Robust Optimization and Simulation of Complex Stochastic Systems
Institute for Computational and Experimental Research in Mathematics (ICERM)
September 13, 2024  September 15, 2024
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Friday, September 13, 2024

8:30  8:50 am EDTCheck In11th Floor Collaborative Space

8:50  9:00 am EDTWelcome11th Floor Lecture Hall
 Session Chair
 Brendan Hassett, ICERM/Brown University

9:00  9:45 am EDTPaul Dupuis's Contributions to Large Deviations, Stochastic Control, and Numerical Methods11th Floor Lecture Hall
 Speaker
 Amarjit Budhiraja, University of North Carolina at Chapel Hill
 Session Chair
 Petr Plechac, University of Delaware
Abstract
In this talk I will present some highlights of Paul Dupuis' contributions to the areas of large deviations, stochastic control, and numerical methods.

10:00  10:30 am EDTCoffee Break11th Floor Collaborative Space

10:30  11:15 am EDTLarge Deviations for Stochastic Approximations11th Floor Lecture Hall
 Speaker
 Pierre Nyquist, Chalmers University of Techology
 Session Chair
 Petr Plechac, University of Delaware
Abstract
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 statedependent 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 WangLandau algorithm.

11:30 am  12:15 pm EDTA MeanField Games Laboratory for Generative Modeling: Implications for Robust Generative Algorithms11th Floor Lecture Hall
 Speaker
 Markos Katsoulakis, University of Massachusetts Amherst
 Session Chair
 Petr Plechac, University of Delaware
Abstract
the first part of the talk, we demonstrate the versatility of meanfield games (MFGs) as a mathematical framework for explaining, enhancing, and designing generative models. We establish connections between MFGs and major classes of flow and diffusionbased generative models by deriving continuoustime normalizing flows, scorebased 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 scorebased generative models (SGMs) are provably robust to multiple sources of error in their practical implementation. The regularizing properties of HamiltonJacobiBellman equations in the MFG formulation of SGMs are the key ingredient of this analysis.

12:25  12:30 pm EDTGroup Photo (Immediately After Talk)11th Floor Lecture Hall

12:30  2:00 pm EDTLunch/Free Time

2:00  2:45 pm EDTAsymptotic Problems for Stochastic Wave Equations with Constraints11th Floor Lecture Hall
 Speaker
 Sandra Cerrai, University of Maryland
 Session Chair
 Amber Puha, CSU San Marcos
Abstract
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

3:00  3:15 pm EDTCoffee Break11th Floor Collaborative Space

3:15  4:00 pm EDTProximal Optimal Transport Divergences for Generative Modeling11th Floor Lecture Hall
 Speaker
 Luc ReyBellet, UMass Amherst
 Session Chair
 Amber Puha, CSU San Marcos
Abstract
We introduce families of information theoretic divergences between probability measure which are constructed as MoreauYosida regularization of fdivergence (e.g. KLdivergences) with a Wasserstein metrics. We discuss the fundamental properties of those divergence as well as their applications to generative modeling and their analysis using meanfield games tools.

4:15  5:00 pm EDTStochastic Optimization Algorithms with Heavy Tailed Input11th Floor Lecture Hall
 Speaker
 Jose Blanchet, Stanford
 Session Chair
 Amber Puha, CSU San Marcos
Abstract
We first explain why statistical analysis of stochastic optimization algorithms with heavytailed 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 easytomonitor 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

5:00  6:30 pm EDTReception11th Floor Collaborative Space
Saturday, September 14, 2024

9:00  9:45 am EDTCoarse Correlated Equilibria in Mean Field Games11th Floor Lecture Hall
 Speaker
 Markus Fischer, University of Padua
 Session Chair
 Wlodzimierz Bryc, University of Cincinnati
Abstract
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 openloop strategies. For noncooperative manyplayer 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 (nonpublic) lottery outcomes to tell players in private which strategy to play. Players have to decide in advance whether to precommit 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 Nplayer 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").

10:00  11:00 am EDT

11:00  11:45 am EDTAspects of the Cutoff Phenomenon for Diffusions11th Floor Lecture Hall
 Speaker
 Djalil Chafai, Universite ParisDauphine
 Session Chair
 Wlodzimierz Bryc, University of Cincinnati
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.

12:00  12:45 pm EDTLarge Deviations and Calculus of Variations for Some Pure Jump Interacting Particle Systems11th Floor Lecture Hall
 Speaker
 Ruoyu Wu, Iowa State University
 Session Chair
 Wlodzimierz Bryc, University of Cincinnati
Abstract
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 JointheShortestQueue. 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 JointheShortestQueued (powerofd) 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.

1:00  2:30 pm EDTLunch/Free Time

2:30  3:15 pm EDTBiochemical Reaction Networks and Reflected Diffusions11th Floor Lecture Hall
 Virtual Speaker
 Ruth Williams, University of California, San Diego
 Session Chair
 Mickey Salins, Boston University
Abstract
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 ddimensional 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 jumpdiffusion 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.

3:30  4:00 pm EDTCoffee Break11th Floor Collaborative Space

4:00  4:45 pm EDTParticleBased Stochastic ReactionDiffusion Models: Mean field limits and fluctuation corrections.11th Floor Lecture Hall
 Speaker
 Konstantinos Spiliopoulos, Boston University
 Session Chair
 Mickey Salins, Boston University
Abstract
Particlebased stochastic reactiondiffusion (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 coarsegrained deterministic partial integrodifferential 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 measurevalued 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 largepopulation (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.

6:00  8:30 pm EDTDinner Venue TBD
Abstract
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. Cutoff to RSVP is September 5
Sunday, September 15, 2024

9:00  9:45 am EDTIn Search of Markov Solutions11th Floor Lecture Hall
 Speaker
 Vivek Borkar, IIT Bombay, India
 Session Chair
 Arnab Ganguly, Louisiana State University
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 nonunique solutions (also known as Markov selection).

10:00  10:30 am EDTCoffee Break11th Floor Collaborative Space

10:30  11:15 am EDTEstimating Rare Event Probabilities in Reflecting Brownian Motion11th Floor Lecture Hall
 Speaker
 Kevin Leder, University of Minnesota
 Session Chair
 Arnab Ganguly, Louisiana State University
Abstract
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.

11:30 am  12:15 pm EDTReflecting Diffusions in Curved Nonsmooth Domains11th Floor Lecture Hall
 Speaker
 Cristina Costantini, University of ChietiPescara
 Session Chair
 Arnab Ganguly, Louisiana State University
Abstract
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.

12:30  2:00 pm EDTLunch/Free Time

2:00  2:45 pm EDTUsing Weak Convergence Methods to Establish Exit Time Asymptotics for a Small Noise Infinitedimensional System11th Floor Lecture Hall
 Speaker
 David Lipshutz, Flatiron Institute
 Session Chair
 Tao Pang, North Carolina State University
Abstract
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 finitedimensional systems, these "exit times" are typically established by proving large deviation principles that are uniform over compact sets. However, for infinitedimensional 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 infinitedimensional, 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.

3:00  3:45 pm EDTRenormalized relative entropies and an Htheorem for interacting diffusions on sparse graphs11th Floor Lecture Hall
 Speaker
 Kavita Ramanan, Brown University
 Session Chair
 Tao Pang, North Carolina State University
Abstract
We consider a class of conditional McKeanVlasov processes that arise in the study of interacting Langevin diffusions on regular trees. We establish an Htheorem that characterizes the longtime 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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