arXiv subjects

Matthieu Jonckheere

Publications and source records attributed to Matthieu Jonckheere.

At least 19 recordsLinked to original sources

arXiv2026

On the Slow Convergence to Trivial Solutions of Algorithms for Hard Optimization Problems

Hard combinatorial optimization problems, many of which are NP-hard, present fundamental algorithmic challenges. Average-case analysis on random instances has emerged as a powerful framework for understanding typical algorithmic performance beyond worst-case guarantees. A substantial body of work has established negative results: for sufficiently hard instances (often controlled by the underlying graph connectivity/constraints density), no known polynomial-time algorithm can significantly outperform naive heuristics in the double asymptotic limit where both problem size and constraints density tend to infinity. We revisit this picture by studying the finite-size behavior of some optimization algorithms across easy, intermediate, and hard regimes. Through rigorous analysis of large-graph asymptotics combined with numerical experiments on canonical problems (maximum independent set and maximum $K$-SAT), we demonstrate that while algorithms do eventually converge to theoretically predicted bounds, this convergence can be remarkably slow. In the intermediate regime where instances are already highly constrained, local algorithms achieve solutions substantially better than their predicted performance in the high-constraint-density limit. This gap between finite-regime and asymptotic behavior has important practical implications: sophisticated algorithmic design remains crucial even when asymptotic theory predicts inevitable failure.

cs.LG↗

arXiv2026

Waves Everywhere: A Distributional Equation Approach to Front Propagation

We study reaction-diffusion particle systems with several interaction mechanisms. As the number of particles tends to infinity, the system admits a mean-field limit describing the bulk behaviour. We focus on determining the propagation speed and the particle distribution around the centre of mass, which corresponds to the travelling wave of the limiting equation. We introduce a probabilistic method to characterise these waves via tagged particle distributional equations. Our key technique connects these to linear distributional equations solvable using martingale limits from branching processes. We first demonstrate our approach on a general model where particles move via L\'evy processes and synchronise at interaction moments (the lower particle jumps to the position of the higher one). Assuming the mean-field limit holds, we characterise its travelling-wave solutions. We then apply the method to two specific models with established mean-field limits. For Brownian particles, we recover known travelling-wave solutions of the F-KPP equation. For the compound Poisson model studied in \cite{baryshnikov2025large} where particles perform random walks with exponential holding times and copy positions at interactions, we extend previous results beyond exponential jumps to arbitrary jump distributions. Finally, we analyse the power-of-2 growth model, where interactions add a random value to the lower particle. We characterise its travelling waves and discover a surprising connection to the synchronisation models. The distributional equations at the heart of our technique are of independent interest, and we identify connections and differences with related equations studied extensively in the literature.

math.PR↗

arXiv2026

Optimization Trade-offs in Asynchronous Federated Learning: A Stochastic Networks Approach

Synchronous federated learning scales poorly due to the straggler effect. Asynchronous algorithms increase the update throughput by processing updates upon arrival, but they introduce two fundamental challenges: gradient staleness, which degrades convergence, and bias toward faster clients under heterogeneous data distributions. Although algorithms such as AsyncSGD and Generalized AsyncSGD mitigate this bias via client-side task queues, most existing analyses neglect the underlying queueing dynamics and lack closed-form characterizations of the update throughput and gradient staleness. To close this gap, we develop a stochastic queueing-network framework for Generalized AsyncSGD that jointly models random computation times at the clients and the central server, as well as random uplink and downlink communication delays. Leveraging product-form network theory, we derive a closed-form expression for the update throughput, alongside closed-form upper bounds for both the communication round complexity and the expected wall-clock time required to reach an $\epsilon$-stationary point. These results formally characterize the trade-off between gradient staleness and wall-clock convergence speed. We further extend the framework to quantify energy consumption under stochastic timing, revealing an additional trade-off between convergence speed and energy efficiency. Building on these analytical results, we propose gradient-based optimization strategies to jointly optimize routing and concurrency. Experiments on EMNIST demonstrate reductions of 29%--46% in convergence time and 36%--49% in energy consumption compared to AsyncSGD.

cs.LG↗

arXiv2026

$K-$means with learned metrics

We study the Fr\'echet $k-$means of a metric measure space when both the measure and the distance are unknown and have to be estimated. We prove a general result that states that the $k-$means are continuous with respect to the measured Gromov-Hausdorff topology. In this situation, we also prove a stability result for the Voronoi clusters they determine. We do not assume uniqueness of the set of $k-$means, but when it is unique, the results are stronger. This framework provides a unified approach to proving consistency for a wide range of metric learning procedures. As concrete applications, we obtain new consistency results for several important estimators that were previously unestablished, even when $k=1$. These include $k-$means based on: (i) Isomap and Fermat geodesic distances on manifolds, (ii) difussion distances, (iii) Wasserstein distances computed with respect to learned ground metrics. Finally, we consider applications beyond the statistical inference paradigm like (iv) first passage percolation and (v) discrete approximations of length spaces.

math.ST↗

arXiv2025

On the instability of local learning algorithms: Q-learning can fail in infinite state spaces

We investigate the challenges of applying model-free reinforcement learning algorithms, like online Q-learning, to infinite state space Markov Decision Processes (MDPs). We first introduce the notion of Local Learning Processes (LLPs), where agents make decisions based solely on local information, and we show that Q-learning can be seen as a specific instance of an LLP. Using renewal techniques, we analyze LLPs and demonstrate their instability under certain drift and initial conditions, revealing fundamental limitations in infinite state spaces. In particular, we show that while asymptotically optimal in finite settings, Q-learning can face instability and strict sub-optimality in infinite spaces. Our findings are illustrated through queueing system examples drawn from load balancing and server allocation. The study underscores the need for new theoretical frameworks and suggests future research into nonlocal Q-learning variants.

math.PR↗

arXiv2025

Online Matching via Reinforcement Learning: An Expert Policy Orchestration Strategy

Online matching problems arise in many complex systems, from cloud services and online marketplaces to organ exchange networks, where timely, principled decisions are critical for maintaining high system performance. Traditional heuristics in these settings are simple and interpretable but typically tailored to specific operating regimes, which can lead to inefficiencies when conditions change. We propose a reinforcement learning (RL) approach that learns to orchestrate a set of such expert policies, leveraging their complementary strengths in a data-driven, adaptive manner. Building on the Adv2 framework (Jonckheere et al., 2024), our method combines expert decisions through advantage-based weight updates and extends naturally to settings where only estimated value functions are available. We establish both expectation and high-probability regret guarantees and derive a novel finite-time bias bound for temporal-difference learning, enabling reliable advantage estimation even under constant step size and non-stationary dynamics. To support scalability, we introduce a neural actor-critic architecture that generalizes across large state spaces while preserving interpretability. Simulations on stochastic matching models, including an organ exchange scenario, show that the orchestrated policy converges faster and yields higher system level efficiency than both individual experts and conventional RL baselines. Our results highlight how structured, adaptive learning can improve the modeling and management of complex resource allocation and decision-making processes.

stat.ML↗

arXiv2025

Efficiency of Parallel and Restart Exploration Strategies in Model Free Stochastic Simulations

We analyze the efficiency of parallelization and restart mechanisms for stochastic simulations in model-free settings, where the underlying system dynamics are unknown. Such settings are common in Reinforcement Learning (RL) and rare event estimation, where standard variance-reduction techniques like importance sampling are inapplicable. Focusing on the challenge of reaching rare states under a finite computational budget, we model exploration via random walks and L\'evy processes. Based on rigorous probability analysis, our work reveals a phase transition in the success probability as a function of the number of parallel simulations: an optimal number $N^*$ exists, balancing exploration diversity and time allocation per simulation. Beyond this threshold, performance degrades exponentially. Furthermore, we demonstrate that a restart strategy, which reallocates resources from stagnant trajectories to promising regions, can yield an exponential improvement in success probability. In the context of RL, these strategies can improve policy gradient methods by enabling more efficient state-space exploration, leading to more accurate policy gradient estimates.

math.PR↗

arXiv2025

Optimizing Asynchronous Federated Learning: A Delicate Trade-Off Between Model-Parameter Staleness and Update Frequency

Synchronous federated learning (FL) scales poorly with the number of clients due to the straggler effect. Algorithms like FedAsync and GeneralizedFedAsync address this limitation by enabling asynchronous communication between clients and the central server. In this work, we rely on stochastic modeling and analysis to better understand the impact of design choices in asynchronous FL algorithms, such as the concurrency level and routing probabilities, and we leverage this knowledge to optimize loss. Compared to most existing studies, we account for the joint impact of heterogeneous and variable service speeds and heterogeneous datasets at the clients. We characterize in particular a fundamental trade-off for optimizing asynchronous FL: minimizing gradient estimation errors by avoiding model parameter staleness, while also speeding up the system by increasing the throughput of model updates. Our two main contributions can be summarized as follows. First, we prove a discrete variant of Little's law to derive a closed-form expression for relative delay, a metric that quantifies staleness. This allows us to efficiently minimize the average loss per model update, which has been the gold standard in literature to date, using the upper-bound of Leconte et al. as a proxy. Second, we observe that naively optimizing this metric drastically slows down the system by overemphasizing staleness at the expense of throughput. This motivates us to introduce an alternative metric that also accounts for speed, for which we derive a tractable upper-bound that can be minimized numerically. Extensive numerical results show these optimizations enhance accuracy by 10% to 30%.

cs.LG↗

arXiv2024

Decision-Epoch Matters: Unveiling its Impact on the Stability of Scheduling with Randomly Varying Connectivity

A classical queuing theory result states that in a parallel-queue single-server model, the maximum stability region does not depend on the scheduling decision epochs, and in particular is the same for preemptive and non-preemptive systems. We consider here the case in which each of the queues may be connected to the server or not, depending on an exogenous process. In our main result, we show that the maximum stability region now does strongly depend on how the decision epochs are defined. We compare the setting where decisions can be made at any moment in time (the unconstrained setting), to two other settings: decisions are taken either (i) at moments of a departure (non-preemptive scheduling), or (ii) when an exponentially clock rings with rate $\gamma$. We characterise the maximum stability region for the two constrained configurations, allowing us to observe a reduction compared to the unconstrained configuration. In the non-preemptive setting, the maximum stability region is drastically reduced compared to the unconstrained setting and we conclude that a non-preemptive scheduler cannot take opportunistically advantage (in terms of stability) of the random varying connectivity. Instead, for the $\gamma$ decision epochs, we observe that the maximum stability region is monotone in the rate of the decision moments $\gamma$, and that one can be arbitrarily close to the maximum stability region in the unconstrained setting if we choose $\gamma$ large enough. We further show that Serve Longest Connected (SLC) queue is maximum stable in both constrained settings, within the set of policies that select a queue among the connected ones. From a methodological viewpoint, we introduce a novel theoretical tool termed a ``test for fluid limits'' (TFL) that might be of independent interest. TFL is a simple test that, if satisfied by the fluid limit, allows us to conclude for stability.

math.PR↗

arXiv2024

Queuing dynamics of asynchronous Federated Learning

We study asynchronous federated learning mechanisms with nodes having potentially different computational speeds. In such an environment, each node is allowed to work on models with potential delays and contribute to updates to the central server at its own pace. Existing analyses of such algorithms typically depend on intractable quantities such as the maximum node delay and do not consider the underlying queuing dynamics of the system. In this paper, we propose a non-uniform sampling scheme for the central server that allows for lower delays with better complexity, taking into account the closed Jackson network structure of the associated computational graph. Our experiments clearly show a significant improvement of our method over current state-of-the-art asynchronous algorithms on an image classification problem.

cs.DC↗

arXiv2023

Score-Aware Policy-Gradient and Performance Guarantees using Local Lyapunov Stability

In this paper, we introduce a policy-gradient method for model-based reinforcement learning (RL) that exploits a type of stationary distributions commonly obtained from Markov decision processes (MDPs) in stochastic networks, queueing systems, and statistical mechanics. Specifically, when the stationary distribution of the MDP belongs to an exponential family that is parametrized by policy parameters, we can improve existing policy gradient methods for average-reward RL. Our key identification is a family of gradient estimators, called score-aware gradient estimators (SAGEs), that enable policy gradient estimation without relying on value-function estimation in the aforementioned setting. We show that SAGE-based policy-gradient locally converges, and we obtain its regret. This includes cases when the state space of the MDP is countable and unstable policies can exist. Under appropriate assumptions such as starting sufficiently close to a maximizer and the existence of a local Lyapunov function, the policy under SAGE-based stochastic gradient ascent has an overwhelming probability of converging to the associated optimal policy. Furthermore, we conduct a numerical comparison between a SAGE-based policy-gradient method and an actor-critic method on several examples inspired from stochastic networks, queueing systems, and models derived from statistical physics. Our results demonstrate that a SAGE-based method finds close-to-optimal policies faster than an actor-critic method.

cs.LG↗

arXiv2023

Choosing the parameter of the Fermat distance: navigating geometry and noise

The Fermat distance has been recently established as a useful tool for machine learning tasks when a natural distance is not directly available to the practitioner or to improve the results given by Euclidean distances by exploding the geometrical and statistical properties of the dataset. This distance depends on a parameter $\alpha$ that greatly impacts the performance of subsequent tasks. Ideally, the value of $\alpha$ should be large enough to navigate the geometric intricacies inherent to the problem. At the same, it should remain restrained enough to sidestep any deleterious ramifications stemming from noise during the process of distance estimation. We study both theoretically and through simulations how to select this parameter.

stat.ML↗

arXiv2023

FEMDA: a unified framework for discriminant analysis

Although linear and quadratic discriminant analysis are widely recognized classical methods, they can encounter significant challenges when dealing with non-Gaussian distributions or contaminated datasets. This is primarily due to their reliance on the Gaussian assumption, which lacks robustness. We first explain and review the classical methods to address this limitation and then present a novel approach that overcomes these issues. In this new approach, the model considered is an arbitrary Elliptically Symmetrical (ES) distribution per cluster with its own arbitrary scale parameter. This flexible model allows for potentially diverse and independent samples that may not follow identical distributions. By deriving a new decision rule, we demonstrate that maximum-likelihood parameter estimation and classification are simple, efficient, and robust compared to state-of-the-art methods.

stat.ML↗

arXiv2023

Policy Optimization via Adv2: Adversarial Learning on Advantage Functions

We revisit the reduction of learning in adversarial Markov decision processes [MDPs] to adversarial learning based on $Q$--values; this reduction has been considered in a number of recent articles as one building block to perform policy optimization. Namely, we first consider and extend this reduction in an ideal setting where an oracle provides value functions: it may involve any adversarial learning strategy (not just exponential weights) and it may be based indifferently on $Q$--values or on advantage functions. We then present two extensions: on the one hand, convergence of the last iterate for a vast class of adversarial learning strategies (again, not just exponential weights), satisfying a property called monotonicity of weights; on the other hand, stronger regret criteria for learning in MDPs, inherited from the stronger regret criteria of adversarial learning called strongly adaptive regret and tracking regret. Third, we demonstrate how adversarial learning, also referred to as aggregation of experts, relates to aggregation (orchestration) of expert policies: we obtain stronger forms of performance guarantees in this setting than existing ones, via yet another, simple reduction. Finally, we discuss the impact of the reduction of learning in adversarial MDPs to adversarial learning in the practical scenarios where transition kernels are unknown and value functions must be learned. In particular, we review the literature and note that many strategies for policy optimization feature a policy-improvement step based on exponential weights with estimated $Q$--values. Our main message is that this step may be replaced by the application of any adversarial learning strategy on estimated $Q$--values or on estimated advantage functions. We leave the empirical evaluation of these twists for future research.

cs.LG↗

arXiv2023

FEMDA: Une méthode de classification robuste et flexible

Linear and Quadratic Discriminant Analysis (LDA and QDA) are well-known classical methods but can heavily suffer from non-Gaussian distributions and/or contaminated datasets, mainly because of the underlying Gaussian assumption that is not robust. This paper studies the robustness to scale changes in the data of a new discriminant analysis technique where each data point is drawn by its own arbitrary Elliptically Symmetrical (ES) distribution and its own arbitrary scale parameter. Such a model allows for possibly very heterogeneous, independent but non-identically distributed samples. The new decision rule derived is simple, fast, and robust to scale changes in the data compared to other state-of-the-art method

stat.ML↗

arXiv2023

Online matching for the multiclass stochastic block model

We consider the problem of sequential matching in a stochastic block model with several classes of nodes and generic compatibility constraints. When the probabilities of connections do not scale with the size of the graph, we show that under the NCOND condition, a simple max-weight type policy allows to attain an asymptotically perfect matching while no sequential algorithm attain perfect matching otherwise. The proof relies on a specific Markovian representation of the dynamics associated with Lyapunov techniques.

math.PR↗

arXiv2022

Parametrized Power-Iteration Clustering for Directed Graphs

Vertex-level clustering for directed graphs (digraphs) remains challenging as edge directionality breaks the key assumptions underlying popular spectral methods, which also incur the overhead of eigen-decomposition. This paper proposes Parametrized Power-Iteration Clustering (ParPIC), a random-walk-based clustering method for weakly connected digraphs. This builds over the Power-Iteration Clustering paradigm, which uses the rows of the iterated diffusion operator as a data embedding. ParPIC has three important features: the use of parametrized reversible random walk operators, the automatic tuning of the diffusion time, and the efficient truncation of the final embedding, which produces low-dimensional data representations and reduces complexity. Empirical results on synthetic and real-world graphs demonstrate that ParPIC achieves competitive clustering accuracy with improved scalability relative to spectral and teleportation-based methods.

cs.LG↗

arXiv2022

Generalized Dirichlet Energy and Graph Laplacians for Clustering Directed and Undirected Graphs

Clustering in directed graphs remains a fundamental challenge due to the asymmetry in edge connectivity, which limits the applicability of classical spectral methods originally designed for undirected graphs. A common workaround is to symmetrize the adjacency matrix, but this often leads to losing critical directional information. In this work, we introduce the generalized Dirichlet energy (GDE), a novel energy functional that extends the classical Dirichlet energy to handle arbitrary positive vertex measures and Markov transition matrices. GDE provides a unified framework applicable to both directed and undirected graphs, and is closely tied to the diffusion dynamics of random walks. Building on this framework, we propose the generalized spectral clustering (GSC) method that enables the principled clustering of weakly connected digraphs without resorting to the introduction of teleportation to the random walk transition matrix. A key component of our approach is the utilization of a parametrized vertex measure encoding graph directionality and density. Experiments on real-world point-cloud datasets demonstrate that GSC consistently outperforms existing spectral clustering approaches in terms of clustering accuracy and robustness, offering a powerful new tool for graph-based data analysis.

stat.ML↗