On the Efficiency of Entropic Regularized Algorithms for Optimal Transport
University of California, Berkeley（加州大学伯克利分校）；University of Texas, Austin（德克萨斯大学奥斯汀分校）
Tianyi Lin, Nhat Ho, Michael I. Jordan
备注：
Accepted by Journal of Machine Learning Research; A preliminary version [arXiv:1901.06482] of this paper, with a subset of the results that are presented here, was presented at ICML 2019; 39 pages, 21 figures
We present several new complexity results for the entropic regularized algorithms that approximately solve the optimal transport (OT) problem between two discrete probability measures with at most $n$ atoms. First, we improve the complexity bound of a greedy variant of Sinkhorn, known as \textit{Greenkhorn}, from $\widetilde{O}(n^2\varepsilon^{-3})$ to $\widetilde{O}(n^2\varepsilon^{-2})$. Notably, our result can match the best known complexity bound of Sinkhorn and help clarify why Greenkhorn significantly outperforms Sinkhorn in practice in terms of row/column updates as observed by~\citet{Altschuler-2017-Near}. Second, we propose a new algorithm, which we refer to as \textit{APDAMD} and which generalizes an adaptive primal-dual accelerated gradient descent (APDAGD) algorithm~\citep{Dvurechensky-2018-Computational} with a prespecified mirror mapping $ϕ$. We prove that APDAMD achieves the complexity bound of $\widetilde{O}(n^2\sqrtδ\varepsilon^{-1})$ in which $δ>0$ stands for the regularity of $ϕ$. In addition, we show by a counterexample that the complexity bound of $\widetilde{O}(\min\{n^{9/4}\varepsilon^{-1}, n^2\varepsilon^{-2}\})$ proved for APDAGD before is invalid and give a refined complexity bound of $\widetilde{O}(n^{5/2}\varepsilon^{-1})$. Further, we develop a \textit{deterministic} accelerated variant of Sinkhorn via appeal to estimated sequence and prove the complexity bound of $\widetilde{O}(n^{7/3}\varepsilon^{-4/3})$. As such, we see that accelerated variant of Sinkhorn outperforms Sinkhorn and Greenkhorn in terms of $1/\varepsilon$ and APDAGD and accelerated alternating minimization (AAM)~\citep{Guminov-2021-Combination} in terms of $n$. Finally, we conduct the experiments on synthetic and real data and the numerical results show the efficiency of Greenkhorn, APDAMD and accelerated Sinkhorn in practice.
An improved central limit theorem and fast convergence rates for entropic transportation costs
IMUVA, Universidad de Valladolid（巴利亚多利德大学IMUVA数学研究所）；IMT, Université de Toulouse（图卢兹大学图卢兹数学研究所）；Center for Data Science, New York University（纽约大学数据科学中心）
Gaussian entropic optimal transport: Schrödinger bridges and the Sinkhorn algorithm
Department of Mathematics, Imperial College London, UK；Centre de Recherche Inria Bordeaux Sud-Ouest, Talence, France；Department of Signal Theory \& Communications, Universidad Carlos III de Madrid, Spain
Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem
Harvard University（哈佛大学）；Massachusetts Institute of Technology（麻省理工学院）
我们证明了几个基本的统计界熵OT的平方欧几里德成本之间的亚高斯概率测度在任意维。首先，通过一个新的样本复杂度的结果，我们建立了收敛速率的熵OT的经验措施。我们的分析在Genevay et al.（2019）的边界上呈指数级改进，并将他们的工作扩展到无界测量。其次，我们基于Del Barrio和Loubes（2019）开发的技术建立了熵OT的中心极限定理。以前，这样的结果只在有限度量空间中才知道。作为我们的结果的应用，我们开发和分析了一种新的技术，用于估计被高斯噪声破坏的随机变量的熵。
Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier maps
CalTech（加州理工学院）；Center for Data Science, New York University（纽约大学数据科学中心）；Massachusets Institute of Technology（麻省理工学院）；Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）
Decentralized Entropic Optimal Transport for Distributed Distribution Comparison
School of Computer Science and Technology, East China Normal University（华东师范大学计算机科学与技术学院）；Gaoling School of Artificial Intelligence, Renmin University of China（中国人民大学高瓴人工智能学院）
