Wasserstein distributionally robust optimization is a recent emerging modeling paradigm for decision making under data uncertainty. Because of its computational tractability and interpretability, it has achieved great empirical successes across several application domains in operations research, computer science, engineering, and business analytics. Despite its recent empirical success, existing performance guarantees for generic problems are not yet satisfactory. In this paper, we develop the first finite-sample guarantee without suffering from the curse of dimensionality, which describes how the out-of-sample performance of a robust solution depends on the sample size, dimension of the uncertainty, and the complexity of the loss function class in a nearly optimal way. 

Finite-Sample Guarantees for Wasserstein Distributionally Robust Optimization: Breaking the Curse of Dimensionality

We study information matrices for statistical models by the $L^2$-Wasserstein metric. We call them Wasserstein information matrices (WIMs), which are analogs of classical Fisher information matrices. We introduce Wasserstein score functions and study covariance operators in statistical models. Using them, we establish Wasserstein-Cramer-Rao bounds for estimations and explore their comparisons with classical results. We next consider the asymptotic behaviors and efficiency of estimators. We derive the on-line asymptotic efficiency for Wasserstein natural gradient. Besides, we study a Poincare efficiency for Wasserstein natural gradient of maximal likelihood estimation. Several analytical examples of WIMs are presented, including location-scale families, independent families, and rectified linear unit (ReLU) generative models.

Wasserstein information matrix

Statistical inference can be performed by minimizing, over the parameter space, the Wasserstein distance between model distributions and the empirical distribution of the data. We study asymptotic properties of such minimum Wasserstein distance estimators, complementing results derived by Bassetti, Bodini and Regazzini in 2006. In particular, our results cover the misspecified setting, in which the data-generating process is not assumed to be part of the family of distributions described by the model. Our results are motivated by recent applications of minimum Wasserstein estimators to complex generative models. We discuss some difficulties arising in the approximation of these estimators and illustrate their behavior in several numerical experiments. Two of our examples are taken from the literature on approximate Bayesian computation and have likelihood functions that are not analytically tractable. Two other examples involve misspecified models.

On parameter estimation with the Wasserstein distance

We consider doing Bayesian inference by minimizing the KL divergence on the 2-Wasserstein space $\mathcal{P}_2$. By exploring the Riemannian structure of $\mathcal{P}_2$, we develop two inference methods by simulating the gradient flow on $\mathcal{P}_2$ via updating particles, and an acceleration method that speeds up all such particle-simulation-based inference methods. Moreover we analyze the approximation flexibility of such methods, and conceive a novel bandwidth selection method for the kernel that they use. We note that $\mathcal{P}_2$ is quite abstract and general so that our methods can make closer approximation, while it still has a rich structure that enables practical implementation. Experiments show the effectiveness of the two proposed methods and the improvement of convergence by the acceleration method.

pdf/accelerated-first-order-methods-on-the-wasserstein-space-for-57ni3sz15c.pdf

Accelerated First-order Methods on the Wasserstein Space for Bayesian Inference.

Information geometry and optimal transport are two distinct geometric frameworks for modeling families of probability measures. During the recent years, there has been a surge of research endeavors that cut across these two areas and explore their links and interactions. This paper is intended to provide an (incomplete) survey of these works, including entropy-regularized transport, divergence functions arising from c-duality, density manifolds and transport information geometry, the para-Kähler and Kähler geometries underlying optimal transport and the regularity theory for its solutions. Some outstanding questions that would be of interest to audience of both these two disciplines are posed. Our piece also serves as an introduction to the Special Issue on Optimal Transport of the journal Information Geometry. 

/pdf/when-optimal-transport-meets-information-geometry-24wj5too.pdf

When optimal transport meets information geometry

We study the Wasserstein natural gradient in parametric statistical models with continuous sample spaces. Our approach is to pull back the $$L^2$$
 -Wasserstein metric tensor in the probability density space to a parameter space, equipping the latter with a positive definite metric tensor, under which it becomes a Riemannian manifold, named the Wasserstein statistical manifold. In general, it is not a totally geodesic sub-manifold of the density space, and therefore its geodesics will differ from the Wasserstein geodesics, except for the well-known Gaussian distribution case, a fact which can also be validated under our framework. We use the sub-manifold geometry to derive a gradient flow and natural gradient descent method in the parameter space. When parametrized densities lie in $$\mathbb {R}$$
 , the induced metric tensor establishes an explicit formula. In optimization problems, we observe that the natural gradient descent outperforms the standard gradient descent when the Wasserstein distance is the objective function. In such a case, we prove that the resulting algorithm behaves similarly to the Newton method in the asymptotic regime. The proof calculates the exact Hessian formula for the Wasserstein distance, which further motivates another preconditioner for the optimization process. To the end, we present examples to illustrate the effectiveness of the natural gradient in several parametric statistical models, including the Gaussian measure, Gaussian mixture, Gamma distribution, and Laplace distribution.

https://link.springer.com/content/pdf/10.1007/s41884-020-00028-0.pdf

Optimal transport natural gradient for statistical manifolds with continuous sample space

We study the Wasserstein natural gradient in parametric statistical models with continuous sample spaces. Our approach is to pull back the $L^2$-Wasserstein metric tensor in the probability density space to a parameter space, equipping the latter with a positive definite metric tensor, under which it becomes a Riemannian manifold, named the Wasserstein statistical manifold. In general, it is not a totally geodesic sub-manifold of the density space, and therefore its geodesics will differ from the Wasserstein geodesics, except for the well-known Gaussian distribution case, a fact which can also be validated under our framework. We use the sub-manifold geometry to derive a gradient flow and natural gradient descent method in the parameter space. When parametrized densities lie in $\bR$, the induced metric tensor establishes an explicit formula. In optimization problems, we observe that the natural gradient descent outperforms the standard gradient descent when the Wasserstein distance is the objective function. In such a case, we prove that the resulting algorithm behaves similarly to the Newton method in the asymptotic regime. The proof calculates the exact Hessian formula for the Wasserstein distance, which further motivates another preconditioner for the optimization process. To the end, we present examples to illustrate the effectiveness of the natural gradient in several parametric statistical models, including the Gaussian measure, Gaussian mixture, Gamma distribution, and Laplace distribution.

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Optimal transport natural gradient for statistical manifolds with continuous sample space

Optimal transport natural gradient for statistical manifolds with continuous sample space