the ambiguous chance-constrained programs (CCPs) may be reformulated as the original CCP with an adjusted confidence level. A number of examples and special cases are also discussed in the paper to show that the reformulated problems may take simple forms that can be solved easily. The main contribution of the paper is to show that the KL divergence constrained DRO problems are often of the same complexity as their original stochastic programming problems and, thus, KL divergence appears a good candidate in modeling distribution ambiguities in mathematical programming. We propose a general purpose variational inference algorithm that forms a natural counterpart of gradient descent for optimization. Our method iteratively transports a set of particles to match the target distribution, by applying a form of functional gradient descent that minimizes the KL divergence. Empirical studies are performed on various real world models and datasets, on which our method is competitive with existing state-of-the-art methods. The derivation of our method is based on a new theoretical result that connects the derivative of KL divergence under smooth transforms with Stein’s identity and a recently proposed kernelized Stein discrepancy, which is of independent interest. We consider the problem of projecting a vector onto the simplex Δ = x ∈ ℝ+d : Σi=1d xi = 1 , using a Bregman projection. This is a common problem in first-order methods for convex optimization and online-learning algorithms, such as mirror descent. We derive the KKT conditions of the projection problem, and show that for Bregman divergences induced by ω-potentials, one can efficiently compute the solution using a bisection method. More precisely, an ω-approximate projection can be obtained in O(d log 1 ω). We also consider a class of exponential potentials for which the exact solution can be computed efficiently, and give a O(d log d) deterministic algorithm and O(d) randomized algorithm to compute the projection. In particular, we show that one can generalize the KL divergence to a Bregman divergence which is bounded on the simplex (unlike the KL divergence), strongly convex with respect to the l1 norm, and for which one can still solve the projection in expected linear time. Determining the structure of dependencies among a set of variables is a common task in many signal and image processing applications, including multitarget tracking and computer vision. In this paper, we present an information-theoretic, machine learning approach to problems of this type. We cast this problem as a hypothesis test between factorizations of variables into mutually independent subsets. We show that the likelihood ratio can be written as sums of two sets of Kullback-Leibler (KL) divergence terms. The first set captures the structure of the statistical dependencies within each hypothesis, whereas the second set measures the details of model differences between hypotheses. We then consider the case when the signal prior models are unknown, so that the distributions of interest must be estimated directly from data, showing that the second set of terms is (asymptotically) negligible and quantifying the loss in hypothesis separability when the models are completely unknown. We demonstrate the utility of nonparametric estimation methods for such problems, providing a general framework for determining and distinguishing between dependency structures in highly uncertain environments. Additionally, we develop a machine learning approach for estimating lower bounds on KL divergence and mutual information from samples of high-dimensional random variables for which direct density estimation is infeasible. We present empirical results in the context of three prototypical applications: association of signals generated by sources possessing harmonic behavior, scene correspondence using video imagery, and detection of coherent behavior among sets of moving objects.