Séminaire commun Mathrisk- INRIA/ LPSM Paris-Diderot

Jeudi 17 octobre : 9h 12:30

Salle 209 couloir 16-26 à Jussieu.

Zhenjie Ren (CEREMADE, Univ. Paris-Dauphine)

Titre : Mean-field Langevin system, optimal control and deep neural networks

Résumé : The deep neural network has achieved impressive results in various applications, and is involved in more and more branches of science. However, there are still few theories supporting its empirical success. In particular, we miss the mathematical tool to analyse the advantage of certain structures of the network, and to have quantitive error bounds. In our recent work, we used a regularised relaxed control problem to model the deep neural network. We managed to characterise its optimal control by the invariant measure of a mean-field Langevin system, which can be approximated by the marginal laws. Through this study we understand the importance of the pooling for the deep nets, and are capable of computing an exponential convergence rate for the (stochastic) gradient descent algorithm.

Stéphane Menozzi (LAMME, Univ. Evry)

Titre : Well-Posedness of Some Non-Linear Stable Driven SDES

Résumé : We prove the well-posedness of some non-linear stochastic differential equations in the sense of McKean- Vlasov driven by non-degenerate symmetric α-stable Lévy processes with values in Rd under some mild Hölder regularity assumptions on the drift

and diffusion coefficients with respect to both space and measure variables. The methodology developed here allows to consider unbounded drift terms even in the so-called super-critical case, i.e. when the stability index α ∈ (0, 1).

New strong well-posedness results are also derived from the previous analysis.

(joint work with N. Firkha and V. Konakov)

Frédéric Bonnans (INRIA/CMAP)

Titre : Schauder Estimates for a Class of Potential Mean Field Games of Controls

Travail commun avec Saeed Hadikhanloo et Laurent Pfeiffer (Cmap et Inria)

Résumé : An existence result for a class of mean field games of controls is provided. In the considered model, the cost functional to be minimized by each agent involves a price depending at a given time on the controls of all agents and a congestion term. The existence of a classical solution is demonstrated with the Leray–Schauder theorem; the proof relies in particular on a priori bounds for the solution, which are obtained with the help of a potential formulation of the problem.

Thibaut Mastrolia (CMAP, Ecole Polytechnique)

Titre : Regulation of natural resource exploitation

Résumé : We investigate the impact of a regulation policy imposed on an agent exploiting a possibly renewable natural resource. We adopt a principal-agent model in which the Principal looks for a contract, i.e. taxes/compensations, leading the Agent to a certain level of exploitation. For a given contract, we first describe the Agent’s optimal harvest using the BSDE theory. Under regularity and boundedness assumptions on the coefficients, we express almost optimal contracts as solutions to HJB equations. We then extend the result to coefficients with less regularity and logistic dynamics for the natural resource. We end by numerical examples to illustrate the impact of the regulation in our model.