Marc Lelarge

Matthieu Blanke, Marc Lelarge

Model-based reinforcement learning is a powerful tool, but collecting data to fit an accurate model of the system can be costly. Exploring an unknown environment in a sample-efficient manner is hence of great importance. However, the complexity of dynamics and the computational limitations of real systems make this task challenging. In this work, we introduce FLEX, an exploration algorithm for nonlinear dynamics based on optimal experimental design. Our policy maximizes the information of the next step and results in an adaptive exploration algorithm, compatible with generic parametric learning models and requiring minimal resources. We test our method on a number of nonlinear environments covering different settings, including time-varying dynamics. Keeping in mind that exploration is intended to serve an exploitation objective, we also test our algorithm on downstream model-based classical control tasks and compare it to other state-of-the-art model-based and model-free approaches. The performance achieved by FLEX is competitive and its computational cost is low.

David A. R. Robin, Kevin Scaman, Marc Lelarge

In this paper, we present a new strategy to prove the convergence of deep learning architectures to a zero training (or even testing) loss by gradient flow. Our analysis is centered on the notion of Rayleigh quotients in order to prove Kurdyka-Łojasiewicz inequalities for a broader set of neural network architectures and loss functions. We show that Rayleigh quotients provide a unified view for several convergence analysis techniques in the literature. Our strategy produces a proof of convergence for various examples of parametric learning. In particular, our analysis does not require the number of parameters to tend to infinity, nor the number of samples to be finite, thus extending to test loss minimization and beyond the over-parameterized regime.

Matthieu Blanke, Marc Lelarge

This work addresses the problem of exploration in an unknown environment. For linear dynamical systems, we use an experimental design framework and introduce an online greedy policy where the control maximizes the information of the next step. In a setting with a limited number of experimental trials, our algorithm has low complexity and shows experimentally competitive performances compared to more elaborate gradient-based methods.

Guillaume Baudart, Marc Lelarge, Tristan Stérin et al.

We report on an experiment in which Claude Opus~4.6, equipped with a suite of Model Context Protocol (MCP) tools for the Rocq proof assistant, autonomously proved 10 of 12 problems from the 2025 Putnam Mathematical Competition. The MCP tools, designed with Claude by analyzing logs from a prior experiment on miniF2F-Rocq, encode a "compile-first, interactive-fallback" strategy. Running on an isolated VM with no internet access, the agent deployed 141 subagents over 17.7 hours of active compute (51.6h wall-clock), consuming approximately 1.9 billion tokens. All proofs are publicly available.

Eric Daoud, Luca Ganassali, Antoine Baker et al.

For a very broad range of problems, recommendation algorithms have been increasingly used over the past decade. In most of these algorithms, the predictions are built upon user-item affinity scores which are obtained from high-dimensional embeddings of items and users. In more complex scenarios, with geometrical or capacity constraints, prediction based on embeddings may not be sufficient and some additional features should be considered in the design of the algorithm. In this work, we study the recommendation problem in the setting where affinities between users and items are based both on their embeddings in a latent space and on their geographical distance in their underlying euclidean space (e.g., $\mathbb{R}^2$), together with item capacity constraints. This framework is motivated by some real-world applications, for instance in healthcare: the task is to recommend hospitals to patients based on their location, pathology, and hospital capacities. In these applications, there is somewhat of an asymmetry between users and items: items are viewed as static points, their embeddings, capacities and locations constraining the allocation. Upon the observation of an optimal allocation, user embeddings, items capacities, and their positions in their underlying euclidean space, our aim is to recover item embeddings in the latent space; doing so, we are then able to use this estimate e.g. in order to predict future allocations. We propose an algorithm (SiMCa) based on matrix factorization enhanced with optimal transport steps to model user-item affinities and learn item embeddings from observed data. We then illustrate and discuss the results of such an approach for hospital recommendation on synthetic data.

Adrien Lagesse, Marc Lelarge

We propose a novel benchmarking methodology for graph neural networks (GNNs) based on the graph alignment problem, a combinatorial optimization task that generalizes graph isomorphism by aligning two unlabeled graphs to maximize overlapping edges. We frame this problem as a self-supervised learning task and present several methods to generate graph alignment datasets using synthetic random graphs and real-world graph datasets from multiple domains. For a given graph dataset, we generate a family of graph alignment datasets with increasing difficulty, allowing us to rank the performance of various architectures. Our experiments indicate that anisotropic graph neural networks outperform standard convolutional architectures. To further demonstrate the utility of the graph alignment task, we show its effectiveness for unsupervised GNN pre-training, where the learned node embeddings outperform other positional encodings on three molecular regression tasks and achieve state-of-the-art results on the PCQM4Mv2 dataset with significantly fewer parameters. To support reproducibility and further research, we provide an open-source Python package to generate graph alignment datasets and benchmark new GNN architectures.

Jules Viennot, Guillaume Baudart, Emilio Jesùs Gallego Arias et al.

In this work, we conduct an experiment using state-of-the-art LLMs to translate MiniF2F into Rocq. The translation task focuses on generating a Rocq theorem based on three sources: a natural language description, the Lean formalization, and the Isabelle formalization. We conducted our experiment in 3 stages of increasing complexity, from basic one-shot prompting to multi-turn conversations that incorporate feedback from unsuccessful attempts. At each stage, we perform multiple rounds of translation using increasingly advanced models: GPT-4o mini, Claude 3.5 Sonnet, o1 mini, and o1. We successfully translated 478 out of 488 theorems. The dataset is opensource: https://github.com/LLM4Rocq/miniF2F-rocq.

Waïss Azizian, Guillaume Baudart, Marc Lelarge

Exact Bayesian inference on state-space models~(SSM) is in general untractable, and unfortunately, basic Sequential Monte Carlo~(SMC) methods do not yield correct approximations for complex models. In this paper, we propose a mixed inference algorithm that computes closed-form solutions using belief propagation as much as possible, and falls back to sampling-based SMC methods when exact computations fail. This algorithm thus implements automatic Rao-Blackwellization and is even exact for Gaussian tree models.

Marc Lelarge

Graph neural networks (GNNs) have struggled to outperform traditional optimization methods on combinatorial problems, limiting their practical impact. We address this gap by introducing a novel chaining procedure for the graph alignment problem, a fundamental NP-hard task of finding optimal node correspondences between unlabeled graphs using only structural information. Our method trains a sequence of GNNs where each network learns to iteratively refine similarity matrices produced by previous networks. During inference, this creates a bootstrap effect: each GNN improves upon partial solutions by incorporating discrete ranking information about node alignment quality from prior iterations. We combine this with a powerful architecture that operates on node pairs rather than individual nodes, capturing global structural patterns essential for alignment that standard message-passing networks cannot represent. Extensive experiments on synthetic benchmarks demonstrate substantial improvements: our chained GNNs achieve over 3x better accuracy than existing methods on challenging instances, and uniquely solve regular graphs where all competing approaches fail. When combined with traditional optimization as post-processing, our method substantially outperforms state-of-the-art solvers on the graph alignment benchmark.

Matthieu Blanke, Ronan Fablet, Marc Lelarge

Data assimilation is a central problem in many geophysical applications, such as weather forecasting. It aims to estimate the state of a potentially large system, such as the atmosphere, from sparse observations, supplemented by prior physical knowledge. The size of the systems involved and the complexity of the underlying physical equations make it a challenging task from a computational point of view. Neural networks represent a promising method of emulating the physics at low cost, and therefore have the potential to considerably improve and accelerate data assimilation. In this work, we introduce a deep learning approach where the physical system is modeled as a sequence of coarse-to-fine Gaussian prior distributions parametrized by a neural network. This allows us to define an assimilation operator, which is trained in an end-to-end fashion to minimize the reconstruction error on a dataset with different observation processes. We illustrate our approach on chaotic dynamical physical systems with sparse observations, and compare it to traditional variational data assimilation methods.

Luca Ganassali, Laurent Massoulié, Marc Lelarge

Motivated by alignment of correlated sparse random graphs, we introduce a hypothesis testing problem of deciding whether or not two random trees are correlated. We obtain sufficient conditions under which this testing is impossible or feasible. We propose MPAlign, a message-passing algorithm for graph alignment inspired by the tree correlation detection problem. We prove MPAlign to succeed in polynomial time at partial alignment whenever tree detection is feasible. As a result our analysis of tree detection reveals new ranges of parameters for which partial alignment of sparse random graphs is feasible in polynomial time. We then conjecture that graph alignment is not feasible in polynomial time when the associated tree detection problem is impossible. If true, this conjecture together with our sufficient conditions on tree detection impossibility would imply the existence of a hard phase for graph alignment, i.e. a parameter range where alignment cannot be done in polynomial time even though it is known to be feasible in non-polynomial time.

Luca Ganassali, Laurent Massoulié, Marc Lelarge

Random graph alignment refers to recovering the underlying vertex correspondence between two random graphs with correlated edges. This can be viewed as an average-case and noisy version of the well-known graph isomorphism problem. For the correlated Erdös-Rényi model, we prove an impossibility result for partial recovery in the sparse regime, with constant average degree and correlation, as well as a general bound on the maximal reachable overlap. Our bound is tight in the noiseless case (the graph isomorphism problem) and we conjecture that it is still tight with noise. Our proof technique relies on a careful application of the probabilistic method to build automorphisms between tree components of a subcritical Erdös-Rényi graph.

Stéphane d'Ascoli, Alice Coucke, Francesco Caltagirone et al.

Scarcity of training data for task-oriented dialogue systems is a well known problem that is usually tackled with costly and time-consuming manual data annotation. An alternative solution is to rely on automatic text generation which, although less accurate than human supervision, has the advantage of being cheap and fast. Our contribution is twofold. First we show how to optimally train and control the generation of intent-specific sentences using a conditional variational autoencoder. Then we introduce a new protocol called query transfer that allows to leverage a large unlabelled dataset, possibly containing irrelevant queries, to extract relevant information. Comparison with two different baselines shows that this method, in the appropriate regime, consistently improves the diversity of the generated queries without compromising their quality. We also demonstrate the effectiveness of our generation method as a data augmentation technique for language modelling tasks.

Waïss Azizian, Marc Lelarge

Various classes of Graph Neural Networks (GNN) have been proposed and shown to be successful in a wide range of applications with graph structured data. In this paper, we propose a theoretical framework able to compare the expressive power of these GNN architectures. The current universality theorems only apply to intractable classes of GNNs. Here, we prove the first approximation guarantees for practical GNNs, paving the way for a better understanding of their generalization. Our theoretical results are proved for invariant GNNs computing a graph embedding (permutation of the nodes of the input graph does not affect the output) and equivariant GNNs computing an embedding of the nodes (permutation of the input permutes the output). We show that Folklore Graph Neural Networks (FGNN), which are tensor based GNNs augmented with matrix multiplication are the most expressive architectures proposed so far for a given tensor order. We illustrate our results on the Quadratic Assignment Problem (a NP-Hard combinatorial problem) by showing that FGNNs are able to learn how to solve the problem, leading to much better average performances than existing algorithms (based on spectral, SDP or other GNNs architectures). On a practical side, we also implement masked tensors to handle batches of graphs of varying sizes.

Stéphane d'Ascoli, Alice Coucke, Francesco Caltagirone et al.

Scarcity of training data for task-oriented dialogue systems is a well known problem that is usually tackled with costly and time-consuming manual data annotation. An alternative solution is to rely on automatic text generation which, although less accurate than human supervision, has the advantage of being cheap and fast. In this paper we propose a novel controlled data generation method that could be used as a training augmentation framework for closed-domain dialogue. Our contribution is twofold. First we show how to optimally train and control the generation of intent-specific sentences using a conditional variational autoencoder. Then we introduce a novel protocol called query transfer that allows to leverage a broad, unlabelled dataset to extract relevant information. Comparison with two different baselines shows that our method, in the appropriate regime, consistently improves the diversity of the generated queries without compromising their quality.

Marc Lelarge, Leo Miolane

Semi-supervised learning (SSL) uses unlabeled data for training and has been shown to greatly improve performance when compared to a supervised approach on the labeled data available. This claim depends both on the amount of labeled data available and on the algorithm used. In this paper, we compute analytically the gap between the best fully-supervised approach using only labeled data and the best semi-supervised approach using both labeled and unlabeled data. We quantify the best possible increase in performance obtained thanks to the unlabeled data, i.e. we compute the accuracy increase due to the information contained in the unlabeled data. Our work deals with a simple high-dimensional Gaussian mixture model for the data in a Bayesian setting. Our rigorous analysis builds on recent theoretical breakthroughs in high-dimensional inference and a large body of mathematical tools from statistical physics initially developed for spin glasses.

Thomas Bonald, Alexandre Hollocou, Marc Lelarge

We present a novel spectral embedding of graphs that incorporates weights assigned to the nodes, quantifying their relative importance. This spectral embedding is based on the first eigenvectors of some properly normalized version of the Laplacian. We prove that these eigenvectors correspond to the configurations of lowest energy of an equivalent physical system, either mechanical or electrical, in which the weight of each node can be interpreted as its mass or its capacitance, respectively. Experiments on a real dataset illustrate the impact of weighting on the embedding.

Edouard Pineau, Marc Lelarge

This paper describes InfoCatVAE, an extension of the variational autoencoder that enables unsupervised disentangled representation learning. InfoCatVAE uses multimodal distributions for the prior and the inference network and then maximizes the evidence lower bound objective (ELBO). We connect the new ELBO derived for our model with a natural soft clustering objective which explains the robustness of our approach. We then adapt the InfoGANs method to our setting in order to maximize the mutual information between the categorical code and the generated inputs and obtain an improved model.

Jean-Baptiste Escudié, Alaa Saade, Alice Coucke et al.

We show how to learn low-dimensional representations (embeddings) of patient visits from the corresponding electronic health record (EHR) where International Classification of Diseases (ICD) diagnosis codes are removed. We expect that these embeddings will be useful for the construction of predictive statistical models anticipated to drive personalized medicine and improve healthcare quality. These embeddings are learned using a deep neural network trained to predict ICD diagnosis categories. We show that our embeddings capture relevant clinical informations and can be used directly as input to standard machine learning algorithms like multi-output classifiers for ICD code prediction. We also show that important medical informations correspond to particular directions in our embedding space.

Alexandre Hollocou, Julien Maudet, Thomas Bonald et al.

We introduce a novel algorithm to perform graph clustering in the edge streaming setting. In this model, the graph is presented as a sequence of edges that can be processed strictly once. Our streaming algorithm has an extremely low memory footprint as it stores only three integers per node and does not keep any edge in memory. We provide a theoretical justification of the design of the algorithm based on the modularity function, which is a usual metric to evaluate the quality of a graph partition. We perform experiments on massive real-life graphs ranging from one million to more than one billion edges and we show that this new algorithm runs more than ten times faster than existing algorithms and leads to similar or better detection scores on the largest graphs.

Lennart Gulikers, Marc Lelarge, Laurent Massoulié

Motivated by community detection, we characterise the spectrum of the non-backtracking matrix $B$ in the Degree-Corrected Stochastic Block Model. Specifically, we consider a random graph on $n$ vertices partitioned into two equal-sized clusters. The vertices have i.i.d. weights $\{ φ_u \}_{u=1}^n$ with second moment $Φ^{(2)}$. The intra-cluster connection probability for vertices $u$ and $v$ is $\frac{φ_u φ_v}{n}a$ and the inter-cluster connection probability is $\frac{φ_u φ_v}{n}b$. We show that with high probability, the following holds: The leading eigenvalue of the non-backtracking matrix $B$ is asymptotic to $ρ= \frac{a+b}{2} Φ^{(2)}$. The second eigenvalue is asymptotic to $μ_2 = \frac{a-b}{2} Φ^{(2)}$ when $μ_2^2 > ρ$, but asymptotically bounded by $\sqrtρ$ when $μ_2^2 \leq ρ$. All the remaining eigenvalues are asymptotically bounded by $\sqrtρ$. As a result, a clustering positively-correlated with the true communities can be obtained based on the second eigenvector of $B$ in the regime where $μ_2^2 > ρ.$ In a previous work we obtained that detection is impossible when $μ_2^2 < ρ,$ meaning that there occurs a phase-transition in the sparse regime of the Degree-Corrected Stochastic Block Model. As a corollary, we obtain that Degree-Corrected Erdős-Rényi graphs asymptotically satisfy the graph Riemann hypothesis, a quasi-Ramanujan property. A by-product of our proof is a weak law of large numbers for local-functionals on Degree-Corrected Stochastic Block Models, which could be of independent interest.

Alaa Saade, Florent Krzakala, Marc Lelarge et al.

We consider the problem of clustering partially labeled data from a minimal number of randomly chosen pairwise comparisons between the items. We introduce an efficient local algorithm based on a power iteration of the non-backtracking operator and study its performance on a simple model. For the case of two clusters, we give bounds on the classification error and show that a small error can be achieved from $O(n)$ randomly chosen measurements, where $n$ is the number of items in the dataset. Our algorithm is therefore efficient both in terms of time and space complexities. We also investigate numerically the performance of the algorithm on synthetic and real world data.

Alaa Saade, Marc Lelarge, Florent Krzakala et al.

We consider the problem of grouping items into clusters based on few random pairwise comparisons between the items. We introduce three closely related algorithms for this task: a belief propagation algorithm approximating the Bayes optimal solution, and two spectral algorithms based on the non-backtracking and Bethe Hessian operators. For the case of two symmetric clusters, we conjecture that these algorithms are asymptotically optimal in that they detect the clusters as soon as it is information theoretically possible to do so. We substantiate this claim for one of the spectral approaches we introduce.

Lennart Gulikers, Marc Lelarge, Laurent Massoulié

We consider the Degree-Corrected Stochastic Block Model (DC-SBM): a random graph on $n$ nodes, having i.i.d. weights $(φ_u)_{u=1}^n$ (possibly heavy-tailed), partitioned into $q \geq 2$ asymptotically equal-sized clusters. The model parameters are two constants $a,b > 0$ and the finite second moment of the weights $Φ^{(2)}$. Vertices $u$ and $v$ are connected by an edge with probability $\frac{φ_u φ_v}{n}a$ when they are in the same class and with probability $\frac{φ_u φ_v}{n}b$ otherwise. We prove that it is information-theoretically impossible to estimate the clusters in a way positively correlated with the true community structure when $(a-b)^2 Φ^{(2)} \leq q(a+b)$. As by-products of our proof we obtain $(1)$ a precise coupling result for local neighbourhoods in DC-SBM's, that we use in a follow up paper [Gulikers et al., 2017] to establish a law of large numbers for local-functionals and $(2)$ that long-range interactions are weak in (power-law) DC-SBM's.

Lennart Gulikers, Marc Lelarge, Laurent Massoulié

We consider community detection in Degree-Corrected Stochastic Block Models (DC-SBM). We propose a spectral clustering algorithm based on a suitably normalized adjacency matrix. We show that this algorithm consistently recovers the block-membership of all but a vanishing fraction of nodes, in the regime where the lowest degree is of order log$(n)$ or higher. Recovery succeeds even for very heterogeneous degree-distributions. The used algorithm does not rely on parameters as input. In particular, it does not need to know the number of communities.

Emilie Kaufmann, Thomas Bonald, Marc Lelarge

This paper presents a novel spectral algorithm with additive clustering designed to identify overlapping communities in networks. The algorithm is based on geometric properties of the spectrum of the expected adjacency matrix in a random graph model that we call stochastic blockmodel with overlap (SBMO). An adaptive version of the algorithm, that does not require the knowledge of the number of hidden communities, is proved to be consistent under the SBMO when the degrees in the graph are (slightly more than) logarithmic. The algorithm is shown to perform well on simulated data and on real-world graphs with known overlapping communities.

Se-Young Yun, Marc Lelarge, Alexandre Proutiere

In this paper, we consider the streaming memory-limited matrix completion problem when the observed entries are noisy versions of a small random fraction of the original entries. We are interested in scenarios where the matrix size is very large so the matrix is very hard to store and manipulate. Here, columns of the observed matrix are presented sequentially and the goal is to complete the missing entries after one pass on the data with limited memory space and limited computational complexity. We propose a streaming algorithm which produces an estimate of the original matrix with a vanishing mean square error, uses memory space scaling linearly with the ambient dimension of the matrix, i.e. the memory required to store the output alone, and spends computations as much as the number of non-zero entries of the input matrix.

Rui Wu, Jiaming Xu, R. Srikant et al.

Given a set of pairwise comparisons, the classical ranking problem computes a single ranking that best represents the preferences of all users. In this paper, we study the problem of inferring individual preferences, arising in the context of making personalized recommendations. In particular, we assume that there are $n$ users of $r$ types; users of the same type provide similar pairwise comparisons for $m$ items according to the Bradley-Terry model. We propose an efficient algorithm that accurately estimates the individual preferences for almost all users, if there are $r \max \{m, n\}\log m \log^2 n$ pairwise comparisons per type, which is near optimal in sample complexity when $r$ only grows logarithmically with $m$ or $n$. Our algorithm has three steps: first, for each user, compute the \emph{net-win} vector which is a projection of its $\binom{m}{2}$-dimensional vector of pairwise comparisons onto an $m$-dimensional linear subspace; second, cluster the users based on the net-win vectors; third, estimate a single preference for each cluster separately. The net-win vectors are much less noisy than the high dimensional vectors of pairwise comparisons and clustering is more accurate after the projection as confirmed by numerical experiments. Moreover, we show that, when a cluster is only approximately correct, the maximum likelihood estimation for the Bradley-Terry model is still close to the true preference.

Richard Combes, M. Sadegh Talebi, Alexandre Proutiere et al.

This paper investigates stochastic and adversarial combinatorial multi-armed bandit problems. In the stochastic setting under semi-bandit feedback, we derive a problem-specific regret lower bound, and discuss its scaling with the dimension of the decision space. We propose ESCB, an algorithm that efficiently exploits the structure of the problem and provide a finite-time analysis of its regret. ESCB has better performance guarantees than existing algorithms, and significantly outperforms these algorithms in practice. In the adversarial setting under bandit feedback, we propose \textsc{CombEXP}, an algorithm with the same regret scaling as state-of-the-art algorithms, but with lower computational complexity for some combinatorial problems.

Marc Lelarge, Laurent Massoulié, Jiaming Xu

The labeled stochastic block model is a random graph model representing networks with community structure and interactions of multiple types. In its simplest form, it consists of two communities of approximately equal size, and the edges are drawn and labeled at random with probability depending on whether their two endpoints belong to the same community or not. It has been conjectured in \cite{Heimlicher12} that correlated reconstruction (i.e.\ identification of a partition correlated with the true partition into the underlying communities) would be feasible if and only if a model parameter exceeds a threshold. We prove one half of this conjecture, i.e., reconstruction is impossible when below the threshold. In the positive direction, we introduce a weighted graph to exploit the label information. With a suitable choice of weight function, we show that when above the threshold by a specific constant, reconstruction is achieved by (1) minimum bisection, (2) a semidefinite relaxation of minimum bisection, and (3) a spectral method combined with removal of edges incident to vertices of high degree. Furthermore, we show that hypothesis testing between the labeled stochastic block model and the labeled Erdős-Rényi random graph model exhibits a phase transition at the conjectured reconstruction threshold.

Alaa Saade, Florent Krzakala, Marc Lelarge et al.

We consider the problem of partially recovering hidden binary variables from the observation of (few) censored edge weights, a problem with applications in community detection, correlation clustering and synchronization. We describe two spectral algorithms for this task based on the non-backtracking and the Bethe Hessian operators. These algorithms are shown to be asymptotically optimal for the partial recovery problem, in that they detect the hidden assignment as soon as it is information theoretically possible to do so.

Jiaming Xu, Laurent Massoulié, Marc Lelarge

The classical setting of community detection consists of networks exhibiting a clustered structure. To more accurately model real systems we consider a class of networks (i) whose edges may carry labels and (ii) which may lack a clustered structure. Specifically we assume that nodes possess latent attributes drawn from a general compact space and edges between two nodes are randomly generated and labeled according to some unknown distribution as a function of their latent attributes. Our goal is then to infer the edge label distributions from a partially observed network. We propose a computationally efficient spectral algorithm and show it allows for asymptotically correct inference when the average node degree could be as low as logarithmic in the total number of nodes. Conversely, if the average node degree is below a specific constant threshold, we show that no algorithm can achieve better inference than guessing without using the observations. As a byproduct of our analysis, we show that our model provides a general procedure to construct random graph models with a spectrum asymptotic to a pre-specified eigenvalue distribution such as a power-law distribution.

Marc Lelarge, Alexandre Proutiere, M. Sadegh Talebi

We consider the problem of allocating radio channels to links in a wireless network. Links interact through interference, modelled as a conflict graph (i.e., two interfering links cannot be simultaneously active on the same channel). We aim at identifying the channel allocation maximizing the total network throughput over a finite time horizon. Should we know the average radio conditions on each channel and on each link, an optimal allocation would be obtained by solving an Integer Linear Program (ILP). When radio conditions are unknown a priori, we look for a sequential channel allocation policy that converges to the optimal allocation while minimizing on the way the throughput loss or {\it regret} due to the need for exploring sub-optimal allocations. We formulate this problem as a generic linear bandit problem, and analyze it first in a stochastic setting where radio conditions are driven by a stationary stochastic process, and then in an adversarial setting where radio conditions can evolve arbitrarily. We provide new algorithms in both settings and derive upper bounds on their regrets.

Stephane Caron, Branislav Kveton, Marc Lelarge et al.

This paper considers stochastic bandits with side observations, a model that accounts for both the exploration/exploitation dilemma and relationships between arms. In this setting, after pulling an arm i, the decision maker also observes the rewards for some other actions related to i. We will see that this model is suited to content recommendation in social networks, where users' reactions may be endorsed or not by their friends. We provide efficient algorithms based on upper confidence bounds (UCBs) to leverage this additional information and derive new bounds improving on standard regret guarantees. We also evaluate these policies in the context of movie recommendation in social networks: experiments on real datasets show substantial learning rate speedups ranging from 2.2x to 14x on dense networks.

Simon Heimlicher, Marc Lelarge, Laurent Massoulié

We consider the problem of community detection from observed interactions between individuals, in the context where multiple types of interaction are possible. We use labelled stochastic block models to represent the observed data, where labels correspond to interaction types. Focusing on a two-community scenario, we conjecture a threshold for the problem of reconstructing the hidden communities in a way that is correlated with the true partition. To substantiate the conjecture, we prove that the given threshold correctly identifies a transition on the behaviour of belief propagation from insensitive to sensitive. We further prove that the same threshold corresponds to the transition in a related inference problem on a tree model from infeasible to feasible. Finally, numerical results using belief propagation for community detection give further support to the conjecture.