Publications

2022

• Sliding window strategy for convolutional spike sorting with Lasso Algorithm, theoretical guarantees and complexity
  
  • Dragoni Laurent
  • Flamary Rémi
  • Lounici Karim
  • Reynaud-Bouret Patricia
  Acta Applicandae Mathematicae, Springer Verlag, 2022. We present a fast algorithm for the resolution of the Lasso for convolutional models in high dimension, with a particular focus on the problem of spike sorting in neuroscience. Making use of biological properties related to neurons, we explain how the particular structure of the problem allows several optimizations, leading to an algorithm with a temporal complexity which grows linearly with respect to the size of the recorded signal and can be performed online. Moreover the spatial separability of the initial problem allows to break it into subproblems, further reducing the complexity and making possible its application on the latest recording devices which comprise a large number of sensors. We provide several mathematical results: the size and numerical complexity of the subproblems can be estimated mathematically by using percolation theory. We also show under reasonable assumptions that the Lasso estimator retrieves the true support with large probability. Finally the theoretical time complexity of the algorithm is given. Numerical simulations are also provided in order to illustrate the efficiency of our approach. (10.1007/s10440-022-00494-x)
  DOI : 10.1007/s10440-022-00494-x
• Growth properties of the infinite-parent spatial Lambda-Fleming-Viot process
  
  • Louvet Apolline
  • Véber Amandine
  , 2022. The infinite-parent spatial Lambda-Fleming Viot process, or infinite-parent SLFV, is a model for spatially expanding populations in which empty areas are filled with "ghost" individuals. The interest of this process lies in the fact that it is akin to a continuous-space version of the classical Eden growth model, while being associated to a dual process encoding ancestry and allowing one to study the evolution of the genetic diversity in such a population. In this article, we focus on the growth properties of the infinite-parent SLFV in two dimensions. To do so, we first define the quantity that we shall use to quantify the speed of growth of the area covered with the subpopulation of real individuals. Using the associated dual process and a comparison with a first-passage percolation problem, we show that the growth of the "occupied" region in the infinite-parent SLFV is linear in time. We use numerical simulations to approximate the growth speed, and conjecture that the actual speed is higher than the speed expected from simple first-moment calculations due to the characteristic front dynamics. We then study a toy model of two interacting growing piles of cubes in order to understand how the growth dynamics at the front edge can increase the global growth speed of the "occupied" region. We obtain an explicit formula for this speed of growth in our toy model, using the invariant distribution of a discretised version of the model. This study is of interest on its own right, and its implications are not restricted to the case of the infinite-parent SLFV.
• Validation of a data-driven fast numerical model to simulate the Immersion Cooling of a Lithium-ion Battery Pack
  
  • Solai Elie
  • Guadagnini Maxime
  • Beaugendre Heloise
  • Daccord Rémi
  • Congedo Pietro Marco
  Energy, Elsevier, 2022, 249, pp.123633. Thermal management of Lithium-ion batteries is a key element to the widespread of elec- tric vehicles. In this study, we illustrate the validation of a data-driven numerical method permitting to evaluate fast the behavior of the Immersion Cooling of a Lithium-ion Battery Pack. First, we illustrate an experiment using a set up of immersion cooling battery pack, where the temperatures, voltage and electrical current evolution of the Li-ion batteries are monitored. The impact of different charging/discharging cycles on the thermal behavior of the battery pack is investigated. Secondly, we introduce a numerical model, that simulates the heat transfer and electrical behavior of an immersion cooling Battery Thermal Management System. The deterministic numerical model is compared against the experimental measure- ments of temperatures. Then, we perform a Bayesian calibration of the multi-physics input parameters using the experimental measurements directly. The informative distributions outcoming of this process are used to validate the model in different experimental conditions and reduce the uncertainty in the model’s temperatures predictions. Finally, the learned distributions of inputs and the numerical model are used to design the system under realistic conditions representing a realistic racing car operation. A Sobol indices based sensitivity analysis is performed to get further analysis elements on the behavior of the BTMS. (10.1016/j.energy.2022.123633)
  DOI : 10.1016/j.energy.2022.123633
• Black-box optimization under constraints: simple selection algorithms and heuristic criteria
  
  • Bellin Étienne
  • Chauleur Quentin
  • Guilluy Samuel
  • Khouja Rima
  • Sylla Ahmadou
  , 2022. This report was done during the Semaine d' Études Mathématiques et Entreprises (SEME) at the Institut de recherche mathématique de Rennes (IRMAR). The subject, proposed by the research centers Cenearo and IRT Saint Exupéry, is about the characterization of optimization problems which can only be tested under a black-box procedure. We propose some selection criteria in order to categorize such problems.
• Stagnation point heat flux characterization under numerical error and boundary conditions uncertainty
  
  • Capriati Michele
  • Cortesi Andrea
  • Magin Thierry E.
  • Congedo Pietro Marco
  European Journal of Mechanics - B/Fluids, Elsevier, 2022. The numerical simulation of hypersonic atmospheric entry flows is a challenging problem. Prediction of quantities of interest, such as surface heat flux and pressure, is strongly influenced by the mesh quality using conventional second-order spatial accuracy schemes, while depending on the boundary conditions, which generally suffer from uncertainty. This paper explores these two aspects, illustrating a CFD study on the forebody of the EXPERT vehicle of the European Space Agency employing the US3D solver.
• Modeling in-flight ice accretion under uncertain conditions
  
  • Gori Giulio
  • Congedo Pietro Marco
  • Le Maitre Olivier
  • Bellosta Tommaso
  • Guardone Alberto
  Journal of Aircraft, American Institute of Aeronautics and Astronautics, 2022, 59 (3). In-flight ice accretion under parametric uncertainty is investigated. Three test cases are presented which reproduce experiments carried out at the NASA's Glenn Icing Research Tunnel (IRT) facility. A preliminary accuracy assessment, achieved comparing numerical predictions against experimental observations, confirm the robustness and the predictiveness of the computerized icing model. Besides, sensitivity analyses highlight the variance of the targeted outputs with respect to the different uncertain inputs. In rime icing conditions, a predominant role is played by the uncertainty affecting the airfoil angle of attack, the cloud liquid water content and the droplets’ mean volume diameter. In glaze icing condition, the sensitivity analysis shows instead that the output variability is due mainly to the ambient temperature uncertainty. Results expose a major criticality of standard uncertainty quantification techniques. The issue is inherent the approximation of the full icing model behavior in domain regions scarcely affected by ice build up. To mitigate the issue, a non-linear regression method is proposed and applied. (10.2514/1.C036545)
  DOI : 10.2514/1.C036545
• An ensemble model based on early predictors to forecast COVID-19 health care demand in France
  
  • Paireau Juliette
  • Andronico Alessio
  • El Hoz Nathanaël
  • Layan Maylis
  • Crepey Pascal
  • Roumagnac Alix
  • Lavielle Marc
  • Boëlle Pierre-Yves
  • Cauchemez Simon
  Proceedings of the National Academy of Sciences of the United States of America, National Academy of Sciences, 2022, 119 (18), pp.e2103302119. Significance The COVID-19 pandemic is inducing significant stress on health care structures, which can be quickly saturated with negative consequences for patients. As hospitalization comes late in the infection history of a patient, early predictors—such as the number of cases, mobility, climate, and vaccine coverage—could improve forecasts of health care demand. Predictive models taken individually have their pros and cons, and it is advantageous to combine the predictions in an ensemble model. Here, we design an ensemble that combines several models to anticipate French COVID-19 health care needs up to 14 days ahead. We retrospectively test this model, identify the best predictors of the growth rate of hospital admissions, and propose a promising approach to facilitate the planning of hospital activity. (10.1073/pnas.2103302119)
  DOI : 10.1073/pnas.2103302119
• Crime pays; homogenized wave equations for long times
  
  • Allaire Grégoire
  • Lamacz-Keymling Agnes
  • Rauch Jeffrey
  Asymptotic Analysis, IOS Press, 2022, 128 (3), pp.295-336. This article examines the accuracy for large times of asymptotic expansions from periodic homogenization of wave equations. As usual, ϵ denotes the small period of the coefficients in the wave equation. We first prove that the standard two scale asymptotic expansion provides an accurate approximation of the exact solution for times t of order ϵ − 2 + δ for any δ > 0. Second, for longer times, we show that a different algorithm, that is called criminal because it mixes different powers of ϵ, yields an approximation of the exact solution with error O ( ϵ N ) for times ϵ − N with N as large as one likes. The criminal algorithm involves high order homogenized equations that, in the context of the wave equation, were first proposed by Santosa and Symes and analyzed by Lamacz. The high order homogenized equations yield dispersive corrections for moderate wave numbers. We give a systematic analysis for all time scales and all high order corrective terms. (10.3233/ASY-211707)
  DOI : 10.3233/ASY-211707
• Optimal Transport for Conditional Domain Matching and Label Shift
  
  • Rakotomamonjy Alain
  • Flamary Rémi
  • Gasso Gilles
  • Alaya M. El
  • Berar Maxime
  • Courty Nicolas
  Machine Learning, Springer Verlag, 2022, 111 (5), pp.1651-1670. We address the problem of unsupervised domain adaptation under the setting of generalized target shift (both class-conditional and label shifts occur). We show that in that setting, for good generalization, it is necessary to learn with similar source and target label distributions and to match the class-conditional probabilities. For this purpose, we propose an estimation of target label proportion by blending mixture estimation and optimal transport. This estimation comes with theoretical guarantees of correctness. Based on the estimation, we learn a model by minimizing a importance weighted loss and a Wasserstein distance between weighted marginals. We prove that this minimization allows to match class-conditionals given mild assumptions on their geometry. Our experimental results show that our method performs better on average than competitors accross a range domain adaptation problems including digits,VisDA and Office. Code for this paper is available at \url{https://github.com/arakotom/mars_domain_adaptation}. (10.1007/s10994-021-06088-2)
  DOI : 10.1007/s10994-021-06088-2
• Monotone discretization of the Monge–Ampère equation of optimal transport
  
  • Bonnet Guillaume
  • Mirebeau Jean-Marie
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2022, 56 (3), pp.815-865. We design a monotone finite difference discretization of the second boundary value problem for the Monge–Ampère equation, whose main application is optimal transport. We prove the existence of solutions to a class of monotone numerical schemes for degenerate elliptic equations whose sets of solutions are stable by addition of a constant, and we show that the scheme that we introduce for the Monge–Ampère equation belongs to this class. We prove the convergence of this scheme, although only in the setting of quadratic optimal transport. The scheme is based on a reformulation of the Monge–Ampère operator as a maximum of semilinear operators. In dimension two, we recommend to use Selling’s formula, a tool originating from low-dimensional lattice geometry, in order to choose the parameters of the discretization. We show that this approach yields a closed-form formula for the maximum that appears in the discretized operator, which allows the scheme to be solved particularly efficiently. We present some numerical results that we obtained by applying the scheme to quadratic optimal transport problems as well as to the far field refractor problem in nonimaging optics. (10.1051/m2an/2022029)
  DOI : 10.1051/m2an/2022029
• Part and supports optimization in metal powder bed additive manufacturing using simplified process simulation
  
  • Bihr Martin
  • Allaire Grégoire
  • Betbeder-Lauque Xavier
  • Bogosel Beniamin
  • Bordeu Felipe
  • Querois Julie
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2022, 395, pp.114975. This paper is concerned with shape and topology optimization of parts and their supports, taking into account constraints coming from the metal powder bed additive manufacturing process. Despite the high complexity of this process, it is represented by the simple inherent strain model, which has the advantage of being computationally cheap. Three optimization criteria, evaluated with this model, are proposed to minimize defects caused by additive manufacturing: vertical displacements, residual stresses and deflection of the part after baseplate separation. Combining these criteria with a constraint on the compliance for the final use of the part leads to optimization problems which deliver optimized manufacturable shapes with only a slight loss on the final use performance. The numerical results are assessed by manufacturing some optimized and reference geometries. These experimental results are also used to calibrate the inherent strain model by an inverse analysis. The same type of optimization is applied to supports in the case of a fixed non-optimizable part. For our 3-d numerical tests we rely on the level set method, the notion of shape derivatives and an augmented Lagrangian algorithm for optimization. (10.1016/j.cma.2022.114975)
  DOI : 10.1016/j.cma.2022.114975
• Modal formulation and paraxial approximation for acoustic wave propagation in waveguides with surface perturbations
  
  • Garnier Josselin
  • Roux Philippe
  Journal of the Acoustical Society of America, Acoustical Society of America, 2022, 151 (5), pp.3239-3254. We propose a modal approach developed in the framework of the paraxial approximation to investigate the effects of deterministic surface perturbations in a planar waveguide. In the first part, the sensitivity of the modal amplitudes is theoretically formulated for a three-dimensional perturbation at the air–water interface. When applied to a broadband ultrasonic signal in a laboratory tank experiment, this approach results in travel-time and amplitude fluctuations that are successfully compared to experimental data recorded between two vertical source–receiver arrays that span the ultrasonic waveguide. The nonlinear shape of the modal amplitude fluctuations is of particular interest and is due to the three-dimensional nature of the surface perturbation. In the second part, a time-harmonic inversion method is built in the paraxial single-scattering approximation to image the dynamic surface perturbation from the modal transmission matrix between two source–receiver arrays. Again, the inversion results for capillary-gravity surface perturbations are successfully compared to similar inversions performed from experimental data processed with a complete set of eigenbeams extracted between the two arrays. (10.1121/10.0010533)
  DOI : 10.1121/10.0010533
• Estimation of seismic fragility curves by sequential design of experiments
  
  • Gauchy Clement
  • Feau Cyril
  • Garnier Josselin
  , 2022. Seismic probabilistic risk assessment studies consist in evaluating the probabilities of failure of mechanical structures when submitted to seismic ground motions. These studies are often concentrated on fragility curve estimation. The fragility curve is the probability of failure of the structure conditionally to a seismic intensity measure. However, its estimation requires computer experiments involving huge computation time. Such a computational burden makes crude Monte Carlo methods untractable, fragility curves estimation must then be economical in terms of sample size. We propose an algorithm of sequential planning of experiments by supposing a Gaussian process prior on the output of the mechanical computer model.
• Semi-relaxed Gromov Wasserstein divergence with applications on graphs
  
  • Vincent-Cuaz Cédric
  • Flamary Rémi
  • Corneli Marco
  • Vayer Titouan
  • Courty Nicolas
  , 2022, pp.1-28. Comparing structured objects such as graphs is a fundamental operation involved in many learning tasks. To this end, the Gromov-Wasserstein (GW) distance, based on Optimal Transport (OT), has proven to be successful in handling the specific nature of the associated objects. More specifically, through the nodes connectivity relations, GW operates on graphs, seen as probability measures over specific spaces. At the core of OT is the idea of conservation of mass, which imposes a coupling between all the nodes from the two considered graphs. We argue in this paper that this property can be detrimental for tasks such as graph dictionary or partition learning, and we relax it by proposing a new semi-relaxed Gromov-Wasserstein divergence. Aside from immediate computational benefits, we discuss its properties, and show that it can lead to an efficient graph dictionary learning algorithm. We empirically demonstrate its relevance for complex tasks on graphs such as partitioning, clustering and completion.
• Biobjective Hypervolume Based HV-ISOOMOO Algorithms Converge with At Least Sublinear Speed to the Entire Pareto Front
  
  • Marescaux Eugénie
  • Auger Anne
  , 2022. In multiobjective optimization, one is interested in finding a good approximation of the Pareto set and the Pareto front, i.e the sets of best compromises in the decision and objective spaces, respectively. In this context, we introduce a new algorithm framework based on the hypervolume and called HyperVolume based Incremental Single-Objective Optimization for MultiObjective Optimization (HV-ISOOMOO) for approximating the Pareto front with an increasing number of points. The hypervolume is a set-quality indicator which is widely used for algorithms design and performance assessment. The class of HV-ISOOMOO algorithms approximate the Pareto front by greedily maximizing this indicator. At each meta-iteration of HV-ISOOMOO algorithms, a single-objective subproblem is solved. We study the convergence to the entire Pareto front of HV-ISOOMOO under the assumption that these subproblems are solved perfectly. The convergence is defined as the convergence of the hypervolume of the sets of all metaiterations incumbents towards the hypervolume of the Pareto front. We prove tight lower bounds on the convergence-speed for convex and bilipschitz Pareto fronts in O(1/n c) with n being the number of metaiterations and c = 1 and c ≤ 1, respectively. For convex Pareto fronts, the convergence speed is in Θ(1/n), namely the highest convergence rate achievable by a biobjective optimization algorithm. These are the first results on speed of convergence of multiobjective optimization algorithms towards the entire Pareto front. We also analyze theoretically the asymptotic convergence behavior.
• RandomizedQuasiMonteCarlo.jl
  
  • Métivier David
  , 2022. Some randomization methods for Randomized Quasi Monte Carlo e.g. scrambling, shift
• Bayesian inference of model error for the calibration of two-phase CFD codes
  
  • Leoni Nicolas
  , 2022. The calibration of a computer code consists in the comparison of its predictions to the experimental data, so that the best input parameter values can be inferred. We work in a Bayesian framework, where the parameters are treated as random variables to accurately represent the uncertainty around their values, and the subsequent model predictions. The focus is on situation where, regardless of parameter values, there remains an irreducible discrepancy between model predictions and the data, which reveals the presence of model error. Every model is an imperfect representation of reality, and its model error is caused by the approximating assumptions it is based on.In the first part of the manuscript we present the theoretical and numerical contributions developed in this thesis, and the principal one is the development of a new technique for estimating model error, named Full Maximum a Posteriori (FMP), based on a new parameterization of the model discrepancy distribution, and the introduction of optimal hyperparameters. This technique is replaced in the context of traditional estimation technique, notably the one of Kennedy-O'Hagan (KOH) which is the standard in the domain. Under an assumption of normality of the posterior density, we show that the FMP method outperforms the KOH method, since it does not underestimate the parameter uncertainty and it reveals all possible explanations of the data when the posterior is multimodal. The relevance of this normality assumption is examined in situations where the number of experimental observations tends to infinity, in three asymptotic frameworks. A second chapter is dedicated to numerical techniques that are used in calibration, notably the sampling of densities using Markov Chain Monte-Carlo techniques, and the construction of a surrogate model following a dimensionality reduction of the code output using a Principal Component Analysis. We introduce a variant of the resampling algorithm applied to the FMP sample to improve its quality. In order to accelerate the FMP technique, we also propose an algorithm of construction of surrogate models for the optimal hyperparameters, with training points drawn from samples of successive Markov Chains, weighted according to the predictive uncertainties.In a second part we present two applications of the FMP calibration, where it is compared to the KOH method and the reference solution when it is attainable. First we consider the study of a liquid boiling at a wall, with a model that predicts the partitioning of the wall heat flux. The calibration is performed using data from multiple experimental configurations, which constitutes a significant number of hyperparameters to be treated simulataneously. In a second chapter we treat both experimental uncertainty and model error in the code Neptune_CFD, specifically considering the model of interfacial area transport based on bubble interaction. The calibration is performed using the data obtained from the DEBORA experimental facility. On both applications we show that the FMP method is more robust than the KOH method since it avoids overconfident estimation of model parameters and falsely narrow confidence intervals on the predictions. Furthermore, the FMP method is performed at lower cost than the reference solution, so it is attainable in situations where the number of unknowns is high. The robust and cheap character of this technique makes it relevant for nuclear applications, but also in many more domains where we require the reproduction of experimental data with numerical simulations.
• A Two-Step Procedure for Time-Dependent Reliability-Based Design Optimization Involving Piece-Wise Stationary Gaussian Processes
  
  • Cousin Alexis
  • Garnier Josselin
  • Guiton Martin
  • Munoz Zuniga Miguel
  Structural and Multidisciplinary Optimization, Springer Verlag, 2022, 65 (4), pp.120. We consider in this paper a time-dependent reliability-based design optimization (RBDO) problem with constraints involving the maximum and/or the integral of a random process over a time interval. We focus especially on problems where the process is a stationary or a piece-wise stationary Gaussian process. A two-step procedure is proposed to solve the problem. First, we use ergodic theory and extreme value theory to reformulate the original constraints into time-independent ones. We obtain an equivalent RBDO problem for which classical algorithms perform poorly. The second step of the procedure is to solve the reformulated problem with a new method introduced in this paper and based on an adaptive kriging strategy well suited to the reformulated constraints called AK-ECO for adaptive kriging for expectation constraints optimization. The procedure is applied to two toy examples involving a harmonic oscillator subjected to random forces. It is then applied to an optimal design problem for a floating offshore wind turbine. (10.1007/s00158-022-03212-1)
  DOI : 10.1007/s00158-022-03212-1
• Sensitivity Analysis Methodology for Extreme Financial Risks Using Splitting Methods based on Reversible Transformations
  
  • Gobet Emmanuel
  • Agarwal Ankush
  • Liu Gang
  • de Marco Stefano
  , 2022.
• SHAFF: Fast and consistent SHApley eFfect estimates via random Forests
  
  • Bénard Clément
  • Biau Gérard
  • da Veiga Sébastien
  • Scornet Erwan
  Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, Camps-Valls, Gustau and Ruiz, Francisco J. R. and Valera, Isabel, 2022, 151, pp.5563-5582. Interpretability of learning algorithms is crucial for applications involving critical decisions, and variable importance is one of the main interpretation tools. Shapley effects are now widely used to interpret both tree ensembles and neural networks, as they can efficiently handle dependence and interactions in the data, as opposed to most other variable importance measures. However, estimating Shapley effects is a challenging task, because of the computational complexity and the conditional expectation estimates. Accordingly, existing Shapley algorithms have flaws: a costly running time, or a bias when input variables are dependent. Therefore, we introduce SHAFF, SHApley eFfects via random Forests, a fast and accurate Shapley effect estimate, even when input variables are dependent. We show SHAFF efficiency through both a theoretical analysis of its consistency, and the practical performance improvements over competitors with extensive experiments. An implementation of SHAFF in C++ and R is available online.
• The Multivariate price formation process and cross-impact
  
  • Tomas Mehdi
  , 2022. This thesis comprises six parts. The first relates anonymous order flow and price changes using static, linear cross-impact models. We list desirable properties of such models, characterise those which satisfy them and test their predictions on different markets. The second part extends this approach to derivatives to obtain a tractable estimation method for cross-impact which is applied to SP500 options and VIX futures. In the third part, we generalise the previous setup to derive and estimate cross-impact models which account for the influence of past trades on current prices. The fourth part uses meta-order databases on stocks and futures to propose a formula for cross-impact which generalises the square-root law of market impact. In the fifth part, we propose a tick-by-tick model for price dynamics using Hawkes processes. We investigate scaling limits of prices in the high endogeneity regime to derive multivariate macroscopic price dynamics of rough Heston type. Finally, the last part solves the calibration problem of volatility models using neural networks.In the first part, we study linear cross-impact models which relate asset prices to anonymous order flow. These models are functions of the covariances of these variables. We introduce properties models should satisfy to behave well across market conditions and show that there exists a unique model which satisfies all such properties. We apply models on stocks and futures and find that the latter model is one of two robust across markets. Thus, it is a good candidate model for a unifying view of the price formation process on stocks and futures.The second part leverages the candidate model identified in the first part to extend the previous setup to derivatives. We derive an estimation method for the large cross-impact matrix which depends on low-dimensional covariances. Using SP500 options and VIX futures data, we show cross-impact captures salient features of the price formation process on derivatives.The second part examines cross-impact kernels, which account for the lasting influence of past trades on current prices. We focus on two kernel classes: kernels that anticipate future order flow to set martingale prices and those that prevent statistical arbitrage. We show that there is at most one kernel belonging to both classes. This kernel sets martingale prices but may not prevent arbitrage. To fix this, we introduce a methodology to obtain a second kernel which prevents statistical arbitrage and is the closest to setting martingale prices. Finally, we derive a calibration methodology for both kernels and apply it to futures data.The third part measures cross-impact from using two databases of proprietary orders sent by asset managers on U.S stocks and futures. These databases allow us to study the cross-impact of individual investor orders. We propose a formula for cross-impact which generalises the square-root law to account for price and order correlations. On both stocks and futures, we find that this generalisation gives more precise predictions than the square-root law.In the fourth part, we model the tick-by-tick price process using Hawkes processes. To capture the high endogeneity of financial markets, we investigate the limit where the L¹ norm of the spectral radius of the Hawkes kernel goes to one. We show that some multivariate rough volatility models emerge as the macroscopic limit of the microscopic price dynamics. In these models, volatility is a combination of underlying variance factors, each driven by a fractional Brownian motion of common Hurst index.Finally, the last part examines the calibration of volatility models by using neural networks. We first approximate the map from model parameters to contract prices using neural networks. This approximation can then be used to recover model parameters given market prices of contracts. We highlight the applicability of the method using synthetic and real market data.
• Minimization by Incremental Stochastic Surrogate Optimization for Large Scale Nonconvex Problems
  
  • Karimi Belhal
  • Wai Hoi-To
  • Moulines Eric
  • Li Ping
  , 2022. Many constrained, nonconvex and nonsmooth optimization problems can be tackled using the Majorization-Minimization (MM) method which alternates between constructing a surrogate function which upper bounds the objective function, and then minimizing this surrogate. For problems which minimize a finite sum of functions, a stochastic version of the MM method selects a batch of functions at random at each iteration and optimizes the accumulated surrogate. However, in many cases of interest such as variational inference for latent variable models, the surrogate functions are expressed as an expectation. In this contribution, we propose a doubly stochastic MM method based on Monte Carlo approximation of these stochastic surrogates. We establish asymptotic and non-asymptotic convergence of our scheme in a constrained, nonconvex, nonsmooth optimization setting. We apply our new framework for inference of logistic regression model with missing data and for variational inference of Bayesian variants of LeNet-5 and Resnet-18 on benchmark datasets.
• Super-Acceleration with Cyclical Step-sizes
  
  • Goujaud Baptiste
  • Scieur Damien
  • Dieuleveut Aymeric
  • Taylor Adrien
  • Pedregosa Fabian
  , 2022. Cyclical step-sizes are becoming increasingly popular in the optimization of deep learning problems. Motivated by recent observations on the spectral gaps of Hessians in machine learning, we show that these step-size schedules offer a simple way to exploit them. More precisely, we develop a convergence rate analysis for quadratic objectives that provides optimal parameters and shows that cyclical learning rates can improve upon traditional lower complexity bounds. We further propose a systematic approach to design optimal first order methods for quadratic minimization with a given spectral structure. Finally, we provide a local convergence rate analysis beyond quadratic minimization for the proposed methods and illustrate our findings through benchmarks on least squares and logistic regression problems.
• Wavelet-Based Multiscale Initial Flow For Improved Atlas Estimation in the Large Diffeomorphic Deformation Model Framework
  
  • Gaudfernau Fleur
  • Blondiaux Eléonore
  • Allassonnière Stéphanie
  • Le Pennec Erwan
  , 2022. Modelling the mean and variability in a population of images, a task referred to as atlas estimation, remains very challenging, especially in a clinical setting where deformations between images can occur at multiple scales. In this paper, we introduce a coarse-to-fine strategy for atlas estimation in the Large Deformation Diffeomorphic Metric Mapping framework, based on a finite parametrization of the subjects' velocity field. Using the Haar Wavelet Transform, a multiscale representation of the initial velocity fields is computed in order to optimize the template-to-subject deformations in a coarse-to-fine fashion. This reparametrization preserves the reproducing kernel Hilbert space structure of the velocity fields, enabling the algorithm to perform efficiently gradient descent. Numerical experiments on three different datasets, including a dataset of abnormal fetal brain images, show that compared to the original algorithm, the coarse-to-fine strategy reaches higher performance and yields template images that preserve important details while avoiding unrealistic features.
• A mean-field game of market-making against strategic traders
  
  • Baldacci Bastien
  • Bergault Philippe
  • Possamaï Dylan
  SIAM Journal on Financial Mathematics, Society for Industrial and Applied Mathematics, 2022, 14 (4), pp.1080-1112. We design a market-making model \`a la Avellaneda-Stoikov in which the market-takers act strategically, in the sense that they design their trading strategy based on an exogenous trading signal. The market-maker chooses her quotes based on the average market-takers' behaviour, modelled through a mean-field interaction. We derive, up to the resolution of a coupled HJB--Fokker--Planck system, the optimal controls of the market-maker and the representative market-taker. This approach is flexible enough to incorporate different behaviours for the market-takers and takes into account the impact of their strategies on the price process. (10.1137/22M1486492)
  DOI : 10.1137/22M1486492