Publications

2025

• Surface Waves in Randomly Perturbed Discrete Models
  
  • Garnier Josselin
  • Sharma Basant Lal
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2025, 23 (1), pp.158-186. (10.1137/24M165510X)
  DOI : 10.1137/24M165510X
• An exterior optimal transport problem
  
  • Candau-Tilh Jules
  • Goldman Michael
  • Merlet Benoît
  Calculus of Variations and Partial Differential Equations, Springer Verlag, 2025, 64 (2), pp.45. This paper deals with a variant of the optimal transportation problem. Given f ∈ L 1 (R d , [0, 1]) and a cost function c ∈ C(R d × R d) of the form c(x, y) = k(y − x), we minimise ∫ c dγ among transport plans γ whose first marginal is f and whose second marginal is not prescribed but constrained to be smaller than 1 − f. Denoting by Υ(f) the infimum of this problem, we then consider the maximisation problem sup{Υ(f) : ∫ f = m} where m > 0 is given. We prove that maximisers exist under general assumptions on k, and that for k radial, increasing and coercive these maximisers are the characteristic functions of the balls of volume m. (10.1007/s00526-024-02900-8)
  DOI : 10.1007/s00526-024-02900-8
• Polynomial approximations in a generalized Nyman–Beurling criterion
  
  • Alouges François
  • Darses Sébastien
  • Hillion Erwan
  , 2023, pp.767 - 785. The Nyman-Beurling criterion, equivalent to the Riemann hypothesis, is an approximation problem in the space of square integrable functions on $(0,\infty)$, involving dilations of the fractional part function by factors $\theta_k\in(0,1)$, $k\ge1$. Randomizing the $\theta_k$ generates new structures and criteria. One of them is a sufficient condition that splits into (i) showing that the indicator function can be approximated by convolution with the fractional part, (ii) a control on the coefficients of the approximation. This self-contained paper aims at identifying functions for which (i) holds unconditionally, by means of polynomial approximations. This yields in passing a short probabilistic proof of a known consequence of Wiener's Tauberian theorem. In order to tackle (ii) in the future, we give some expressions of the scalar products. New and remarkable structures arise for the Gram matrix, in particular moment matrices for a suitable weight that may be the squared $\Xi$-function for instance. (10.5802/jtnb.1227)
  DOI : 10.5802/jtnb.1227
• Inexact subgradient methods for semialgebraic functions
  
  • Bolte Jérôme
  • Le Tam
  • Moulines Éric
  • Pauwels Edouard
  Mathematical Programming, Series A, Springer, 2025. Motivated by the extensive application of approximate gradients in machine learning and optimization, we investigate inexact subgradient methods subject to persistent additive errors. Within a nonconvex semialgebraic framework, assuming boundedness or coercivity, we establish that the method yields iterates that eventually fluctuate near the critical set at a proximity characterized by an $O(\epsilon^\rho)$ distance, where $\epsilon$ denotes the magnitude of subgradient evaluation errors, and $\rho$ encapsulates geometric characteristics of the underlying problem. Our analysis comprehensively addresses both vanishing and constant step-size regimes. Notably, the latter regime inherently enlarges the fluctuation region, yet this enlargement remains on the order of $\epsilon^\rho$. In the convex scenario, employing a universal error bound applicable to coercive semialgebraic functions, we derive novel complexity results concerning averaged iterates. Additionally, our study produces auxiliary results of independent interest, including descent-type lemmas for nonsmooth nonconvex functions and an invariance principle governing the behavior of algorithmic sequences under small-step limits. (10.1007/s10107-025-02245-w)
  DOI : 10.1007/s10107-025-02245-w
• L'IA générative : une révolution technologique entre promesses et défis
  
  • Moulines Éric
  Réalités industrielles. Annales des mines, Eska, 2025 (2), pp.7–10. (10.3917/rindu1.252.0007)
  DOI : 10.3917/rindu1.252.0007
• Constructive proofs for some semilinear PDEs on $H^2(e^{|x|^2/4},\mathbb{R}^d)$
  
  • Breden Maxime
  • Chu Hugo
  Numerische Mathematik, Springer Verlag, 2025, 157 (6), pp.2097-2143. We develop computer-assisted tools to study semilinear equations of the form \begin{equation*} -\Delta u -\frac{x}{2}\cdot \nabla{u}= f(x,u,\nabla u) ,\quad x\in\mathbb{R}^d. \end{equation*} Such equations appear naturally in several contexts, and in particular when looking for self-similar solutions of parabolic PDEs. We develop a general methodology, allowing us not only to prove the existence of solutions, but also to describe them very precisely. We introduce a spectral approach based on an eigenbasis of $\mathcal{L}:= -\Delta -\frac{x}{2}\cdot \nabla$ in spherical coordinates, together with a quadrature rule allowing to deal with nonlinearities, in order to get accurate approximate solutions. We then use a Newton--Kantorovich argument, in an appropriate weighted Sobolev space, to prove the existence of a nearby exact solution. We apply our approach to nonlinear heat equations, to nonlinear Schrödinger equations and to a generalised viscous Burgers equation, and obtain both radial and non-radial self-similar profiles. (10.1007/s00211-025-01504-4)
  DOI : 10.1007/s00211-025-01504-4
• Kinetic theory and moment models of electrons in a reactive weakly-ionized non-equilibrium plasma
  
  • Laguna Alejandro Alvarez
  • Pichard Teddy
  Kinetic and Related Models, AIMS, 2025. <div><p>We study the electrons in a multi-component weakly-ionized plasma with an external electric field under conditions that are far from thermodynamic equilibrium, representative of a gas discharge plasma. Our starting point is the generalized Boltzmann equation with elastic, inelastic and reactive collisions. We perform a dimensional analysis of the equation and an asymptotic analysis of the collision operators for small electron-to-atom mass ratios and small ionization levels. The dimensional analysis leads to a diffusive scaling for the electron transport. We perform a Hilbert expansion of the electron distribution function that, in the asymptotic limit, results in a reduced model characterized by a spherically symmetric distribution function in the velocity space with a small anisotropic perturbation. We show that the spherical-harmonics expansion model, widely used in low-temperature plasmas, is a particular case of our approach. We approximate the solution of our kinetic model with a truncated moment hierarchy. Finally, we study the moment problem for a particular case: a Langevin collision (equivalent to Maxwell molecules) for the electron-gas elastic collisions. The resulting Stieltjes moment problem leads to an advection-diffusion-reaction system of equations that is approximated with two different closures: the quadrature method of moments and a Hermitian moment closure. A special focus is given along the derivations and approximations to the notion of entropy dissipation.</p></div> (10.3934/krm.2025007)
  DOI : 10.3934/krm.2025007
• Macroscopic limit from a structured population model to the Kirkpatrick-Barton model
  
  • Raoul Gaël
  Bulletin des Sciences Mathématiques, Elsevier, 2025, 205, pp.103697. We consider an ecology model in which the population is structured by a spatial variable and a phenotypic trait. The model combines a parabolic operator on the spatial variable with a kinetic operator on the trait variable. We prove the existence of solutions to that model, and show that these solutions are unique. The kinetic operator present in the model, that represents the effect of sexual reproductions, satisfies a Tanaka-type inequality: it implies a contraction of the Wasserstein distance in the space of phenotypic traits. We combine this contraction argument with parabolic estimates controlling the spatial regularity of solutions to prove the convergence of the population size and the mean phenotypic trait to solutions of the Kirkpatrick-Barton model, which is a well-established model in evolutionary ecology. Specifically, at high reproductive rates, we provide explicit convergence estimates for the moments of solutions of the kinetic model. (10.48550/arXiv.1706.04094)
  DOI : 10.48550/arXiv.1706.04094
• An analysis of the noise schedule for score-based generative models
  
  • Strasman Stanislas
  • Ocello Antonio
  • Boyer Claire
  • Le Corff Sylvain
  • Lemaire Vincent
  Transactions on Machine Learning Research, [Amherst Massachusetts]: OpenReview.net, 2022, 2025. Score-based generative models (SGMs) aim at estimating a target data distribution by learning score functions using only noise-perturbed samples from the target. Recent literature has focused extensively on assessing the error between the target and estimated distributions, gauging the generative quality through the Kullback-Leibler (KL) divergence and Wasserstein distances. Under mild assumptions on the data distribution, we establish an upper bound for the KL divergence between the target and the estimated distributions, explicitly depending on any time-dependent noise schedule. Under additional regularity assumptions, taking advantage of favorable underlying contraction mechanisms, we provide a tighter error bound in Wasserstein distance compared to state-of-the-art results. In addition to being tractable, this upper bound jointly incorporates properties of the target distribution and SGM hyperparameters that need to be tuned during training. Finally, we illustrate these bounds through numerical experiments using simulated and CIFAR-10 datasets, identifying an optimal range of noise schedules within a parametric family.
• Development of a rabbit model of uterine rupture after caesarean section, Histological, biomechanical and polarimetric analysis of the uterine tissue
  
  • Debras Elodie
  • Maudot Constance
  • Allain Jean-Marc
  • Pierangelo Angelo
  • Courilleau Aymeric
  • Rivière Julie
  • Dahirel Michèle
  • Richard Christophe
  • Gelin Valérie
  • Morin Gwendoline
  • Capmas Perrine
  • Chavatte-Palmer Pascale
  Reproduction & Fertility, Bioscientifica Ltd, 2025, 6 (4), pp.e-250018. Uterine rupture is a major complication of caesarean section (CS) associated with a high foetal and maternal morbidity. The objective is to develop an in-vivo model of uterus healing and rupture after CS in order to analyse histological phenomena controlling scarring tissue development and potential cause of defects. Eighteen pregnant primiparous female rabbits were bred naturally. At caesarean, after 28 days of gestation, foetuses were either extracted through a longitudinal incision in one of the uterine horns (“CS horn”) or via a short incision at the tip of the contralateral horn (“control horn”). The uterine horns were sutured by single layer, all by the same surgeon. They were mated again 14 days later and euthanized at G28. Genital tracts were collected for histological, biomechanical and polarimetric analyses. Macroscopically, 2/18 presented a dehiscence and 1/18 a spontaneous rupture. The mean thickness of the scarred area was significantly lower 0.9 mm [0.7-1.4] that the non-scarring area on CS horns 2.2 [1.6-2.3] or control horns 2 [1.5-2.3] (p&lt;0.0001). The scar zone was statistically more fibrous (p&lt;0.0001), containing fewer vessels (p=0.03) and oestrogen (p&lt;0.001) and progesterone receptors (p&lt;0.0001). After balloon inflation, ruptured occurred in the scar zone in 8 out of 17 cases (47%). Polarimetry revealed that the scar zone was statistically inhomogeneous (73%). Multifactorial analysis allowed to identify groups with poor uterine healing and less resistant to rupture (balloon inflation) mostly in case of thin myometrium in the scar and a group with strong resistant to rupture and correct healing characteristics. Lay summary Caesarean section rates are rising across the world. When a caesarean section is carried out, it can lead to scarring on the uterus that can affect its resistance to pressure. During the next pregnancy, the uterus can tear, increasing risks to the mother and baby. We carried out caesarean sections in a rabbit, allowing us to analyse the scar on the uterus, the healing and tissue resistance. The scarred part of the uterus was statistically thinner, more fibrous and contained fewer vessels and hormone receptors than the area without scarring. Under similar conditions, poor healing was observed in some animals, reducing resistance in following pregnancies. These results suggest that individual and genetic factor have an effect on healing after a caesarean section. This study may enable us to improve our knowledge and management care for patients who have a caesarean section in order to reduce complications. (10.1530/raf-25-0018)
  DOI : 10.1530/raf-25-0018
• Solving inverse source wave problem from Carleman estimates to observer design
  
  • Boulakia Muriel
  • de Buhan Maya
  • Delaunay Tiphaine
  • Imperiale Sébastien
  • Moireau Philippe
  Mathematical Control and Related Fields, AIMS, 2025. In this work, we are interested by the identification in a wave equation of a space dependent source term multiplied by a known time and space dependent function, from internal velocity or field measurements. The first part of the work consists in proving stability inequalities associated with this inverse problem from adapted Carleman estimates. Then, we present a sequential reconstruction strategy which is proved to be equivalent to the minimization of a cost functional with Tikhonov regularization. Based on the obtained stability estimates, the reconstruction error is evaluated with respect to the noise intensity. Finally, the proposed method is illustrated with numerical simulations, both in the case of regular source terms and of piecewise constant source terms. (10.3934/mcrf.2025007)
  DOI : 10.3934/mcrf.2025007
• From Stochastic Zakharov System to Multiplicative Stochastic Nonlinear Schrödinger Equation
  
  • Barrué Grégoire
  • de Bouard Anne
  • Debussche Arnaud
  Stochastics and Partial Differential Equations: Analysis and Computations, Springer US, 2025, pp.1-40. We study the convergence of a Zakharov system driven by a time white noise, colored in space, to a multiplicative stochastic nonlinear Schrödinger equation, as the ion-sound speed tends to infinity. In the absence of noise, the conservation of energy gives bounds on the solutions, but this evolution becomes singular in the presence of the noise. To overcome this difficulty, we show that the problem may be recasted in the diffusion-approximation framework, and make use of the perturbed test-function method. We also obtain convergence in probability. The result is limited to dimension one, to avoid too much technicalities. As a prerequisite, we prove the existence and uniqueness of regular solutions of the stochastic Zakharov system.
• Dynamics of a kinetic model describing protein exchanges in a cell population
  
  • Magal Pierre
  • Raoul Gaël
  Journal of Mathematical Biology, Springer, 2025, 91 (6), pp.76. We consider a cell population structured by a positive real number, describing the number of P-glycoproteins carried by the cell. We are interested in the effect of those proteins on the growth of the population: those proteins are indeed involve in the resistance of cancer cells to chemotherapy drugs. To describe this dynamics, we introduce a kinetic model. We then introduce a rigorous hydrodynamic limit, showing that if the exchanges are frequent, then the dynamics of the model can be described by a system of two coupled differential equations. Finally, we also show that the kinetic model converges to a unique limit in large times. The main idea of this analysis is to use Wasserstein distance estimates to describe the effect of the kinetic operator, combined to more classical estimates on the macroscopic quantities. (10.1007/s00285-025-02295-w)
  DOI : 10.1007/s00285-025-02295-w
• A holographic global uniqueness in passive imaging
  
  • Novikov Roman
  Journal de l'École polytechnique — Mathématiques, École polytechnique, 2025, 12, pp.1069-1081. We consider a radiation solution $\psi$ for the Helmholtz equation in an exterior region in $\mathbb R^3$. We show that the restriction of $\psi$ to any ray $L$ in the exterior region is uniquely determined by its imaginary part Im $\psi$ on an interval of this ray. As a corollary, the restriction of $\psi$ to any plane $X$ in the exterior region is uniquely determined by Im $\psi$ on an open domain in this plane. These results have holographic prototypes in the recent work Novikov (2024, Proc. Steklov Inst. Math. 325, 218-223). In particular, these and known results imply a holographic type global uniqueness in passive imaging and for the Gelfand-Krein-Levitan inverse problem (from boundary values of the spectral measure in the whole space) in the monochromatic case. Some other surfaces for measurements instead of the planes $X$ are also considered. (10.5802/jep.306)
  DOI : 10.5802/jep.306
• Integrating Aggregated Electric Vehicle Flexibilities in Unit Commitment Models using Submodular Optimization
  
  • Arvis Hélène
  • Beaude Olivier
  • Gast Nicolas
  • Gaubert Stéphane
  • Gaujal Bruno
  , 2025. <div><p>The Unit Commitment (UC) problem consists in controlling a large fleet of heterogeneous electricity production units in order to minimize the total production cost while satisfying consumer demand. Electric Vehicles (EVs) are used as a source of flexibility and are often aggregated for problem tractability. We develop a new approach to integrate EV flexibilities in the UC problem and exploit the generalized polymatroid structure of aggregated flexibilities of a large population of users to develop an exact optimization algorithm, combining a cutting-plane approach and submodular optimization. We show in particular that the UC can be solved exactly in a time which scales linearly, up to a logarithmic factor, in the number of EV users when each production unit is subject to convex constraints. We illustrate our approach by solving a real instance of a long-term UC problem, combining open-source data of the European grid (European Resource Adequacy Assessment project) and data originating from a survey of user behavior of the French EV fleet.</p></div>
• Parameters estimation of a Threshold CKLS process from continuous and discrete observations
  
  • Mazzonetto Sara
  • Nieto Benoît
  Scandinavian Journal of Statistics, Wiley, 2025, 52 (4), pp.1670-1707. We consider a continuous time process that is self-exciting and ergodic, called threshold Chan–Karolyi–Longstaff–Sanders (CKLS) process. This process is a generalization of various models in econometrics, such as Vasicek model, Cox-Ingersoll-Ross, and Black-Scholes, allowing for the presence of several thresholds which determine changes in the dynamics. We study the asymptotic behavior of maximum-likelihood and quasi-maximum-likelihood estimators of the drift parameters in the case of continuous time and discrete time observations. We show that for high frequency observations and infinite horizon the estimators satisfy the same asymptotic normality property as in the case of continuous time observations. We also discuss diffusion coefficient estimation. Finally, we apply our estimators to simulated and real data to motivate considering (multiple) thresholds. (10.1111/sjos.70005)
  DOI : 10.1111/sjos.70005
• From random matrices to systems of particles in interaction
  
  • Pesce Valentin
  , 2025. The goal of these expository notes is to give an introduction to random matrices for non-specialist of this topic focusing on the link between random matrices and systems of particles in interaction. We first recall some general results about the random matrix theory that create a link between random matrices and systems of particles through the knowledge of the law of the eigenvalues of certain random matrices models. We next focus on a continuous in time approach of random matrices called the Dyson Brownian motion. We detail some general methods to study the existence of system of particles in singular interaction and the existence of a mean field limit for these systems of particles. Finally, we present the main result of large deviations when studying the eigenvalues of random matrices. This method is based on the fact that the eigenvalues of certain models of random matrices can be viewed as log gases in dimension 1 or 2.
• Tensor rectifiable G-flat chains
  
  • Goldman Michael
  • Merlet Benoît
  Transactions of the American Mathematical Society, American Mathematical Society, 2025. A rigidity result for normal rectifiable $k$-chains in $\mathbb{R}^n$ with coefficients in an Abelian normed group is established. Given some decompositions $k=k_1+k_2$, $n=n_1+n_2$ and some rectifiable $k$-chain $A$ in $\mathbb{R}^n$, we consider the properties: (1) The tangent planes to $\mu_A$ split as $T_x\mu_A=L^1(x)\times L^2(x)$ for some $k_1$-plane $L^1(x)\subset\mathbb{R}^{n_1}$ and some $k_2$-plane $L^2(x)\subset\mathbb{R}^{n_2}$. (2) $A=A_{\vert\Sigma^1\times\Sigma^2}$ for some sets $\Sigma^1\subset\mathbb{R}^{n_1}$, $\Sigma^2\subset\mathbb{R}^{n_2}$ such that $\Sigma^1$ is $k_1$-rectifiable and $\Sigma^2$ is $k_2$-rectifiable (we say that $A$ is $(k_1,k_2)$-rectifiable). The main result is that for normal chains, (1) implies (2), the converse is immediate. In the proof we introduce the new groups of tensor flat chains (or $(k_1,k_2)$-chains) in $\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}$ which generalize Fleming's $G$-flat chains. The other main tool is White's rectifiable slices theorem. We show that on the one hand any normal rectifiable chain satisfying~(1) identifies with a normal rectifiable $(k_1,k_2)$-chain and that on the other hand any normal rectifiable $(k_1,k_2)$-chain is $(k_1,k_2)$-rectifiable. (10.1090/tran/9392)
  DOI : 10.1090/tran/9392
• Smoothness estimation for Whittle–Matérn processes on closed Riemannian manifolds
  
  • Korte-Stapff Moritz
  • Karvonen Toni
  • Moulines Éric
  Stochastic Processes and their Applications, Elsevier, 2025, 189, pp.104685. (10.1016/j.spa.2025.104685)
  DOI : 10.1016/j.spa.2025.104685
• Universal complexity bounds based on value iteration for stochastic mean payoff games and entropy games
  
  • Allamigeon Xavier
  • Gaubert Stéphane
  • Katz Ricardo
  • Skomra Mateusz
  Information and Computation, Elsevier, 2025, 302, pp.105236. We develop value iteration-based algorithms to solve in a unified manner different classes of combinatorial zero-sum games with mean-payoff type rewards. These algorithms rely on an oracle, evaluating the dynamic programming operator up to a given precision. We show that the number of calls to the oracle needed to determine exact optimal (positional) strategies is, up to a factor polynomial in the dimension, of order R/sep, where the “separation” sep is defined as the minimal difference between distinct values arising from strategies, and R is a metric estimate, involving the norm of approximate sub and super-eigenvectors of the dynamic programming operator. We illustrate this method by two applications. The first one is a new proof, leading to improved complexity estimates, of a theorem of Boros, Elbassioni, Gurvich and Makino, showing that turn-based mean-payoff games with a fixed number of random positions can be solved in pseudo-polynomial time. The second one concerns entropy games, a model introduced by Asarin, Cervelle, Degorre, Dima, Horn and Kozyakin. The rank of an entropy game is defined as the maximal rank among all the ambiguity matrices determined by strategies of the two players. We show that entropy games with a fixed rank, in their original formulation, can be solved in polynomial time, and that an extension of entropy games incorporating weights can be solved in pseudo-polynomial time under the same fixed rank condition. (10.1016/j.ic.2024.105236)
  DOI : 10.1016/j.ic.2024.105236
• Optimal Liquidation with Signals: the General Propagator Case
  
  • Abi Jaber Eduardo
  • Neuman Eyal
  Mathematical Finance, Wiley, 2025, 35 (4), pp.841–866. We consider a class of optimal liquidation problems where the agent's transactions create transient price impact driven by a Volterra-type propagator along with temporary price impact. We formulate these problems as minimization of a revenue-risk functionals, where the agent also exploits available information on a progressively measurable price predicting signal. By using an infinite dimensional stochastic control approach, we characterize the value function in terms of a solution to a free-boundary $L^2$-valued backward stochastic differential equation and an operator-valued Riccati equation. We then derive analytic solutions to these equations which yields an explicit expression for the optimal trading strategy. We show that our formulas can be implemented in a straightforward and efficient way for a large class of price impact kernels with possible singularities such as the power-law kernel.
• Maxwell's equations with hypersingularities at a negative index material conical tip
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Rihani Mahran
  Pure and Applied Analysis, Mathematical Sciences Publishers, 2025, 7 (1), pp.127–169. We study a transmission problem for the time harmonic Maxwell's equations between a classical positive material and a so-called negative index material in which both the permittivity ε and the permeability µ take negative values. Additionally, we assume that the interface between the two domains is smooth everywhere except at a point where it coincides locally with a conical tip. In this context, it is known that for certain critical values of the contrasts in ε and in µ, the corresponding scalar operators are not of Fredholm type in the usual H^1 spaces. In this work, we show that in these situations, the Maxwell's equations are not well-posed in the classical L^2 framework due to existence of hypersingular fields which are of infinite energy at the tip. By combining the T-coercivity approach and the Kondratiev theory, we explain how to construct new functional frameworks to recover well-posedness of the Maxwell's problem. We also explain how to select the setting which is consistent with the limiting absorption principle. From a technical point of view, the fields as well as their curls decompose as the sum of an explicit singular part, related to the black hole singularities of the scalar operators, and a smooth part belonging to some weighted spaces. The analysis we propose rely in particular on the proof of new key results of scalar and vector potential representations of singular fields.
• A new approach for the unitary Dyson Brownian motion through the theory of viscosity solutions
  
  • Bertucci Charles
  • Pesce Valentin
  , 2025. In this paper, we study the unitary Dyson Brownian motion through a partial differential equation approach recently introduced for the real Dyson case. The main difference with the real Dyson case is that the spectrum is now on the circle and not on the real line, which leads to particular attention to comparison principles. First we recall why the system of particles which are the eigenvalues of unitary Dyson Brownian motion is well posed thanks to a containment function. Then we proved that the primitive of the limit spectral measure of the unitary Dyson Brownian motion is the unique solution to a viscosity equation obtained by primitive the Dyson equation on the circle. Finally, we study some properties of solutions of Dyson's equation on the circle. We prove a L ∞ regularization. We also look at the long time behaviour in law of a solution through a study of the so-called free entropy of the system. We conclude by discussing the uniform convergence towards the uniform measure on the circle of a solution of the Dyson equation.
• Ergodic behavior of products of random positive operators
  
  • Ligonnière Maxime
  ALEA : Latin American Journal of Probability and Mathematical Statistics, Instituto Nacional de Matemática Pura e Aplicada (Rio de Janeiro, Brasil) [2006-....], 2025, XXII, pp.93-129. This article is devoted to the study of products of random operators of the form $M_{0,n}=M_0\cdots M_{n-1}$, where $(M_{n})_{n\geq 0}$ is an ergodic sequence of positive operators acting on the space of signed measures on some set $\XX$. Under suitable conditions, in particular, a Doeblin-type minoration suited for non conservative operators, we obtain asymptotic results of the form \[ \mu M_{0,n} \simeq \mu({h}) r_n \pi_n,\] for any positive measure $\mu$, where $\tilde{h}$ is a random bounded function, $(r_n)_{n\geq 0}$ is a random non negative sequence and $(\pi_n)$ is a random sequence of probability measures on $\XX$. Moreover, $\tilde{h}$, $(r_n)$ and $(\pi_n)$ do not depend on the choice of the measure $\mu$. We prove additionally that $n^{-1} \log (r_n)$ converges almost surely to the Lyapunov exponent $\lambda$ of the process $(M_{0,n})_{n\geq 0}$ and that the sequence of random probability measures $(\pi_n)$ converges weakly towards a random probability measure. These results are analogous to previous estimates from Hennion in the case of $d\times d$ matrices, that were obtained with different techniques, based on a projective contraction in Hilbert distance. In the case where the sequence $(M_n)$ is i.i.d, we additionally exhibit an expression of the Lyapunov exponent $\lambda$ as an integral with respect to the weak limit of the sequence of random probability measures $(\pi_n)$ and exhibit an oscillation behavior of $r_n$ and $\Vert \mu M_{0,n} \Vert$ when $\lambda=0$. We provide a detailed comparison of our assumptions with the ones of Hennion and present an example of application of our results to the modelling of an age structured population. (10.30757/ALEA.v22-03)
  DOI : 10.30757/ALEA.v22-03
• Proximal Point Nash Learning from Human Feedback
  
  • Tiapkin Daniil
  • Calandriello Daniele
  • Belomestny Denis
  • Moulines Eric
  • Naumov Alexey
  • Rasul Kashif
  • Valko Michal
  • Menard Pierre
  , 2025. Traditional Reinforcement Learning from Human Feedback (RLHF) often relies on reward models, frequently assuming preference structures like the Bradley--Terry model, which may not accurately capture the complexities of real human preferences (e.g., intransitivity). Nash Learning from Human Feedback (NLHF) offers a more direct alternative by framing the problem as finding a Nash equilibrium of a game defined by these preferences. While many works study the Nash learning problem directly in the policy space, we instead consider it under a more realistic policy parametrization setting. We first analyze a simple self-play policy gradient method, which is equivalent to Online IPO. We establish high-probability last-iterate convergence guarantees for this method, but our analysis also reveals a possible stability limitation of the underlying dynamics. Motivated by this, we embed the self-play updates into a proximal point framework, yielding a stabilized algorithm. For this combined method, we prove high-probability last-iterate convergence and discuss its more practical version, which we call Nash Prox. Finally, we apply this method to post-training of large language models and validate its empirical performance. (10.48550/arXiv.2505.19731v2)
  DOI : 10.48550/arXiv.2505.19731v2