Arnaud Guillin

Arnaud Descours, Arnaud Guillin, Geoffrey Lacour et al.

Uncertainty quantification in neural networks prediction is a main issue for usual applications. Our approach seeks at reducing computation costs by directly evaluating uncertainty using PDE's information on the asymptotic variance, rather than the deep ensemble method which may be seen as a Monte Carlo estimation of the prediction, requiring the training of multiple networks. We thus study the law of the limiting process describing the random fluctuations around the mean-field limit of wide two-layer neural networks trained by stochastic gradient descent in a weak-noise regime. Building on a recent trajectorial central limit theorem, in which this limit is characterized as the weak solution of a linear stochastic evolution equation, we identify its law explicitly. More precisely, we show that it is a centered Gaussian process in the dual of a weighted Sobolev space, and we derive a closed covariance representation for the finite-dimensional distributions obtained by testing it against smooth functions. This covariance is expressed through the solution of a backward transport equation with a nonlocal source term, whose coefficients are driven by the mean-field trajectory. As a consequence, by testing against the activation function at a fixed input, we obtain an expression for the limiting variance of the corresponding network-output fluctuations. We illustrate this result numerically on a one-dimensional regression example.

Vincent Souveton, Arnaud Guillin, Jens Jasche et al.

Normalizing Flows (NF) are Generative models which transform a simple prior distribution into the desired target. They however require the design of an invertible mapping whose Jacobian determinant has to be computable. Recently introduced, Neural Hamiltonian Flows (NHF) are Hamiltonian dynamics-based flows, which are continuous, volume-preserving and invertible and thus make for natural candidates for robust NF architectures. In particular, their similarity to classical Mechanics could lead to easier interpretability of the learned mapping. In this paper, we show that the current NHF architecture may still pose a challenge to interpretability. Inspired by Physics, we introduce a fixed-kinetic energy version of the model. This approach improves interpretability and robustness while requiring fewer parameters than the original model. We illustrate that on a 2D Gaussian mixture and on the MNIST and Fashion-MNIST datasets. Finally, we show how to adapt NHF to the context of Bayesian inference and illustrate the method on an example from cosmology.

Arnaud Guillin, Yu Wang, Lihu Xu et al.

Stochastic gradient descent with momentum is a popular variant of stochastic gradient descent, which has recently been reported to have a close relationship with the underdamped Langevin diffusion. In this paper, we establish a quantitative error estimate between them in the 1-Wasserstein and total variation distances.

Aho Yapi, Pierre Latouche, Arnaud Guillin et al.

We propose a generic framework for short-term occupational accident forecasting that leverages safety inspections and models accident occurrences as binary time series. The approach generates daily predictions, which are then aggregated into weekly safety assessments to better inform decision making. To ensure the reliability and operational applicability of the forecasts, we apply a sliding-window cross-validation procedure specifically designed for time series data, combined with an evaluation based on aggregated period-level metrics. Several machine learning algorithms, including logistic regression, tree-based models, and neural networks, are trained and systematically compared within this framework. Unlike the other approaches, the long short-term memory (LSTM) network outperforms the other approaches and detects the upcoming high-risk periods with a balanced accuracy of 0.86, confirming the robustness of our methodology and demonstrating that a binary time series model can anticipate these critical periods based on safety inspections. The proposed methodology converts routine safety inspection data into clear weekly risk scores, detecting the periods when accidents are most likely. Decision-makers can integrate these scores into their planning tools to classify inspection priorities, schedule targeted interventions, and funnel resources to the sites or shifts classified as highest risk, stepping in before incidents occur and getting the greatest return on safety investments.

Arnaud Descours, Tom Huix, Arnaud Guillin et al.

In this paper, we rigorously derive Central Limit Theorems (CLT) for Bayesian two-layerneural networks in the infinite-width limit and trained by variational inference on a regression task. The different networks are trained via different maximization schemes of the regularized evidence lower bound: (i) the idealized case with exact estimation of a multiple Gaussian integral from the reparametrization trick, (ii) a minibatch scheme using Monte Carlo sampling, commonly known as Bayes-by-Backprop, and (iii) a computationally cheaper algorithm named Minimal VI. The latter was recently introduced by leveraging the information obtained at the level of the mean-field limit. Laws of large numbers are already rigorously proven for the three schemes that admits the same asymptotic limit. By deriving CLT, this work shows that the idealized and Bayes-by-Backprop schemes have similar fluctuation behavior, that is different from the Minimal VI one. Numerical experiments then illustrate that the Minimal VI scheme is still more efficient, in spite of bigger variances, thanks to its important gain in computational complexity.