On the Complexity of Approximating Wasserstein Barycenter

We study the complexity of approximating Wassertein barycenter of $m$ discrete measures, or histograms of size $n$ by contrasting two alternative approaches, both using entropic regularization. The first approach is based on the Iterative Bregman Projections (IBP) algorithm for which our novel analysis gives a complexity bound proportional to $\frac{mn^2}{\varepsilon^2}$ to approximate the original non-regularized barycenter. Using an alternative accelerated-gradient-descent-based approach, we obtain a complexity proportional to $\frac{mn^{2.5}}{\varepsilon}$. As a byproduct, we show that the regularization parameter in both approaches has to be proportional to $\varepsilon$, which causes instability of both algorithms when the desired accuracy is high. To overcome this issue, we propose a novel proximal-IBP algorithm, which can be seen as a proximal gradient method, which uses IBP on each iteration to make a proximal step. We also consider the question of scalability of these algorithms using approaches from distributed optimization and show that the first algorithm can be implemented in a centralized distributed setting (master/slave), while the second one is amenable to a more general decentralized distributed setting with an arbitrary network topology.Comment: Corrected misprints. Added a reference to accelerated Iterative Bregman Projections introduced in arXiv:1906.0362

On the complexity of approximating Wasserstein barycenter

We study the complexity of approximating Wassertein barycenter of discrete measures, or histograms by contrasting two alternative approaches, both using entropic regularization. We provide a novel analysis for our approach based on the Iterative Bregman Projections (IBP) algorithm to approximate the original non-regularized barycenter. We also get the complexity bound for alternative accelerated-gradient-descent-based approach and compare it with the bound obtained for IBP. As a byproduct, we show that the regularization parameter in both approaches has to be proportional to ", which causes instability of both algorithms when the desired accuracy is high. To overcome this issue, we propose a novel proximal-IBP algorithm, which can be seen as a proximal gradient method, which uses IBP on each iteration to make a proximal step. We also consider the question of scalability of these algorithms using approaches from distributed optimization and show that the first algorithm can be implemented in a centralized distributed setting (master/slave), while the second one is amenable to a more general decentralized distributed setting with an arbitrary network topology

On the complexity of approximating Wasserstein barycenter

We study the complexity of approximating Wassertein barycenter of discrete measures, or histograms by contrasting two alternative approaches, both using entropic regularization. We provide a novel analysis for our approach based on the Iterative Bregman Projections (IBP) algorithm to approximate the original non-regularized barycenter. We also get the complexity bound for alternative accelerated-gradient-descent-based approach and compare it with the bound obtained for IBP. As a byproduct, we show that the regularization parameter in both approaches has to be proportional to ", which causes instability of both algorithms when the desired accuracy is high. To overcome this issue, we propose a novel proximal-IBP algorithm, which can be seen as a proximal gradient method, which uses IBP on each iteration to make a proximal step. We also consider the question of scalability of these algorithms using approaches from distributed optimization and show that the first algorithm can be implemented in a centralized distributed setting (master/slave), while the second one is amenable to a more general decentralized distributed setting with an arbitrary network topology

Матеріали VI Міжнародного семінару з професійної перепідготовки та навчання впродовж життя з використанням ІКТ: Особистісно-орієнтований підхід (3L-Person 2021), який відбувся одночасно з 17-ю Міжнародною конференцією "ІКТ в освіті, науці та промисловості

Proceedings of the VI International Workshop on Professional Retraining and Life-Long Learning using ICT: Person-oriented Approach (3L-Person 2021) co-located with 17th International Conference on ICT in Education, Research, and Industrial Applications: Integration, Harmonization, and Knowledge Transfer (ICTERI 2021), Kherson, Ukraine, October 1, 2021.Матеріали VI Міжнародного семінару з професійної перепідготовки та навчання впродовж життя з використанням ІКТ: Особистісно-орієнтований підхід (3L-Person 2021), який відбувся одночасно з 17-ю Міжнародною конференцією "ІКТ в освіті, наукових дослідженнях та промисловому застосуванні": Інтеграція, гармонізація та передача знань (ICTERI 2021), м. Херсон, Україна, 1 жовтня 2021 р