Publications

2022

• Membership Inference Attacks via Adversarial Examples
  
  • Jalalzai Hamid
  • Kadoche Elie
  • Leluc Rémi
  • Plassier Vincent
  , 2022. The raise of machine learning and deep learning led to significant improvement in several domains. This change is supported by both the dramatic rise in computation power and the collection of large datasets. Such massive datasets often include personal data which can represent a threat to privacy. Membership inference attacks are a novel direction of research which aims at recovering training data used by a learning algorithm. In this paper, we develop a mean to measure the leakage of training data leveraging a quantity appearing as a proxy of the total variation of a trained model near its training samples. We extend our work by providing a novel defense mechanism. Our contributions are supported by empirical evidence through convincing numerical experiments.
• Algorithms for non-linear constrained continuous optimization: a comparison between gradient-based methods and evolution strategies, and applications to radar antenna design
  
  • Dufossé Paul
  , 2022. This thesis proposes several contributions to the problem of optimizing a nonlinear function of several variables in a continuous search space in the presence of constraints. The approach is called black box in that the considered algorithms have only access to the objective and constraint functions for the search of an optimal solution. The first contribution is the establishment of a suite of test problems for continuous constrained optimization. The method used to construct these test problems is based on optimality conditions that are general enough to allow defining non-convex problems where the optimum is known. The construction has other practical interests, such as the possibility to vary the number of variables or the number of constraints. We also propose a method for monitoring the performance of an optimization algorithm that can be applied to any such problem. The second contribution is a thorough study and improvement of a stochastic algorithm for the optimization of a multivariate function. This algorithm combines the principles of an evolution strategy with covariance matrix adaptation (CMA-ES) for non-convex optimization with the augmented Lagrangian (AL) technique to handle constraints. Based on recent work, our experiments reveal that other values of the hyperparameters, and even alternative mechanisms for adapting the penalty parameters, are competitive in a black-box context. We also propose a heuristic for the initialization of the adaptive parameters, and a preliminary study of the use of linear models to replace the constraints. The previously introduced suite of test problems is used as a tool to develop and validate the performance of these algorithmic alternatives. The third contribution is the generation of extensive experimental data with many popular algorithms for nonlinear optimization operating on our test problem suite. These data highlight concrete use cases where one algorithm is more suitable than the other competitors. Finally, we illustrate the diversity of optimization problems encountered in practice in the case of Radar signal processing. We address two classical Radar antenna design problems as numerical optimization problems. The first problem is the optimal configuration of a sensor array for the estimation of the direction of arrival of a detected signal. The second one is the phase modulation of an antenna array in order to obtain a radiation pattern with minimum unwanted emission losses. These 2 problems differ in many aspects (regularity of the objective function, dimension, number of constraints) and lead to the selection of different algorithms as the most suitable for each problem. In addition, some aspects of computer programming for handling and solving optimization problems will be discussed, as well as some avenues for future work in the quest for the best optimization algorithm to solve a given problem.
• Bridging socioeconomic pathways of CO2 emission and credit risk
  
  • Bourgey Florian
  • Gobet Emmanuel
  • Jiao Ying
  Annals of Operations Research, Springer Verlag, 2022. This paper investigates the impact of transition risk on a firm's low-carbon production. As the world is facing global climate changes, the Intergovernmental Panel on Climate Change (IPCC) has set the idealized carbon-neutral scenario around 2050. In the meantime, many carbon reduction scenarios, known as Shared Socioeconomic Pathways (SSPs) have been proposed in the literature for different production sectors in more comprehensive socioeconomic context. In this paper, we consider, on the one hand, a firm that aims to optimize its emission level under the double objectives of maximizing its production profit and respecting the emission mitigation scenarios. Solving the penalized optimization problem provides the optimal emission according to a given SSP benchmark. On the other hand, such transitions affect the firm's credit risk. We model the default time by using the structural default approach. We are particularly concerned with how the adopted strategies by following different SSPs scenarios may influence the firm's default probability. (10.1007/s10479-022-05135-y)
  DOI : 10.1007/s10479-022-05135-y
• Estimation of extreme quantiles from heavy-tailed distributions with neural networks
  
  • Allouche Michaël
  • Girard Stéphane
  • Gobet Emmanuel
  , 2022. New parametrizations for neural networks are proposed in order to estimate extreme quantiles in both non-conditional and conditional heavy-tailed settings. All proposed neural network estimators feature a bias correction based on an extension of the usual second-order condition to an arbitrary order. The convergence rate of the uniform error between extreme log-quantiles and their neural network approximation is established. The finite sample performances of the non-conditional neural network estimator are compared to other bias-reduced extreme-value competitors on simulated data. It is shown that our method outperforms them in difficult heavy-tailed situations where other estimators almost all fail. The source code is available at github. Finally, conditional neural network estimators are implemented to investigate the behavior of extreme rainfalls as functions of their geographical location in the southern part of France.
• Contributions to volatility modeling and risk management and to efficient numerical simulation in finance
  
  • de Marco Stefano
  , 2022.
• Entropic optimal planning for path-dependent mean field games
  
  • Ren Zhenjie
  • Tan Xiaolu
  • Touzi Nizar
  • Yang Junjian
  , 2022. In the context of mean field games, with possible control of the diffusion coefficient, we consider a path-dependent version of the planning problem introduced by P.L. Lions: given a pair of marginal distributions $(\mu_0, \mu_1)$, find a specification of the game problem starting from the initial distribution $\mu_0$, and inducing the target distribution $\mu_1$ at the mean field game equilibrium. Our main result reduces the path-dependent planning problem into an embedding problem, that is, constructing a McKean-Vlasov dynamics with given marginals $(\mu_0,\mu_1)$. Some sufficient conditions on $(\mu_0,\mu_1)$ are provided to guarantee the existence of solutions. We also characterize, up to integrability, the minimum entropy solution of the planning problem. In particular, as uniqueness does not hold anymore in our path-dependent setting, one can naturally introduce an optimal planning problem which would be reduced to an optimal transport problem along with controlled McKean-Vlasov dynamics.
• Viscosity solutions for obstacle problems on Wasserstein space
  
  • Talbi Mehdi
  • Touzi Nizar
  • Zhang Jianfeng
  , 2022. This paper is a continuation of our accompanying paper [Talbi, Touzi and Zhang (2021)], where we characterized the mean field optimal stopping problem by an obstacle equation on the Wasserstein space of probability measures, provided that the value function is smooth. Our purpose here is to establish this characterization under weaker regularity requirements. We shall define a notion of viscosity solutions for such equation, and prove existence, stability, and comparison principle.
• Dynamic programming equation for the mean field optimal stopping problem
  
  • Talbi Mehdi
  • Touzi Nizar
  • Zhang Jianfeng
  , 2022. We study the optimal stopping problem of McKean-Vlasov diffusions when the criterion is a function of the law of the stopped process. A remarkable new feature in this setting is that the stopping time also impacts the dynamics of the stopped process through the dependence of the coefficients on the law. The mean field stopping problem is introduced in weak formulation in terms of the joint marginal law of the stopped underlying process and the survival process. This specification satisfies a dynamic programming principle. The corresponding dynamic programming equation is an obstacle problem on the Wasserstein space, and is obtained by means of a general It\^o formula for flows of marginal laws of c\`adl\`ag semimartingales. Our verification result characterizes the nature of optimal stopping policies, highlighting the crucial need to randomized stopping. The effectiveness of our dynamic programming equation is illustrated by various examples including the mean-variance optimal stopping problem.
• From finite population optimal stopping to mean field optimal stopping
  
  • Talbi Mehdi
  • Touzi Nizar
  • Zhang Jianfeng
  , 2022. This paper analyzes the convergence of the finite population optimal stopping problem towards the corresponding mean field limit. Building on the viscosity solution characterization of the mean field optimal stopping problem of our previous papers [Talbi, Touzi & Zhang 2021 & 2022], we prove the convergence of the value functions by adapting the Barles-Souganidis [1991] monotone scheme method to our context. We next characterize the optimal stopping policies of the mean field problem by the accumulation points of the finite population optimal stopping strategies. In particular, if the limiting problem has a unique optimal stopping policy, then the finite population optimal stopping strategies do converge towards this solution. As a by-product of our analysis, we provide an extension of the standard propagation of chaos to the context of stopped McKean-Vlasov diffusions.
• Contributions to generative modeling and dictionary learning : theory and application
  
  • Allouche Michaël
  , 2022. This thesis aims at investigating data-based methods in the paradigms of Artificial Intelligence and Machine Learning. Although very popular, those methods are mainly used in empirical works. Therefore, providing theoretical guidelines for building such models is of primal importance.In the first part, we study generative modeling using neural networks in two different settings: the simulation of a fractional Brownian motion, and of heavy-tailed distributions in both conditional and non-conditional cases. In all works, we analyze the convergence rate of the uniform error between the function of interest and its neural network approximation. The performance of our models are illustrated on simulations and real practical problems in finance and in meteorology: generating extreme negative returns of financial indexes and rainfalls as functions of their geographical location.In the second part, we propose a new method based on dictionary learning for modeling financial rating migration matrices. We have to deal with small amount of data, close to the dimension of the problem, a fast evolution in time of the matrices and a collection of linear constraints. We present a numerical test with real data and show the performance of the model as an economic sentiment indicator.
• Inverse problems for stochastic neutronics
  
  • Houpert Corentin
  , 2022. The aim of this work is the estimation of nuclear parameters fromneutron correlation measurements. It is an inverse problem with noised observations, this is not an exception to Feynman's quote. The physics of the neutronic system provides some hunches about the behaviour of the observations, then we will be uncertain as suggested by the remark. As this viewpoint indicates, it is necessary to quantify the level of certainty: uncertainty quantification appears as a good choice.The experimental data is the list of number of neutrons detected duringtime intervals of same duration. A statistical analysis based on the moments of the number of detection is performed for parameter inference.Fission neutrons are produced by bunches (between 2 and 3 on average).The neutrons originating from the same fission are time correlated. Neutron source emission is a compound Poisson process. In the detections, there will be an excess of variance with respect to a Poisson process. This fact is exploited in the Feynman method. In general, due to the correlations, moments of higher order than the mean contain information about the system.Since we are looking for not only point estimates but also the fullprobability distribution of the parameters, we will consider Bayesian inference and Monte Carlo Markov Chain (MCMC) sampling of the a posterioridistribution.Regarding the direct calculation of the parameters, a simple model wherethe phase space is reduced to a single point is implemented. With this pointmodel, the moments have analytic expressions and can be calculated efficiently and quickly.The thesis is structured as follows.First, we recall the state of the art about basics probability, neutron pointmodel and neutron equations, uncertainty quantification and inverse problem.Then in a second part we will establish the expressions of the observationsthat we get from the detection times: the empirical moments of thedistribution of the number of detected neutrons.Then in a third part we will study the associated inverse problem i.e. knowing the observations what are the parameters and their uncertainties. This will be done by the use of MCMC methods with the Metropolis algorithm and covariance matrix adaptation.Finally, we will conclude about the improvements provided by the thesisand what could be continued after this work.
• Market making by an FX dealer: tiers, pricing ladders and hedging rates for optimal risk control
  
  • Barzykin Alexander
  • Bergault Philippe
  • Guéant Olivier
  , 2022. Dealers make money by providing liquidity to clients but face flow uncertainty and thus price risk. They can efficiently skew their prices and wait for clients to mitigate risk (internalization), or trade with other dealers in the open market to hedge their position and reduce their inventory (externalization). Of course, the better control associated with externalization comes with transaction costs and market impact. The internalization vs. externalization dilemma has been a topic of recent active discussion within the foreign exchange (FX) community. This paper offers an optimal control framework for market making tackling both pricing and hedging, thus answering a question well known to dealers: `to hedge, or not to hedge?
• Ergodic behaviour of a multi-type growth-fragmentation process modelling the mycelial network of a filamentous fungus
  
  • Tomasevic Milica
  • Bansaye Vincent
  • Véber Amandine
  ESAIM: Probability and Statistics, EDP Sciences, 2022, 26, pp.397-435. In this work, we introduce a stochastic growth-fragmentation model for the expansion of the network of filaments, or mycelium, of a filamentous fungus. In this model, each individual is described by a discrete type e ∈ {0, 1} indicating whether the individual corresponds to an internal or terminal segment of filament, and a continuous trait x ≥ 0 corresponding to the length of this segment. The length of internal segments cannot grow, while the length of terminal segments increases at a deterministic speed v. Both types of individuals/segment branch according to a type-dependent mechanism. After constructing the stochastic bi-type growth-fragmentation process of interest, we analyse the corresponding mean measure (or first moment semigroup). We show that its ergodic behaviour is, as expected, governed by the maximal eigenelements. In the long run, the total mass of the mean measure increases exponentially fast while the type-dependent density in trait converges to an explicit distribution N, independent of the initial condition, at some exponential speed. We then obtain a law of large numbers that relates the long term behaviour of the stochastic process to the limiting distribution N. In the particular model we consider, which depends on only 3 parameters, all the quantities needed to describe this asymptotic behaviour are explicit, which paves the way for parameter inference based on data collected in lab experiments. (10.1051/ps/2022013)
  DOI : 10.1051/ps/2022013
• Multi-fidelity surrogate modeling adapted to functional outputs for uncertainty quantification of complex models
  
  • Kerleguer Baptiste
  , 2022. This thesis focuses on approximating the output of a complex computational code in a multi-fidelity framework, i.e. the code can be run at different levels of accuracy with different computational costs. Thus, the predictive qualities of the surrogate model of the output of a complex code can be improved by using in addition less accurate but more numerous (because less expensive) simulations.This work aims to extend multi-fidelity surrogate modeling methods when the code outputs are functional. First, an approach allowing to combine Bayesian neural networks and Gaussian processes is proposed. This model is suitable when the relationship between the low- and high-fidelity codes is non-linear, but is not yet suitable for functional outputs.In a second step, an approach using wavelet decomposition and Gaussian processes is proposed. This approach allows to develop a Gaussian process in the wavelet space which is equivalent to a Gaussian process in the time space. This method is naturally adapted to functional outputs.Finally, the third proposed method combines an output dimension reduction method with a covariance tensorised Gaussian process regression method. This approach is developed in the context of time-series output. The analytical expressions for the predictive mean and variance are also introduced.
• A generative model for fBm with deep ReLU neural networks
  
  • Allouche Michaël
  • Girard Stéphane
  • Gobet Emmanuel
  Journal of Complexity, Elsevier, 2022, 73, pp.101667. We provide a large probability bound on the uniform approximation of fractional Brownian motion $(B^H(t) : t ∈ [0,1])$ with Hurst parameter $H$, by a deep-feedforward ReLU neural network fed with a $N$-dimensional Gaussian vector, with bounds on the network construction (number of hidden layers and total number of neurons). Essentially, up to log terms, achieving an uniform error of $\mathcal{O}(N^{-H})$ is possible with log$(N)$ hidden layers and $\mathcal{O} (N \log N)$ parameters. Our analysis relies, in the standard Brownian motion case $(H = 1/2)$, on the Levy construction of $B^H$ and in the general fractional Brownian motion case $(H \ne 1/2)$, on the Lemarié-Meyer wavelet representation of $B^H$. This work gives theoretical support on new generative models based on neural networks for simulating continuous-time processes. (10.1016/j.jco.2022.101667)
  DOI : 10.1016/j.jco.2022.101667
• A few aspects of large random maps
  
  • Marzouk Cyril
  , 2022. The aim of this document is to present my research work carried out since my PhD, compare the papers to each others, and put them into the scientific literature. It is not meant to be self-contained at all! The reader can find the detailed arguments in the articles, which are listed inside.
• FLamby: Datasets and Benchmarks for Cross-Silo Federated Learning in Realistic Healthcare Settings
  
  • Terrail Jean Ogier Du
  • Ayed Samy-Safwan
  • Cyffers Edwige
  • Grimberg Felix
  • He Chaoyang
  • Loeb Regis
  • Mangold Paul
  • Marchand Tanguy
  • Marfoq Othmane
  • Mushtaq Erum
  • Muzellec Boris
  • Philippenko Constantin
  • Silva Santiago
  • Teleńczuk Maria
  • Albarqouni Shadi
  • Avestimehr Salman
  • Bellet Aurélien
  • Dieuleveut Aymeric
  • Jaggi Martin
  • Karimireddy Sai Praneeth
  • Lorenzi Marco
  • Neglia Giovanni
  • Tommasi Marc
  • Andreux Mathieu
  , 2022. Federated Learning (FL) is a novel approach enabling several clients holding sensitive data to collaboratively train machine learning models, without centralizing data. The cross-silo FL setting corresponds to the case of few ($2$--$50$) reliable clients, each holding medium to large datasets, and is typically found in applications such as healthcare, finance, or industry. While previous works have proposed representative datasets for cross-device FL, few realistic healthcare cross-silo FL datasets exist, thereby slowing algorithmic research in this critical application. In this work, we propose a novel cross-silo dataset suite focused on healthcare, FLamby (Federated Learning AMple Benchmark of Your cross-silo strategies), to bridge the gap between theory and practice of cross-silo FL. FLamby encompasses 7 healthcare datasets with natural splits, covering multiple tasks, modalities, and data volumes, each accompanied with baseline training code. As an illustration, we additionally benchmark standard FL algorithms on all datasets. Our flexible and modular suite allows researchers to easily download datasets, reproduce results and re-use the different components for their research. FLamby is available at~\url{www.github.com/owkin/flamby}.
• Template based Graph Neural Network with Optimal Transport Distances
  
  • Vincent-Cuaz Cédric
  • Flamary Rémi
  • Corneli Marco
  • Vayer Titouan
  • Courty Nicolas
  , 2022. Current Graph Neural Networks (GNN) architectures generally rely on two important components: node features embedding through message passing, and aggregation with a specialized form of pooling. The structural (or topological) information is implicitly taken into account in these two steps. We propose in this work a novel point of view, which places distances to some learnable graph templates at the core of the graph representation. This distance embedding is constructed thanks to an optimal transport distance: the Fused Gromov-Wasserstein (FGW) distance, which encodes simultaneously feature and structure dissimilarities by solving a soft graph-matching problem. We postulate that the vector of FGW distances to a set of template graphs has a strong discriminative power, which is then fed to a non-linear classifier for final predictions. Distance embedding can be seen as a new layer, and can leverage on existing message passing techniques to promote sensible feature representations. Interestingly enough, in our work the optimal set of template graphs is also learnt in an end-to-end fashion by differentiating through this layer. After describing the corresponding learning procedure, we empirically validate our claim on several synthetic and real life graph classification datasets, where our method is competitive or surpasses kernel and GNN state-of-the-art approaches. We complete our experiments by an ablation study and a sensitivity analysis to parameters.
• Aligning individual brains with Fused Unbalanced Gromov-Wasserstein
  
  • Thual Alexis
  • Tran Huy
  • Zemskova Tatiana
  • Courty Nicolas
  • Flamary Rémi
  • Dehaene Stanislas
  • Thirion Bertrand
  , 2022. Individual brains vary in both anatomy and functional organization, even within a given species. Inter-individual variability is a major impediment when trying to draw generalizable conclusions from neuroimaging data collected on groups of subjects. Current co-registration procedures rely on limited data, and thus lead to very coarse inter-subject alignments. In this work, we present a novel method for inter-subject alignment based on Optimal Transport, denoted as Fused Unbalanced Gromov Wasserstein (FUGW). The method aligns cortical surfaces based on the similarity of their functional signatures in response to a variety of stimulation settings, while penalizing large deformations of individual topographic organization. We demonstrate that FUGW is well-suited for whole-brain landmark-free alignment. The unbalanced feature allows to deal with the fact that functional areas vary in size across subjects. Our results show that FUGW alignment significantly increases between-subject correlation of activity for independent functional data, and leads to more precise mapping at the group level.
• Estimation of extreme quantiles with neural networks, application to extreme rainfalls
  
  • Allouche Michaël
  • Girard Stéphane
  • Gobet Emmanuel
  , 2022.
• Statistical learning and causal inference for energy production
  
  • Acharki Naoufal
  , 2022. With the significant growth of the data generated by the sector, energy companies are relying more on Artificial Intelligence for their business and development. Indeed, applying Machine Learning algorithms to this data can help them to predict energy demand and consumption and anticipate its failures efficiently, with less time and at low cost. Machine Learning presents a powerful tool to search for new sustainable energy sources and optimize the use of current traditional sources.In recent years, Machine Learning has seen many successful applications and advances in the energy field. However, several difficulties arise despite its accuracy: Machine Learning models' predictions are sometimes unreliable and lack interpretability. Indeed, most Machine Learning models are black boxes. We have no idea of (i) the uncertainty of the prediction nor (ii) the real impact of changes in variables and interventions through these black boxes. This may produce an over/underestimation of the model uncertainty or misleading predictions that contradict engineers' and experts' knowledge. This problem is quite critical in energy systems where risk management and interpretability of predictions are vital for economic, environmental and operational reasons.In the first part of the thesis, we consider the problem of Uncertainty Quantification. The Gaussian Process model is known to be one of the most powerful Bayesian Machine Learning methods for quantifying the uncertainty of predictions. Maximum Likelihood estimation or Cross-Validation methods are widely used to fit parameters. Nevertheless, they may fail to fit the optimal model that estimates Prediction Intervals correctly if some assumptions do not hold, typically the well-specification of the Gaussian Process model.Concerning the problem of Gaussian process misspecified models, a robust two-step approach is developed to adjust and calibrate Prediction Intervals for Gaussian Processes Regression. The method gives prediction Intervals with appropriate coverage probabilities and small widths. It uses the Cross-Validation and the Leave-One-Out Coverage Probability as a metric to fit covariance hyperparameters and assess the Coverage Probability to a nominal level.In the second part, we consider the problem of Causal Inference of interventions. The Neyman-Rubin Causal model is widely used by statisticians to make Causal Inference and estimate the effects of a treatment on the outcome. Unfortunately, most considerations of this model are limited to the setting of a binary treatment. In many real-world applications, the variable of interest can be multi-valued or even continuous. Furthermore, treatment effects vary across units with different characteristics. The heterogeneity should be explored to personalize the intervention policy and optimize the outcome.A well-known framework of statistical estimators, called meta-learners, is extended to multiple and continuous treatments to solve the problem of heterogeneous treatment effects. The discussion about the consistency of meta-learners and the analysis of their bias and variance gives an overview of the advantages and disadvantages of each meta-learner. Finally, some recommendations and limits are highlighted about the use of meta-learners for continuous treatments.The proposed methods and contributions of the thesis are generic and can be applied to any industrial problem. The actual applications include, but are not limited to, unconventional gas wells, batteries and enhanced geothermal systems.
• Tight Bound for Sum of Heterogeneous Random Variables: Application to Chance Constrained Programming
  
  • Jacquet Quentin
  • Zorgati Riadh
  , 2022. We study a tight Bennett-type concentration inequality for sums of heterogeneous and independent variables, defined as a one-dimensional minimization. We show that this refinement, which outperforms the standard known bounds, remains computationally tractable: we develop a polynomial-time algorithm to compute confidence bounds, proved to terminate with an epsilon-solution. From the proposed inequality, we deduce tight distributionally robust bounds to Chance-Constrained Programming problems. To illustrate the efficiency of our approach, we consider two use cases. First, we study the chance-constrained binary knapsack problem and highlight the efficiency of our cutting-plane approach by obtaining stronger solution than classical inequalities (such as Chebyshev-Cantelli or Hoeffding). Second, we deal with the Support Vector Machine problem, where the convex conservative approximation we obtain improves the robustness of the separation hyperplane, while staying computationally tractable.
• Multi-output Gaussian processes for inverse uncertainty quantification in neutron noise analysis
  
  • Lartaud Paul
  • Humbert Philippe
  • Garnier Josselin
  , 2022. In a fissile material, the inherent multiplicity of neutrons born through induced fissions leads to correlations in their detection statistics. The correlations between neutrons can be used to trace back some characteristics of the fissile material. This technique known as neutron noise analysis has applications in nuclear safeguards or waste identification. It provides a non-destructive examination method for an unknown fissile material. This is an example of an inverse problem where the cause is inferred from observations of the consequences. However, neutron correlation measurements are often noisy because of the stochastic nature of the underlying processes. This makes the resolution of the inverse problem more complex since the measurements are strongly dependent on the material characteristics. A minor change in the material properties can lead to very different outputs. Such an inverse problem is said to be ill-posed. For an ill-posed inverse problem the inverse uncertainty quantification is crucial. Indeed, seemingly low noise in the data can lead to strong uncertainties in the estimation of the material properties. Moreover, the analytical framework commonly used to describe neutron correlations relies on strong physical assumptions and is thus inherently biased. This paper addresses dual goals. Firstly, surrogate models are used to improve neutron correlations predictions and quantify the errors on those predictions. Then, the inverse uncertainty quantification is performed to include the impact of measurement error alongside the residual model bias.
• Stochastic measure-valued models for populations expanding in a continuum
  
  • Louvet Apolline
  , 2022. We model spatially expanding populations by means of two spatial Λ-Fleming Viot processes (or SLFVs) with selection: the k-parent SLFV and the ∞-parent SLFV. In order to do so, we fill empty areas with type 0 "ghost" individuals with a strong selective disadvantage against "real" type 1 individuals, quantified by a parameter k. The reproduction of ghost individuals is interpreted as local extinction events due to stochasticity in reproduction. When k → +∞, the limiting process, corresponding to the ∞-parent SLFV, is reminiscent of stochastic growth models from percolation theory, but is associated to tools making it possible to investigate the genetic diversity in a population sample. In this article, we provide a rigorous construction of the ∞-parent SLFV, and show that it corresponds to the limit of the k-parent SLFV when k → +∞. In order to do so, we introduce an alternative construction of the k-parent SLFV which allows us to couple SLFVs with different selection strengths and is of interest in its own right. We exhibit three different characterizations of the ∞-parent SLFV, which are valid in different settings and link together population genetics models and stochastic growth models.
• Convergence analysis and novel algorithms in multi-objective optimization
  
  • Marescaux Eugénie
  , 2022. Optimization is the field of applied mathematics concerned with minimizing (or maxi-mizing) one or more objective functions. It has many applications in science and indus-try, from scheduling power outages to designing cars to describing physical phenomena.Multi-objective optimization aims at finding a good approximation of the Pareto set,that is the set of feasible solutions which cannot be improved in all objectives simultane-ously, and the Pareto front, its image in the objective space. In this Ph.D. thesis, we seekto improve the understanding of the convergence speed of multi-objective optimizationalgorithms towards the entire Pareto front. We rely on the hypervolume, a widely usedset quality indicator, to quantify the optimality gap between the Pareto front and itsapproximation provided by an algorithm. The computational cost is measured by thenumber of evaluations of the objective functions.First, we derive a theoretical upper bound on the speed of convergence. We provethat for a wide class of Pareto fronts, the smallest optimality gap associated to a set ofn points is higher than 1/(n + 1) multiplied by a constant. The constant depends onthe Pareto front and the reference point used in the definition of the hypervolume. Thisresult shows that the speed of convergence is sublinear at best, which is worse than theconvergence rates that can be achieved by single-objective optimization algorithms.Then, we introduce an algorithmic framework, HyperVolume-Based IncrementalSingle-Objective Optimization for Multi-Objective Optimization (HV-ISOOMOO). Ameta-iteration of HV-ISOOMOO corresponds to the solving of a single-objective op-timization subproblem. We provide lower bounds on the convergence speed of thePareto front approximation formed by the solutions returned by the single-objectivesolver (which we call the final incumbents Pareto front approximation) under the as-sumption that the solver returns a global optimum with infinite precision. For convexPareto fronts, this ideal algorithm reaches the best achievable convergence rate: θ(1/n).Finally, we provide an implementation of HV-ISOOMOO coupled with CMA-ES,a state-of-the-art single-objective optimization algorithm. We call it Multi-ObjectiveCovariance Matrix Adaptation 2 (MO-CMA-2). The MO-CMA-2 algorithm presentsstate-of-the-art performance. On a simple convex problem, we empirically analyze theevolution of the optimality gap of its final incumbents Pareto front approximation withrespect to the number of meta-iterations of HV-ISOOMOO and iterations of CMA-ES