Seminars

The Departement of Mathematical Engineering organizes a series of seminars. The seminars are usually held on Tuesday from 2:00pm to 3:00pm in the Euler lecture room, Building EULER, av. Georges Lemaître 4-6, Louvain-la-Neuve (Parking 13). Be mindful that exceptions may occur; see the talk annoucements.

If you wish to receive the seminar announcements by email, please send an email to Pascale Premereur.

Master students can take this seminar for credit in either of the two semesters; see LINMA2120 for more information.

Seminars to come

23/04/2024 (14h) [Location: Maxwell building (Shannon room)]
Jean-Charles Delvenne (UCLouvain)
Markov chains, transport theory and statistical physics

We look at the following motivating problem: how to move an electronic memory from a 'zero' state to a 'one' state, at minimal energy cost? Mathematically, this amounts to design a Markov chain that drives a certain probability measure (encoding a 'zero') towards another one (a 'one') through an 'optimal' path --- an avatar of Gaspard Monge's so-called 'earth mover problem', at the core of transport theory. We explore various recent results and conjectures around this theme at the interface of statistical physics and Markov chain theory. We support these by illustrations on realistic simulations on electronic memories.

07/05/2024 (14h) [Location: Euler building (room A.002)]
Giuseppe Notarstefano (Univ. of Bologna)
System theory for optimization and learning in complex and distributed systems

In this talk I will address control and learning scenarios requiring the solution of (possibly) distributed optimization problems. I will first highlight some key challenges arising in addressing these problems as, e.g., large scale nature, structure of the required policy, local communication requirements, online and closed-loop nature of the learning strategy. Then I will show how system theory tools can be used to design and analyze solution strategies and gain insights on the algorithm performance properties. Energy and robotic systems are key sources of concrete scenarios in which these challenges arise. Applications to these key domains will be shown along with future perspectives.

13/05/2024 (11h) [Location: Euler building (room A.002)]
Damien Scieur (Samsung SAIL Montreal)
Adaptive Quasi-Newton and Anderson Acceleration Framework with Explicit Global (Accelerated) Convergence Rates

Despite the impressive numerical performance of the quasi-Newton and Anderson/nonlinear acceleration methods, their global convergence rates have remained elusive for over 50 years. This study addresses this long-standing issue by introducing a framework that derives novel, adaptive quasi-Newton and nonlinear/Anderson acceleration schemes. Under mild assumptions, the proposed iterative methods exhibit explicit, non-asymptotic convergence rates that blend those of the gradient descent and Cubic Regularized Newton's methods. The proposed approach also includes an accelerated version for convex functions. Notably, these rates are achieved adaptively without prior knowledge of the function's parameters. The framework presented in this study is generic, and its special cases include algorithms such as Newton's method with random subspaces, finite-differences, or lazy Hessian. Numerical experiments demonstrated the efficiency of the proposed framework, even compared to the l-BFGS algorithm with Wolfe line-search. See https://arxiv.org/abs/2305.19179 for more information

Previous seminars

16/04/2024 (14h) [Location: Euler building (room A.002)]
Guillaume Berger (UCLouvain)
Safety in Systems and Control: From Theory to Implementation

Modern control systems are increasingly deployed in safety-critical applications, such as autonomous vehicles, medical devices, energy grids, etc. Therefore, verifying and guaranteeing the safety of these systems is of paramount importance. Nevertheless, it remains a big challenge where NP-hardness and undecidability are the rule rather than the exception. We review the state-of-the-art theoretical tools for the formal verification and correct-by-design control of dynamical systems such as reachability analysis and barrier functions. We also discuss the practical implementation of these tools relying on recent advances in computational geometry and optimization. Finally, we present our contribution to the field by proposing a new approach for safety verification based on multiple barrier functions, called vector barrier functions. We present the theoretical foundations of this approach as well as the practical implementation leveraging ideas from reachability analysis and convex optimization. We conclude with some examples showing the efficiency of the approach on benchmark systems

02/04/2024 (14h) [Location: Euler building (room A.002)]
Ian K. Craig (University of Pretoria)
Trends and opportunities in advanced industrial process control

This presentation will look at some trends in industrial process control and opportunities that arise from these. Trends that will be discussed include how the establishment of advanced process controllers are changing, how engineers are increasingly automating tasks related to the maintenance of advanced process controllers, and how operations are transitioning from local to remote operation and control. Some of the driving factors for these trends will be discussed, as well as the challenges and opportunities that arise from them. Examples will be given to illustrate the concepts

26/03/2024 (14h) [Location: Euler building (room A.002)]
Luca Laurenti (TU Delft)
Safety of stochastic systems: from stochastic barrier functions to uncertain abstractions

Providing safety guarantees for stochastic dynamical systems has become a central problem in many fields, including control theory, machine learning, and robotics. In this talk I will present our recent work on providing safety guarantees for non-linear stochastic dynamical systems, including dynamical systems with neural networks in the loop. I will focus on two different approaches to quantify safety for stochastic systems: Stochastic Barrier Functions (SBFs) and abstractions to uncertain Markov models. While SBFs are analogous to Lyapunov functions to prove (probabilistic) set invariance, abstraction-based approaches approximate the stochastic system into a finite model for the computation of safety probability bounds. I will illustrate pros and cons both methods. I will then conclude the talk illustrating how recent results from optimal transport and stochastic approximation could be employed to complement both methods to finally provide scalable safety guarantees for non-linear uncertain systems.

19/03/2024 (14h) [Location: Euler building (room A.002)]
Maureen Heymans (Google)
Harnessing AI: Solutions for Climate Change and Learning Innovation

The recent explosion of AI has led to remarkable advancements and a wave of innovative applications. In this talk, we will explore the immense potential of AI to address two critical challenges facing our world: education and climate change. We will quickly discuss the transformative potential of GenAI in revolutionizing education, from providing personalized learning experiences, empowering teachers with useful tools, and bridging gaps in access to quality education. We will also uncover how AI can be a powerful force in the fight against climate change and learn about AI-driven solutions for predicting climate scenarios and mitigating risks, helping individuals make sustainable decisions, and building a more sustainable future for our planet.

12/03/2024 (14h) [Location: Euler building (room A.002)]
Emanuele Garone (ULB)
Active Monitoring of Large Scale Phenomena

The capability to acquire relevant measurements is a key aspect for the monitoring of phenomena of any kind. However, measurements come at a cost. In this seminar we will focus on some recent results on the “smart measurement” for those systems where it is unrealistic “to measure everything at every time” and decisions on “what to measure and when”. Case studies coming from precision agriculture will be shown and discussed.

05/03/2024 (14h) [Location: Euler building (room A.002)]
Laurent Jacques (INMA,UCLouvain)
Learning to Reconstruct Signals From Binary Measurements

Recent advances in unsupervised learning have highlighted the possibility of learning to reconstruct signals from noisy and incomplete linear measurements alone. These methods play a key role in medical and scientific imaging and sensing, where ground truth data is often scarce or difficult to obtain. However, in practice measurements are not only noisy and incomplete but also quantized. Here we explore the extreme case of learning from binary observations and provide necessary and sufficient conditions on the number of measurements required for identifying a set of signals from incomplete binary data. Our results are complementary to existing bounds on signal recovery from binary measurements. Furthermore, we introduce a novel self-supervised learning approach that only requires binary data for training. We demonstrate in a series of experiments with real datasets that our approach is on par with supervised learning and outperforms sparse reconstruction methods with a fixed wavelet basis by a large margin. This is joint work with Julián Tachella.

27/02/2024 (14h) [Location: Euler building (room A.002)]
Jemima Tabeart (TU Eindhoven)
Numerical linear algebra for weather forecasting

The quality of a weather forecast is strongly determined by the accuracy of the initial condition. Data assimilation methods allow us to combine prior forecast information with new measurements in order to obtain the best estimate of the true initial condition. However, many of these approaches require the solution an enormous least-squares problem. In this talk I will discuss some mathematical and computational challenges associated with data assimilation for numerical weather prediction, and show how structure-exploiting numerical linear algebra approaches can lead to theoretical and computational improvements. This work was conducted together with John Pearson (Edinburgh) and Davide Palitta (Bologna).

20/02/2024 (14h) [Location: Euler building (room a.002)]
Newcomers seminars

Title: Analysis and development of almost-linear time Laplacian solvers for finite element problems.
Speaker: Christophe Heneffe(PhD UCLouvain/INMA)
Abstract: Many applications require solving Laplacian systems as fast as possible. It is the case, for example, with finite element simulations. There exist many methods to solve such systems that work well in practice, but they don't enjoy strong theoretical guarantees except in very specific cases. In the last decades, many theoretical algorithms have been created which allows to solve any Laplacian systems in almost linear time. However, most have never been implemented in practice. These algorithms are very general since they work for any Laplacian system. Adapting them by taking advantage of the properties of the finite element meshes could lead to very efficient solvers for FEM problems. The purpose of this project is the development, analysis and implementation of almost-linear time algorithms specifically adapted to solve FEM problems.

Title: Efficient, Complete G-Invariance for G-Equivariant Networks via Algorithmic reduction.
Speaker: Simon Mataigne(PhD UCLouvain/INMA)
Abstract: Group-Equivariant Convolutional Neural Networks (G-CNNs) generalize the translation-equivariance of traditional CNNs to group-equivariance, using more general symmetry transformations such as rotation for weight tying. For tasks such as classification, such transformations are removed at the end of the network, to achieve group-invariance, typically by taking a maximum over the group. While this is indeed invariant, it is excessively so; two inputs that are non-identical up to group action can yield the same output, resulting in a general lack of robustness to adversarial attacks. Sanborn and Miolane (2023) proposed an alternative method for achieving invariance without loss of signal structure, called the $G$-triple correlation ($G$-TC). While this method yields demonstrable gains in accuracy and robustness, it comes with a significant increase in computational cost. In this paper, we introduce a new invariant layer based on the Fourier transform of the $G$-TC: the $G$-bispectrum. Operating in Fourier space allows us to significantly reduce the computational cost. Our main theoretical result provides a reduction of the $G$-bispectrum that conserves the selective invariance of the $G$-TC, while only requiring $\mathcal{O}(|G|)$ coefficients. In a suite of experiments, we demonstrate that our approach retains all of the advantages of the $G$-TC, while significantly reducing the computational cost

Title: An event-triggered data-driven predictive control
Speaker: Amir Mehrnoosh(PhD UCLouvain/INMA)
Abstract: We develop a data-driven model predictive control (MPC) design procedure to control unknown linear time-invariant systems. This algorithm only requires measured input-output data to drive the system to the reference signal. We add filters on desired inputs and outputs in the cost function to improve the transient response. Moreover, the Hankel matrices are updated online based on a multi-step event-triggered MPC scheme to deal with the uncertainties. This also reduces the computational cost and balances it with the closed-loop performance. Simulation results illustrate the effectiveness of the proposed approach.

13/02/2024 (14h) [Location: Euler building (room A.002)]
Stéphane Chretien (University of Lyon 2)
Relationship between sample size and architecture for the estimation of Sobolev functions using deep neural networks

Beyond the many successes of Deep Learning based techniques in various branches of data analytics, medicine, business, engineering and the human sciences, a sound understanding of the generalisation properties of these techniques is still elusive. Central to these successes are the availability of huge datasets and the availability of huge computational ressources and some of the most recent trends have given paramount importance to the necessity of building huge neural networks with millions of parameters, and most often, of several orders of magnitude larger than the size of the training set. This set-up has however led to many surprises and counterintuitive discoveries. Overprametrisation was recently shown to favour connectivity in a weak sense of the set of stationary points, hence permitting stochastic gradient type methods to potentially reach good minimisers in several stages despite the wild nonconvexity of the training problem as demonstrated by Kuditipudi et al. Relating generalisation to stability, recent theoretical breakthroughs have been able to provide a better understanding of why generalisation cannot even happen without overparametrisation as shown by Bubeck et al. Following the ideas developed by Belkin, a substantial amount of work has also been undertaken in order to study the double descent phenomenon, and the associated benign overfitting property which holds for least norm estimators in linear and mildly non-linear regression, as well as in for certain kernel based methods. In the present paper, we aim at studying the generalisation properties of overparametrised deep neural networks using a novel approach based on Neuberger's theorem.

06/02/2024 (14h) [Location: Euler building (room A.002)]
Hemant Tyagi (INRIA Lille - Nord Europe)
Dynamic ranking and translation synchronization

In many applications such as recommendation systems or sports tournaments we are given outcomes of pairwise comparisons within a collection of $n$ items, the goal being to estimate the latent strengths and/or global ranking of the items. The Bradley-Terry-Luce (BTL) model is a popular statistical model which has been studied extensively in the literature from a theoretical perspective. Existing results for this problem predominantly focus on the setting consisting of a single comparison graph $G$. However, there exist scenarios (e.g., sports tournaments) where the pairwise comparison data (both the graph and the outcomes) evolves with time, and the data is made available at $T$ grid points in the time domain. Theoretical results for this dynamic setting are relatively limited in the literature. In this talk, I will first describe a dynamic BTL model where the latent strengths of the items evolve in a Lipschitz manner over time. Given access to a sequence of $T$ comparison graphs and the associated pairwise outcomes, our goal is to estimate the latent strength of the items ($w_t \in R^n$) at any given time point $t$. To this end we propose a simple nearest neighbor based estimator combined with an existing spectral method for ranking (namely Rank Centrality). When the graphs are Erd\"os-Renyi graphs, $\ell_2$ and $\ell_{\infty}$ error bounds are obtained for estimating $w_t$ which in particular establishes the consistency of this method in terms of $T$. Next, we will look at a dynamic version of a related problem, namely Translation Synchronization, where the latent strengths of the items satisfy a weaker global smoothness assumption over the grid. I will describe two estimators for jointly estimating the latent strengths (over all grid points), and show $\ell_2$ error bounds which establish the (weak) consistency of the estimators with respect to the grid size $T$. Based on joint work with Eglantine Karle and Ernesto Araya.

30/01/2024 (14h) [Location: Euler building (room A.002)]
Mihai Florea (INMA,UCLouvain)
Gradient Methods with Memory featuring quadratic functional growth estimation for minimizing composite functions

The recently introduced Gradient Methods with Memory use a subset of the past oracle information to create a model of the objective function, whose accuracy enables them to surpass the traditional Gradient Methods in practical performance. The model introduces an overhead that is substantial, unless dealing with smooth unconstrained problems. In this work, we introduce several Gradient Methods with Memory that can solve composite problems efficiently, including unconstrained problems with non-smooth objectives. The inexactness of the auxiliary problem does not degrade the convergence guarantees. Actually, we dynamically increase the guarantees as to provably surpass those of their memory-less counterparts. These strengths are preserved when applying acceleration and the containment of inexactness further prevents error accumulation. Our methods are able to estimate key geometry parameters to attain state-of-the-art worst-case rates on many important subclasses of composite problems, where the objective smooth part satisfies a strong convexity condition or a relaxation thereof. In particular, we formulate a restart strategy applicable to optimization methods with sublinear convergence guarantees of any order. Preliminary computational results on a synthetic benchmark of signal processing and machine learning problems show that the combined use of memory and dynamic estimation of quadratic functional growth attains 9 digits of objective accuracy in fewer than 200 iterations, even when strong convexity is absent. This research was mostly performed in the Department of Electronics and Computers, Transilvania University of Brasov, Romania.

19/12/2023 (14h) [Location: Euler building (room A.002)]
Luc Rocher (University of Oxford)
Decoding deceptive algorithms: risks, public accountability, and regulation

Dramatic claims about the potential and risks of Artificial Intelligence are now legion. Computers could soon replace most jobs or become sentient warlords. In this talk, we will take a step back and examine the inherent brittleness and vulnerabilities of algorithms and data-centric technologies. Luc Rocher, a researcher and lecturer at the Oxford Internet Institute, will present an overview of their research investigating harms in socio-technical systems infused with algorithms, using a human-centered computing lens. Their research in privacy and security helped uncover risks in privacy technologies, from de-identification to query-based algorithms and differential privacy . Their work on adversarial manipulation in algorithmic markets shows that pricing algorithms can be exploited by stronger competitors, leading to sustained collusion that harms consumers and may fall outside of current competition laws. They lead the Observatory of Anonymity (https://ooa.world), an interactive website in 89 countries where visitors can find out what makes them more vulnerable to re-identification and where researchers can test the anonymity of their research data.

12/12/2023 (14h) [Location: Euler building (room A.002)]
Frédéric Crevecoeur (UCLouvain)
Closed loop control model of human reaching movements

Current research in motor control aims to understand how the brain transforms sensory information into motor commands. This question is surprisingly difficult when one considers the complexity of all the components of the biological system: the human body is a complex structure with intricate geometry, non-linear dynamics, delays and noise, while the nervous system consists of distributed processing in a network of billions of cells, dealing with information from multiple senses with their own reference frames and statistical properties. In the face of this complexity, researchers have relied on simplifying assumptions and on the theory of optimal control, enabling simple movements to be modeled and macroscopic properties of behavior to be described. Recently, our lab has focused on how healthy humans respond to changes in the environment that are consistent with model parameters and motor costs. I will present examples of results showing that the human brain updates the online controller following changes in the environment. We will discuss the implications of these experimental results from the point of view of theoretical models of control and present how they can be exploited to gain new insight into the neural basis of motor dysfunctions in clinical populations.

05/12/2023 (14h) [Location: Euler building (room A.002)]
Two talks

Speaker: Moslem Zamani(UCLouvain)
Title:Exact convergence rate of the last iterate in subgradient methods
Abstract:Subgradient methods are widely employed for addressing non-differentiable optimization problems. In this talk, we study the convergence of the last iterate in subgradient methods applied to the minimization of a nonsmooth convex function with bounded subgradients. We propose a novel proof technique that expands upon the conventional analysis of subgradient methods. We then derive convergence rates for two variants of the subgradient method, with either fixed step size and fixed step length. We show that these rates are exact by constructing functions for which the subgradient method matches the proven rate. Finally, we introduce an optimized subgradient method, based on a new sequence of stepsizes, which achieves a last-iterate convergence rate matching the established lower bounds for non-differentiable convex optimization problems.
Speaker: Simon Vary(UCLouvain)
Title:Optimization without retraction on the random generalized Stiefel manifold for canonical correlation analysis
Abstract: Optimization over the set of matrices that satisfy X^T B X = I_p , referred to as the generalized Stiefel manifold, appears in many applications such as canonical correlation analysis (CCA) and the generalized eigenvalue problem (GEVP). Solving these problems for large-scale datasets is computationally expensive and is typically done by either computing the closed-form solution with subsampled data or by iterative methods such as Riemannian approaches. Building on the work of Ablin and Peyre (2022), we propose an inexpensive iterative method that does not enforce the constraint in every iteration exactly, but instead it produces iterations that converge to the generalized Stiefel manifold. We also tackle the random case, where the matrix B is an expectation. Our method requires only efficient matrix multiplications, and has the same sublinear convergence rate as its Riemannian counterpart. Experiments demonstrate its effectiveness in various machine learning applications involving generalized orthogonality constraints, including CCA for measuring model representation similarity.

28/11/2023 (14h) [Location: Euler building (room A.002)]
Giordano Pola (University of L'Aquila)
A Formal Methods approach to the design of Cyber Physical Systems

A challenging paradigm in the design of modern engineered systems are Cyber–Physical Systems (CPS). CPS are complex, heterogeneous, spatially distributed systems where physical processes interact with distributed computing units through non-ideal communication networks. Key features of CPS are among many others, heterogeneity and complexity. Indeed, while physical processes are generally described by e.g. differential equations, computing units are generally described by finite state machines. To complicate things, communication infrastructures, conveying information in sub–systems of CPS, are characterized by a number of non–idealities that are needed to be considered towards a robust control design of such systems. The paradigm of symbolic models is promising of being appropriate in coping with the inherent heterogeneity of CPS. Symbolic models are abstract descriptions of control systems where any state corresponds to an aggregate of continuous states and any control label to an aggregate of control inputs. Since symbolic models are of the same nature as the mathematical models of the computing units, they offer a sound approach to solve control problems where software and hardware interact with the physical world, as in the case of CPS. Furthermore, by using symbolic models, one can address a wealth of logic specifications that are difficult to enforce by means of conventional control design methods. In this talk, I will give an overview of my research in this area and show an approach based on symbolic models for the control design of CPS. I will first show how a symbolic model can be constructed for approximating a nonlinear control system with any desired accuracy. I then will show how this symbolic model can be used to design digital and quantized controllers for enforcing complex logic specifications on the original nonlinear system. I will finally briefly discuss possible extensions to more realistic scenarios including disturbance inputs, time-delays in the state and control variables evolution, nonideal communication infrastructures, etc. Techniques to tame computational complexity of the approach taken will be also highlighted.

27/11/2023 (11h) [Location: Euler building (room A.002)]
Andrew McRae (EPFL)
Benign nonconvexity in group synchronization and graph clustering

I consider an optimization problem arising in orthogonal group synchronization, in which we seek to estimate orthogonal matrices from (noisy) relative measurements. The least-squares estimator over orthogonal matrices is a nonconvex program that, in general, has many spurious local minima. We show that adding a small number of degrees of freedom (specifically, relaxing to optimization over slightly “wider” Stiefel manifold matrices) makes the nonconvexity benign and still yields an optimal solution to the original problem. The general matrix case is studied in our preprint. Time permitting, I will discuss how these results can be strengthened for $Z_2$ synchronization and can be extended to the graph clustering problem under the stochastic block model; our nonconvex approach yields exact recovery for these problems up to the information-theoretic SNR threshold.

21/11/2023 (14h) [Location: Euler building (room A.002)]
Benoît Delhaye (UCLouvain)
Experimental and modeling approaches to tap into tactile feedback during manipulation

The sense of touch originates from the skin, where the deformations caused by our interactions with objects and surfaces are transduced into neural signals by the mechanoreceptors. Those signals then travel the nerves and reach the central nervous system where they are processed to give rise to a sensation related to the touched surface, or a specific motor response to act on the object. My research aims to better understand the sense of touch by studying three aspects involved in tactile feedback: First, the characterization of skin biomechanics, second, computational modeling of neural activity in touch neurons, and third, quantifying sensory perception and motor behavior. In this talk, I will provide an example of those three important aspects in the context of the tactile mechanisms that signal friction during manipulation. Then, I will provide an overview of the recent developments made in our group to better understand the sense of touch

14/11/2023 (14h) [Location: Euler building (room A.002)]
Charles Monnoyer de Galland de Carnières (UCLouvain)
Online Riverflow Forecasting with Hydromax: Background and Last Developments

For almost 30 years, the « Hydromax » application has been used for online river flow forecasting in Wallonia with the aim of warning of extreme hydrological events (such as floods and low waters). In this talk, we first provide a general description of Hydromax, and in particular of the mathematical models used to perform both short-term and long-term forecasting, respectively based on past rainfall and river flow measurements and on meteorological forecasts. We then overview the last developments recently brought to Hydromax, focusing on the one hand on new statistical corrections added to the forecasts to better deal with low-water episodes for small stations, and on the other hand on the addition of a model to forecast the level of water in the lake at the Eupen dam, with an alternative modelling able to handle temporary lacks of river flow data (such as observed during the flood of July 2021).

07/11/2023 (14h) [Location: Euler building (room A.002)]
Denis Dochain (UCLouvain)
Modelling, analysis and control of counterflow heat exchangers

Heat exchangers are one of the most largely used devices in industry. Almost all the produced or collected thermal energy passes at least once through a heat exchanger. The objective of this presentation is to give a survey of the results obtained during the PhD thesis of Jacques Kadima. The dynamics of counterflow heat exchangers are described by a set of partial differential equations for both fluids involved in the heat exchange. The presentation will provide identification results for the heat exchanger model, analysis results (including a thermodynamic perspective) and control design results.

31/10/2023 (14h) [Location: Euler building (room A.002)]
Martina Vanelli (Politecnico di Torino)
On mixed network coordination/anticoordination games

Whilst network coordination games and network anti-coordination games have received a considerable amount of attention in the literature, network games with coexisting coordinating and anti-coordinating players are known to exhibit more complex behaviors. In fact, depending on the network structure, such games may even fail to have pure-strategy Nash equilibria. An example is represented by the well-known matching pennies (discoordination) game. We derive graph-theoretic conditions for the existence of pure-strategy Nash equilibria in mixed network coordination/anti-coordination games of arbitrary size. For the case where such conditions are met, we study the asymptotic behavior of best-response dynamics and provide sufficient conditions for finite-time convergence to the set of Nash equilibria. These results build on an extension and refinement of the notion of network cohesiveness and on the formulation of the new concept of network indecomposability. The findings are extended to directed graphs and employed to prove necessary and sufficient conditions for global stability of consensus equilibria in linear threshold dynamics, robustly with respect to a (constant or time-varying) external field.

24/10/2023 (14h) [Location: Euler building (room A.002)]
Karel Devriendt (max planck institute, Leipzig)
Tropical toric maximum likelihood estimation

Many common statistical models are parametrized by polynomial maps; some examples are log-linear and graphical models. To study such statistical models, applied algebraic geometry can be used in an approach which is known as algebraic statistics. One well-studied problem in this setting is maximum likelihood estimation (MLE): given a model and some data, which points in your model best explain the data? In this talk, we consider the MLE problem when our data depends on a parameter and we ask what can be said about the convergence rates of the solution as the parameter goes to zero. This problem was solved for linear models by Agostini et al. (2021) and Ardila-Eur-Penaguiao (2022). Here we consider the problem for log-linear models, also called toric models, where the MLE problem comes down to intersecting a toric variety with a linear space. Using tools from tropical geometry -- a combinatorical shadow of algebraic geometry -- the problem simplies to intersecting the tropical toric variety (which is a linear space) with a tropical linear space (which is a polyhedral complex). I will present some preliminary results which show that the tropical MLE points, i.e. the convergence rates, are given by simple linear transformations of the data, and that the different MLE points are labeled by simplices in a certain triangulation. This is joint work with Erick Boniface and Serkan Hosten.

17/10/2023 (14h) [Location: Euler building (room a.002)]
Newcomers seminar

Title: Derivative-free optimization methods based on finite-difference gradient approximations
Speaker: Dânâ Davar(PhD UCLouvain/INMA)
Abstract: Many applications require the solution of optimization problems, however, it can be difficult to access the gradient of the objective function. This issue is very common, especially when the function values come from a computer simulation that is realized through a black-box software. In such case, derivative-free optimization methods are required, i.e., methods that only rely on function evaluations. The purpose of this project is the development and worst-case complexity analysis of derivative-free optimization methods based on finite-difference gradient approximations, for large-scale nonconvex problems with possibly inexact function values.

Title: Unleashing the power of neural networks for derivative-free optimization
Speaker: Timothé Taminiau(PhD UCLouvain/INMA)
Abstract:Derivative-free methods are useful in problems where the analytical form of the objective is either hidden or too intricate. In this setup, we consider the objective function as a black box which can only be evaluated in some points. The framework "Learning-to-Optimize" is a promising way for designing such algorithms where a new method is learned thanks to a deep neural network model. These methods showed good practical performances although theoretical results of complexity have npt been proved yet. The purpose of this project is to explore empirically and theoretically the possible benefits of deep learning for designing derivative-free optimization algorithms.

Title: Chance-Constrained Optimization Probablistic Upper bounds Applied for the JSR
Speaker: Alexis Vuille(PhD UCLouvain/INMA)
Abstract: Chance-Constrained Optimization - where only a subset of the constraints are sampled - allow under regularized and structural assumptions to compute probablistic upper bounds on the violation probability. This theory can be applied to the Joint Spectral Radius to analyse the data-driven stability of switched linear systems.

10/10/2023 (14h) [Location: Euler building (room A.002)]
Nicolas Boulle (University of Cambridge)
Elliptic PDE learning is provably data-efficient

PDE learning is an emerging field at the intersection of machine learning, physics, and mathematics, that aims to discover properties of unknown physical systems from experimental data. Popular techniques exploit the approximation power of deep learning to learn solution operators, which map source terms to solutions of the underlying PDE. Solution operators can then produce surrogate data for data-intensive machine learning approaches such as learning reduced order models for design optimization in engineering and PDE recovery. In most deep learning applications, a large amount of training data is needed, which is often unrealistic in engineering and biology. However, PDE learning is shockingly data-efficient in practice. We provide a theoretical explanation for this behavior by constructing an algorithm that recovers solution operators associated with elliptic PDEs and achieves an exponential convergence rate with respect to the size of the training dataset. The proof technique combines prior knowledge of PDE theory and randomized numerical linear algebra techniques and may lead to practical benefits such as improving dataset and neural network architecture designs.

03/10/2023 (14h) [Location: Euler building (room A.002)]
Bryan Van Scoy (Miami University, Ohio)
Systematic Analysis of Iterative Black-Box Optimization Algorithms using Control

Iterative algorithms are used to solve optimization problems throughout control, robotics, statistics and estimation, signal processing, communication, networks, machine learning, and data science. Recent work from both optimization and control communities has developed a systematic methodology to analyze the worst-case performance of a black-box algorithm over a class of problems. In this talk, we first describe this systematic methodology from a controls perspective and then show how it can be used to analyze and design algorithms in various contexts, such as trading off convergence rate and robustness to gradient noise with noisy first-order oracles, consensus optimization for a multi-agent system, and primal-dual algorithms.

26/09/2023 (14h) [Location: Euler building (room A.002)]
Somya Singh (UCLouvain)
Reinforcement Models via Pólya Urns

Classical Pólya urns have been widely used in the field of applied mathematics to model social and contagion networks. In this talk, I will illustrate three distinct random reinforcement models constructed via modified Pólya urns, through which an epidemic spread model, a consensus-achieving network of agents and a randomly growing preferential attachment graph is developed. The former two models are devised via interacting two-color finite memory Pólya urns, where the underlying draw process is Markovian in nature, while the latter is constructed via an expanding color Pólya urn.

19/09/2023 (14h) [Location: Euler building (room A.002)]
Estelle Massart (UCLouvain)
Global optimization using random embeddings

We address global high-dimensional optimization problems in which the objective is mostly varying along a low-dimensional subspace of the search space. These problems appear, e.g., in complex engineering and physical simulation/inverse problems. We propose a random-subspace algorithmic framework (referred to as X-REGO) that randomly projects, in a sequential or simultaneous manner, the high-dimensional original problem into low-dimensional subproblems that can then be solved with any global, or even local, optimization solver. For Lipschitz-continuous objectives, we analyse its convergence using novel tools from probability theory as well as conic integral geometry; our analysis relies on an estimation of the probability that the randomly-embedded subproblem shares (approximately) the same global optimum as the original problem. This success probability is then used to show almost sure convergence of X-REGO to an approximate global solution of the original problem, under weak assumptions on the problem (having a strictly feasible global solution) and on the solver (guaranteed to find an approximate global solution of the reduced problem with sufficiently high probability). This is joint work with C. Cartis (University of Oxford) and A. Otemissov (Nazarbayev University).