Publications

2025

• Benchmarking Powell's Legacy: Performance of Five Derivative-Free Solvers in pdfo on the bbob Test Suite
  
  • Brockhoff Dimo
  • Villain Tanguy
  , 2025, pp.1833-1841. The pdfo library by Tom M. Ragonneau and Zaikun Zhang makes the five derivative-free solvers BOBYQA, COBYLA, LINCOA, NEWUOA, and UOBYQA—originally written by Michael J. D. Powell—available in Python. In this paper, we are comparing their performance on the bbob test suite with three other solvers from the COCO data archive: CMA-ES from pycma, SLSQP and BFGS from scipy. We also compare the original solvers, written by Powell in Fortran 77, with the current pdfo versions, which saw multiple bug fixes and code improvements by Ragonneau and Zhang. For the latter comparison, we do not see large effects on performance between the Fortran 77 version and the current pdfo version. The only notable exception is the Bent Cigar function where we observe differences by a factor of 2–5 for BOBYQA, LINCOA, and NEWUOA. Compared to the other baseline algorithms, BOBYQA, LINCOA and NEWUOA perform very similarly over all bbob functions, being about a factor of 5 slower than SLSQP and BFGS while UOBYQA—as the best-performing pdfo solver—outperforms SLSQP and BFGS for larger budgets when compared over all 24 bbob functions. The linear surrogate of COBYLA, on the contrary, is clearly worse over all functions than the other algorithms. (10.1145/3712255.3734343)
  DOI : 10.1145/3712255.3734343
• A Continuation Method Based on CMA-ES
  
  • Vu Hoang Nguyen
  • Brockhoff Dimo
  , 2025, pp.439 - 442. In this poster, we showcase a new algorithm for approximating the Pareto set of two-objective (unconstrained) optimization problems based on the idea of continuation. The algorithm tries to move "along" the Pareto set from one single-objective optimum to the other and back via a single-objective reformulation of the two-objective problem and the well-known CMA-ES as single-objective solver. The introduced algorithm BOG-CMA-ES (standing for bi-objective gradient based CMA-ES) is visually analyzed on simple convex-quadratic objective functions and extensively benchmarked on the bbob-biobj test suite of the COCO platform, including comparisons with current state-of-the-art algorithms. (10.1145/3712255.3726645)
  DOI : 10.1145/3712255.3726645
• On the Robustness of Nelder-Mead to Positive and Negative Noise Outliers with Heavy-Tails on the BBOB Test Suite
  
  • Chotard Alexandre
  • Auger Anne
  , 2025, pp.1859-1866. We investigate the robustness to noise and outliers of the Nelder-Mead derivative-free optimization algorithm implemented in the scipy.optimize Python library. Using the noisifier module from the COCO platform, we investigate the impact of adding or subtracting a positive Cauchy noise to the function value with a certain probability p. This probability is varied from 0 till 0.4 allowing to appraise the impact of the noise. We find that Nelder-Mead has highly asymmetric performances with respect to adding a positive noise (i.e. degrading artificially possibly good solutions) or adding a negative noise (i.e. making possibly bad solutions appear good). When adding positive noise, Nelder-Mead shows some robustness in low dimensions (≤ 5). However adding negative noise has very detrimental consequences for the performances of Nelder-Mead, even with a low value of p or in low dimension. (10.1145/3712255.3734335)
  DOI : 10.1145/3712255.3734335
• On the Robustness of BFGS to Positive and Negative Noise Outliers on the BBOB Test Suite
  
  • Chotard Alexandre
  • Auger Anne
  , 2025, pp.1850-1858. We investigate the impact of outlier noise on the performance of the scipy.optimize implementation of the quasi-Newton BFGS solver. Using the BBOB testbed corrupted with positive—making solution look worse than what they are—or negative—making solutions look better than what they are—outliers simulated with a Cauchy distribution with a probability p, we analyze how the performance is impacted. We show that the impact of positive or negative noise outliers is almost symmetric, that on simple problems BFGS has some robustness to noise, and that for ill-conditionned problems BFGS appears to fail when p or the dimension is too high. (10.1145/3712255.3734333)
  DOI : 10.1145/3712255.3734333
• Rank-based Linear-Quadratic Surrogate Assisted CMA-ES
  
  • Gharafi Mohamed
  • Hansen Nikolaus
  • Le Riche Rodolphe
  • Brockhoff Dimo
  , 2025. In this poster, we introduce a rank-based surrogate-assisted variant of CMA-ES. Unlike previous methods that employ rank information as constraints to train an SVM classifier, our approach employs a linear-quadratic regression on the ranks. We investigate the method's invariance empirically. While this first algorithm outperforms CMA-ES with a few exceptions, it falls short to entirely meet the lq-CMA-ES performance levels. To address this, we propose an enhanced variant that handles together two alternative surrogates, one based on the ranks and one based on the original function values. Although this variant sacrifices strict invariance, it gains in robustness and achieves performance comparable to, or even exceeding, lq-CMA-ES on transformed problems. This last algorithm shows how simply incorporating new transformations of rank values could improve any surrogate-based CMA-ES variant.
• How Robust is UOBYQA to Worsening, Frozen Noise? Investigations on the bbob Test Suite With Outliers
  
  • Brockhoff Dimo
  • Villain Tanguy
  , 2025, pp.1842 - 1849. UOBYQA, short for Unconstrained Optimization By Quadratic Approximation, is one of the well-known solvers derived and implemented by Michael J. D. Powell. In each step, the algorithm builds a quadratic surrogate of the objective function, interpolating quadratically many points for which the true function values are known. The model is optimized within the so-called trust region and the resulting solution is evaluated next. Adaptation of the trust region radius allows for fast convergence on a wide range of (noiseless) functions without the need for derivatives. In this workshop paper, we investigate the effect of (frozen) nonnegative, i.e., worsening noise on UOBYQA with varying probability of solutions being affected by the noise. To this end, we use the COCO platform and its newest addition, the noiser, applied to the classical bbob functions. The numerical benchmarking experiments showcase that UOBYQA is negatively affected by the noise, but surprisingly little over a wide range of noise strengths for some of the bbob functions. (10.1145/3712255.3734354)
  DOI : 10.1145/3712255.3734354
• On the Pareto Set and Front of Multiobjective Spherical Functions with Convex Constraints
  
  • Auger Anne
  • Brockhoff Dimo
  • Cork Jordan
  • Tušar Tea
  , 2025, pp.527-535. We analyze a fundamental class of multiobjective constrained problems where the objectives are spherical functions and the constraints are convex. As an application from the projection theorem on closed convex sets, we prove that the constrained Pareto set corresponds to the orthogonal projection of the unconstrained Pareto set onto the feasible region. We establish this fundamental geometric property and illustrate its implications using visualizations of Pareto sets and fronts under various constraint configurations. Furthermore, we assess the performance of NSGA-II on these problems, examining its ability to approximate the constrained Pareto set across different dimensions. Our findings highlight the importance of theoretically grounded and understood benchmark problems for assessing algorithmic behavior and contribute to a deeper understanding of constrained multiobjective landscapes. (10.1145/3712256.3726432)
  DOI : 10.1145/3712256.3726432
• Classification-Based Linear Surrogate Modeling of Constraints for AL-CMA-ES
  
  • Girardin Oskar
  • Hansen Nikolaus
  • Brockhoff Dimo
  • Auger Anne
  , 2025, pp.728-736. We introduce linear surrogate functions for modeling inequality constraints to solve constrained blackbox optimization problems with the Augmented Lagrangian CMA-ES. Each surrogate is constructed from a binary classifier that predicts the sign of the constraint value. The classifier, and consequently the resulting algorithm, is invariant under sign preserving transformations of the constraint values and can handle binary, flat, and deceptive constraints. Somewhat surprisingly, we find that adopting a sign-based classification model of the constraints allows to solve classes of constrained problems which can not be solved with the original Augmented Lagrangian method using the true constraint value. (10.1145/3712256.3726435)
  DOI : 10.1145/3712256.3726435
• Benchmarking CMA-ES under Additive and Subtractive Noise on the BBOB Testbed
  
  • Girardin Oskar
  , 2025, pp.1867-1874. We benchmark a non-elitist CMA-ES algorithm on the BBOB testbed with additive and subtractive noise. In particular, we consider the case where re-evaluated solutions produce the same observed function value. As a comparison, we benchmark a version of CMA-ES with resampling, which aims at reducing the effective noise level. We find CMA-ES to be more sensitive to subtractive noise than to additive noise in dimensions 2, 3, 5, 10, 20 and 40. Resampling for CMA-ES appears to be detrimental for low noise levels, while it is beneficial for high noise levels. (10.1145/3712255.3734332)
  DOI : 10.1145/3712255.3734332
• Prediction-Aware Learning in Multi-Agent Systems
  
  • Capitaine Aymeric
  • Boursier Etienne
  • Moulines Eric
  • Jordan Michael I.
  • Durmus Alain
  , 2025, PMLR 267. The framework of uncoupled online learning in multiplayer games has made significant progress in recent years. In particular, the development of time-varying games has considerably expanded its modeling capabilities. However, current regret bounds quickly become vacuous when the game undergoes significant variations over time, even when these variations are easy to predict. Intuitively, the ability of players to forecast future payoffs should lead to tighter guarantees, yet existing approaches fail to incorporate this aspect. This work aims to fill this gap by introducing a novel prediction-aware framework for time-varying games, where agents can forecast future payoffs and adapt their strategies accordingly. In this framework, payoffs depend on an underlying state of nature that agents predict in an online manner. To leverage these predictions, we propose the POWMU algorithm, a contextual extension of the optimistic Multiplicative Weight Update algorithm, for which we establish theoretical guarantees on social welfare and convergence to equilibrium. Our results demonstrate that, under bounded prediction errors, the proposed framework achieves performance comparable to the static setting. Finally, we empirically demonstrate the effectiveness of POWMU in a traffic routing experiment.
• Revisiting Non-Acyclic GFlowNets in Discrete Environments
  
  • Morozov Nikita
  • Maksimov Ian
  • Tiapkin Daniil
  • Samsonov Sergey
  , 2025. Generative Flow Networks (GFlowNets) are a family of generative models that learn to sample objects from a given probability distribution, potentially known up to a normalizing constant. Instead of working in the object space, GFlowNets proceed by sampling trajectories in an appropriately constructed directed acyclic graph environment, greatly relying on the acyclicity of the graph. In our paper, we revisit the theory that relaxes the acyclicity assumption and present a simpler theoretical framework for non-acyclic GFlowNets in discrete environments. Moreover, we provide various novel theoretical insights related to training with fixed backward policies, the nature of flow functions, and connections between entropy-regularized RL and non-acyclic GFlowNets, which naturally generalize the respective concepts and theoretical results from the acyclic setting. In addition, we experimentally re-examine the concept of loss stability in non-acyclic GFlowNet training, as well as validate our own theoretical findings. (10.48550/arXiv.2502.07735)
  DOI : 10.48550/arXiv.2502.07735
• On Teacher Hacking in Language Model Distillation
  
  • Tiapkin Daniil
  • Calandriello Daniele
  • Ferret Johan
  • Perrin Sarah
  • Vieillard Nino
  • Ramé Alexandre
  • Blondel Mathieu
  , 2025. Post-training of language models (LMs) increasingly relies on the following two stages: (i) knowledge distillation, where the LM is trained to imitate a larger teacher LM, and (ii) reinforcement learning from human feedback (RLHF), where the LM is aligned by optimizing a reward model. In the second RLHF stage, a well-known challenge is reward hacking, where the LM over-optimizes the reward model. Such phenomenon is in line with Goodhart's law and can lead to degraded performance on the true objective. In this paper, we investigate whether a similar phenomenon, that we call teacher hacking, can occur during knowledge distillation. This could arise because the teacher LM is itself an imperfect approximation of the true distribution. To study this, we propose a controlled experimental setup involving: (i) an oracle LM representing the ground-truth distribution, (ii) a teacher LM distilled from the oracle, and (iii) a student LM distilled from the teacher. Our experiments reveal the following insights. When using a fixed offline dataset for distillation, teacher hacking occurs; moreover, we can detect it by observing when the optimization process deviates from polynomial convergence laws. In contrast, employing online data generation techniques effectively mitigates teacher hacking. More precisely, we identify data diversity as the key factor in preventing hacking. Overall, our findings provide a deeper understanding of the benefits and limitations of distillation for building robust and efficient LMs.
• Discrete Markov Probabilistic Models
  
  • Pham Le-Tuyet-Nhi
  • Shariatian Dario
  • Ocello Antonio
  • Conforti Giovanni
  • Durmus Alain
  , 2025. This paper introduces the Discrete Markov Probabilistic Model (DMPM), a novel algorithm for discrete data generation. The algorithm operates in the space of bits {0, 1} d , where the noising process is a continuous-time Markov chain that can be sampled exactly via a Poissonian clock that flips labels uniformly at random. The time-reversal process, like the forward noise process, is a jump process, with its intensity governed by a discrete analogue of the classical score function. Crucially, this intensity is proven to be the conditional expectation of a function of the forward process, strengthening its theoretical alignment with score-based generative models while ensuring robustness and efficiency. We further establish convergence bounds for the algorithm under minimal assumptions and demonstrate its effectiveness through experiments on low-dimensional Bernoulli-distributed datasets and high-dimensional binary MNIST data. The results highlight its strong performance in generating discrete structures. This work bridges theoretical foundations and practical applications, advancing the development of effective and theoretically grounded discrete generative modeling.
• Unified Breakdown Analysis for Byzantine Robust Gossip
  
  • Gaucher Renaud
  • Dieuleveut Aymeric
  • Hendrikx Hadrien
  , 2025, 267, pp.18868-18896. In decentralized machine learning, different devices communicate in a peer-to-peer manner to collaboratively learn from each other's data. Such approaches are vulnerable to misbehaving (or Byzantine) devices. We introduce F-RG, a general framework for building robust decentralized algorithms with guarantees arising from robust-sum-like aggregation rules F. We then investigate the notion of breakdown point, and show an upper bound on the number of adversaries that decentralized algorithms can tolerate. We introduce a practical robust aggregation rule, coined CSours, such that CSours-RG has a near-optimal breakdown. Other choices of aggregation rules lead to existing algorithms such as ClippedGossip or NNA. We give experimental evidence to validate the effectiveness of CSours-RG and highlight the gap with NNA, in particular against a novel attack tailored to decentralized communications.
• Finite-Sample Convergence Bounds for Trust Region Policy Optimization in Mean-Field Games
  
  • Ocello Antonio
  • Tiapkin Daniil
  • Mancini Lorenzo
  • Laurière Mathieu
  • Moulines Eric
  , 2025. We introduce Mean-Field Trust Region Policy Optimization (MF-TRPO), a novel algorithm designed to compute approximate Nash equilibria for ergodic Mean-Field Games (MFG) in finite state-action spaces. Building on the well-established performance of TRPO in the reinforcement learning (RL) setting, we extend its methodology to the MFG framework, leveraging its stability and robustness in policy optimization. Under standard assumptions in the MFG literature, we provide a rigorous analysis of MF-TRPO, establishing theoretical guarantees on its convergence. Our results cover both the exact formulation of the algorithm and its sample-based counterpart, where we derive high-probability guarantees and finite sample complexity. This work advances MFG optimization by bridging RL techniques with mean-field decision-making, offering a theoretically grounded approach to solving complex multi-agent problems. (10.48550/arXiv.2505.22781)
  DOI : 10.48550/arXiv.2505.22781
• High Performance Parallel Solvers for the time-harmonic Maxwell Equations
  
  • Fressart Elise
  • Dubois Sébastien
  • Gouarin Loïc
  • Massot Marc
  • Nowak Michel
  • Spillane Nicole
  , 2025. We consider the numerical solution of large scale time-harmonic Maxwell equations. To this day, this problem remains difficult, in particular because the equations are neither Hermitian nor semi-definite. Our approach is to compare different strategies for solving this set of equations with preconditioners that are available either in PETSc, MUMPS, or in hypre. Four different preconditioners are considered. The first is the sparse approximate inverse, which is often applied to electromagnetic problems. The second is Restricted Additive Schwarz, a domain decomposition preconditioner. The third is the Hiptmair-Xu preconditioner which is tailored to the positive Maxwell equations, a nearby problem. The final preconditioner is MUMPS's Block Low-Rank method, a compressed block procedure. We also compare the performance of this method to the standard LU factorization technique, which is a direct solver. Performance with respect to the mesh size, the number of CPU cores, the wavelength and the physical size of the domain are considered. This work in progress yields temporary conclusions in favour of the Hiptmair-Xu and the Block Low-Rank preconditioners.
• Signed tropicalization of polar cones
  
  • Akian Marianne
  • Allamigeon Xavier
  • Gaubert Stéphane
  • Sergeev Sergei
  Journal of Optimization Theory and Applications, Springer Verlag, 2025, 207 (10). We study the tropical analogue of the notion of polar of a cone, working over the semiring of tropical numbers with signs. We characterize the cones which arise as polars of sets of tropically nonnegative vectors by an invariance property with respect to a tropical analogue of Fourier-Motzkin elimination. We also relate tropical polars with images by the nonarchimedean valuation of classical polars over real closed nonarchimedean fields and show, in particular, that for semi-algebraic sets over such fields, the operation of taking the polar commutes with the operation of signed valuation (keeping track both of the nonarchimedean valuation and sign). We apply these results to characterize images by the signed valuation of classical cones of matrices, including the cones of positive semidefinite matrices, completely positive matrices, completely positive semidefinite matrices, and their polars, including the cone of co-positive matrices, showing that hierarchies of classical cones collapse under tropicalization. We finally discuss an application of these ideas to optimization with signed tropical numbers. (10.1007/s10957-025-02732-2)
  DOI : 10.1007/s10957-025-02732-2
• On Global Convergence Rates for Federated Policy Gradient under Heterogeneous Environment
  
  • Labbi Safwan
  • Mangold Paul
  • Tiapkin Daniil
  • Moulines Eric
  , 2025. Ensuring convergence of policy gradient methods in federated reinforcement learning (FRL) under environment heterogeneity remains a major challenge. In this work, we first establish that heterogeneity, perhaps counter-intuitively, can necessitate optimal policies to be non-deterministic or even time-varying, even in tabular environments. Subsequently, we prove global convergence results for federated policy gradient (FedPG) algorithms employing local updates, under a Łojasiewicz condition that holds only for each individual agent, in both entropy-regularized and non-regularized scenarios. Crucially, our theoretical analysis shows that FedPG attains linear speed-up with respect to the number of agents, a property central to efficient federated learning. Leveraging insights from our theoretical findings, we introduce b-RS-FedPG, a novel policy gradient method that employs a carefully constructed softmax-inspired parameterization coupled with an appropriate regularization scheme. We further demonstrate explicit convergence rates for b-RS-FedPG toward near-optimal stationary policies. Finally, we demonstrate that empirically both FedPG and b-RS-FedPG consistently outperform federated Q-learning on heterogeneous settings. (10.48550/arXiv.2505.23459)
  DOI : 10.48550/arXiv.2505.23459
• Large-Eddy simulation of solid/fluid heat and mass transfer applied to the thermal degradation of composite materials
  
  • Grenouilloux Adrien
  • Bioche Kévin
  • Dellinger Nicolas
  • Letournel Roxane
  • Bechane Yacine
  • Moureau Vincent
  , 2025.
• Improving GFlowNets with Monte Carlo Tree Search
  
  • Morozov Nikita
  • Tiapkin Daniil
  • Samsonov Sergey
  • Naumov Alexey
  • Vetrov Dmitry
  , 2024. Generative Flow Networks (GFlowNets) treat sampling from distributions over compositional discrete spaces as a sequential decision-making problem, training a stochastic policy to construct objects step by step. Recent studies have revealed strong connections between GFlowNets and entropy-regularized reinforcement learning. Building on these insights, we propose to enhance planning capabilities of GFlowNets by applying Monte Carlo Tree Search (MCTS). Specifically, we show how the MENTS algorithm (Xiao et al., 2019) can be adapted for GFlowNets and used during both training and inference. Our experiments demonstrate that this approach improves the sample efficiency of GFlowNet training and the generation fidelity of pre-trained GFlowNet models. (10.48550/arXiv.2406.13655)
  DOI : 10.48550/arXiv.2406.13655
• Contribution to the study of the mean field games master equation
  
  • Meynard Charles
  , 2025. This thesis focuses on the problems of uniqueness and existence to the mean field games master equation. The first part is devoted to the study of this equation for mean field games in which the optimization problem solved by an individual player depends on the realization of a stochastic process common to all players. This process may be autonomous, but it can also depend on the evolution of the game through the distribution of players. We provide sufficient conditions ensuring the existence and uniqueness of a solution to the master equation associated with such games, first in the context of finite state space mean field games. We then extend these results to master equations defined on the space of probability measures under several notions of monotonicity. In the second part, we introduce the notion of monotone solution for so-called displacement monotone mean field games. This notion of weak solution to the master equation does not require any differentiability assumptions on the solution with respect to probability measures. Under suitable monotonicity conditions, we establish uniqueness, stability, and finally existence of such solutions. (10.70675/0388d2d1z7036z40b0zae52zbf67dd78a45e)
  DOI : 10.70675/0388d2d1z7036z40b0zae52zbf67dd78a45e
• Mathematical Analysis of the Gross-Pitaevskii equation in disordered media
  
  • Mackowiak Samaël
  , 2025. This thesis aims to study the Gross-Pitaevskii equation in the presence of a spatial white noise potential. The work in this thesis is threefold. First, Schrödinger-type operators with both a confining potential and an irregular potential (i.e. a negative regularity distribution) are constructed in dimensions 1 and 2. In dimension 1, a quadratic form approach is used to construct these operators for large classes of confining potentials (covering the case of the harmonic potential) and irregular potentials (covering the case of the white noise potential). In dimension 2, a renormalisation procedure is required in order to deal with a white noise potential. This renormalisation procedure is obtained for general confining potentials, thus extending the methods previously developed for the case of harmonic potentials. Once these operators have been constructed, we study the solution of the associated non-linear Schrödinger equations. In dimension 1, the local existence of solutions is obtained by the usual fixed point method, using the algebra property of Sobolev spaces. In dimension 2, this direct approach is no longer effective. Thus, we start by constructing regular solutions using a compactness method. Then, a paracontrolled approach to the confining Anderson operator is used to obtain Strichartz estimates and to deduce the local existence of solutions in dimension 2. Finally, we look at the stationary waves of these Schrödinger equations. The existence of such solutions is obtained by adapting the usual constrained minimisation methods. We also show that standing waves have a limited local regularity in Sobolev spaces, based on the optimal regularity of white noise. (10.70675/3ec6de1az3d40z4451zba32z575574591eff)
  DOI : 10.70675/3ec6de1az3d40z4451zba32z575574591eff
• High-order adaptive multi-domain time integration scheme for microscale lithium-ion batteries simulations
  
  • Asad Ali
  • de Loubens Romain
  • François Laurent
  • Massot Marc
  SMAI Journal of Computational Mathematics, Société de Mathématiques Appliquées et Industrielles (SMAI), 2025, 11, pp.369-404. We investigate the modeling and simulation of ionic transport and charge conservation in lithium-ion batteries (LIBs) at the microscale. It is a multiphysics problem that involves a wide range of time scales. The associated computational challenges motivate the investigation of numerical techniques that can decouple the time integration of the governing equations in the liquid electrolyte and the solid phase (active materials and current collectors). First, it is shown that semi-discretization in space of the non-dimensionalized governing equations leads to a system of index-1 semi-explicit differential algebraic equations (DAEs). Then, a new generation of strategies for multi-domain integration is presented, enabling high-order adaptive coupling of both domains in time, with efficient and potentially different domain integrators. They reach a high level of flexibility for real applications, beyond the limitations of multirate methods. A simple 1D LIB half-cell code is implemented as a demonstrator of the new strategy for the simulation of different modes of cell operation. The integration of the decoupled subsystems is performed with high-order accurate implicit nonlinear solvers. The accuracy of the space discretization is assessed by comparing the numerical results to the analytical solutions. Then, temporal convergence studies demonstrate the accuracy of the new multi-domain coupling approach. Finally, the accuracy and computational efficiency of the adaptive coupling strategy are discussed in the light of the conditioning of the decoupled subproblems compared to the one of the fully-coupled problem. This new approach will constitute a key ingredient for the high-fidelity 3D LIB simulations based on actual electrode microstructures. (10.5802/smai-jcm.128)
  DOI : 10.5802/smai-jcm.128
• Longest increasing subsequences for distributions with atoms, and an inhomogeneous Hammersley process
  
  • Basdevant Anne-Laure
  • Gerin Lucas
  • Marivain Maxime
  , 2025. A famous result by Hammersley and Versik-Kerov states that the length $L_n$ of the longest increasing subsequence among $n$ iid continuous random variables grows like $2\sqrt{n}$. We investigate here the asymptotic behavior of $L_n$ for distributions with atoms. For purely discrete random variables, we characterize the asymptotic order of $L_n$ through a variational problem and provide explicit estimates for classical distributions. The proofs rely on a coupling with an inhomogeneous version of the discrete-time continuous-space Hammersley process. This reveals that, in contrast to the continuous case, the discrete setting exhibits a wide range of growth rates between $\mathcal{O}(1)$ and $o(\sqrt{n})$, depending on the tail behavior of the distribution. We can then easily deduce the asymptotics of $L_n$ for a completely arbitrary distribution.
• Exponentially Fading Memory Signature
  
  • Abi Jaber Eduardo
  • Sotnikov Dimitri
  , 2025. We introduce the exponentially fading memory (EFM) signature, a time-invariant transformation of an infinite (possibly rough) path that serves as a mean-reverting analogue of the classical path signature. We construct the EFM-signature via rough path theory, carefully adapted to accommodate improper integration from minus infinity. The EFM-signature retains many of the key algebraic and analytical properties of classical signatures, including a suitably modified Chen identity, the linearization property, path-determinacy, and the universal approximation property. From the probabilistic perspective, the EFM-signature provides a "stationarized" representation, making it particularly well-suited for timeseries analysis and signal processing overcoming the shortcomings of the standard signature. In particular, the EFM-signature of time-augmented Brownian motion evolves as a group-valued Ornstein-Uhlenbeck process. We establish its stationarity, Markov property, and exponential ergodicity in the Wasserstein distance, and we derive an explicit formula à la Fawcett for its expected value in terms of Magnus expansions. We also study linear combinations of EFM-signature elements and the computation of associated characteristic functions in terms of a mean-reverting infinite dimensional Riccati equation.