Institute for Computational and Experimental Research in Mathematics (ICERM)

Abstract

In this talk I will present some highlights of Paul Dupuis' contributions to the areas of large deviations, stochastic control, and numerical methods.

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The theory of stochastic approximations form the theoretical foundation for studying convergence properties of many popular recursive learning algorithms in statistics, machine learning and statistical physics. Large deviations for stochastic approximations provide asymptotic estimates of the probability that the learning algorithm deviates from its expected path, given by a limit ODE, and the large deviation rate function gives insights to the most likely way that such deviations occur. In this paper, the weak convergence approach to large deviations is developed for general stochastic approximations with state-dependent Markovian noise and decreasing step size. The results generalize previous results on large deviations for stochastic approximations and identifies the appropriate scaling sequence. A new representation of the rate function is given, where the rate function is expressed as an action functional involving the family of Markov transition kernels. Learning algorithms that satisfy the assumptions of the large deviation principle include stochastic gradient descent, persistent contrastive divergence and the Wang-Landau algorithm.

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the first part of the talk, we demonstrate the versatility of mean-field games (MFGs) as a mathematical framework for explaining, enhancing, and designing generative models. We establish connections between MFGs and major classes of flow- and diffusion-based generative models by deriving continuous-time normalizing flows, score-based models, and Wasserstein gradient flows through different choices of particle dynamics and cost functions. Second, from a UQ perspective, relying on a new Wasserstein uncertainty propagation theorem, we show that score-based generative models (SGMs) are provably robust to multiple sources of error in their practical implementation. The regularizing properties of Hamilton-Jacobi-Bellman equations in the MFG formulation of SGMs are the key ingredient of this analysis.

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I will s bytart reviewing some results obtained in collaboration with Z. Brzezniak about the small mass limit for stochastic wave equations subject to suitable functional constraints. Then, I will outline more recent developments achieved in collaboration with M. Xie on constrained systems of stochastic partial differential equations

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We introduce families of information theoretic divergences between probability measure which are constructed as Moreau-Yosida regularization of f-divergence (e.g. KL-divergences) with a Wasserstein metrics. We discuss the fundamental properties of those divergence as well as their applications to generative modeling and their analysis using mean-field games tools.

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We first explain why statistical analysis of stochastic optimization algorithms with heavy-tailed input arises naturally in applications. In fact, we will argue that models that assume infinite variance gradient estimators in stochastic gradient descent are appropriate depending on easy-to-monitor features of historical data and on the spatial and temporal scales over which the algorithm will be deployed (even if models have finite variance in theory). We will then discuss inference tools that can be applied to monitor convergence of stochastic optimization algorithms based on several asymptotic statistics. The results that we will present include the first weak convergence analysis of stochastic gradient descent with infinite variance, extending results which assume finite variance or homogeneous and additive gradient noise. Based on joint work with Aleks Mijatovic, Wenhao Yang, Joost Jorritsma, Bert Zwart

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In the context of finite horizon mean field games with continuous time dynamics driven by additive Wiener noise, we introduce a notion of coarse correlated equilibrium in open-loop strategies. For non-cooperative many-player games, a coarse correlated equilibrium can be seen as a lottery on strategy profiles run according to a publicly known mechanism by a moderator who uses the (non-public) lottery outcomes to tell players in private which strategy to play. Players have to decide in advance whether to pre-commit to the mediator's recommendations or to play without seeing them. We justify our definition by showing that any coarse correlated solution of the mean field game induces approximate coarse correlated equilibria for the underlying N-player games. An existence result for coarse correlated mean field game solutions, not relying on the existence of classical solutions, will be given; an explicitly solvable example will be discussed as well. Joint work with Luciano Campi and Federico Cannerozzi (University of Milan "La Statale").

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• Slides

Abstract

The cutoff phenomenon, conceptualized in the context of finite Markov chains, states that for certain linear evolution equations, started from a point, the distance towards a long time equilibrium may become more and more abrupt in high dimensional state spaces and for certain choices of initial conditions. This can be seen as a critical competition between trend to equilibrium and initial condition. This talk is about the cutoff phenomenon for few classes of linear and nonlinear diffusions.

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Sample path large deviation principles (LDP) are established for some pure jump stochastic processes arising from queueing systems, including M/M/1+M, M/M/n+M, and Join-the-Shortest-Queue. The process of evolution suffers from features of infinite dimensional dynamics, vanishing jump rates (near the boundary), and/or discontinuous statistics (at the boundary), which pose challenges on the variational representation and weak convergence approach, in particular the uniqueness argument. Rare events of interest are analyzed by solving the related calculus of variations problems written in terms of the LDP rate functions. If time permits, the Join-the-Shortest-Queue-d (power-of-d) queueing system will be discussed, including conjectures on the LDP and establishment of moderate deviation principles. This is based on several joint works with Rami Atar, Amarjit Budhiraja, Paul Dupuis, Eric Friedlander, and Zhenhua Wang.

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Leite and Williams have proposed certain reflected diffusion processes as approximations to continuous time Markov chain models frequently used to model biochemical reaction networks. These diffusions live in the positive orthant of a d-dimensional space and are confined there by a smoothly varying oblique reflection field on the boundary. Leite and Williams showed that, under mild conditions, these diffusions can be obtained as weak limits of certain jump-diffusion extensions of the traditional Langevin approximations, and therefore called these constrained Langevin approximations. In this talk, we will review this approximation and describe some progress on proving error estimates for strong versions of this approximation and also describe some remaining open problems. Part of this work is joint with Felipe Campos.

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Particle-based stochastic reaction-diffusion (PBSRD) models are a popular approach for studying biological systems involving both noise in the reaction process and diffusive transport. In this work we derive coarse-grained deterministic partial integro-differential equation (PIDE) models that provide a mean field approximation to the volume reactivity PBSRD model, a model commonly used for studying cellular processes. We formulate a weak measure-valued stochastic process (MVSP) representation for the volume reactivity PBSRD model, demonstrating for a simplified but representative system that it is consistent with the commonly used Doi Fock Space representation of the corresponding forward equation. We then prove, (a): the convergence of the general volume reactivity model MVSP to the mean field PIDEs in the large-population (i.e. thermodynamic) limit, and (b): the next order fluctuation correction to the mean field limit, which satisfies systems of stochastic PIDEs with Gaussian noise. Numerical examples are presented to illustrate how such approximations can enable the accurate estimation of higher order statistics of the underlying PBSRD model. Time permitting, I will also discuss the reduction of the underdamped to the overdamped case in the setting PBSRD models. This is joint work with Samuel Isaacson, Max Heldman, Jingwei Ma, Qianhan Liu and Chen Yao.

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This dinner has an associated cost and is by RSVP only. Please contact mariam_han@brown.edu if you would like to register for the dinner, and she will provide you with a link to the payment portal. Cut-off to RSVP is September 5

Abstract

This talk will summarise recent contributions to two strands of research - mimicking one dimensional marginals of random processes by Markov processes, and selecting a Markov family of solutions for degenerate diffusions with non-unique solutions (also known as Markov selection).

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Reflecting Brownian motion (RBM) is a stochastic process that behaves like a Brownian motion in the interior of its domain and is pushed into the interior whenever it reaches the boundary of its domain. RBM’s in the positive orthant were first introduced by Harrison and Reiman 1981, and arise naturally in a wide variety of settings, e.g., heavy traffic queueing networks. A difficult question that has received a significant amount of attention is identifying the asymptotics for tail probabilities associated with RBM in the positive orthant. In this work we focus on the specific tail probability that a stable RBM started near the origin exits a large box before returning to the origin. We develop particle based algorithms to estimate this probability. Using results of Dean and Dupuis 2008 we are able to develop algorithms that efficiently estimate this tail probability in two dimensions. In three and higher dimensions, we are not able to construct an efficient estimator, but we do construct estimators that are provably superior (in an asymptotic sense) to standard Monte Carlo. Numerical results show the benefits of our algorithm to standard Monte Carlo. This is based on joint work with Xin Liu and Zicheng Wang.

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Reflecting diffusions arise in many applications: from stochastic networks, to singular stochastic control, to the motion of physical particles, etc.. In many examples the domain in which the reflecting diffusion is to be confined is nonsmooth or the direction of reflection varies nonsmoothly: in these cases it is not obvious that a reflecting diffusion with the prescribed direction of reflection exists and is uniquely characterized. After reviewing the literature, in which the works by Dupuis and Ishii play a central role, I will discuss how a recent ergodic theorem for inhomogeneous killed Markov chains allows to extend the 1993 Dupuis and Ishii results to some new classes of domains and directions of reflection. In particular one can obtain existence and uniqueness of a semimartingale reflecting diffusion in a piecewise smooth domain in dimension 2, possibly with cusps, under optimal conditions on the directions of reflection, and one can uniquely characterize obliquely reflecting Brownian motion in some piecewise smooth cones. The talk is based on joint works with T.G. Kurtz.

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Dynamical systems with small noise arise in a wide variety of applications. In such applications, we'd like to know how long it takes the small noise system to escape stable equilibra of the corresponding deterministic system. For finite-dimensional systems, these "exit times" are typically established by proving large deviation principles that are uniform over compact sets. However, for infinite-dimensional systems, bounded sets are no long compact and this approach does not naturally extend. Here, in the context of a small noise stochastic delay differential equation with strong solutions whose natural state space is infinite-dimensional, we show how the weak convergence method that was pioneered by Paul Dupuis can be used to prove a large deviation principle that is uniform over bounded sets. In turn, this enables us to prove exit time asymptotics for the small noise stochastic delay differential equation.

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We consider a class of conditional McKean-Vlasov processes that arise in the study of interacting Langevin diffusions on regular trees. We establish an H-theorem that characterizes the long-time behavior of these processes. Specifically, we show that a certain function that arises when taking the limit of certain renormalized relative entropies serves as a global Lyapunov function for the associated measure flow. We also provide counterexamples to highlight some subtleties in the approach, and discuss some open problems. This is joint work with Kevin Hu.

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