Nicolas Vayatis

Antoine de Mathelin, Francois Deheeger, Mathilde Mougeot et al.

We consider the problem of uncertainty quantification in high dimensional regression and classification for which deep ensemble have proven to be promising methods. Recent observations have shown that deep ensemble often return overconfident estimates outside the training domain, which is a major limitation because shifted distributions are often encountered in real-life scenarios. The principal challenge for this problem is to solve the trade-off between increasing the diversity of the ensemble outputs and making accurate in-distribution predictions. In this work, we show that an ensemble of networks with large weights fitting the training data are likely to meet these two objectives. We derive a simple and practical approach to produce such ensembles, based on an original anti-regularization term penalizing small weights and a control process of the weight increase which maintains the in-distribution loss under an acceptable threshold. The developed approach does not require any out-of-distribution training data neither any trade-off hyper-parameter calibration. We derive a theoretical framework for this approach and show that the proposed optimization can be seen as a "water-filling" problem. Several experiments in both regression and classification settings highlight that Deep Anti-Regularized Ensembles (DARE) significantly improve uncertainty quantification outside the training domain in comparison to recent deep ensembles and out-of-distribution detection methods. All the conducted experiments are reproducible and the source code is available at \url{https://github.com/antoinedemathelin/DARE}.

Antoine de Mathelin, Francois Deheeger, Mathilde Mougeot et al.

Bias in datasets can be very detrimental for appropriate statistical estimation. In response to this problem, importance weighting methods have been developed to match any biased distribution to its corresponding target unbiased distribution. The seminal Kernel Mean Matching (KMM) method is, nowadays, still considered as state of the art in this research field. However, one of the main drawbacks of this method is the computational burden for large datasets. Building on previous works by Huang et al. (2007) and de Mathelin et al. (2021), we derive a novel importance weighting algorithm which scales to large datasets by using a neural network to predict the instance weights. We show, on multiple public datasets, under various sample biases, that our proposed approach drastically reduces the computational time on large dataset while maintaining similar sample bias correction performance compared to other importance weighting methods. The proposed approach appears to be the only one able to give relevant reweighting in a reasonable time for large dataset with up to two million data.

Antoine de Mathelin, François Deheeger, Mathilde Mougeot et al.

This paper deals with uncertainty quantification and out-of-distribution detection in deep learning using Bayesian and ensemble methods. It proposes a practical solution to the lack of prediction diversity observed recently for standard approaches when used out-of-distribution (Ovadia et al., 2019; Liu et al., 2021). Considering that this issue is mainly related to a lack of weight diversity, we claim that standard methods sample in "over-restricted" regions of the weight space due to the use of "over-regularization" processes, such as weight decay and zero-mean centered Gaussian priors. We propose to solve the problem by adopting the maximum entropy principle for the weight distribution, with the underlying idea to maximize the weight diversity. Under this paradigm, the epistemic uncertainty is described by the weight distribution of maximal entropy that produces neural networks "consistent" with the training observations. Considering stochastic neural networks, a practical optimization is derived to build such a distribution, defined as a trade-off between the average empirical risk and the weight distribution entropy. We develop a novel weight parameterization for the stochastic model, based on the singular value decomposition of the neural network's hidden representations, which enables a large increase of the weight entropy for a small empirical risk penalization. We provide both theoretical and numerical results to assess the efficiency of the approach. In particular, the proposed algorithm appears in the top three best methods in all configurations of an extensive out-of-distribution detection benchmark including more than thirty competitors.

Alejandro de la Concha, Nicolas Vayatis, Argyris Kalogeratos

Quantifying the difference between two probability density functions, $p$ and $q$, using available data, is a fundamental problem in Statistics and Machine Learning. A usual approach for addressing this problem is the likelihood-ratio estimation (LRE) between $p$ and $q$, which -- to our best knowledge -- has been investigated mainly for the offline case. This paper contributes by introducing a new framework for online non-parametric LRE (OLRE) for the setting where pairs of iid observations $(x_t \sim p, x'_t \sim q)$ are observed over time. The non-parametric nature of our approach has the advantage of being agnostic to the forms of $p$ and $q$. Moreover, we capitalize on the recent advances in Kernel Methods and functional minimization to develop an estimator that can be efficiently updated online. We provide theoretical guarantees for the performance of the OLRE method along with empirical validation in synthetic experiments.

Alejandro de la Concha, Nicolas Vayatis, Argyris Kalogeratos

Assuming we have iid observations from two unknown probability density functions (pdfs), $p$ and $q$, the likelihood-ratio estimation (LRE) is an elegant approach to compare the two pdfs only by relying on the available data. In this paper, we introduce the first -to the best of our knowledge-graph-based extension of this problem, which reads as follows: Suppose each node $v$ of a fixed graph has access to observations coming from two unknown node-specific pdfs, $p_v$ and $q_v$, and the goal is to estimate for each node the likelihood-ratio between both pdfs by also taking into account the information provided by the graph structure. The node-level estimation tasks are supposed to exhibit similarities conveyed by the graph, which suggests that the nodes could collaborate to solve them more efficiently. We develop this idea in a concrete non-parametric method that we call Graph-based Relative Unconstrained Least-squares Importance Fitting (GRULSIF). We derive convergence rates for our collaborative approach that highlights the role played by variables such as the number of available observations per node, the size of the graph, and how accurately the graph structure encodes the similarity between tasks. These theoretical results explicit the situations where collaborative estimation effectively leads to an improvement in performance compared to solving each problem independently. Finally, in a series of experiments, we illustrate how GRULSIF infers the likelihood-ratios at the nodes of the graph more accurately compared to state-of-the art LRE methods, which would operate independently at each node, and we also verify that the behavior of GRULSIF is aligned with our previous theoretical analysis.

Alejandro de la Concha, Argyris Kalogeratos, Nicolas Vayatis

Consider each node of a graph to be generating a data stream that is synchronized and observed at near real-time. At a change-point $τ$, a change occurs at a subset of nodes $C$, which affects the probability distribution of their associated node streams. In this paper, we propose a novel kernel-based method to both detect $τ$ and localize $C$, based on the direct estimation of the likelihood-ratio between the post-change and the pre-change distributions of the node streams. Our main working hypothesis is the smoothness of the likelihood-ratio estimates over the graph, i.e connected nodes are expected to have similar likelihood-ratios. The quality of the proposed method is demonstrated on extensive experiments on synthetic scenarios.

Ioannis Bargiotas, Argyris Kalogeratos, Nicolas Vayatis

The standard paired-sample testing approach in the multidimensional setting applies multiple univariate tests on the individual features, followed by p-value adjustments. Such an approach suffers when the data carry numerous features. A number of studies have shown that classification accuracy can be seen as a proxy for two-sample testing. However, neither theoretical foundations nor practical recipes have been proposed so far on how this strategy could be extended to multidimensional paired-sample testing. In this work, we put forward the idea that scoring functions can be produced by the decision rules defined by the perpendicular bisecting hyperplanes of the line segments connecting each pair of instances. Then, the optimal scoring function can be obtained by the pseudomedian of those rules, which we estimate by extending naturally the Hodges-Lehmann estimator. We accordingly propose a framework of a two-step testing procedure. First, we estimate the bisecting hyperplanes for each pair of instances and an aggregated rule derived through the Hodges-Lehmann estimator. The paired samples are scored by this aggregated rule to produce a unidimensional representation. Second, we perform a Wilcoxon signed-rank test on the obtained representation. Our experiments indicate that our approach has substantial performance gains in testing accuracy compared to the traditional multivariate and multiple testing, while at the same time estimates each feature's contribution to the final result.

Gabriel Singer, Samuel Gruffaz, Olivier Vo Van et al.

As acquiring reliable ground-truth labels is usually costly, or infeasible, crowdsourcing and aggregation of noisy human annotations is the typical resort. Aggregating subjective labels, though, may amplify individual biases, particularly regarding sensitive features, raising fairness concerns. Nonetheless, fairness in crowdsourced aggregation remains largely unexplored, with no existing convergence guarantees and only limited post-processing approaches for enforcing $\varepsilon$-fairness under demographic parity. We address this gap by analyzing the fairness s of crowdsourced aggregation methods within the $\varepsilon$-fairness framework, for Majority Vote and Optimal Bayesian aggregation. In the small-crowd regime, we derive an upper bound on the fairness gap of Majority Vote in terms of the fairness gaps of the individual annotators. We further show that the fairness gap of the aggregated consensus converges exponentially fast to that of the ground-truth under interpretable conditions. Since ground-truth itself may still be unfair, we generalize a state-of-the-art multiclass fairness post-processing algorithm from the continuous to the discrete setting, which enforces strict demographic parity constraints to any aggregation rule. Experiments on synthetic and real datasets demonstrate the effectiveness of our approach and corroborate the theoretical insights.

Antoine de Mathelin, Mounir Atiq, Guillaume Richard et al.

In this paper, we introduce the ADAPT library, an open source Python API providing the implementation of the main transfer learning and domain adaptation methods. The library is designed with a user friendly approach to facilitate the access to domain adaptation for a wide public. ADAPT is compatible with scikit-learn and TensorFlow and a full documentation is proposed online https://adapt-python.github.io/adapt/ with a substantial gallery of examples.

Antoine de Mathelin, Nicolas Enrique Cecchi, François Deheeger et al.

This paper proposes a novel k-medoids approximation algorithm to handle large-scale datasets with reasonable computational time and memory complexity. We develop a local-search algorithm that iteratively improves the medoid selection based on the estimation of the k-medoids objective. A single batch of size m << n provides the estimation, which reduces the required memory size and the number of pairwise dissimilarities computations to O(mn), instead of O(n^2) compared to most k-medoids baselines. We obtain theoretical results highlighting that a batch of size m = O(log(n)) is sufficient to guarantee, with strong probability, the same performance as the original local-search algorithm. Multiple experiments conducted on real datasets of various sizes and dimensions show that our algorithm provides similar performances as state-of-the-art methods such as FasterPAM and BanditPAM++ with a drastically reduced running time.

Alejandro de la Concha, Nicolas Vayatis, Argyris Kalogeratos

This paper addresses the multiple two-sample test problem in a graph-structured setting, which is a common scenario in fields such as Spatial Statistics and Neuroscience. Each node $v$ in fixed graph deals with a two-sample testing problem between two node-specific probability density functions (pdfs), $p_v$ and $q_v$. The goal is to identify nodes where the null hypothesis $p_v = q_v$ should be rejected, under the assumption that connected nodes would yield similar test outcomes. We propose the non-parametric collaborative two-sample testing (CTST) framework that efficiently leverages the graph structure and minimizes the assumptions over $p_v$ and $q_v$. Our methodology integrates elements from f-divergence estimation, Kernel Methods, and Multitask Learning. We use synthetic experiments and a real sensor network detecting seismic activity to demonstrate that CTST outperforms state-of-the-art non-parametric statistical tests that apply at each node independently, hence disregard the geometry of the problem.

Perceval Beja-Battais, Alain Grossetête, Nicolas Vayatis

In recent years, there has been an increasing need for Nuclear Power Plants (NPPs) to improve flexibility in order to match the rapid growth of renewable energies. The Operator Assistance Predictive System (OAPS) developed by Framatome addresses this problem through Model Predictive Control (MPC). In this work, we aim to improve MPC methods through data-driven simulation schemes. Thus, from a set of nonlinear stiff ordinary differential equations (ODEs), this paper introduces two surrogate models acting as alternative simulation schemes to enhance nuclear reactor core simulation. We show that both data-driven and physics-informed models can rapidly integrate complex dynamics, with a very low computational time (up to 1000x time reduction).

Jean Vassoyan, Anan Schütt, Jill-Jênn Vie et al.

Massive Open Online Courses (MOOCs) have greatly contributed to making education more accessible. However, many MOOCs maintain a rigid, one-size-fits-all structure that fails to address the diverse needs and backgrounds of individual learners. Learning path personalization aims to address this limitation, by tailoring sequences of educational content to optimize individual student learning outcomes. Existing approaches, however, often require either massive student interaction data or extensive expert annotation, limiting their broad application. In this study, we introduce a novel data-efficient framework for learning path personalization that operates without expert annotation. Our method employs a flexible recommender system pre-trained with reinforcement learning on a dataset of raw course materials. Through experiments on semi-synthetic data, we show that this pre-training stage substantially improves data-efficiency in a range of adaptive learning scenarios featuring new educational materials. This opens up new perspectives for the design of foundation models for adaptive learning.

Alejandro de la Concha, Argyris Kalogeratos, Nicolas Vayatis

Consider a heterogeneous data stream being generated by the nodes of a graph. The data stream is in essence composed by multiple streams, possibly of different nature that depends on each node. At a given moment $τ$, a change-point occurs for a subset of nodes $C$, signifying the change in the probability distribution of their associated streams. In this paper we propose an online non-parametric method to infer $τ$ based on the direct estimation of the likelihood-ratio between the post-change and the pre-change distribution associated with the data stream of each node. We propose a kernel-based method, under the hypothesis that connected nodes of the graph are expected to have similar likelihood-ratio estimates when there is no change-point. We demonstrate the quality of our method on synthetic experiments and real-world applications.

Stéphan Clémençon, Myrto Limnios, Nicolas Vayatis

The ROC curve is the gold standard for measuring the performance of a test/scoring statistic regarding its capacity to discriminate between two statistical populations in a wide variety of applications, ranging from anomaly detection in signal processing to information retrieval, through medical diagnosis. Most practical performance measures used in scoring/ranking applications such as the AUC, the local AUC, the p-norm push, the DCG and others, can be viewed as summaries of the ROC curve. In this paper, the fact that most of these empirical criteria can be expressed as two-sample linear rank statistics is highlighted and concentration inequalities for collections of such random variables, referred to as two-sample rank processes here, are proved, when indexed by VC classes of scoring functions. Based on these nonasymptotic bounds, the generalization capacity of empirical maximizers of a wide class of ranking performance criteria is next investigated from a theoretical perspective. It is also supported by empirical evidence through convincing numerical experiments.

Antoine de Mathelin, Francois Deheeger, Mathilde Mougeot et al.

The goal of the paper is to design active learning strategies which lead to domain adaptation under an assumption of Lipschitz functions. Building on previous work by Mansour et al. (2009) we adapt the concept of discrepancy distance between source and target distributions to restrict the maximization over the hypothesis class to a localized class of functions which are performing accurate labeling on the source domain. We derive generalization error bounds for such active learning strategies in terms of Rademacher average and localized discrepancy for general loss functions which satisfy a regularity condition. A practical K-medoids algorithm that can address the case of large data set is inferred from the theoretical bounds. Our numerical experiments show that the proposed algorithm is competitive against other state-of-the-art active learning techniques in the context of domain adaptation, in particular on large data sets of around one hundred thousand images.

Pierre Humbert, Batiste Le Bars, Ludovic Minvielle et al.

In this paper, we introduce a robust nonparametric density estimator combining the popular Kernel Density Estimation method and the Median-of-Means principle (MoM-KDE). This estimator is shown to achieve robustness to any kind of anomalous data, even in the case of adversarial contamination. In particular, while previous works only prove consistency results under known contamination model, this work provides finite-sample high-probability error-bounds without a priori knowledge on the outliers. Finally, when compared with other robust kernel estimators, we show that MoM-KDE achieves competitive results while having significant lower computational complexity.

Alejandro de la Concha, Nicolas Vayatis, Argyris Kalogeratos

This paper addresses the problem of segmenting a stream of graph signals: we aim to detect changes in the mean of a multivariate signal defined over the nodes of a known graph. We propose an offline method that relies on the concept of graph signal stationarity and allows the convenient translation of the problem from the original vertex domain to the spectral domain (Graph Fourier Transform), where it is much easier to solve. Although the obtained spectral representation is sparse in real applications, to the best of our knowledge this property has not been sufficiently exploited in the existing related literature. Our change-point detection method adopts a model selection approach that takes into account the sparsity of the spectral representation and determines automatically the number of change-points. Our detector comes with a proof of a non-asymptotic oracle inequality. Numerical experiments demonstrate the performance of the proposed method.

Antoine de Mathelin, Guillaume Richard, Francois Deheeger et al.

We present a novel instance-based approach to handle regression tasks in the context of supervised domain adaptation under an assumption of covariate shift. The approach developed in this paper is based on the assumption that the task on the target domain can be efficiently learned by adequately reweighting the source instances during training phase. We introduce a novel formulation of the optimization objective for domain adaptation which relies on a discrepancy distance characterizing the difference between domains according to a specific task and a class of hypotheses. To solve this problem, we develop an adversarial network algorithm which learns both the source weighting scheme and the task in one feed-forward gradient descent. We provide numerical evidence of the relevance of the method on public data sets for regression domain adaptation through reproducible experiments.

Mathilde Fekom, Nicolas Vayatis, Argyris Kalogeratos

In this paper we present the Warm-starting Dynamic Thresholding algorithm, developed using dynamic programming, for a variant of the standard online selection problem. The problem allows job positions to be either free or already occupied at the beginning of the process. Throughout the selection process, the decision maker interviews one after the other the new candidates and reveals a quality score for each of them. Based on that information, she can (re)assign each job at most once by taking immediate and irrevocable decisions. We relax the hard requirement of the class of dynamic programming algorithms to perfectly know the distribution from which the scores of candidates are drawn, by presenting extensions for the partial and no-information cases, in which the decision maker can learn the underlying score distribution sequentially while interviewing candidates.

Batiste Le Bars, Pierre Humbert, Argyris Kalogeratos et al.

This work focuses on the estimation of multiple change-points in a time-varying Ising model that evolves piece-wise constantly. The aim is to identify both the moments at which significant changes occur in the Ising model, as well as the underlying graph structures. For this purpose, we propose to estimate the neighborhood of each node by maximizing a penalized version of its conditional log-likelihood. The objective of the penalization is twofold: it imposes sparsity in the learned graphs and, thanks to a fused-type penalty, it also enforces them to evolve piece-wise constantly. Using few assumptions, we provide two change-points consistency theorems. Those are the first in the context of unknown number of change-points detection in time-varying Ising model. Finally, experimental results on several synthetic datasets and a real-world dataset demonstrate the performance of our method.

Mathilde Fekom, Nicolas Vayatis, Argyris Kalogeratos

Under the Dynamic Resource Allocation (DRA) model, an administrator has the mission to allocate dynamically a limited budget of resources to the nodes of a network in order to reduce a diffusion process (DP) (e.g. an epidemic). The standard DRA assumes that the administrator has constantly full information and instantaneous access to the entire network. Towards bringing such strategies closer to real-life constraints, we first present the Restricted DRA model extension where, at each intervention round, the access is restricted to only a fraction of the network nodes, called sample. Then, inspired by sequential selection problems such as the well-known Secretary Problem, we propose the Sequential DRA (SDRA) model. Our model introduces a sequential aspect in the decision process over the sample of each round, offering a completely new perspective to the dynamic DP control. Finally, we incorporate several sequential selection algorithms to SDRA control strategies and compare their performance in SIS epidemic simulations.

Pierre Humbert, Julien Audiffren, Laurent Oudre et al.

This paper introduces a new multivariate convolutional sparse coding based on tensor algebra with a general model enforcing both element-wise sparsity and low-rankness of the activations tensors. By using the CP decomposition, this model achieves a significantly more efficient encoding of the multivariate signal-particularly in the high order/ dimension setting-resulting in better performance. We prove that our model is closely related to the Kruskal tensor regression problem, offering interesting theoretical guarantees to our setting. Furthermore, we provide an efficient optimization algorithm based on alternating optimization to solve this model. Finally, we evaluate our algorithm with a large range of experiments, highlighting its advantages and limitations.

Ioannis Bargiotas, Argyris Kalogeratos, Myrto Limnios et al.

Falling in Parkinsonian syndromes (PS) is associated with postural instability and consists a common cause of disability among PS patients. Current posturographic practices record the body's center-of-pressure displacement (statokinesigram) while the patient stands on a force platform. Statokinesigrams, after appropriate signal processing, can offer numerous posturographic features, which however challenges the efforts for valid statistics via standard univariate approaches. In this work, we present the ts-AUC, a non-parametric multivariate two-sample test, which we employ to analyze statokinesigram differences among PS patients that are fallers (PSf) and non-fallers (PSNF). We included 123 PS patients who were classified into PSF or PSNF based on clinical assessment and underwent simple Romberg Test (eyes open/eyes closed). We analyzed posturographic features using both multiple testing with p-value adjustment and the ts-AUC. While the ts-AUC showed significant difference between groups (p-value = 0.01), multiple testing did not show any such difference. Interestingly, significant difference between the two groups was found only using the open-eyes protocol. PSF showed significantly increased antero-posterior movements as well as increased posturographic area, compared to PSNF. Our study demonstrates the superiority of the ts-AUC test compared to standard statistical tools in distinguishing PSF and PSNF in the multidimensional feature space. This result highlights more generally the fact that machine learning-based statistical tests can be seen as a natural extension of classical statistical approaches and should be considered, especially when dealing with multifactorial assessments.

Kevin Scaman, Argyris Kalogeratos, Luca Corinzia et al.

Information Cascades Model captures dynamical properties of user activity in a social network. In this work, we develop a novel framework for activity shaping under the Continuous-Time Information Cascades Model which allows the administrator for local control actions by allocating targeted resources that can alter the spread of the process. Our framework employs the optimization of the spectral radius of the Hazard matrix, a quantity that has been shown to drive the maximum influence in a network, while enjoying a simple convex relaxation when used to minimize the influence of the cascade. In addition, use-cases such as quarantine and node immunization are discussed to highlight the generality of the proposed activity shaping framework. Finally, we present the NetShape influence minimization method which is compared favorably to baseline and state-of-the-art approaches through simulations on real social networks.

Thomas Moreau, Laurent Oudre, Nicolas Vayatis

In this paper, we introduce DICOD, a convolutional sparse coding algorithm which builds shift invariant representations for long signals. This algorithm is designed to run in a distributed setting, with local message passing, making it communication efficient. It is based on coordinate descent and uses locally greedy updates which accelerate the resolution compared to greedy coordinate selection. We prove the convergence of this algorithm and highlight its computational speed-up which is super-linear in the number of cores used. We also provide empirical evidence for the acceleration properties of our algorithm compared to state-of-the-art methods.

Cédric Malherbe, Nicolas Vayatis

The goal of the paper is to design sequential strategies which lead to efficient optimization of an unknown function under the only assumption that it has a finite Lipschitz constant. We first identify sufficient conditions for the consistency of generic sequential algorithms and formulate the expected minimax rate for their performance. We introduce and analyze a first algorithm called LIPO which assumes the Lipschitz constant to be known. Consistency, minimax rates for LIPO are proved, as well as fast rates under an additional Hölder like condition. An adaptive version of LIPO is also introduced for the more realistic setup where the Lipschitz constant is unknown and has to be estimated along with the optimization. Similar theoretical guarantees are shown to hold for the adaptive LIPO algorithm and a numerical assessment is provided at the end of the paper to illustrate the potential of this strategy with respect to state-of-the-art methods over typical benchmark problems for global optimization.

Cédric Malherbe, Nicolas Vayatis

We consider the problem of maximizing an unknown function over a compact and convex set using as few observations as possible. We observe that the optimization of the function essentially relies on learning the induced bipartite ranking rule of f. Based on this idea, we relate global optimization to bipartite ranking which allows to address problems with high dimensional input space, as well as cases of functions with weak regularity properties. The paper introduces novel meta-algorithms for global optimization which rely on the choice of any bipartite ranking method. Theoretical properties are provided as well as convergence guarantees and equivalences between various optimization methods are obtained as a by-product. Eventually, numerical evidence is given to show that the main algorithm of the paper which adapts empirically to the underlying ranking structure essentially outperforms existing state-of-the-art global optimization algorithms in typical benchmarks.

Emile Contal, Nicolas Vayatis

The paper considers the problem of global optimization in the setup of stochastic process bandits. We introduce an UCB algorithm which builds a cascade of discretization trees based on generic chaining in order to render possible his operability over a continuous domain. The theoretical framework applies to functions under weak probabilistic smoothness assumptions and also extends significantly the spectrum of application of UCB strategies. Moreover generic regret bounds are derived which are then specialized to Gaussian processes indexed on infinite-dimensional spaces as well as to quadratic forms of Gaussian processes. Lower bounds are also proved in the case of Gaussian processes to assess the optimality of the proposed algorithm.

Emile Contal, Cédric Malherbe, Nicolas Vayatis

In this paper, we consider the problem of stochastic optimization under a bandit feedback model. We generalize the GP-UCB algorithm [Srinivas and al., 2012] to arbitrary kernels and search spaces. To do so, we use a notion of localized chaining to control the supremum of a Gaussian process, and provide a novel optimization scheme based on the computation of covering numbers. The theoretical bounds we obtain on the cumulative regret are more generic and present the same convergence rates as the GP-UCB algorithm. Finally, the algorithm is shown to be empirically more efficient than its natural competitors on simple and complex input spaces.

Emile Contal, Vianney Perchet, Nicolas Vayatis

In this paper, we analyze a generic algorithm scheme for sequential global optimization using Gaussian processes. The upper bounds we derive on the cumulative regret for this generic algorithm improve by an exponential factor the previously known bounds for algorithms like GP-UCB. We also introduce the novel Gaussian Process Mutual Information algorithm (GP-MI), which significantly improves further these upper bounds for the cumulative regret. We confirm the efficiency of this algorithm on synthetic and real tasks against the natural competitor, GP-UCB, and also the Expected Improvement heuristic.

Emile Contal, David Buffoni, Alexandre Robicquet et al.

In this paper, we consider the challenge of maximizing an unknown function f for which evaluations are noisy and are acquired with high cost. An iterative procedure uses the previous measures to actively select the next estimation of f which is predicted to be the most useful. We focus on the case where the function can be evaluated in parallel with batches of fixed size and analyze the benefit compared to the purely sequential procedure in terms of cumulative regret. We introduce the Gaussian Process Upper Confidence Bound and Pure Exploration algorithm (GP-UCB-PE) which combines the UCB strategy and Pure Exploration in the same batch of evaluations along the parallel iterations. We prove theoretical upper bounds on the regret with batches of size K for this procedure which show the improvement of the order of sqrt{K} for fixed iteration cost over purely sequential versions. Moreover, the multiplicative constants involved have the property of being dimension-free. We also confirm empirically the efficiency of GP-UCB-PE on real and synthetic problems compared to state-of-the-art competitors.

Emile Richard, Stephane Gaiffas, Nicolas Vayatis

In the paper, we consider the problem of link prediction in time-evolving graphs. We assume that certain graph features, such as the node degree, follow a vector autoregressive (VAR) model and we propose to use this information to improve the accuracy of prediction. Our strategy involves a joint optimization procedure over the space of adjacency matrices and VAR matrices which takes into account both sparsity and low rank properties of the matrices. Oracle inequalities are derived and illustrate the trade-offs in the choice of smoothing parameters when modeling the joint effect of sparsity and low rank property. The estimate is computed efficiently using proximal methods through a generalized forward-backward agorithm.

Emile Richard, Pierre-Andre Savalle, Nicolas Vayatis

The paper introduces a penalized matrix estimation procedure aiming at solutions which are sparse and low-rank at the same time. Such structures arise in the context of social networks or protein interactions where underlying graphs have adjacency matrices which are block-diagonal in the appropriate basis. We introduce a convex mixed penalty which involves $\ell_1$-norm and trace norm simultaneously. We obtain an oracle inequality which indicates how the two effects interact according to the nature of the target matrix. We bound generalization error in the link prediction problem. We also develop proximal descent strategies to solve the optimization problem efficiently and evaluate performance on synthetic and real data sets.

Emile Richard, Pierre-Andre Savalle, Nicolas Vayatis

We study the problem of prediction for evolving graph data. We formulate the problem as the minimization of a convex objective encouraging sparsity and low-rank of the solution, that reflect natural graph properties. The convex formulation allows to obtain oracle inequalities and efficient solvers. We provide empirical results for our algorithm and comparison with competing methods, and point out two open questions related to compressed sensing and algebra of low-rank and sparse matrices.

Emile Richard, Andreas Argyriou, Theodoros Evgeniou et al.

We consider the two problems of predicting links in a dynamic graph sequence and predicting functions defined at each node of the graph. In many applications, the solution of one problem is useful for solving the other. Indeed, if these functions reflect node features, then they are related through the graph structure. In this paper, we formulate a hybrid approach that simultaneously learns the structure of the graph and predicts the values of the node-related functions. Our approach is based on the optimization of a joint regularization objective. We empirically test the benefits of the proposed method with both synthetic and real data. The results indicate that joint regularization improves prediction performance over the graph evolution and the node features.