Publications

2021

• A generative model for fBm with deep ReLU neural networks
  
  • Allouche Michaël
  • Girard Stéphane
  • Gobet Emmanuel
  , 2021.
• On the Stability of Random Matrix Product with Markovian Noise: Application to Linear Stochastic Approximation and TD Learning
  
  • Durmus Alain
  • Moulines Eric
  • Naumov Alexey
  • Samsonov Sergey
  • Wai Hoi-To
  , 2021, 134. This paper studies the exponential stability of random matrix products driven by a general (possibly unbounded) state space Markov chain. It is a cornerstone in the analysis of stochastic algorithms in machine learning (e.g. for parameter tracking in online learning or reinforcement learning). The existing results impose strong conditions such as uniform boundedness of the matrix-valued functions and uniform ergodicity of the Markov chains. Our main contribution is an exponential stability result for the $p$-th moment of random matrix product, provided that (i) the underlying Markov chain satisfies a super-Lyapunov drift condition, (ii) the growth of the matrix-valued functions is controlled by an appropriately defined function (related to the drift condition). Using this result, we give finite-time $p$-th moment bounds for constant and decreasing stepsize linear stochastic approximation schemes with Markovian noise on general state space. We illustrate these findings for linear value-function estimation in reinforcement learning. We provide finite-time $p$-th moment bound for various members of temporal difference (TD) family of algorithms.
• A new simulation strategy for solid rocket motor ignition: coupling a CFD code with a one-dimensional boundary flame model, verification against a fully resolved approach
  
  • François Laurent
  • Dupays Joël
  • Massot Marc
  , 2021. A new framework is presented for the simulation of ignition transients in complete solid rocket motors. The methodology is based on the coupling of a CFD solver for the combustion chamber flow field and a 1D solver for the unsteady combustion of the propellant, locally at each propellant boundary face of the CFD domain. Two approaches are presented. The first approach solves the propellant flame with a 1D approach, encapsulating all the numerical and multiscale modelling difficulties (surface processes, kinetics for the flame) within the boundary model. It enables a dynamic and physics-based transition from the initial inert heating of the propellant by the igniter flow to established burning, whereas traditional ignition models switch from an inert material behaviour to a quasi-steady empirical burning behaviour once a predefined ignition temperature has been reached, restricting the ability to reproduce fine ignition dynamics and unsteady combustion response. The second approach solves the propellant flame within the CFD domain, while only the surface and solid phase thermal profile are solved within the boundary model. While being more demanding in terms of mesh refinement, it allows for a detailed interaction between the ignition and the internal flow field to be obtained. Both approaches are compared for the laser-induced ignition of a propellant in one-dimension, and on a simplified yet demanding 2D test case where a propellant surface is impinged by a hot igniter jet flow. Despite the sensitivity of the configuration, the results compare very favourably, showing that accounting for the propellant flame only in the boundary model is acceptable and leads to accurate first ignition times. Yet, potential limitations and pitfalls of this approach are discussed, as well as subsequent improvements. (10.2514/6.2021-3695)
  DOI : 10.2514/6.2021-3695
• Backward Nonlinear Smoothing Diffusions
  
  • Anderson B.
  • Bishop A.
  • Moral P. Del
  • Palmier Camille
  Theory of Probability and its Applications, Society for Industrial and Applied Mathematics, 2021, 66 (2), pp.245-262. We present a backward diffusion flow (i.e., a backward-in-time stochastic differential equation) whose marginal distribution at any (earlier) time is equal to the smoothing distribution when the terminal state (at a later time) is distributed according to the filtering distribution. This is a novel interpretation of the smoothing solution in terms of a nonlinear diffusion (stochastic) flow. This solution contrasts with, and complements, the (backward) deterministic flow of probability distributions (viz. a type of Kushner smoothing equation) studied in a number of prior works. A number of corollaries of our main result are given, including a derivation of the time-reversal of a stochastic differential equation, and an immediate derivation of the classical Rauch--Tung--Striebel smoothing equations in the linear setting. (10.1137/S0040585X97T99037X)
  DOI : 10.1137/S0040585X97T99037X
• Uncertainty quantification for in-flight ice accretion under Appendix-C and Appendix-O conditions
  
  • Gori Giulio
  • Bellosta Tommaso
  • Guardone Alberto
  • Congedo Pietro
  • Le Maitre Olivier
  , 2021. This paper introduces a simulation framework for the investigation of in-flight ice accretion under parametric uncertainty. The focus of this work is to extend current uncertainty quantification approaches to the ice accretion phenomenon for Supercooled Large Droplets conditions. Forward uncertainty propagation techniques are implemented to establish uncertainty estimates associated with numerical predictions. Different test cases are presented with the aim of reproducing real experiments carried out at the NASA's Glenn Icing Research Tunnel (IRT) facility. Both Appendix C and Appendix O cases are reproduced and their results compared. Sensitivity analysis highlights the dependency of the targeted model outputs with respect to the different uncertain inputs for different icing conditions. (10.2514/6.2021-2645)
  DOI : 10.2514/6.2021-2645
• Federated mean-field stochastic control with common noise of numerous heterogeneous energy storage systems
  
  • Gobet Emmanuel
  • Grangereau Maxime
  , 2021.
• Inverse potential problems in divergence form for measures in the plane
  
  • Baratchart Laurent
  • Villalobos Guillén Cristobal
  • Hardin Doug
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2021, 27 (87). The present document is a slightly extended version of the article published in ESAIM: COCV. We show that a divergence-free measure on the plane is a continuous sum of unit tangent vector fields on rectifiable Jordan curves. This loop decomposition is more precise than the general decomposition in terms of elementary solenoids given by S.K. Smirnov when applied to the planar case. The proof involves extending the Fleming-Rishel formula to homogeneous BV functions (in any dimension), and establishing for such functions approximate continuity of measure theoretic connected components of suplevel sets as functions of the level. We apply these results to inverse potential problems whose source term is the divergence of some unknown (vector-valued) measure. A prototypical case is that of inverse magnetization problems when magnetizations are modeled by R 3-valued Borel measures. We investigate methods for recovering a magnetization µ by penalizing the measure theoretic total variation norm µ T V. In particular, we show that if a magnetization is supported in a plane, then T V-regularization schemes always have a unique minimizer, even in the presence of noise. It is further shown that T V-norm minimization (among magnetizations generating the same field) uniquely recovers planar magnetizations in the following cases: when the magnetization is carried by a collection of sufficiently separated line segments and a set that is purely 1-unrectifiable, or when a superset of the support is tree-like. We note that such magnetizations can be recovered via T V-regularization schemes in the zero noise limit by taking the regularization parameter to zero. This suggests definitions of sparsity in the present infinite dimensional context, that generate results akin to compressed sensing. (10.1051/cocv/2021082)
  DOI : 10.1051/cocv/2021082
• A tangent linear approximation of the ignition delay time. I: Sensitivity to rate parameters
  
  • Almohammadi Saja
  • Hantouche Mireille
  • Le Maitre Olivier
  • Knio Omar M
  Combustion and Flame, Elsevier, 2021, 230, pp.111426. A tangent linear approximation is developed to estimate the sensitivity of the ignition delay time with respect to individual rate parameters in a detailed chemical mechanism. Attention is focused on a gas mixture reacting under adiabatic, constant-volume conditions. The uncertainty in the rates of elementary reactions is described in terms of uncertainty factors, and are parametrized using independent canonical random variables. The approach is based on integrating the linearized system of equations governing the evolution of the partial derivatives of the state vector with respect to individual random variables, and a linearized approximation is developed to relate the ignition delay sensitivity to the scaled partial derivatives of temperature. The efficiency of the approach is demonstrated through applications to chemical mechanisms of different sizes. In particular, the computations indicate that for detailed reaction mechanisms the TLA leads to robust local sensitivity predictions at a computational cost that is order-of-magnitude smaller than that incurred by finite-difference approaches based on one-at-a-time rate perturbations. (10.1016/j.combustflame.2021.111426)
  DOI : 10.1016/j.combustflame.2021.111426
• Causal and Interpretable Rules for Time Series Analysis
  
  • Dhaou Amin
  • Bertoncello Antoine
  • Gourvénec Sébastien
  • Garnier Josselin
  • Le Pennec Erwan
  , 2021, pp.2764-2772. (10.1145/3447548.3467161)
  DOI : 10.1145/3447548.3467161
• Generalization bounds for nonparametric regression with β−mixing samples
  
  • Barrera David
  • Gobet Emmanuel
  , 2021. In this paper we present a series of results that permit to extend in a direct manner uniform deviation inequalities of the empirical process from the independent to the dependent case characterizing the additional error in terms of beta-mixing coefficients associated to the training sample. We then apply these results to some previously obtained inequalities for independent samples associated to the deviation of the least-squared error in nonparametric regression to derive corresponding generalization bounds for regression schemes in which the training sample may not be independent. These results provide a framework to analyze the error associated to regression schemes whose training sample comes from a large class of β−mixing sequences, including geometrically ergodic Markov samples, using only the independent case. More generally, they permit a meaningful extension of the Vapnik-Chervonenkis and similar theories for independent training samples to this class of β−mixing samples.
• Beyond the chemical master equation: stochastic chemical kinetics coupled with auxiliary processes
  
  • Lunz Davin
  • Batt Gregory
  • Ruess Jakob
  • Frédéric Bonnans Joseph
  PLoS Computational Biology, PLOS, 2021. The chemical master equation and its continuum approximations are indispensable tools in the modeling of chemical reaction networks. These are routinely used to capture complex nonlinear phenomena such as multimodality as well as transient events such as first-passage times, that accurately characterise a plethora of biological and chemical processes. However, some mechanisms, such as heterogeneous cellular growth or phenotypic selection at the population level, cannot be represented by the master equation and thus have been tackled separately. In this work, we propose a unifying framework that augments the chemical master equation to capture such auxiliary dynamics, and we develop and analyse a numerical solver that accurately simulates the system dynamics. We showcase these contributions by casting a diverse array of examples from the literature within this framework, and apply the solver to both match and extend previous studies. Analytical calculations performed for each example validate our numerical results and benchmark the solver implementation. (10.1371/journal.pcbi.1009214)
  DOI : 10.1371/journal.pcbi.1009214
• Uncovering the structure of the French media ecosystem
  
  • Cointet Jean-Philippe
  • Cardon Dominique
  • Mogoutov Andreï
  • Ooghe Benjamin
  • Plique Guillaume
  • Ramaciotti Morales Pedro
  , 2021. This study provides a large-scale mapping of the French media space using digital methods to estimate political polarization and to study information circuits. We collect data about the production and circulation of online news stories in France over the course of one year, adopting a multi-layer perspective on the media ecosystem. We source our data from websites, Twitter and Facebook. We also identify a certain number of important structural features. A stochastic block model of the hyperlinks structure shows the systematic rejection of counter-informational press in a separate cluster which hardly receives any attention from the mainstream media. Counter-informational sub-spaces are also peripheral on the consumption side. We measure their respective audiences on Twitter and Facebook and do not observe a large discrepancy between both social networks, with counter-information space, far right and far left media gathering limited audiences. Finally, we also measure the ideological distribution of news stories using Twitter data, which also suggests that the French media landscape is quite balanced. We therefore conclude that the French media ecosystem does not suffer from the same level of polarization as the US media ecosystem. The comparison with the American situation also allows us to consolidate a result from studies on disinformation: the polarization of the journalistic space and the circulation of fake news are phenomena that only become more widespread when dominant and influential actors in the political or journalistic space spread topics and dubious content originally circulating in the fringe of the information space. (10.48550/arXiv.2107.12073)
  DOI : 10.48550/arXiv.2107.12073
• Accuracy Assessment of Li-ion Batteries Internal Resistance Model Through CFD Simulations, Experimental Measurements And Uncertainties
  
  • Solai Elie
  • Beaugendre Héloïse
  • Bieder Ulrich
  • Congedo Pietro Marco
  , 2021. Internal resistance is a critical parameter of the thermal behavior of Li-ion battery cells. This paper proposes an innovative way to deal with the uncertainties related to this physical parameter using experimental data and numerical simulation. First, a CFD model is validated against an experimental configuration representing the behavior of heated Li-ion battery cells under constant discharging current conditions. Secondly, an Uncertainty Quantification based methodology is proposed to represent the internal resistance and its inherent uncertainties. Thanks to an accurate and fast to compute surrogate model, the impact of those uncertainties on the temperature evolution of Li-ion cells is quantified. Finally, a Bayesian inference of the internal resistance model parameters using experimental measurements is performed, permitting to reduce the prediction uncertainty by almost 95% for some temperatures of interest.
• Modelling and numerical simulation of compressible multicomponent flows
  
  • Rai Pratik
  , 2021. This thesis involves two main objectives: the modelling of compressible multiphase and multicomponent flows, and the design of novel numerical schemes for their accurate simulation.In regards to the modelling of multiphase flows, we focus on a Baer-Nunziato-like non-equilibrium model and propose a novel for reactive gas-liquid flows. Our model is hyperbolic and accounts for the mass transfer, the interfacial drag, the mechanical disequilibrium as well as thermal transfer between the phases. The model is Galilean invariant and entropy dissipative and exhibits these properties at complete non-equilibrium.With respect to the design of novel schemes for compressible flows, we focus on hyperbolic multiphase and multicomponent flow models in nonconservative form. Our discretization framework is based on the discontinuous Galerkin spectral elements method (DGSEM), based on collocation of quadrature and interpolation points. The DGSEM uses summation-by-parts (SBP) operators with simultaneous approximation terms (SAT) in the numerical quadrature for approximating the integrals over discretization elements. For our purpose, we modify the integral over cell elements by replacing the physical fluxes with entropy conservative fluxes in fluctuation form while applying entropy stable fluxes at the interface. This modification along with the SBP-SAT operators allows us to establish a semi-discrete scheme which is high-order accurate and that satisfies a semi-discrete entropy inequality. For high-order integration in time, we rely on explicit strong-stability preserving Runge-Kutta schemes that retain the properties of first order time integration schemes.We apply this scheme for the discretization of the homogeneous Baer-Nunziato model, where we derive entropy conservative and entropy stable fluxes for the model. The design of the numerical fluxes are also shown to, formally, preserve the kinetic energy at the discrete level. By analysing the fully discrete scheme, we impose conditions on the numerical parameters that restricts the time step and guarantees the positivity of the cell averaged solutions. The positivity of cell-averaged solution is further enforced at the nodal values by applying a posteriori limiters.The semi-discrete DG scheme is also applied for the discretization of the gamma-based multicomponent model of Shyue (1998). Here we aim to propose a novel high-order entropy stable scheme which allows for a sharp resolution of the material discontinuities. To this purpose, we derive entropy conservative fluxes and contact-preserving fluxes that are applied in the volume integral, based on a troubled-cell indicator function. For the numerical fluxes at the interface we design a HLLC-like solver for the multicomponent model. We show that the DG scheme satisfies a semi-discrete entropy inequality for shock solutions, preserves uniform profiles across contacts and material interfaces and maintains positivity of the solution.Numerical tests are performed in one and two space dimensions that highlight the properties of our numerical schemes.
• Spatial dynamics of a population in a heterogeneous environment
  
  • Maillard Pascal
  • Raoul Gaël
  • Tourniaire Julie
  , 2021. We consider a certain lattice branching random walk with on-site competition and in an environment which is heterogeneous at a macroscopic scale $1/\varepsilon$ in space and time. This can be seen as a model for the spatial dynamics of a biological population in a habitat which is heterogeneous at a large scale (mountains, temperature or precipitation gradient...). The model incorporates another parameter, $K$, which is a measure of the local population density. We study the model in the limit when first $\varepsilon\to 0$ and then $K\to\infty$. In this asymptotic regime, we show that the rescaled position of the front as a function of time converges to the solution of an explicit ODE. We further discuss the relation with another popular model of population dynamics, the Fisher-KPP equation, which arises in the limit $K\to\infty$. Combined with known results on the Fisher-KPP equation, our results show in particular that the limits $\varepsilon\to0$ and $K\to\infty$ do not commute in general. We conjecture that an interpolating regime appears when $\log K$ and $1/\varepsilon$ are of the same order.
• Generative model for fbm with deep ReLU neural networks
  
  • Allouche Michaël
  • Girard Stéphane
  • Gobet Emmanuel
  , 2021. Over the last few years, a new paradigm of generative models based on neural networks have shown impressive results to simulate – with high fidelity – objects in high-dimension, while being fast in the simulation phase. In this work, we focus on the simulation of continuous-time processes (infinite dimensional objects) based on Generative Adversarial Networks (GANs) setting. More precisely, we focus on fractional Brownian motion, which is a centered Gaussian process with specific covariance function. Since its stochastic simulation is known to be quite delicate, having at hand a generative model for full path is really appealing for practical use. However, designing the architecture of such neural networks models is a very difficult question and therefore often left to empirical search. We provide a high confidence bound on the uniform approximation of fractional Brownian motion B^H(t), with Hurst parameter H, by a deep-feedforward ReLU neural network fed with a Z-dimensional Gaussian vector, with bounds on the network construction (number of hidden layers and total number of neurons). 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 ≠ 1/2 ), on the Lemarié-Meyer wavelet representation of B^H. This work gives theoretical support to use, and guidelines to construct, new generative models based on neural networks for simulating stochastic processes. It may well open the way to handle more complicated stochastic models written as a Stochastic Differential Equation driven by fractional Brownian motion.
• Role of H 3 + ions in deposition of silicon thin films from SiH 4 /H 2 discharges: modeling and experiments
  
  • Zhang Tinghui
  • Orlac’h Jean-Maxime
  • Ghosh Monalisa
  • Giovangigli Vincent
  • Roca I Cabarrocas Pere
  • Novikova Tatiana
  Plasma Sources Science and Technology, IOP Publishing, 2021, 30 (7), pp.075024. (10.1088/1361-6595/ac0da2)
  DOI : 10.1088/1361-6595/ac0da2
• Analyzing the tree-layer structure of Deep Forests
  
  • Arnould Ludovic
  • Boyer Claire
  • Scornet Erwan
  • Lpsm Sorbonne
  , 2021. Random forests on the one hand, and neural networks on the other hand, have met great success in the machine learning community for their predictive performance. Combinations of both have been proposed in the literature, notably leading to the so-called deep forests (DF) (Zhou & Feng,2019). In this paper, our aim is not to benchmark DF performances but to investigate instead their underlying mechanisms. Additionally, DF architecture can be generally simplified into more simple and computationally efficient shallow forest networks. Despite some instability, the latter may outperform standard predictive tree-based methods. We exhibit a theoretical framework in which a shallow tree network is shown to enhance the performance of classical decision trees. In such a setting, we provide tight theoretical lower and upper bounds on its excess risk. These theoretical results show the interest of tree-network architectures for well-structured data provided that the first layer, acting as a data encoder, is rich enough. (10.48550/arXiv.2010.15690)
  DOI : 10.48550/arXiv.2010.15690
• Online Graph Dictionary Learning
  
  • Vincent-Cuaz Cédric
  • Vayer Titouan
  • Flamary Rémi
  • Corneli Marco
  • Courty Nicolas
  , 2021. Dictionary learning is a key tool for representation learning, that explains the data as linear combination of few basic elements. Yet, this analysis is not amenable in the context of graph learning, as graphs usually belong to different metric spaces. We fill this gap by proposing a new online Graph Dictionary Learning approach, which uses the Gromov Wasserstein divergence for the data fitting term. In our work, graphs are encoded through their nodes’ pairwise relations and modeled as convex combination of graph atoms, i.e. dictionary elements, estimated thanks to an online stochastic algorithm, which operates on a dataset of unregistered graphs with potentially different number of nodes. Our approach naturally extends to labeled graphs, and is completed by a novel upper bound that can be used as a fast approximation of Gromov Wasserstein in the embedding space. We provide numerical evidences showing the interest of our approach for unsupervised embedding of graph datasets and for online graph subspace estimation and tracking. (10.48550/arXiv.2102.06555)
  DOI : 10.48550/arXiv.2102.06555
• Unbalanced minibatch Optimal Transport; applications to Domain Adaptation
  
  • Fatras Kilian
  • Séjourné Thibault
  • Courty Nicolas
  • Flamary Rémi
  , 2021. Optimal transport distances have found many applications in machine learning for their capacity to compare non-parametric probability distributions. Yet their algorithmic complexity generally prevents their direct use on large scale datasets. Among the possible strategies to alleviate this issue, practitioners can rely on computing estimates of these distances over subsets of data, {\em i.e.} minibatches. While computationally appealing, we highlight in this paper some limits of this strategy, arguing it can lead to undesirable smoothing effects. As an alternative, we suggest that the same minibatch strategy coupled with unbalanced optimal transport can yield more robust behavior. We discuss the associated theoretical properties, such as unbiased estimators, existence of gradients and concentration bounds. Our experimental study shows that in challenging problems associated to domain adaptation, the use of unbalanced optimal transport leads to significantly better results, competing with or surpassing recent baselines. (10.48550/arXiv.2103.03606)
  DOI : 10.48550/arXiv.2103.03606
• Second order monotone finite differences discretization of linear anisotropic differential operators
  
  • Bonnans Frédéric
  • Bonnet Guillaume
  • Mirebeau Jean-Marie
  Mathematics of Computation, American Mathematical Society, 2021, 90, pp.2671-2703. We design adaptive finite differences discretizations, which are degenerate elliptic and second order consistent, of linear and quasi-linear partial differential operators featuring both a first order term and an anisotropic second order term. Our approach requires the domain to be discretized on a Cartesian grid, and takes advantage of techniques from the field of low-dimensional lattice geometry. We prove that the stencil of our numerical scheme is optimally compact, in dimension two, and that our approach is quasi-optimal in terms of the compatibility condition required of the first and second order operators, in dimension two and three. Numerical experiments illustrate the efficiency of our method in several contexts. (10.1090/mcom/3671)
  DOI : 10.1090/mcom/3671
• Existence of traveling wave solutions for the Diffusion Poisson Coupled Model: a computer-assisted proof
  
  • Breden Maxime
  • Chainais-Hillairet Claire
  • Zurek Antoine
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2021, 55 (4), pp.1669 - 1697. The Diffusion Poisson Coupled Model describes the evolution of a dense oxide layer appearing at the surface of carbon steel canisters in contact with a claystone formation. This model is a one dimensional free boundary problem involving drift-diffusion equations on the density of species (electrons, ferric cations and oxygen vacancies), coupled with a Poisson equation on the electrostatic potential and with moving boundary equations, which describe the evolution of the position of each unknown interfaces of the spatial domain. Numerical simulations suggest the existence of traveling wave solutions for this model. These solutions are defined by stationary profiles on a fixed size domain with interfaces moving both at the same velocity. In this paper, we present and apply a computer-assisted method in order to prove the existence of these traveling wave solutions. We also establish a precise and certified description of the solutions. (10.1051/m2an/2021037)
  DOI : 10.1051/m2an/2021037
• The perturbed prox-preconditioned spider algorithm: non-asymptotic convergence bounds
  
  • Fort Gersende
  • Moulines E.
  , 2021, pp.96-100. A novel algorithm named Perturbed Prox-Preconditioned SPIDER (3P-SPIDER) is introduced. It is a stochastic variancereduced proximal-gradient type algorithm built on Stochastic Path Integral Differential EstimatoR (SPIDER), an algorithm known to achieve near-optimal first-order oracle inequality for nonconvex and nonsmooth optimization. Compared to the vanilla prox-SPIDER, 3P-SPIDER uses preconditioned gradient estimators. Preconditioning can either be applied "explicitly" to a gradient estimator or be introduced "implicitly" as in applications to the EM algorithm. 3P-SPIDER also assumes that the preconditioned gradients may (possibly) be not known in closed analytical form and therefore must be approximated which adds an additional degree of perturbation. Studying the convergence in expectation, we show that 3P-SPIDER achieves a near-optimal oracle inequality O(n^(1/2) /epsilon) where n is the number of observations and epsilon the target precision even when the gradient is estimated by Monte Carlo methods. We illustrate the algorithm on an application to the minimization of a penalized empirical loss. (10.1109/SSP49050.2021.9513846)
  DOI : 10.1109/SSP49050.2021.9513846
• The Perturbed Prox-Preconditioned SPIDER algorithm for EM-based large scale learning
  
  • Fort Gersende
  • Moulines Eric
  , 2021, pp.316-320. Incremental Expectation Maximization (EM) algorithms were introduced to design EM for the large scale learning framework by avoiding the full data set to be processed at each iteration. Nevertheless, these algorithms all assume that the conditional expectations of the sufficient statistics are explicit. In this paper, we propose a novel algorithm named Perturbed Prox-Preconditioned SPIDER (3P-SPIDER), which builds on the Stochastic Path Integral Differential EstimatoR EM (SPIDER-EM) algorithm. The 3P-SPIDER algorithm addresses many intractabilities of the E-step of EM; it also deals with non-smooth regularization and convex constraint set. Numerical experiments show that 3P-SPIDER outperforms other incremental EM methods and discuss the role of some design parameters. (10.1109/SSP49050.2021.9513769)
  DOI : 10.1109/SSP49050.2021.9513769
• Hypervolume in Biobjective Optimization Cannot Converge Faster Than $\Omega(1/p)$
  
  • Marescaux Eugénie
  • Hansen Nikolaus
  , 2021. The hypervolume indicator is widely used by multi-objective optimization algorithms and for assessing their performance. We investigate a set of $p$ vectors in the biobjective space that maximizes the hypervolume indicator with respect to some reference point, referred to as $p$-<I>optimal distribution</I>. We prove explicit lower and upper bounds on the gap between the hypervolumes of the $p$-optimal distribution and the $\infty$-optimal distribution (the Pareto front) as a function of $p$, of the reference point, and of some Lipschitz constants. On a wide class of functions, this optimality gap can not be smaller than $\Omega(1/p)$, thereby establishing a bound on the optimal convergence speed of any algorithm. For functions with either bilipschitz or convex Pareto fronts, we also establish an upper bound and the gap is hence $\Theta(1/p)$. The presented bounds are not only asymptotic. In particular, functions with a linear Pareto front have the normalized exact gap of $1/(p + 1)$ for any reference point dominating the nadir point. We empirically investigate on a small set of Pareto fronts the exact optimality gap for values of $p$ up to 1000 and find in all cases a dependency resembling $1/(p + CONST)$. (10.1145/3449639.3459371)
  DOI : 10.1145/3449639.3459371