Séminaire Images Optimisation et Probabilités

Dans cet exposé, je vais m'intéresser au mouvement Brownien dans des cadres simples de géométrie sous riemannienne: le groupe de Heisenberg et les groupes de Carnot de rang 2. Nous proposons une construction d'un couplage de deux mouvement Browniens à un temps fixe. Cette construction est basée sur une décomposition de Legendre du mouvement Brownien standard et de son aire de Lévy. Nous déduisons alors des estimées précises de la décroissance en variation totale entre les lois des mouvements Browniens 

et par une technique de changement de probabilité une formule d'intégration par partie de type Bismut ainsi des estimées de régularisation de type Poincaré inverse pour le semi-groupe associé. Travail en commun avec Marc Arnaudon, Magalie Bénéfice et Delphine Féral

Many problems, especially in machine learning, can be formulated as optimization problems. Using optimization algorithms, such as stochastic gradient descent or ADAM, has become a cornerstone to solve these optimization problems. However for many practical cases, theoretical proofs of their efficiency are lacking. In particular, it has been empirically observed that adding a momentum mechanism to the stochastic gradient descent often allows solving these optimization problems more efficiently. In this talk, we introduce a condition linked to a measure of the gradient correlation that allows to theoretically characterize the possibility to observe this acceleration.

À préciser

Abstact: We examine the long-run distribution of stochastic gradient descent (SGD) in general, non-convex problems. Specifically, we seek to understand which regions of the problem's state space are more likely to be visited by SGD, and by how much. Using an approach based on the theory of large deviations and randomly perturbed dynamical systems, we show that the long-run distribution of SGD resembles the Boltzmann-Gibbs distribution of equilibrium thermodynamics with temperature equal to the method's step-size and energy levels determined by the problem's objective and the statistics of the noise. Joint work w/ W. Azizian, J. Malick, P. Mertikopoulos

https://arxiv.org/abs/2406.09241 published at ICML 2024

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In this presentation, a response matrix (here, species abundances) is assumed to depend on explanatory variables (here, environmental variables) supposed many and redundant, thus demanding dimension reduction. The Supervised Component-based Generalized Linear Regression (SCGLR), a Partial Least Squares-type method, is designed to extract from the explanatory variables several components jointly supervised by the set of responses. However, this methodology still has some limitations we aim to overcome in this work. The first limitation comes from the assumption that all the responses are predicted by the same explanatory space. As a second limitation, the previous works involving SCGLR assume the responses independent conditional on the explanatory variables. Again, this is not very likely in practice, especially in situations like those in ecology, where a non-negligible part of the explanatory variables could not be measured. To overcome the first limitation, we assume that the responses are partitioned into several unknown groups. We suppose that the responses in each group are predictable from an appropriate number of specific orthogonal supervised components of the explanatory variables. The second work relaxes the conditional independence assumption. A set of few latent factors models the residual covariance matrix of the responses conditional on the components. The approaches presented in this work are tested on simulation schemes, and then applied on ecology datasets.

Séminaire joint avec OptimAI.

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