Séminaire des doctorant·es SO

par Iyad Zekhnini (IMT), Jeremy Boyer (IMT), Lucas Monteiro (IMT & ONERA)

mardi 9 déc. 2025, 11:15 → 12:15 Europe/Paris

Salle K. Johnson (1R3, 1er étage)

Lucas Monteiro (IMT & ONERA)

Reliability-based Shapley effects estimation with Normalizing Flows 

When studying critical systems, such as those used in nuclear or aerospace engineering, it is crucial to understand why and how failures occur. At the intersection of sensitivity analysis and reliability analysis lies reliability sensitivity analysis, which aims to quantify the role played by each variable in the occurrence of a failure. In cases where the variables are correlated, reliability-based Shapley indices are used. However, their estimation is limited to systems of dimension less than 10. We propose a new estimation scheme for these indices to higher dimensions using Normalizing Flows, a powerful tool derived from generative modeling. In addition, to manage the dimension, we use a particular writing of Shapley indices as an expectation based on permutations, allowing approximation. To quantify the error made by the different approximations, we propose a procedure providing bounds. Finally, we illustrate the performance our method on numerical applications.

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Iyad Zekhnini (IMT)

How to handle non-convex optimization problems to obtain concentration guarantees? 

Training machine learning models often results in minimizing the error between the observed data and the model's predictions. Such a problem is called Empirical Risk Minimization and a huge literature provides guarantees that the solutions of this problem get close to the ones of the True Risk (i.e. the expected error over the unknown data distribution) for convex losses. However, many modern machine learning models lead to non-convex risks (e.g. neural networks), for which very few concentration guarantees exist. 

Recent works made progress in that direction by relying on restrictive assumptions over the geometry of the risk functions. In this talk, we will provide concentrations results for the empirical risk minimizers in the non-convex setting by combining tools from functional analysis and non-parametric statistics.

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Jeremy Boyer (IMT)

Non stationary empirical processes : Detection and Estimation of time dependent mixtures

It is standard in inferential statistics to assume that our sample $X_1, ..., X_n$ consists of independent and identically distributed random variables. The objective of this work is to remove the assumption of stationarity of the distribution while preserving independence. We assume that $X_i$ has distribution $\mu_{i/n}$ and we are interested, for a given class of functions $\mathcal{F}$, in the empirical measure $P_n(f)=n^{-1}\sum_{i=1}^n f(X_i)$. Under regularity assumptions on $t \mapsto \mu_t$ and entropy conditions for $\mathcal{F}$, we obtain the weak convergence of the associated empirical process, $\Gamma_n=\sqrt{n}(P_n-\overline{\mu}_n)$, centered at $\overline{\mu}_n=n^{-1}\sum_{i=1}^n \mu_{i/n}$, towards an explicit Gaussian process. The aim of this presentation is to apply these results to the study of time-dependent mixture models. We propose a test to determine the order of the mixture and a second test to estimate the mixture coefficients.