Doctoral Workshop on Decision Mathematics and Statistics

Background and objective 

The objective of the workshop, organized by the Department of Mathematics and Statistics is to foster the interest in the emerging developments in decision mathematics. The workshop provides an opportunity for PhD students, post-doctoral researchers, and faculty members to interact and discuss both theoretical and empirical contributions, with a special focus this year on deep learning, games, optimization, probabilities, and statistics. It will consist in 9 talks and will be held on Thursday June 04, 2026, 9:55-16:30 (UTC+2).

Louison Aubert 
Title: Controlled Swarm Gradient Dynamics for Non-Convex Optimization
Abstract: Global optimization of non-convex potentials remains a fundamental challenge, as classical gradient descent frequently fails in the presence of local minima. While simulated annealing stands out as one of the few methods with rigorous convergence guarantees, it is notoriously slow due to metastability phenomena. A recent approach is Swarm Gradient Dynamics, a Langevin-type diffusion whose noise intensity depends locally on the marginal density of the process, naturally amplifying stochasticity where particles tend to accumulate, in particular near local minima. As the inverse temperature parameter tends to infinity, this family of invariant measures converges weakly to a measure supported on the set of global minimizers of the potential. Building on the controlled simulated annealing framework of Molin et al. (2026), we construct a velocity field satisfying the continuity equation associated with the resulting curve of invariant measures, which can be characterized as the limit of optimal transport maps. Superimposing this vector field onto the original swarm dynamics yields a controlled process whose marginal law exactly follows this curve, with convergence rates governed by the choice of cooling schedule. We propose a particle-based implementation and illustrate its behavior through numerical experiments on non-convex functions.

Colombe Becquart 
Title: Revisiting and Enhancing the Univariate Folding Test of Unimodality
Abstract:  Determining whether a distribution is unimodal or multimodal is a fundamental problem in statistical analysis, particularly in the context of clustering, where the presence of multiple modes is often interpreted as evidence of underlying cluster structure. In the univariate case, a standard reference for testing unimodality is Hartigan’s dip test, which is based on the cumulative distribution function. Another commonly used test is Silverman’s test, relying on kernel density estimation. The Folding Test of Unimodality (FTU) proposed by Siffer et al. offers an appealing alternative: it is computationally efficient, easy to implement, and particularly well suited to streaming data. Its rationale is that folding a multimodal distribution greatly reduces itsvariance, whereas the reduction is much smaller for a unimodal distribution. This presentation addresses a key limitation of the Folding Test of Unimodality (FTU).
In specific univariate mixture settings, the folding-based criterion can systematically fail, misclassifying clearly multimodal distributions as unimodal. We fully characterize these failures for Dirac mixtures and extend the analysis to Gaussian mixtures. We then introduce a double-folding procedure that captures complementary information, leading to a new test, the Double Folding Test of Unimodality. It resolves the FTU failures and improves multimodality detection power in simulations.

Margot Ferré 
Title: Estimation of Sobol' Indices Using Stochastic Algorithms
Abstract: Sobol' indices make it possible to quantify the sensitivity of a random model with respect to its different inputs. The problem of their estimation falls within the broader framework of global sensitivity analysis. This presentation will discuss several approaches for estimating Sobol' indices. We will compare the convergence properties of the classical Pick-Freeze estimator and the estimator obtained via stochastic gradient descent, illustrating them numerically on a few examples. We will introduce the concept of mirror stochastic gradient descent, which enables the simultaneous estimation of all Sobol' indices, and we will discuss the convergence guarantees of this algorithm.

Melissa Gonzalez Garcia
Title: 1-player Percolation Games
Abstract: Percolation games, introduced in 2023 by Garnier and Ziliotto, are a class of zero-sum stochastic games with an infinite state space and random payoffs revealed to both players in advance. In these games, players alternately move a token along the vertices of a d-dimensional integer lattice. The objective of player 1 in an n-stage game is to maximize the average payoff over n stages. A central question in percolation games is whether the sequence of n-stage game values converges to a deterministic limit value as the number of steps approaches infinity. In this work, we focus on the 1-player version of the model. We prove the existence of a uniform value and 0-optimal strategies and further establish the existence of a stationary 0-optimal strategy in the oriented case. Additionally, we present interesting examples of oriented games that are closely related to classical percolation models. These examples provide valuable insights into the structural complexity of optimal strategies.

Joseph Hachem
Title: A heterogeneous extreme value theory for multivariate heavy-tailed data 
Abstract: In this talk, we present an extension of the derandomization technique, originally developed for univariate heterogeneous heavy-tailed data to estimate several extreme risks jointly, to the multivariate setting where the data differ in nature and are collected from multiple sources. When the same simple sufficient conditions as in the univariate case hold across components, we derive the joint asymptotic distribution of so-called tail empirical excesses and Expected Shortfall with the underlying threshold level. In particular, when the observations are independent across components and their average distribution is heavy-tailed, we establish joint asymptotic normality results for estimators of the tail index and extreme risk measures, including the Hill estimator of the tail index, the Weissman estimator of extreme quantiles, and two extrapolated Expected Shortfall estimators above an extreme level. This novel multivariate heterogeneous extreme value framework requires the heavy-tailed samples across components to have proportional sizes and to verify a classical tail correlation condition. We apply our results to scenarios involving pooled samples from several tail-correlated populations.

Solal Martin
Title: Lyapunov Functions in Optimization: from Verification to Construction
Abstract: Lyapunov functions provide convergence certificates for optimization algorithms: exhibiting a quantity that decreases along trajectories is sufficient to conclude. While verifying a candidate is straightforward, constructing such a function is a difficult task for which no general method exists. This talk explores this asymmetry through the lens of optimization. We first illustrate the variety of Lyapunov structures that arise for different classes of systems, autonomous or not, revealing the complexity that their discovery represents. We then present three approaches to address this construction: understanding the geometry of the problem through Hessian metrics induced by Legendre functions (Alvarez, Bolte, Brahic, 2004); automating the search in the discrete setting by formulating the existence of a quadratic Lyapunov function as a semidefinite feasibility problem, in the spirit of performance estimation problems (Taylor, Van Scoy, Lessard, 2018); and a more recent direction leveraging sequence-to-sequence language models trained on synthetic data, capable of discovering global Lyapunov functions for systems for which no algorithm is known (Alfarano, Charton, Hayat, 2024).

Camille Mondon
Title:  Identifying the discriminant subspace using Invariant Coordinate Selection: a general criterion based on group proportions
Abstract:  Invariant coordinate selection (ICS) is defined by David E. Tyler as the joint diagonalisation of two scatter matrices, such as the covariance and a kurtosis matrix based of fourth-order moments. His main result is that for some mixtures of elliptical distributions, the discriminant subspace is identifiable from the eigenvalues of ICS, provided a ‘threshold’ eigenvalue has the correct multiplicity.
In the noise-free and assuming affine independence of the group centers, we find this threshold eigenvalue and an explicit criterion for its multiplicity to be minimal, based on the group proportions alone. More generally, we perform an analytic solve of ICS using COV-COV4 under these hypotheses, which retrieves numerous particular cases previously studied in the literature.
Using symbolic computations, we add noise within the groups or collinearity between the group centers and observe that in these cases the criterion depends on the locations of groups and the shape of the noise.

Léo Portales
Title: Statistical Estimation of Monge Transport Maps via Brenier Potentials
Abstract: We introduce and analyze a statistical estimator for Monge transport maps: solutions to the quadratic optimal transport problem in the Euclidean space. For absolutely continuous source measures, this map is uniquely defined as the gradient of a convex function, a result known as Brenier’s theorem. Without absolute continuity, the problem is relaxed, maps are replaced by coupling measures, and optimal couplings are supported on the subdifferential of a convex function, a Brenier potential. This characterization is the basis for our Monge transport map statistical estimator, for measures known only through finite samples. The resulting Brenier potential has a simple closed form expression based on the dual solution of the discrete sampled problem. In particular, our methodology does not rely on smoothness or continuity of the Monge transport map and requires no computation beyond primal-dual solutions of the discrete finite dimensional problem.
We exhibit convergence rates for this estimator based on a new error bound for the quadratic optimal transport problem. In the semi-discrete setting, where the target measure is finitely supported, our estimator enjoys sharper convergence rates. Finally, using similar proof techniques, we provide novel convergence rate for empirical couplings.

Claire Lebrun
Title:  Interconnected Narratives: Network Measures of Systemic Risk from Written News
Abstract:  Motivated by the global financial crisis, which demonstrated how distress propagates through interconnected financial networks, this paper studies the interconnectedness of financial institutions in daily news networks to construct a novel systemic risk indicator. Daily written news is modelled as a dynamic, weighted and directed network from a large-scale textual dataset provided by the Causality Link AI platform. Networks represent written news sentences that suggest causal relationships (edges), between key indicators of companies, industries, countries, and events (nodes).  Daily network centrality measures how central a node is in the network, a bank with high centrality in stress periods might indicate it has become entangled in crisis narratives. The systemic risk measure is constructed as the expected change in a bank's narrative centrality conditional on market distress and estimated with a GARCH-DCC model. The indicator closely co-moves with, and leads market-based systemic risk during major stress episodes. Eigenvector centrality emerges as the most informative centrality measure as it identifies institutions connected to other central nodes, capturing banks discussed with macroeconomic indicators, implying the risk is starting to be perceived as global and transmitting across the economy.