Deep Learning and Mean-Field Games: A Stochastic Optimal Control Perspective

We provide a rigorous mathematical formulation of Deep Learning (DL) methodologies through an in-depth analysis of the learning procedures characterizing Neural Network (NN) models within the theoretical frameworks of Stochastic Optimal Control (SOC) and Mean-Field Games (MFGs). In particular, we show how the supervised learning approach can be translated in terms of a (stochastic) mean-field optimal control problem by applying the Hamilton\u2013Jacobi\u2013Bellman (HJB) approach and the mean-field Pontryagin maximum principle. Our contribution sheds new light on a possible theoretical connection between mean-field problems and DL, melting heterogeneous approaches and reporting the state-of-the-art within such fields to show how the latter different perspectives can be indeed fruitfully unified

Variational formulations of ODE-Net as a mean-field optimal control problem and existence results

This paper presents a mathematical analysis of ODE-Net, a continuum model of deep neural networks (DNNs). In recent years, Machine Learning researchers have introduced ideas of replacing the deep structure of DNNs with ODEs as a continuum limit. These studies regard the "learning" of ODE-Net as the minimization of a "loss" constrained by a parametric ODE. Although the existence of a minimizer for this minimization problem needs to be assumed, only a few studies have investigated its existence analytically in detail. In the present paper, the existence of a minimizer is discussed based on a formulation of ODE-Net as a measure-theoretic mean-field optimal control problem. The existence result is proved when a neural network, which describes a vector field of ODE-Net, is linear with respect to learnable parameters. The proof employs the measure-theoretic formulation combined with the direct method of Calculus of Variations. Secondly, an idealized minimization problem is proposed to remove the above linearity assumption. Such a problem is inspired by a kinetic regularization associated with the Benamou--Brenier formula and universal approximation theorems for neural networks. The proofs of these existence results use variational methods, differential equations, and mean-field optimal control theory. They will stand for a new analytic way to investigate the learning process of deep neural networks.Comment: 33 page

Learning Dynamics from Data Using Optimal Transport Techniques and Applications

Optimal Transport has been studied widely in recent years, the concept of Wasserstein distance brings a lot of applications in computational mathematics, machine learning, engineering, even finance areas. Meanwhile, people are gradually realizing that as the amount of data as well as the needs of utilizing data increase vastly, data-driven models have great potentials in real-world applications. In this thesis, we apply the theories of OT and design data-driven algorithms to form and compute various OT problems. We also build a framework to learn inverse OT problem. Furthermore, we develop OT and deep learning based models to solve stochastic differential equations, optimal control, mean field games related problems, all in data-driven settings. In Chapter 2, we provide necessary mathematical concepts and results that form the basis of this thesis. It contains brief surveys of optimal transport, stochastic differential equations, Fokker-Planck equations, deep learning, optimal controls and mean field games. Chapter 3 to Chapter 5 present several scalable algorithms to handle optimal transport problems within different settings. Specifically, Chapter 3 shows a new saddle scheme and learning strategy for computing the Wasserstein geodesic, as well as the Wasserstein distance and OT map between two probability distributions in high dimensions. We parametrize the map and Lagrange multipliers as neural networks. We demonstrate the performance of our algorithms through a series of experiments with both synthetic and realistic data. Chapter 4 presents a scalable algorithm for computing the Monge map between two probability distributions since computing the Monge maps remains challenging, in spite of the rapid developments of the numerical methods for optimal transport problems. Similarly, we formulate the problem as a mini-max problem and solve it via deep learning. The performance of our algorithms is demonstrated through a series of experiments with both synthetic and realistic data. In Chapter 5 we study OT problem in an inverse view, which we also call Inverse OT (IOT) problem. IOT also refers to the problem of learning the cost function for OT from observed transport plan or its samples. We derive an unconstrained convex optimization formulation of the inverse OT problem. We provide a comprehensive characterization of the properties of inverse OT, including uniqueness of solutions. We also develop two numerical algorithms, one is a fast matrix scaling method based on the Sinkhorn-Knopp algorithm for discrete OT, and the other one is a learning based algorithm that parameterizes the cost function as a deep neural network for continuous OT. Our numerical results demonstrate promising efficiency and accuracy advantages of the proposed algorithms over existing state-of-the-art methods. In Chapter 6 we propose a novel method using the weak form of Fokker Planck Equation (FPE) --- a partial differential equation --- to describe the density evolution of data in a sampled form, which is then combined with Wasserstein generative adversarial network (WGAN) in the training process. In such a sample-based framework we are able to learn the nonlinear dynamics from aggregate data without explicitly solving FPE. We demonstrate our approach in the context of a series of synthetic and real-world data sets. Chapter 7 introduces the application of OT and neural networks in optimal density control. Particularly, we parametrize the control strategy via neural networks, and provide an algorithm to learn the strategy that can drive samples following one distribution to new locations following target distribution. We demonstrate our method in both synthetic and realistic experiments, where we also consider perturbation fields. Finally Chapter 8 presents applications of mean field game in generative modeling and finance area. With more details, we build a GAN framework upon mean field game to generate desired distribution starting with white noise, we also investigate its connection to OT. Moreover, we apply mean field game theories to study the equilibrium trading price in stock markets, we demonstrate the theoretical result by conducting experiments on real trading data.Ph.D