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1. Robust Wasserstein profile inference and applications to machine learning

   https://doi.org/10.1017/jpr.2019.49  

   Blanchet, Jose; Kang, Yang; Murthy, Karthyek (September 2019, Journal of Applied Probability)

   Abstract We show that several machine learning estimators, including square-root least absolute shrinkage and selection and regularized logistic regression, can be represented as solutions to distributionally robust optimization problems. The associated uncertainty regions are based on suitably defined Wasserstein distances. Hence, our representations allow us to view regularization as a result of introducing an artificial adversary that perturbs the empirical distribution to account for out-of-sample effects in loss estimation. In addition, we introduce RWPI (robust Wasserstein profile inference), a novel inference methodology which extends the use of methods inspired by empirical likelihood to the setting of optimal transport costs (of which Wasserstein distances are a particular case). We use RWPI to show how to optimally select the size of uncertainty regions, and as a consequence we are able to choose regularization parameters for these machine learning estimators without the use of cross validation. Numerical experiments are also given to validate our theoretical findings. 

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2. A Fast Proximal Point Method for Computing Exact Wasserstein Distance

   Xie, Yujia; Wang, Xiangfeng; Wang, Ruijia; Zha, Hongyuan (January 2019, Uncertainty in artificial intelligence)

   Wasserstein distance plays increasingly important roles in machine learning, stochastic programming and image processing. Major efforts have been under way to address its high computational complexity, some leading to approximate or regularized variations such as Sinkhorn distance. However, as we will demonstrate, regularized variations with large regularization parameter will degradate the performance in several important machine learning applications, and small regularization parameter will fail due to numerical stability issues with existing algorithms. We address this challenge by developing an Inexact Proximal point method for exact Optimal Transport problem (IPOT) with the proximal operator approximately evaluated at each iteration using projections to the probability simplex. The algorithm (a) converges to exact Wasserstein distance with theoretical guarantee and robust regularization parameter selection, (b) alleviates numerical stability issue, (c) has similar computational complexity to Sinkhorn, and (d) avoids the shrinking problem when apply to generative models. Furthermore, a new algorithm is proposed based on IPOT to obtain sharper Wasserstein barycenter. 

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3. Continuous Wasserstein-2 Barycenter Estimation without Minimax Optimization

   Korotin, Alexander; Li, Lingxiao; Solomon, Justin; Burnaev, Evgeny (March 2021, International Conference on Learning Representations)

   Wasserstein barycenters provide a geometric notion of the weighted average of probability measures based on optimal transport. In this paper, we present a scalable algorithm to compute Wasserstein-2 barycenters given sample access to the input measures, which are not restricted to being discrete. While past approaches rely on entropic or quadratic regularization, we employ input convex neural networks and cycle-consistency regularization to avoid introducing bias. As a result, our approach does not resort to minimax optimization. We provide theoretical analysis on error bounds as well as empirical evidence of the effectiveness of the proposed approach in low-dimensional qualitative scenarios and high-dimensional quantitative experiments. 

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4. Optimal Transport Barycenter via Nonconvex-Concave Minimax Optimization

   Kim, Kaheon; Yao, Rentian; Zhu, Changbo; Chen, Xiaohui (July 2025, International Conference on Machine Learning)

   The optimal transport barycenter (a.k.a. Wasserstein barycenter) is a fundamental notion of averaging that extends from the Euclidean space to the Wasserstein space of probability distributions. Computation of the unregularized barycenter for discretized probability distributions on point clouds is a challenging task when the domain dimension d>1. Most practical algorithms for approximating the barycenter problem are based on entropic regularization. In this paper, we introduce a nearly linear time O(mlogm) and linear space complexity O(m) primal-dual algorithm, the Wasserstein-Descent ℍ˙1-Ascent (WDHA) algorithm, for computing the exact barycenter when the input probability density functions are discretized on an m-point grid. The key success of the WDHA algorithm hinges on alternating between two different yet closely related Wasserstein and Sobolev optimization geometries for the primal barycenter and dual Kantorovich potential subproblems. Under reasonable assumptions, we establish the convergence rate and iteration complexity of WDHA to its stationary point when the step size is appropriately chosen. Superior computational efficacy, scalability, and accuracy over the existing Sinkhorn-type algorithms are demonstrated on high-resolution (e.g., 1024×1024 images) 2D synthetic and real data. 

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5. Visual Transfer for Reinforcement Learning via Wasserstein Domain Confusion

   Roy, J; Konidaris, G.D. (February 2021, Proceedings of the Thirty-Fifth AAAI Conference on Artificial Intelligence)

   We introduce Wasserstein Adversarial Proximal Policy Optimization (WAPPO), a novel algorithm for visual transfer in Reinforcement Learning that explicitly learns to align the distributions of extracted features between a source and target task. WAPPO approximates and minimizes the Wasserstein-1 distance between the distributions of features from source and target domains via a novel Wasserstein Confusion objective. WAPPO outperforms the prior state-of-the-art in visual transfer and successfully transfers policies across Visual Cartpole and both the easy and hard settings of of 16 OpenAI Procgen environments. 

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