We prove a rate of convergence for the $N$-particle approximation of a second-order partial differential equation in the space of probability measures, like the Master equation or Bellman equation of mean-field control problem under common noise. The rate is of order $1/N$ for the pathwise error on the solution $v$ and of order $1/\sqrt{N}$ for the $L^2$-error on its $L$-derivative $\partial_μv$. The proof relies on backward stochastic differential equations techniques.
In this note, we provide a smooth variational principle on Wasserstein space by constructing a smooth gauge-type function using the sliced Wasserstein distance. This function is a crucial tool for optimization problems and in viscosity theory of PDEs on Wasserstein space.
p-Wasserstein distances on networks and 3D to 1D convergence
Helmholtz Imaging, Deutsches Elektronen-Synchrotron DESY（亥姆霍兹成像，德国电子同步辐射中心）；Fachbereich Mathematik, Universität Hamburg（汉堡大学数学系）；Applied and Computational Mathematics, Yale University（耶鲁大学应用与计算数学）；Universität Augsburg, Institut für Mathematik and Centre for Advanced Analytics and Predictive Sciences (CAAPS)（奥格斯堡大学数学研究所及高级分析与预测科学中心）
The Ohio State University（俄亥俄州立大学）；Purdue University（普渡大学）
This paper develops a unified framework, based on iterated random operator theory, to analyze the convergence of constant stepsize recursive stochastic algorithms (RSAs). RSAs use randomization to efficiently compute expectations, and so their iterates form a stochastic process. The key idea of our analysis is to lift the RSA into an appropriate higher-dimensional space and then express it as an equivalent Markov chain. Instead of determining the convergence of this Markov chain (which may not converge under constant stepsize), we study the convergence of the distribution of this Markov chain.…
Linear convergence of proximal descent schemes on the Wasserstein space
Center for Advanced Intelligence Project, RIKEN, Tokyo, Japan（日本东京RIKEN先进智能研究中心）；School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, UK（赫瑞-瓦特大学数学与计算机科学学院）；Maxwell Institute for Mathematical Sciences, Edinburgh, UK（爱丁堡数学科学麦克斯韦研究所）；School of Mathematics, University of Edinburgh, UK（爱丁堡大学数学学院）；School of Mathematics, University of Edinburgh, UK, The Alan Turing Institute, UK（爱丁堡大学数学学院、英国艾伦·图灵研究所）
Several issues in machine learning and inverse problems require to generate discrete data, as if sampled from a model probability distribution. A common way to do so relies on the construction of a uniform probability distribution over a set of $N$ points which minimizes the Wasserstein distance to the model distribution. This minimization problem, where the unknowns are the positions of the atoms, is non-convex. Yet, in most cases, a suitably adjusted version of Lloyd's algorithm -- in which Voronoi cells are replaced by Power cells -- leads to configurations with small Wasserstein error. Thi…
Wasserstein Convergence of Score-based Generative Models under Semiconvexity and Discontinuous Gradients
School of Mathematics University of Edinburgh, United Kingdom（爱丁堡大学数学学院）；Department of Industrial Engineering Ulsan National Institute of Science and Technology, Republic of Korea（釜山国立科学技术院工业工程系）；National Technical University of Athens, Greece（雅典国家技术大学）；Archimedes, Athena Research Center, Greece（阿提卡研究中心）
We develop a theoretical foundation for the application of Nesterov's accelerated gradient descent method (AGD) to the approximation of solutions of a wide class of partial differential equations (PDEs). This is achieved by proving the existence of an invariant set and exponential convergence rates when its preconditioned version (PAGD) is applied to minimize locally Lipschitz smooth, strongly convex objective functionals. We introduce a second-order ordinary differential equation (ODE) with a preconditioner built-in and show that PAGD is an explicit time-discretization of this ODE, which requ…
