Abstract: The reliability of machine learning (ML) methods is critical in contexts that involve high-stakes decisions. However, while ML methods have achieved impressive results in a wide range of applications, none of them are able to provide a small error guarantee in any situation. Since models cannot be perfect, they should at least “know that they do not know”. While there have been many efforts in the literature to address this issue, whether in the field of probability calibration, or prediction sets, these solutions are not satisfactory when an actual decision is required. By contrast, a key to keeping the error rate (or risk) below a certain threshold is to make use of a type of abstention option, which amounts to abstain from making a decision when there is too much uncertainty. The goal of this thesis is to propose new methods for risk control, i.e. for keeping the risk below a certain user-specified threshold α, in several learning tasks: novelty detection, clustering and link prediction. Our general idea is to enhance the best existing ML methods by developing an additional layer on top of them that provides an interpretable guarantee on the error rate. This is achieved by formalizing risk control in a certain task as a type of false discovery rate (FDR) control problem, and by using tools from the multiple testing literature on FDR control. Our methods can be seen as wrappers that take as input an off-the-shelf ML technique, designed for a certain learning task, and return a set of decisions such that the FDR is controlled.

In conformal field theory (CFT), the null-vector (or BPZ) equations are a set of PDEs satisfied by correlation functions (and conformal blocks) involving so-called “degenerate primary fields” (i.e. fields associated to degenerate representations of the Virasoro algebra). These equations are parametrised by a pair of positive integers (r,s) labelling the representation (“Kac table”), and by the topological type of the surface on which the CFT lives. The BPZ operator is a partial differential operator of order rs (the “level”) on the Teichmüller space of the surface, and it annihilates the conformal blocks if the corresponding representation is irreducible. In the probabilistic formulation of Liouville CFT, the BPZ equations have been shown to hold in some special cases (for correlation functions at level 2, on the sphere and to some extent in genus one), and they are key inputs in the proofs of certain exact formulae (e.g. the “DOZZ formula”). In this work, we generalise these results to all values of the parameters, and we make an explicit connection with the algebraic structure of the theory. Namely, we construct the degenerate modules for all (r,s) and show that they are irreducible. Then, we use a geometric characterisation of conformal blocks to translate this local information into a PDE on Teichmüller space. The talk will focus on the probabilistic aspects of this work: I will explain how to construct the modules using elementary properties of the Gaussian free field and Gaussian multiplicative chaos. An interesting feature of this construction is a probabilistic interpretation of the Kac table. Ongoing work with Baojun Wu.

The motivation of this presentation comes from the analysis of metastable stochastic process in statistical physics. One way to bridge the scale between full atomistic models and more coarse-grained descriptions is to use Markov State models parameterized by the Eyring Kramers formulas. These formulas give the hopping rates between local minima of the potential energy function. They require to identify the local minima and saddle points of the potential energy function. This approach is for example used in materials science (kinetic Monte Carlo models).

In this talk, I will first present a recent result obtained in collaboration with D. Le Peutrec (Université d'Orléans) and B. Nectoux (Université Clermont Auvergne) about the mathematical foundations of this approach, by deriving these Eyring-Kramers exit rates starting from the overdamped Langevin dynamics [1]. I will then introduce a recent algorithm we proposed together with P. Parpas (Imperial College London) in order to locate saddle points [2]. I will explain why these two works both rely on concentration properties of the eigenvectors of Witten Laplacians, in the small temperature regime.

References: [1] TL, D. Le Peutrec and B. Nectoux, Eyring-Kramers exit rates for the overdamped Langevin dynamics: the case with saddle points on the boundary, https://arxiv.org/abs/2207.09284. [2] TL, P. Parpas /Using Witten Laplacians to locate index-1 saddle points/, https://arxiv.org/abs/2212.10135.

Je présenterai quelques résultats récents concernant les limites d’échelles de systèmes désordonnés et des conséquences que l’on peu en tirer. Je me concentrerai essentiellement sur le modèle de Poland-Scheraga, aussi connu sous le nom de modèle d'accrochage, qui est utilisé pour décrire le phénomène de dénaturation de l’ADN : la question est de savoir si (et comment) le désordre perturbe la transition de dénaturation. Je décrirai notamment les résultats obtenus dans le cadre d’une version généralisée (censée être plus réaliste) du modèle, en collaboration avec Alexandre Legrand.

Obtaining guarantees on the convergence of the minimizers of empirical risks to the ones of the true risk is a fundamental matter in statistical learning.

Instead of deriving guarantees on the usual estimation error, we will explore concentration inequalities on the distance between the sets of minimizers of the risks. We will argue that for a broad spectrum of estimation problems, there exists a regime where optimal concentration rates can be proven. The bounds will be showcased on a selection of estimation problems such as barycenters on metric space with positive or negative curvature, subspaces of covariance matrices, regression problems and entropic-Wasserstein barycenters.