Séance du 13 mars 2023

Séance organisée par Anna Korba et Umut Şimşekli 

Lieu : IHP,  amphi Hermite

14.00 :  Markus Reiss (Humboldt-Universität)

Titre :  Nonparametric Statistics for SPDEs

Résumé : Stochastic partial differential equations (SPDEs) are used more and more often to model real-world phenomena. Currently, statistical methodology for these equations driven by space-time white noise is developing rapidly. Based on the classical spectral method for parametric drift estimation, we shall exhibit fundamental differences with the case of stochastic ordinary differential equations. This method, however, is restricted to simple parametric situations and we discuss the local estimation method in detail, which allows to estimate varying coefficients in the differential operator of a parabolic SPDE nonparametrically with optimal rates. This approach is extended to observations under measurement errors ('static noise'), showing a fundamentally different impact of dynamic and static noise levels.  Finally, we present an abstract minimax lower bound framework for stochastic evolution equations generated by normal operators in Hilbert space and obtain a rich picture of complexity for different SPDE estimation objectives. Some illustrations with cell motility experiments in biophysics are provided.

15.00 :  Gabriel Stoltz (Ecole des Ponts et Inria Paris)

Titre :  Removing the mini-batching error in Bayesian inference using Adaptive Langevin dynamics

Résumé : The computational cost of usual Monte Carlo methods for sampling a posteriori laws in Bayesian inference scales linearly with the number of data points. One option to reduce it to a fraction of this cost is to resort to mini-batching to estimate the gradient. However, this leads to an additional noise in the dynamics and hence a bias on the invariant measure which is sampled by the Markov chain. We advocate using the so-called Adaptive Langevin dynamics, which is a modification of standard inertial Langevin dynamics with a dynamical friction which automatically corrects for the increased noise arising from mini-batching. We show using techniques from hypocoercivity that the law of Adaptive Langevin dynamics converges exponentially fast to equilibrium, with a rate which can be quantified in terms of the key parameters of the dynamics (mass of the extra variable and magnitude of the fluctuation in the Langevin dynamics). This allows us in particular to obtain a Central Limit Theorem on time averages along realizations of the dynamics. We also investigate the practical relevance of the assumptions underpinning Adaptive Langevin (constant covariance for the estimation of the gradient), which are not satisfied in typical models of Bayesian inference; and show how to extend the approach to more general situations. Applications and extensions to Bayesian Neural Networks will also be discussed.

16.00 :  Francesca Crucinio (CREST)

Titre : Solving Fredholm Integral Equations of the First Kind via Wasserstein Gradient Flows

Résumé :  Solving Fredholm equations of the first kind is crucial in many areas of the applied sciences. In this work we adopt a probabilistic and variational point of view by considering a minimization problem in the space of probability measures with an entropic regularization. Contrary to classical approaches which discretize the domain of the solutions, we introduce an algorithm to asymptotically sample from the unique solution of the regularized minimization problem. As a result our estimators do not depend on any underlying grid and have better scalability properties than most existing methods.  Our algorithm is based on a particle approximation of the solution of a McKean--Vlasov stochastic differential equation associated with the Wasserstein gradient flow of our variational formulation. We prove the convergence towards a minimizer and provide practical guidelines for its numerical implementation. Finally, our method is compared with other approaches on several examples including density deconvolution and epidemiology.

Joint work with Valentin De Bortoli, Arnaud Doucet, Adam M. Johansen.