arXiv subjects

Grégoire Ferré

Publications and source records attributed to Grégoire Ferré.

11 recordsLinked to original sources

arXiv2026

Random Matrix Theory of Early-Stopped Gradient Flow: A Transient BBP Scenario

Empirical studies of trained models often report a transient regime in which signal is detectable in a finite gradient descent time window before overfitting dominates. We provide an analytically tractable random-matrix model that reproduces this phenomenon for gradient flow in a linear teacher--student setting. In this framework, learning occurs when an isolated eigenvalue separates from a noisy bulk, before eventually disappearing in the overfitting regime. The key ingredient is anisotropy in the input covariance, which induces fast and slow directions in the learning dynamics. In a two-block covariance model, we derive the full time-dependent bulk spectrum of the symmetrized weight matrix through a $2\times 2$ Dyson equation, and we obtain an explicit outlier condition for a rank-one teacher via a rank-two determinant formula. This yields a transient Baik-Ben Arous-Péché (BBP) transition: depending on signal strength and covariance anisotropy, the teacher spike may never emerge, emerge and persist, or emerge only during an intermediate time interval before being reabsorbed into the bulk. We map the corresponding phase diagrams and validate the theory against finite-size simulations. Our results provide a minimal solvable mechanism for early stopping as a transient spectral effect driven by anisotropy and noise.

stat.ML↗

arXiv2025

A subexponential version of Cramer's theorem

We consider the large deviations associated with the empirical mean of independent and identically distributed random variables under a subexponential moment condition. We show that non-trivial deviations are observable at a subexponential scale in the number of variables, and we provide the associated rate function, which is non-convex and is not derived from a Legendre-Fenchel transform. The proof adapts the one of Cramer's theorem to the case where the fluctuation is generated by a single variable. In particular, we develop a new tilting strategy for the lower bound, which leads us to introduce a condition on the second derivative of the moment generating function. Our results are illustrated by a couple of simple examples.

math.PR↗

arXiv2025

Heavy tailed large deviations for time averages of diffusions: the Ornstein-Uhlenbeck case

We study large deviations for the time average of the Ornstein-Uhlenbeck process raised to an arbitrary power. We prove that beyond a critical value, large deviations are subexponential in time, with a non-convex rate function whose main coefficient is given by the solution to a Hamilton-Jacobi problem. Although a similar problem was addressed in a recent work, the originality of the paper is to provide a short, self-contained proof of this result through a couple of standard large deviations arguments.

math.PR↗

arXiv2023

Stochastic viscosity approximations of Hamilton-Jacobi equations and variance reduction

We consider the computation of free energy-like quantities for diffusions in high dimension, when resorting to Monte Carlo simulation is necessary. Such stochastic computations typically suffer from high variance, in particular in a low noise regime, because the expectation is dominated by rare trajectories for which the observable reaches large values. Although importance sampling, or tilting of trajectories, is now a standard technique for reducing the variance of such estimators, quantitative criteria for proving that a given control reduces variance are scarce, and often do not apply to practical situations. The goal of this work is to provide a quantitative criterion for assessing whether a given bias reduces variance, and at which order. We rely for this on a recently introduced notion of stochastic solution for Hamilton-Jacobi-Bellman (HJB) equations. Based on this tool, we introduce the notion of k-stochastic viscosity approximation (SVA) of a HJB equation. We next prove that such approximate solutions are associated with estimators having a relative variance of order k-1 at log-scale. In particular, a sampling scheme built from a 1-SVA has bounded variance as noise goes to zero. Finally, in order to show that our definition is relevant, we provide examples of stochastic viscosity approximations of order one and two, with a numerical illustration confirming our theoretical findings.

math.NA↗

arXiv2021

Approximate optimal controls via instanton expansion for low temperature free energy computation

The computation of free energies is a common issue in statistical physics. A natural technique to compute such high dimensional integrals is to resort to Monte Carlo simulations. However these techniques generally suffer from a high variance in the low temperature regime, because the expectation is often dominated by high values corresponding to rare system trajectories. A standard way to reduce the variance of the estimator is to modify the drift of the dynamics with a control enhancing the probability of rare event, leading to so-called importance sampling estimators. In theory, the optimal control leads to a zero-variance estimator; it is however defined implicitly and computing it is of the same difficulty as the original problem. We propose here a general strategy to build approximate optimal controls in the small temperature limit for diffusion processes, with the first goal to reduce the variance of free energy Monte Carlo estimators. Our construction builds upon low noise asymptotics by expanding the optimal control around the instanton, which is the path describing most likely fluctuations at low temperature. This technique not only helps reducing variance, but it is also interesting as a theoretical tool since it differs from usual small temperature expansions (WKB ansatz). As a complementary consequence of our expansion, we provide a perturbative formula for computing the free energy in the small temperature regime, which refines the now standard Freidlin--Wentzell asymptotics. We compute this expansion explicitly for lower orders, and explain how our strategy can be extended to an arbitrary order of accuracy. We support our findings with illustrative numerical examples.

cond-mat.stat-mech↗

arXiv2020

Coulomb gases under constraint: some theoretical and numerical results

We consider Coulomb gas models for which the empirical measure typically concentrates, when the number of particles becomes large, on an equilibrium measure minimizing an electrostatic energy. We study the behavior when the gas is conditioned on a rare event. We first show that the special case of quadratic confinement and linear constraint is exactly solvable due to a remarkable factorization, and that the conditioning has then the simple effect of shifting the cloud of particles without deformation. To address more general cases, we perform a theoretical asymptotic analysis relying on a large deviations technique known as the Gibbs conditioning principle. The technical part amounts to establishing that the conditioning ensemble is an I-continuity set of the energy. This leads to characterizing the conditioned equilibrium measure as the solution of a modified variational problem. For simplicity, we focus on linear statistics and on quadratic statistics constraints. Finally, we numerically illustrate our predictions and explore cases in which no explicit solution is known. For this, we use a Generalized Hybrid Monte Carlo algorithm for sampling from the conditioned distribution for a finite but large system.

math.PR↗

arXiv2020

Large deviations of empirical measures of diffusions in weighted topologies

We consider large deviations of empirical measures of diffusion processes. In a first part, we present conditions to obtain a large deviations principle (LDP) for a precise class of unbounded functions. This provides an analogue to the standard Cramér condition in the context of diffusion processes, which turns out to be related to a spectral gap condition for a Witten-Schrödinger operator. Secondly, we study more precisely the properties of the Donsker-Varadhan rate functional associated with the LDP. We revisit and generalize some standard duality results as well as a more original decomposition of the rate functional with respect to the symmetric and antisymmetric parts of the dynamics. Finally, we apply our results to overdamped and underdamped Langevin dynamics, showing the applicability of our framework for degenerate diffusions in unbounded configuration spaces.

math.PR↗

arXiv2020

More on the long time stability of Feynman-Kac semigroups

Feynman-Kac semigroups appear in various areas of mathematics: non-linear filtering, large deviations theory, spectral analysis of Schrodinger operators among others. Their long time behavior provides important information, for example in terms of ground state energy of Schrodinger operators, or scaled cumulant generating function in large deviations theory. In this paper, we propose a simple and natural extension of the stability of Markov chains for these non-linear evolutions. As other classical ergodicity results, it relies on two assumptions: a Lyapunov condition that induces some compactness, and a minorization condition ensuring some mixing. Illustrative examples are provided, where the stability of the non-linear semigroup arises either from the underlying dynamics or from the Feynman-Kac weight function. We also use our technique to provide uniform in the time step convergence estimates for discretizations of stochastic differential equations

math.PR↗

arXiv2019

Error estimates on ergodic properties of discretized Feynman-Kac semigroups

We consider the numerical analysis of the time discretization of Feynman-Kac semigroups associated with diffusion processes. These semigroups naturally appear in several fields, such as large deviation theory, Diffusion Monte Carlo or non-linear filtering. We present errors estimates a la Talay-Tubaro on their invariant measures when the underlying continuous stochastic differential equation is discretized; as well as on the leading eigenvalue of the generator of the dynamics, which corresponds to the rate of creation of probability. This provides criteria to construct efficient integration schemes of Feynman-Kac dynamics, as well as a mathematical justification of numerical results already observed in the Diffusion Monte Carlo community. Our analysis is illustrated by numerical simulations.

math.NA↗

arXiv2019

Simulating Coulomb gases and log-gases with hybrid Monte Carlo algorithms

Coulomb and log-gases are exchangeable singular Boltzmann-Gibbs measures appearing in mathematical physics at many places, in particular in random matrix theory. We explore experimentally an efficient numerical method for simulating such gases. It is an instance of the Hybrid or Hamiltonian Monte Carlo algorithm, in other words a Metropolis-Hastings algorithm with proposals produced by a kinetic or underdamped Langevin dynamics. This algorithm has excellent numerical behavior despite the singular interaction, in particular when the number of particles gets large. It is more efficient than the well known overdamped version previously used for such problems, and allows new numerical explorations. It suggests for instance to conjecture a universality of the Gumbel fluctuation at the edge of beta Ginibre ensembles for all beta.

math.PR↗

arXiv2019

Adaptive sampling of large deviations

We introduce and test an algorithm that adaptively estimates large deviation functions characterizing the fluctuations of additive functionals of Markov processes in the long-time limit. These functions play an important role for predicting the probability and pathways of rare events in stochastic processes, as well as for understanding the physics of nonequilibrium systems driven in steady states by external forces and reservoirs. The algorithm uses methods from risk-sensitive and feedback control to estimate from a single trajectory a new process, called the driven process, known to be efficient for importance sampling. Its advantages compared to other simulation techniques, such as splitting or cloning, are discussed and illustrated with simple equilibrium and nonequilibrium diffusion models.

cond-mat.stat-mech↗