Séance du 9 janvier 2023

Séance organisée par Sylvain Le Corff et Marc Hoffmann

Lieu : IHP, amphi Hermite

14.00 : Chiara Amorino (Université de Luxembourg)

Titre : On estimating interacting particle systems from discrete data

Résumé : We consider the problem of joint parameter estimation for drift and volatility coefficients of a stochastic McKean-Vlasov equation and for the associated system of interacting particles. The analysis is provided in a general framework, as both coefficients depend on the solution of the process and on the law of the solution itself. Starting from discrete observations of the interacting particle system over a fixed interval [0, T], we propose a contrast function based on a pseudo likelihood approach. We show that the associated estimator is consistent when the discretization step ($\Delta_n$) goes to 0 and the number of particles N goes to $\infty$, and asymptotically normal when additionally the condition $\Delta_n N \rightarrow 0$ holds. We will also compare our results (and our condition on the decay of the discretization step) with the results known for classical SDEs.

The talk is based on a joint work with A. Heidari, V. Pilipauskaite and M. Podolskij.

15.00 : Valentin De Bortoli (CNRS - ENS Ulm)

Titre : Diffusion Schrödinger Bridge with Applications to Score-Based Generative Modeling

Résumé : Progressively applying Gaussian noise transforms complex data distributions to approximately Gaussian. Reversing this dynamic defines a generative model. When the forward noising process is given by a Stochastic Differential Equation (SDE), Song et al. (2021) demonstrate how the time inhomogeneous drift of the associated reverse-time SDE may be estimated using score-matching. A limitation of this approach is that the forward-time SDE must be run for a sufficiently long time for the final distribution to be approximately Gaussian. In contrast, solving the Schrödinger Bridge problem (SB), i.e. an entropy-regularized optimal transport problem on path spaces, yields diffusions which generate samples from the data distribution in finite time. We present Diffusion SB (DSB), an original approximation of the Iterative Proportional Fitting (IPF) procedure to solve the SB problem, and provide theoretical analysis along with generative modeling experiments. The first DSB iteration recovers the methodology proposed by Song et al. (2021), with the flexibility of using shorter time intervals, as subsequent DSB iterations reduce the discrepancy between the final-time marginal of the forward (resp. backward) SDE with respect to the prior (resp. data) distribution. Beyond generative modeling, DSB offers a widely applicable computational optimal transport tool as the continuous state-space analogue of the popular Sinkhorn algorithm (Cuturi, 2013).

16.00 : Randal Douc (Telecom SudParis)

Titre : Boost your favorite Markov Chain Monte Carlo sampler using Kac's theorem: the Kick-Kac teleportation algorithm

Résumé : This talk focuses on the problem of sampling from a given target distribution π defined on some general state space. To this end, we introduce a novel class of non-reversible Markov chains, each chain being defined on an extended state space and having an invariant probability measure admitting π as a marginal distribution. The proposed methodology is inspired by a new formulation of Kac's theorem and allows global and local dynamics to be smoothly combined. Under mild conditions, the corresponding Markov transition kernel can be shown to be irreducible and Harris recurrent. In addition, we establish that geometric ergodicity holds under appropriate conditions on the global and local dynamics. Finally, we illustrate numerically the use of the proposed method and its potential benefits in comparison to existing Markov chain Monte Carlo (MCMC) algorithms.