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Quantitative fluctuation analysis of multiscale diffusion systems via Malliavin calculus
Solesne Bourguin, Konstantinos Spiliopoulos
We study fluctuations of small noise multiscale diffusions around their homogenized deterministic limit. We derive quantitative rates of convergence of the fluctuation processes to…
Fluctuation analysis for particle-based stochastic reaction-diffusion models
Max Heldman, Samuel Isaacson, Jingwei Ma +1
Recent works have derived and proven the large-population mean-field limit for several classes of particle-based stochastic reaction-diffusion (PBSRD) models. These limits correspo…
Moderate deviation principle for multiscale systems driven by fractional Brownian motion
Solesne Bourguin, Thanh Dang, Konstantinos Spiliopoulos
In this paper we study the moderate deviations principle (MDP) for slow-fast stochastic dynamical systems where the slow motion is governed by small fractional Brownian motion (fBm…
Importance sampling for stochastic reaction-diffusion equations in the moderate deviation regime
Ioannis Gasteratos, Michael Salins, Konstantinos Spiliopoulos
We develop a provably efficient importance sampling scheme that estimates exit probabilities of solutions to small-noise stochastic reaction-diffusion equations from scaled neighbo…
Rate of homogenization for fully-coupled McKean-Vlasov SDEs
Zachary Bezemek, Konstantinos Spiliopoulos
We consider a fully-coupled slow-fast system of McKean-Vlasov SDEs with full dependence on the slow and fast component and on the law of the slow component and derive convergence r…
Moderate deviations for systems of slow-fast stochastic reaction-diffusion equations
Ioannis Gasteratos, Michael Salins, Konstantinos Spiliopoulos
The goal of this paper is to study the Moderate Deviation Principle (MDP) for a system of stochastic reaction-diffusion equations with a time-scale separation in slow and fast comp…
Large deviations for interacting multiscale particle systems
Zachary Bezemek, Konstantinos Spiliopoulos
We consider a collection of weakly interacting diffusion processes moving in a two-scale locally periodic environment. We study the large deviations principle of the empirical dist…
Mean Field Limits of Particle-Based Stochastic Reaction-Diffusion Models
Samuel A. Isaacson, Jingwei Ma, Konstantinos Spiliopoulos
Particle-based stochastic reaction-diffusion (PBSRD) models are a popular approach for studying biological systems involving both noise in the reaction process and diffusive transp…
Scaling Limit of Neural Networks with the Xavier Initialization and Convergence to a Global Minimum
Justin Sirignano, Konstantinos Spiliopoulos
We analyze single-layer neural networks with the Xavier initialization in the asymptotic regime of large numbers of hidden units and large numbers of stochastic gradient descent tr…
Typical dynamics and fluctuation analysis of slow-fast systems driven by fractional Brownian motion
Solesne Bourguin, Siragan Gailus, Konstantinos Spiliopoulos
This article studies typical dynamics and fluctuations for a slow-fast dynamical system perturbed by a small fractional Brownian noise. Based on an ergodic theorem with explicit ra…
Metastability and exit problems for systems of stochastic reaction-diffusion equations
Michael Salins, Konstantinos Spiliopoulos
In this paper we develop a metastability theory for a class of stochastic reaction-diffusion equations exposed to small multiplicative noise. We consider the case where the unpertu…
Mean Field Analysis of Deep Neural Networks
Justin Sirignano, Konstantinos Spiliopoulos
We analyze multi-layer neural networks in the asymptotic regime of simultaneously (A) large network sizes and (B) large numbers of stochastic gradient descent training iterations.…
Mean Field Analysis of Neural Networks: A Central Limit Theorem
Justin Sirignano, Konstantinos Spiliopoulos
We rigorously prove a central limit theorem for neural network models with a single hidden layer. The central limit theorem is proven in the asymptotic regime of simultaneously (A)…
Importance sampling for slow-fast diffusions based on moderate deviations
Matthew R. Morse, Konstantinos Spiliopoulos
We consider systems of slow--fast diffusions with small noise in the slow component. We construct provably logarithmic asymptotically optimal importance schemes for the estimation…
Mean Field Analysis of Neural Networks: A Law of Large Numbers
Justin Sirignano, Konstantinos Spiliopoulos
Machine learning, and in particular neural network models, have revolutionized fields such as image, text, and speech recognition. Today, many important real-world applications in…
Stochastic Gradient Descent in Continuous Time: A Central Limit Theorem
Justin Sirignano, Konstantinos Spiliopoulos
Stochastic gradient descent in continuous time (SGDCT) provides a computationally efficient method for the statistical learning of continuous-time models, which are widely used in…
Large deviations and averaging for systems of slow--fast stochastic reaction--diffusion equations
Wenqing Hu, Michael Salins, Konstantinos Spiliopoulos
We study a large deviation principle for a system of stochastic reaction--diffusion equations (SRDEs) with a separation of fast and slow components and small noise in the slow comp…
Discrete-Time Statistical Inference for Multiscale Diffusions
Siragan Gailus, Konstantinos Spiliopoulos
We study statistical inference for small-noise-perturbed multiscale dynamical systems under the assumption that we observe a single time series from the slow process only. We const…
Importance sampling for metastable and multiscale dynamical systems
Konstantinos Spiliopoulos
In this article, we address the issues that come up in the design of importance sampling schemes for rare events associated to stochastic dynamical systems. We focus on the issue o…
Moderate deviations principle for systems of slow-fast diffusions
Matthew R. Morse, Konstantinos Spiliopoulos
In this paper, we prove the moderate deviations principle (MDP) for a general system of slow-fast dynamics. We provide a unified approach, based on weak convergence ideas and stoch…
Stochastic Gradient Descent in Continuous Time
Justin Sirignano, Konstantinos Spiliopoulos
Stochastic gradient descent in continuous time (SGDCT) provides a computationally efficient method for the statistical learning of continuous-time models, which are widely used in…
Rare event simulation via importance sampling for linear SPDE's
Michael Salins, Konstantinos Spiliopoulos
The goal of this paper is to develop provably efficient importance sampling Monte Carlo methods for the estimation of rare events within the class of linear stochastic partial diff…
Dimension Reduction in Statistical Estimation of Partially Observed Multiscale Processes
Andrew Papanicolaou, Konstantinos Spiliopoulos
We consider partially observed multiscale diffusion models that are specified up to an unknown vector parameter. We establish for a very general class of test functions that the fi…
Improving the convergence of reversible samplers
Luc Rey-Bellet, Konstantinos Spiliopoulos
In Monte-Carlo methods the Markov processes used to sample a given target distribution usually satisfy detailed balance, i.e. they are time-reversible. However, relatively recent r…