Stéphan Clémençon

Jean-Rémy Conti, Nathan Noiry, Vincent Despiegel et al.

In spite of the high performance and reliability of deep learning algorithms in a wide range of everyday applications, many investigations tend to show that a lot of models exhibit biases, discriminating against specific subgroups of the population (e.g. gender, ethnicity). This urges the practitioner to develop fair systems with a uniform/comparable performance across sensitive groups. In this work, we investigate the gender bias of deep Face Recognition networks. In order to measure this bias, we introduce two new metrics, $\mathrm{BFAR}$ and $\mathrm{BFRR}$, that better reflect the inherent deployment needs of Face Recognition systems. Motivated by geometric considerations, we mitigate gender bias through a new post-processing methodology which transforms the deep embeddings of a pre-trained model to give more representation power to discriminated subgroups. It consists in training a shallow neural network by minimizing a Fair von Mises-Fisher loss whose hyperparameters account for the intra-class variance of each gender. Interestingly, we empirically observe that these hyperparameters are correlated with our fairness metrics. In fact, extensive numerical experiments on a variety of datasets show that a careful selection significantly reduces gender bias. The code used for the experiments can be found at https://github.com/JRConti/EthicalModule_vMF.

Pierre Laforgue, Stephan Clémençon, Patrice Bertail

Tournament procedures, recently introduced in Lugosi & Mendelson (2016), offer an appealing alternative, from a theoretical perspective at least, to the principle of Empirical Risk Minimization in machine learning. Statistical learning by Median-of-Means (MoM) basically consists in segmenting the training data into blocks of equal size and comparing the statistical performance of every pair of candidate decision rules on each data block: that with highest performance on the majority of the blocks is declared as the winner. In the context of nonparametric regression, functions having won all their duels have been shown to outperform empirical risk minimizers w.r.t. the mean squared error under minimal assumptions, while exhibiting robustness properties. It is the purpose of this paper to extend this approach in order to address other learning problems, in particular for which the performance criterion takes the form of an expectation over pairs of observations rather than over one single observation, as may be the case in pairwise ranking, clustering or metric learning. Precisely, it is proved here that the bounds achieved by MoM are essentially conserved when the blocks are built by means of independent sampling without replacement schemes instead of a simple segmentation. These results are next extended to situations where the risk is related to a pairwise loss function and its empirical counterpart is of the form of a $U$-statistic. Beyond theoretical results guaranteeing the performance of the learning/estimation methods proposed, some numerical experiments provide empirical evidence of their relevance in practice.

Stephan Clémençon, Nathan Huet, Anne Sabourin

We establish a statistical learning theoretical framework aimed at extrapolation, or out-of-domain generalization, on the unobserved tails of covariates in continuous regression problems. Our strategy involves performing statistical regression on a subsample of observations with continuous labels that are the furthest away from the origin, focusing specifically on their angular components. The underlying assumptions of our approach are grounded in the theory of multivariate regular variation, a cornerstone of extreme value theory. We address the stylized problem of nonparametric least squares regression with predictors chosen from a Vapnik-Chervonenkis class. This work contributes to a broader initiative to develop statistical learning theoretical foundations for supervised learning strategies that enhance performance on the supposedly heavy tails of covariates. Previous efforts in this area have focused exclusively on binary classification on extreme covariates. Although the continuous target setting necessitates different techniques and regularity assumptions, our main results echo findings from earlier studies. We quantify the predictive performance on tail regions in terms of excess risk, presenting it as a finite sample risk bound with a clear bias-variance decomposition. Numerical experiments with simulated and real data illustrate our theoretical findings.

Igor Colin, Aurélien Bellet, Stephan Clémençon et al.

In the IoT era, information is more and more frequently picked up by connected smart sensors with increasing, though limited, storage, communication and computation abilities. Whether due to privacy constraints or to the structure of the distributed system, the development of statistical learning methods dedicated to data that are shared over a network is now a major issue. Gossip-based algorithms have been developed for the purpose of solving a wide variety of statistical learning tasks, ranging from data aggregation over sensor networks to decentralized multi-agent optimization. Whereas the vast majority of contributions consider situations where the function to be estimated or optimized is a basic average of individual observations, it is the goal of this article to investigate the case where the latter is of pairwise nature, taking the form of a U -statistic of degree two. Motivated by various problems such as similarity learning, ranking or clustering for instance, we revisit gossip algorithms specifically designed for pairwise objective functions and provide a comprehensive theoretical framework for their convergence. This analysis fills a gap in the literature by establishing conditions under which these methods succeed, and by identifying the graph properties that critically affect their efficiency. In particular, a refined analysis of the convergence upper and lower bounds is performed.

Qi Gan, Sao Mai Nguyen, Eric Fenaux et al.

Human pose capture is essential for sports analysis, enabling precise evaluation of athletes' movements. While deep learning-based human pose estimation (HPE) models from RGB videos have achieved impressive performance on public datasets, their effectiveness in real-world sports scenarios is often hindered by motion blur, occlusions, and domain shifts across different pose representations. Fine-tuning these models can partially alleviate such challenges but typically requires large-scale annotated data and still struggles to generalize across diverse sports environments. To address these limitations, we propose a 2D pose prior-guided refinement approach based on Neural Distance Fields (NDF). Unlike existing approaches that rely solely on angular representations of human poses, we introduce a polar coordinate-based representation that explicitly incorporates joint connection lengths, enabling a more accurate correction of erroneous pose estimations. Additionally, we define a novel non-geodesic distance metric that separates angular and radial discrepancies, which we demonstrate is better suited for polar representations than traditional geodesic distances. To mitigate data scarcity, we develop a gradient-based batch-projection augmentation strategy, which synthesizes realistic pose samples through iterative refinement. Our method is evaluated on a long jump dataset, demonstrating its ability to improve 2D pose estimation across multiple pose representations, making it robust across different domains. Experimental results show that our approach enhances pose plausibility while requiring only limited training data. Code is available at: https://github.com/QGAN2019/polar-NDF.

Jean-Rémy Conti, Stéphan Clémençon

The ROC curve is the major tool for assessing not only the performance but also the fairness properties of a similarity scoring function. In order to draw reliable conclusions based on empirical ROC analysis, accurately evaluating the uncertainty level related to statistical versions of the ROC curves of interest is absolutely necessary, especially for applications with considerable societal impact such as Face Recognition. In this article, we prove asymptotic guarantees for empirical ROC curves of similarity functions as well as for by-product metrics useful to assess fairness. We also explain that, because the false acceptance/rejection rates are of the form of U-statistics in the case of similarity scoring, the naive bootstrap approach may jeopardize the assessment procedure. A dedicated recentering technique must be used instead. Beyond the theoretical analysis carried out, various experiments using real face image datasets provide strong empirical evidence of the practical relevance of the methods promoted here, when applied to several ROC-based measures such as popular fairness metrics.

James Cheshire, Stephan Clémençon

In this article, bipartite ranking, a statistical learning problem involved in many applications and widely studied in the passive context, is approached in a much more general \textit{active setting} than the discrete one previously considered in the literature. While the latter assumes that the conditional distribution is piece wise constant, the framework we develop permits in contrast to deal with continuous conditional distributions, provided that they fulfill a Hölder smoothness constraint. We first show that a naive approach based on discretisation at a uniform level, fixed \textit{a priori} and consisting in applying next the active strategy designed for the discrete setting generally fails. Instead, we propose a novel algorithm, referred to as smooth-rank and designed for the continuous setting, which aims to minimise the distance between the ROC curve of the estimated ranking rule and the optimal one w.r.t. the $\sup$ norm. We show that, for a fixed confidence level $ε>0$ and probability $δ\in (0,1)$, smooth-rank is PAC$(ε,δ)$. In addition, we provide a problem dependent upper bound on the expected sampling time of smooth-rank and establish a problem dependent lower bound on the expected sampling time of any PAC$(ε,δ)$ algorithm. Beyond the theoretical analysis carried out, numerical results are presented, providing solid empirical evidence of the performance of the algorithm proposed, which compares favorably with alternative approaches.

Anna Van Elst, Igor Colin, Stephan Clémençon

This paper addresses the problem of robust estimation in gossip algorithms over arbitrary communication graphs. Gossip algorithms are fully decentralized, relying only on local neighbor-to-neighbor communication, making them well-suited for situations where communication is constrained. A fundamental challenge in existing mean-based gossip algorithms is their vulnerability to malicious or corrupted nodes. In this paper, we show that an outlier-robust mean can be computed by globally estimating a robust statistic. More specifically, we propose a novel gossip algorithm for rank estimation, referred to as \textsc{GoRank}, and leverage it to design a gossip procedure dedicated to trimmed mean estimation, coined \textsc{GoTrim}. In addition to a detailed description of the proposed methods, a key contribution of our work is a precise convergence analysis: we establish an $\mathcal{O}(1/t)$ rate for rank estimation and an $\mathcal{O}(1 / {t})$ rate for trimmed mean estimation, where by $t$ is meant the number of iterations. Moreover, we provide a breakdown point analysis of \textsc{GoTrim}. We empirically validate our theoretical results through experiments on diverse network topologies, data distributions and contamination schemes.

Jean-Rémy Conti, Stéphan Clémençon

The urging societal demand for fair AI systems has put pressure on the research community to develop predictive models that are not only globally accurate but also meet new fairness criteria, reflecting the lack of disparate mistreatment with respect to sensitive attributes ($\textit{e.g.}$ gender, ethnicity, age). In particular, the variability of the errors made by certain Facial Recognition (FR) systems across specific segments of the population compromises the deployment of the latter, and was judged unacceptable by regulatory authorities. Designing fair FR systems is a very challenging problem, mainly due to the complex and functional nature of the performance measure used in this domain ($\textit{i.e.}$ ROC curves) and because of the huge heterogeneity of the face image datasets usually available for training. In this paper, we propose a novel post-processing approach to improve the fairness of pre-trained FR models by optimizing a regression loss which acts on centroid-based scores. Beyond the computational advantages of the method, we present numerical experiments providing strong empirical evidence of the gain in fairness and of the ability to preserve global accuracy.

Anna Van Elst, Igor Colin, Stephan Clémençon

As decentralized AI and edge intelligence become increasingly prevalent, ensuring robustness and trustworthiness in such distributed settings has become a critical issue-especially in the presence of corrupted or adversarial data. Traditional decentralized algorithms are vulnerable to data contamination as they typically rely on simple statistics (e.g., means or sum), motivating the need for more robust statistics. In line with recent work on decentralized estimation of trimmed means and ranks, we develop gossip algorithms for computing a broad class of rank-based statistics, including L-statistics and rank statistics-both known for their robustness to outliers. We apply our method to perform robust distributed two-sample hypothesis testing, introducing the first gossip algorithm for Wilcoxon rank-sum tests. We provide rigorous convergence guarantees, including the first convergence rate bound for asynchronous gossip-based rank estimation. We empirically validate our theoretical results through experiments on diverse network topologies.

Emilia Siviero, Guillaume Staerman, Stephan Clémençon et al.

Many modern spatio-temporal data sets, in sociology, epidemiology or seismology, for example, exhibit self-exciting characteristics, triggering and clustering behaviors both at the same time, that a suitable Hawkes space-time process can accurately capture. This paper aims to develop a fast and flexible parametric inference technique to recover the parameters of the kernel functions involved in the intensity function of a space-time Hawkes process based on such data. Our statistical approach combines three key ingredients: 1) kernels with finite support are considered, 2) the space-time domain is appropriately discretized, and 3) (approximate) precomputations are used. The inference technique we propose then consists of a $\ell_2$ gradient-based solver that is fast and statistically accurate. In addition to describing the algorithmic aspects, numerical experiments have been carried out on synthetic and real spatio-temporal data, providing solid empirical evidence of the relevance of the proposed methodology.

Laurence Likforman-Sulem, Anna Esposito, Marcos Faundez-Zanuy et al.

The detection of negative emotions through daily activities such as handwriting is useful for promoting well-being. The spread of human-machine interfaces such as tablets makes the collection of handwriting samples easier. In this context, we present a first publicly available handwriting database which relates emotional states to handwriting, that we call EMOTHAW. This database includes samples of 129 participants whose emotional states, namely anxiety, depression and stress, are assessed by the Depression Anxiety Stress Scales (DASS) questionnaire. Seven tasks are recorded through a digitizing tablet: pentagons and house drawing, words copied in handprint, circles and clock drawing, and one sentence copied in cursive writing. Records consist in pen positions, on-paper and in-air, time stamp, pressure, pen azimuth and altitude. We report our analysis on this database. From collected data, we first compute measurements related to timing and ductus. We compute separate measurements according to the position of the writing device: on paper or in-air. We analyse and classify this set of measurements (referred to as features) using a random forest approach. This latter is a machine learning method [2], based on an ensemble of decision trees, which includes a feature ranking process. We use this ranking process to identify the features which best reveal a targeted emotional state. We then build random forest classifiers associated to each emotional state. Our results, obtained from cross-validation experiments, show that the targeted emotional states can be identified with accuracies ranging from 60% to 71%.

Emilia Siviero, Emilie Chautru, Stephan Clémençon

In the Big Data era, with the ubiquity of geolocation sensors in particular, massive datasets exhibiting a possibly complex spatial dependence structure are becoming increasingly available. In this context, the standard probabilistic theory of statistical learning does not apply directly and guarantees of the generalization capacity of predictive rules learned from such data are left to establish. We analyze here the simple Kriging task from a statistical learning perspective, i.e. by carrying out a nonparametric finite-sample predictive analysis. Given $d\geq 1$ values taken by a realization of a square integrable random field $X=\{X_s\}_{s\in S}$, $S\subset \mathbb{R}^2$, with unknown covariance structure, at sites $s_1,\; \ldots,\; s_d$ in $S$, the goal is to predict the unknown values it takes at any other location $s\in S$ with minimum quadratic risk. The prediction rule being derived from a training spatial dataset: a single realization $X'$ of $X$, independent from those to be predicted, observed at $n\geq 1$ locations $σ_1,\; \ldots,\; σ_n$ in $S$. Despite the connection of this minimization problem with kernel ridge regression, establishing the generalization capacity of empirical risk minimizers is far from straightforward, due to the non independent and identically distributed nature of the training data $X'_{σ_1},\; \ldots,\; X'_{σ_n}$ involved in the learning procedure. In this article, non-asymptotic bounds of order $O_{\mathbb{P}}(1/\sqrt{n})$ are proved for the excess risk of a plug-in predictive rule mimicking the true minimizer in the case of isotropic stationary Gaussian processes, observed at locations forming a regular grid in the learning stage. These theoretical results are illustrated by various numerical experiments, on simulated data and on real-world datasets.

Pierre Colombo, Nathan Noiry, Ekhine Irurozki et al.

In Machine Learning, a benchmark refers to an ensemble of datasets associated with one or multiple metrics together with a way to aggregate different systems performances. They are instrumental in (i) assessing the progress of new methods along different axes and (ii) selecting the best systems for practical use. This is particularly the case for NLP with the development of large pre-trained models (e.g. GPT, BERT) that are expected to generalize well on a variety of tasks. While the community mainly focused on developing new datasets and metrics, there has been little interest in the aggregation procedure, which is often reduced to a simple average over various performance measures. However, this procedure can be problematic when the metrics are on a different scale, which may lead to spurious conclusions. This paper proposes a new procedure to rank systems based on their performance across different tasks. Motivated by the social choice theory, the final system ordering is obtained through aggregating the rankings induced by each task and is theoretically grounded. We conduct extensive numerical experiments (on over 270k scores) to assess the soundness of our approach both on synthetic and real scores (e.g. GLUE, EXTREM, SEVAL, TAC, FLICKR). In particular, we show that our method yields different conclusions on state-of-the-art systems than the mean-aggregation procedure while being both more reliable and robust.

Morgane Goibert, Stéphan Clémençon, Ekhine Irurozki et al.

The concept of median/consensus has been widely investigated in order to provide a statistical summary of ranking data, i.e. realizations of a random permutation $Σ$ of a finite set, $\{1,\; \ldots,\; n\}$ with $n\geq 1$ say. As it sheds light onto only one aspect of $Σ$'s distribution $P$, it may neglect other informative features. It is the purpose of this paper to define analogs of quantiles, ranks and statistical procedures based on such quantities for the analysis of ranking data by means of a metric-based notion of depth function on the symmetric group. Overcoming the absence of vector space structure on $\mathfrak{S}_n$, the latter defines a center-outward ordering of the permutations in the support of $P$ and extends the classic metric-based formulation of consensus ranking (medians corresponding then to the deepest permutations). The axiomatic properties that ranking depths should ideally possess are listed, while computational and generalization issues are studied at length. Beyond the theoretical analysis carried out, the relevance of the novel concepts and methods introduced for a wide variety of statistical tasks are also supported by numerous numerical experiments.

Mathieu Chambefort, Raphaël Butez, Emilie Chautru et al.

In image denoising problems, the increasing density of available images makes an exhaustive visual inspection impossible and therefore automated methods based on machine-learning must be deployed for this purpose. This is particulary the case in seismic signal processing. Engineers/geophysicists have to deal with millions of seismic time series. Finding the sub-surface properties useful for the oil industry may take up to a year and is very costly in terms of computing/human resources. In particular, the data must go through different steps of noise attenuation. Each denoise step is then ideally followed by a quality control (QC) stage performed by means of human expertise. To learn a quality control classifier in a supervised manner, labeled training data must be available, but collecting the labels from human experts is extremely time-consuming. We therefore propose a novel active learning methodology to sequentially select the most relevant data, which are then given back to a human expert for labeling. Beyond the application in geophysics, the technique we promote in this paper, based on estimates of the local error and its uncertainty, is generic. Its performance is supported by strong empirical evidence, as illustrated by the numerical experiments presented in this article, where it is compared to alternative active learning strategies both on synthetic and real seismic datasets.

Guillaume Staerman, Eric Adjakossa, Pavlo Mozharovskyi et al.

The increasing automation in many areas of the Industry expressly demands to design efficient machine-learning solutions for the detection of abnormal events. With the ubiquitous deployment of sensors monitoring nearly continuously the health of complex infrastructures, anomaly detection can now rely on measurements sampled at a very high frequency, providing a very rich representation of the phenomenon under surveillance. In order to exploit fully the information thus collected, the observations cannot be treated as multivariate data anymore and a functional analysis approach is required. It is the purpose of this paper to investigate the performance of recent techniques for anomaly detection in the functional setup on real datasets. After an overview of the state-of-the-art and a visual-descriptive study, a variety of anomaly detection methods are compared. While taxonomies of abnormalities (e.g. shape, location) in the functional setup are documented in the literature, assigning a specific type to the identified anomalies appears to be a challenging task. Thus, strengths and weaknesses of the existing approaches are benchmarked in view of these highlighted types in a simulation study. Anomaly detection methods are next evaluated on two datasets, related to the monitoring of helicopters in flight and to the spectrometry of construction materials namely. The benchmark analysis is concluded by recommendation guidance for practitioners.

Myrto Limnios, Nathan Noiry, Stéphan Clémençon

The ability to collect and store ever more massive databases has been accompanied by the need to process them efficiently. In many cases, most observations have the same behavior, while a probable small proportion of these observations are abnormal. Detecting the latter, defined as outliers, is one of the major challenges for machine learning applications (e.g. in fraud detection or in predictive maintenance). In this paper, we propose a methodology addressing the problem of outlier detection, by learning a data-driven scoring function defined on the feature space which reflects the degree of abnormality of the observations. This scoring function is learnt through a well-designed binary classification problem whose empirical criterion takes the form of a two-sample linear rank statistics on which theoretical results are available. We illustrate our methodology with preliminary encouraging numerical experiments.

Stephan Clémençon, Pierre Laforgue, Robin Vogel

In practice, and especially when training deep neural networks, visual recognition rules are often learned based on various sources of information. On the other hand, the recent deployment of facial recognition systems with uneven performances on different population segments has highlighted the representativeness issues induced by a naive aggregation of the datasets. In this paper, we show how biasing models can remedy these problems. Based on the (approximate) knowledge of the biasing mechanisms at work, our approach consists in reweighting the observations, so as to form a nearly debiased estimator of the target distribution. One key condition is that the supports of the biased distributions must partly overlap, and cover the support of the target distribution. In order to meet this requirement in practice, we propose to use a low dimensional image representation, shared across the image databases. Finally, we provide numerical experiments highlighting the relevance of our approach.

Guillaume Ausset, Tom Ciffreo, Francois Portier et al.

Survival analysis, or time-to-event modelling, is a classical statistical problem that has garnered a lot of interest for its practical use in epidemiology, demographics or actuarial sciences. Recent advances on the subject from the point of view of machine learning have been concerned with precise per-individual predictions instead of population studies, driven by the rise of individualized medicine. We introduce here a conditional normalizing flow based estimate of the time-to-event density as a way to model highly flexible and individualized conditional survival distributions. We use a novel hierarchical formulation of normalizing flows to enable efficient fitting of flexible conditional distributions without overfitting and show how the normalizing flow formulation can be efficiently adapted to the censored setting. We experimentally validate the proposed approach on a synthetic dataset as well as four open medical datasets and an example of a common financial problem.

Guillaume Staerman, Pavlo Mozharovskyi, Stéphan Clémençon

Because it determines a center-outward ordering of observations in $\mathbb{R}^d$ with $d\geq 2$, the concept of statistical depth permits to define quantiles and ranks for multivariate data and use them for various statistical tasks (e.g. inference, hypothesis testing). Whereas many depth functions have been proposed \textit{ad-hoc} in the literature since the seminal contribution of \cite{Tukey75}, not all of them possess the properties desirable to emulate the notion of quantile function for univariate probability distributions. In this paper, we propose an extension of the \textit{integrated rank-weighted} statistical depth (IRW depth in abbreviated form) originally introduced in \cite{IRW}, modified in order to satisfy the property of \textit{affine-invariance}, fulfilling thus all the four key axioms listed in the nomenclature elaborated by \cite{ZuoS00a}. The variant we propose, referred to as the Affine-Invariant IRW depth (AI-IRW in short), involves the covariance/precision matrices of the (supposedly square integrable) $d$-dimensional random vector $X$ under study, in order to take into account the directions along which $X$ is most variable to assign a depth value to any point $x\in \mathbb{R}^d$. The accuracy of the sampling version of the AI-IRW depth is investigated from a nonasymptotic perspective. Namely, a concentration result for the statistical counterpart of the AI-IRW depth is proved. Beyond the theoretical analysis carried out, applications to anomaly detection are considered and numerical results are displayed, providing strong empirical evidence of the relevance of the depth function we propose here.

Stéphan Clémençon, Hamid Jalalzai, Stéphane Lhaut et al.

The angular measure on the unit sphere characterizes the first-order dependence structure of the components of a random vector in extreme regions and is defined in terms of standardized margins. Its statistical recovery is an important step in learning problems involving observations far away from the center. In the common situation that the components of the vector have different distributions, the rank transformation offers a convenient and robust way of standardizing data in order to build an empirical version of the angular measure based on the most extreme observations. However, the study of the sampling distribution of the resulting empirical angular measure is challenging. It is the purpose of the paper to establish finite-sample bounds for the maximal deviations between the empirical and true angular measures, uniformly over classes of Borel sets of controlled combinatorial complexity. The bounds are valid with high probability and, up to logarithmic factors, scale as the square root of the effective sample size. The bounds are applied to provide performance guarantees for two statistical learning procedures tailored to extreme regions of the input space and built upon the empirical angular measure: binary classification in extreme regions through empirical risk minimization and unsupervised anomaly detection through minimum-volume sets of the sphere.

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.

Corentin Larroche, Johan Mazel, Stephan Clémençon

Anomaly detection in event logs is a promising approach for intrusion detection in enterprise networks. By building a statistical model of usual activity, it aims to detect multiple kinds of malicious behavior, including stealthy tactics, techniques and procedures (TTPs) designed to evade signature-based detection systems. However, finding suitable anomaly detection methods for event logs remains an important challenge. This results from the very complex, multi-faceted nature of the data: event logs are not only combinatorial, but also temporal and heterogeneous data, thus they fit poorly in most theoretical frameworks for anomaly detection. Most previous research focuses on either one of these three aspects, building a simplified representation of the data that can be fed to standard anomaly detection algorithms. In contrast, we propose to simultaneously address all three of these characteristics through a specifically tailored statistical model. We introduce \textsc{Decades}, a \underline{d}ynamic, h\underline{e}terogeneous and \underline{c}ombinatorial model for \underline{a}nomaly \underline{d}etection in \underline{e}vent \underline{s}treams, and we demonstrate its effectiveness at detecting malicious behavior through experiments on a real dataset containing labelled red team activity. In particular, we empirically highlight the importance of handling the multiple characteristics of the data by comparing our model with state-of-the-art baselines relying on various data representations.

Guillaume Staerman, Pavlo Mozharovskyi, Pierre Colombo et al.

The design of a metric between probability distributions is a longstanding problem motivated by numerous applications in Machine Learning. Focusing on continuous probability distributions on the Euclidean space $\mathbb{R}^d$, we introduce a novel pseudo-metric between probability distributions by leveraging the extension of univariate quantiles to multivariate spaces. Data depth is a nonparametric statistical tool that measures the centrality of any element $x\in\mathbb{R}^d$ with respect to (w.r.t.) a probability distribution or a data set. It is a natural median-oriented extension of the cumulative distribution function (cdf) to the multivariate case. Thus, its upper-level sets -- the depth-trimmed regions -- give rise to a definition of multivariate quantiles. The new pseudo-metric relies on the average of the Hausdorff distance between the depth-based quantile regions w.r.t. each distribution. Its good behavior w.r.t. major transformation groups, as well as its ability to factor out translations, are depicted. Robustness, an appealing feature of this pseudo-metric, is studied through the finite sample breakdown point. Moreover, we propose an efficient approximation method with linear time complexity w.r.t. the size of the data set and its dimension. The quality of this approximation as well as the performance of the proposed approach are illustrated in numerical experiments.

Guillaume Ausset, Stephan Clémençon, François Portier

Motivated by a wide variety of applications, ranging from stochastic optimization to dimension reduction through variable selection, the problem of estimating gradients accurately is of crucial importance in statistics and learning theory. We consider here the classic regression setup, where a real valued square integrable r.v. $Y$ is to be predicted upon observing a (possibly high dimensional) random vector $X$ by means of a predictive function $f(X)$ as accurately as possible in the mean-squared sense and study a nearest-neighbour-based pointwise estimate of the gradient of the optimal predictive function, the regression function $m(x)=\mathbb{E}[Y\mid X=x]$. Under classic smoothness conditions combined with the assumption that the tails of $Y-m(X)$ are sub-Gaussian, we prove nonasymptotic bounds improving upon those obtained for alternative estimation methods. Beyond the novel theoretical results established, several illustrative numerical experiments have been carried out. The latter provide strong empirical evidence that the estimation method proposed works very well for various statistical problems involving gradient estimation, namely dimensionality reduction, stochastic gradient descent optimization and quantifying disentanglement.

Robin Vogel, Aurélien Bellet, Stephan Clémençon

Many applications of AI involve scoring individuals using a learned function of their attributes. These predictive risk scores are then used to take decisions based on whether the score exceeds a certain threshold, which may vary depending on the context. The level of delegation granted to such systems in critical applications like credit lending and medical diagnosis will heavily depend on how questions of fairness can be answered. In this paper, we study fairness for the problem of learning scoring functions from binary labeled data, a classic learning task known as bipartite ranking. We argue that the functional nature of the ROC curve, the gold standard measure of ranking accuracy in this context, leads to several ways of formulating fairness constraints. We introduce general families of fairness definitions based on the AUC and on ROC curves, and show that our ROC-based constraints can be instantiated such that classifiers obtained by thresholding the scoring function satisfy classification fairness for a desired range of thresholds. We establish generalization bounds for scoring functions learned under such constraints, design practical learning algorithms and show the relevance our approach with numerical experiments on real and synthetic data.

Robin Vogel, Mastane Achab, Stéphan Clémençon et al.

We consider statistical learning problems, when the distribution $P'$ of the training observations $Z'_1,\; \ldots,\; Z'_n$ differs from the distribution $P$ involved in the risk one seeks to minimize (referred to as the test distribution) but is still defined on the same measurable space as $P$ and dominates it. In the unrealistic case where the likelihood ratio $Φ(z)=dP/dP'(z)$ is known, one may straightforwardly extends the Empirical Risk Minimization (ERM) approach to this specific transfer learning setup using the same idea as that behind Importance Sampling, by minimizing a weighted version of the empirical risk functional computed from the 'biased' training data $Z'_i$ with weights $Φ(Z'_i)$. Although the importance function $Φ(z)$ is generally unknown in practice, we show that, in various situations frequently encountered in practice, it takes a simple form and can be directly estimated from the $Z'_i$'s and some auxiliary information on the statistical population $P$. By means of linearization techniques, we then prove that the generalization capacity of the approach aforementioned is preserved when plugging the resulting estimates of the $Φ(Z'_i)$'s into the weighted empirical risk. Beyond these theoretical guarantees, numerical results provide strong empirical evidence of the relevance of the approach promoted in this article.

Guillaume Staerman, Pavlo Mozharovskyi, Stephan Clémençon

With the ubiquity of sensors in the IoT era, statistical observations are becoming increasingly available in the form of massive (multivariate) time-series. Formulated as unsupervised anomaly detection tasks, an abundance of applications like aviation safety management, the health monitoring of complex infrastructures or fraud detection can now rely on such functional data, acquired and stored with an ever finer granularity. The concept of statistical depth, which reflects centrality of an arbitrary observation w.r.t. a statistical population may play a crucial role in this regard, anomalies corresponding to observations with 'small' depth. Supported by sound theoretical and computational developments in the recent decades, it has proven to be extremely useful, in particular in functional spaces. However, most approaches documented in the literature consist in evaluating independently the centrality of each point forming the time series and consequently exhibit a certain insensitivity to possible shape changes. In this paper, we propose a novel notion of functional depth based on the area of the convex hull of sampled curves, capturing gradual departures from centrality, even beyond the envelope of the data, in a natural fashion. We discuss practical relevance of commonly imposed axioms on functional depths and investigate which of them are satisfied by the notion of depth we promote here. Estimation and computational issues are also addressed and various numerical experiments provide empirical evidence of the relevance of the approach proposed.

Maël Chiapino, Stéphan Clémençon, Vincent Feuillard et al.

In a wide variety of situations, anomalies in the behaviour of a complex system, whose health is monitored through the observation of a random vector X = (X1,. .. , X d) valued in R d , correspond to the simultaneous occurrence of extreme values for certain subgroups $α$ $\subset$ {1,. .. , d} of variables Xj. Under the heavy-tail assumption, which is precisely appropriate for modeling these phenomena, statistical methods relying on multivariate extreme value theory have been developed in the past few years for identifying such events/subgroups. This paper exploits this approach much further by means of a novel mixture model that permits to describe the distribution of extremal observations and where the anomaly type $α$ is viewed as a latent variable. One may then take advantage of the model by assigning to any extreme point a posterior probability for each anomaly type $α$, defining implicitly a similarity measure between anomalies. It is explained at length how the latter permits to cluster extreme observations and obtain an informative planar representation of anomalies using standard graph-mining tools. The relevance and usefulness of the clustering and 2-d visual display thus designed is illustrated on simulated datasets and on real observations as well, in the aeronautics application domain.

Stephan Clémençon, Pierre Laforgue

With the deluge of digitized information in the Big Data era, massive datasets are becoming increasingly available for learning predictive models. However, in many practical situations, the poor control of the data acquisition processes may naturally jeopardize the outputs of machine learning algorithms, and selection bias issues are now the subject of much attention in the literature. The present article investigates how to extend Empirical Risk Minimization, the principal paradigm in statistical learning, when training observations are generated from biased models, i.e., from distributions that are different from that in the test/prediction stage, and absolutely continuous with respect to the latter. Precisely, we show how to build a "nearly debiased" training statistical population from biased samples and the related biasing functions, following in the footsteps of the approach originally proposed in Vardi (1985). Furthermore, we study from a nonasymptotic perspective the performance of minimizers of an empirical version of the risk computed from the statistical population thus created. Remarkably, the learning rate achieved by this procedure is of the same order as that attained in absence of selection bias. Beyond the theoretical guarantees, we also present experimental results supporting the relevance of the algorithmic approach promoted in this paper.

Stéphan Clémençon, Robin Vogel

In many situations, the choice of an adequate similarity measure or metric on the feature space dramatically determines the performance of machine learning methods. Building automatically such measures is the specific purpose of metric/similarity learning. In Vogel et al. (2018), similarity learning is formulated as a pairwise bipartite ranking problem: ideally, the larger the probability that two observations in the feature space belong to the same class (or share the same label), the higher the similarity measure between them. From this perspective, the ROC curve is an appropriate performance criterion and it is the goal of this article to extend recursive tree-based ROC optimization techniques in order to propose efficient similarity learning algorithms. The validity of such iterative partitioning procedures in the pairwise setting is established by means of results pertaining to the theory of U-processes and from a practical angle, it is discussed at length how to implement them by means of splitting rules specifically tailored to the similarity learning task. Beyond these theoretical/methodological contributions, numerical experiments are displayed and provide strong empirical evidence of the performance of the algorithmic approaches we propose.

Robin Vogel, Aurélien Bellet, Stephan Clémençon et al.

The development of cluster computing frameworks has allowed practitioners to scale out various statistical estimation and machine learning algorithms with minimal programming effort. This is especially true for machine learning problems whose objective function is nicely separable across individual data points, such as classification and regression. In contrast, statistical learning tasks involving pairs (or more generally tuples) of data points - such as metric learning, clustering or ranking do not lend themselves as easily to data-parallelism and in-memory computing. In this paper, we investigate how to balance between statistical performance and computational efficiency in such distributed tuplewise statistical problems. We first propose a simple strategy based on occasionally repartitioning data across workers between parallel computation stages, where the number of repartitioning steps rules the trade-off between accuracy and runtime. We then present some theoretical results highlighting the benefits brought by the proposed method in terms of variance reduction, and extend our results to design distributed stochastic gradient descent algorithms for tuplewise empirical risk minimization. Our results are supported by numerical experiments in pairwise statistical estimation and learning on synthetic and real-world datasets.

Guillaume Ausset, Stéphan Clémençon, François Portier

We consider the classic supervised learning problem, where a continuous non-negative random label $Y$ (i.e. a random duration) is to be predicted based upon observing a random vector $X$ valued in $\mathbb{R}^d$ with $d\geq 1$ by means of a regression rule with minimum least square error. In various applications, ranging from industrial quality control to public health through credit risk analysis for instance, training observations can be right censored, meaning that, rather than on independent copies of $(X,Y)$, statistical learning relies on a collection of $n\geq 1$ independent realizations of the triplet $(X, \; \min\{Y,\; C\},\; δ)$, where $C$ is a nonnegative r.v. with unknown distribution, modeling censorship and $δ=\mathbb{I}\{Y\leq C\}$ indicates whether the duration is right censored or not. As ignoring censorship in the risk computation may clearly lead to a severe underestimation of the target duration and jeopardize prediction, we propose to consider a plug-in estimate of the true risk based on a Kaplan-Meier estimator of the conditional survival function of the censorship $C$ given $X$, referred to as Kaplan-Meier risk, in order to perform empirical risk minimization. It is established, under mild conditions, that the learning rate of minimizers of this biased/weighted empirical risk functional is of order $O_{\mathbb{P}}(\sqrt{\log(n)/n})$ when ignoring model bias issues inherent to plug-in estimation, as can be attained in absence of censorship. Beyond theoretical results, numerical experiments are presented in order to illustrate the relevance of the approach developed.

Guillaume Staerman, Pavlo Mozharovskyi, Stephan Clémençon et al.

For the purpose of monitoring the behavior of complex infrastructures (e.g. aircrafts, transport or energy networks), high-rate sensors are deployed to capture multivariate data, generally unlabeled, in quasi continuous-time to detect quickly the occurrence of anomalies that may jeopardize the smooth operation of the system of interest. The statistical analysis of such massive data of functional nature raises many challenging methodological questions. The primary goal of this paper is to extend the popular Isolation Forest (IF) approach to Anomaly Detection, originally dedicated to finite dimensional observations, to functional data. The major difficulty lies in the wide variety of topological structures that may equip a space of functions and the great variety of patterns that may characterize abnormal curves. We address the issue of (randomly) splitting the functional space in a flexible manner in order to isolate progressively any trajectory from the others, a key ingredient to the efficiency of the algorithm. Beyond a detailed description of the algorithm, computational complexity and stability issues are investigated at length. From the scoring function measuring the degree of abnormality of an observation provided by the proposed variant of the IF algorithm, a Functional Statistical Depth function is defined and discussed as well as a multivariate functional extension. Numerical experiments provide strong empirical evidence of the accuracy of the extension proposed.

Mastane Achab, Anna Korba, Stephan Clémençon

Whereas most dimensionality reduction techniques (e.g. PCA, ICA, NMF) for multivariate data essentially rely on linear algebra to a certain extent, summarizing ranking data, viewed as realizations of a random permutation $Σ$ on a set of items indexed by $i\in \{1,\ldots,\; n\}$, is a great statistical challenge, due to the absence of vector space structure for the set of permutations $\mathfrak{S}_n$. It is the goal of this article to develop an original framework for possibly reducing the number of parameters required to describe the distribution of a statistical population composed of rankings/permutations, on the premise that the collection of items under study can be partitioned into subsets/buckets, such that, with high probability, items in a certain bucket are either all ranked higher or else all ranked lower than items in another bucket. In this context, $Σ$'s distribution can be hopefully represented in a sparse manner by a bucket distribution, i.e. a bucket ordering plus the ranking distributions within each bucket. More precisely, we introduce a dedicated distortion measure, based on a mass transportation metric, in order to quantify the accuracy of such representations. The performance of buckets minimizing an empirical version of the distortion is investigated through a rate bound analysis. Complexity penalization techniques are also considered to select the shape of a bucket order with minimum expected distortion. Beyond theoretical concepts and results, numerical experiments on real ranking data are displayed in order to provide empirical evidence of the relevance of the approach promoted.

Robin Vogel, Aurélien Bellet, Stéphan Clémençon

The performance of many machine learning techniques depends on the choice of an appropriate similarity or distance measure on the input space. Similarity learning (or metric learning) aims at building such a measure from training data so that observations with the same (resp. different) label are as close (resp. far) as possible. In this paper, similarity learning is investigated from the perspective of pairwise bipartite ranking, where the goal is to rank the elements of a database by decreasing order of the probability that they share the same label with some query data point, based on the similarity scores. A natural performance criterion in this setting is pointwise ROC optimization: maximize the true positive rate under a fixed false positive rate. We study this novel perspective on similarity learning through a rigorous probabilistic framework. The empirical version of the problem gives rise to a constrained optimization formulation involving U-statistics, for which we derive universal learning rates as well as faster rates under a noise assumption on the data distribution. We also address the large-scale setting by analyzing the effect of sampling-based approximations. Our theoretical results are supported by illustrative numerical experiments.

Pierre Laforgue, Stephan Clémençon, Florence d'Alché-Buc

This paper investigates a novel algorithmic approach to data representation based on kernel methods. Assuming that the observations lie in a Hilbert space X, the introduced Kernel Autoencoder (KAE) is the composition of mappings from vector-valued Reproducing Kernel Hilbert Spaces (vv-RKHSs) that minimizes the expected reconstruction error. Beyond a first extension of the autoencoding scheme to possibly infinite dimensional Hilbert spaces, KAE further allows to autoencode any kind of data by choosing X to be itself a RKHS. A theoretical analysis of the model is carried out, providing a generalization bound, and shedding light on its connection with Kernel Principal Component Analysis. The proposed algorithms are then detailed at length: they crucially rely on the form taken by the minimizers, revealed by a dedicated Representer Theorem. Finally, numerical experiments on both simulated data and real labeled graphs (molecules) provide empirical evidence of the KAE performances.

Mastane Achab, Stephan Clémençon, Aurélien Garivier

Originally motivated by default risk management applications, this paper investigates a novel problem, referred to as the profitable bandit problem here. At each step, an agent chooses a subset of the K possible actions. For each action chosen, she then receives the sum of a random number of rewards. Her objective is to maximize her cumulated earnings. We adapt and study three well-known strategies in this purpose, that were proved to be most efficient in other settings: kl-UCB, Bayes-UCB and Thompson Sampling. For each of them, we prove a finite time regret bound which, together with a lower bound we obtain as well, establishes asymptotic optimality. Our goal is also to compare these three strategies from a theoretical and empirical perspective both at the same time. We give simple, self-contained proofs that emphasize their similarities, as well as their differences. While both Bayesian strategies are automatically adapted to the geometry of information, the numerical experiments carried out show a slight advantage for Thompson Sampling in practice.

Stephan Clémençon, Mastane Achab

We formulate a supervised learning problem, referred to as continuous ranking, where a continuous real-valued label Y is assigned to an observable r.v. X taking its values in a feature space $\mathcal{X}$ and the goal is to order all possible observations x in $\mathcal{X}$ by means of a scoring function $s:\mathcal{X}\rightarrow \mathbb{R}$ so that s(X) and Y tend to increase or decrease together with highest probability. This problem generalizes bi/multi-partite ranking to a certain extent and the task of finding optimal scoring functions s(x) can be naturally cast as optimization of a dedicated functional criterion, called the IROC curve here, or as maximization of the Kendall $τ$ related to the pair (s(X), Y ). From the theoretical side, we describe the optimal elements of this problem and provide statistical guarantees for empirical Kendall $τ$ maximization under appropriate conditions for the class of scoring function candidates. We also propose a recursive statistical learning algorithm tailored to empirical IROC curve optimization and producing a piecewise constant scoring function that is fully described by an oriented binary tree. Preliminary numerical experiments highlight the difference in nature between regression and continuous ranking and provide strong empirical evidence of the performance of empirical optimizers of the criteria proposed.

Stephan Clémençon, Anna Korba, Eric Sibony

This article is devoted to the problem of predicting the value taken by a random permutation $Σ$, describing the preferences of an individual over a set of numbered items $\{1,\; \ldots,\; n\}$ say, based on the observation of an input/explanatory r.v. $X$ e.g. characteristics of the individual), when error is measured by the Kendall $τ$ distance. In the probabilistic formulation of the 'Learning to Order' problem we propose, which extends the framework for statistical Kemeny ranking aggregation developped in \citet{CKS17}, this boils down to recovering conditional Kemeny medians of $Σ$ given $X$ from i.i.d. training examples $(X_1, Σ_1),\; \ldots,\; (X_N, Σ_N)$. For this reason, this statistical learning problem is referred to as \textit{ranking median regression} here. Our contribution is twofold. We first propose a probabilistic theory of ranking median regression: the set of optimal elements is characterized, the performance of empirical risk minimizers is investigated in this context and situations where fast learning rates can be achieved are also exhibited. Next we introduce the concept of local consensus/median, in order to derive efficient methods for ranking median regression. The major advantage of this local learning approach lies in its close connection with the widely studied Kemeny aggregation problem. From an algorithmic perspective, this permits to build predictive rules for ranking median regression by implementing efficient techniques for (approximate) Kemeny median computations at a local level in a tractable manner. In particular, versions of $k$-nearest neighbor and tree-based methods, tailored to ranking median regression, are investigated. Accuracy of piecewise constant ranking median regression rules is studied under a specific smoothness assumption for $Σ$'s conditional distribution given $X$.

Mastane Achab, Stephan Clémençon, Aurélien Garivier et al.

This paper is devoted to the study of the max K-armed bandit problem, which consists in sequentially allocating resources in order to detect extreme values. Our contribution is twofold. We first significantly refine the analysis of the ExtremeHunter algorithm carried out in Carpentier and Valko (2014), and next propose an alternative approach, showing that, remarkably, Extreme Bandits can be reduced to a classical version of the bandit problem to a certain extent. Beyond the formal analysis, these two approaches are compared through numerical experiments.

Igor Colin, Aurélien Bellet, Joseph Salmon et al.

In decentralized networks (of sensors, connected objects, etc.), there is an important need for efficient algorithms to optimize a global cost function, for instance to learn a global model from the local data collected by each computing unit. In this paper, we address the problem of decentralized minimization of pairwise functions of the data points, where these points are distributed over the nodes of a graph defining the communication topology of the network. This general problem finds applications in ranking, distance metric learning and graph inference, among others. We propose new gossip algorithms based on dual averaging which aims at solving such problems both in synchronous and asynchronous settings. The proposed framework is flexible enough to deal with constrained and regularized variants of the optimization problem. Our theoretical analysis reveals that the proposed algorithms preserve the convergence rate of centralized dual averaging up to an additive bias term. We present numerical simulations on Area Under the ROC Curve (AUC) maximization and metric learning problems which illustrate the practical interest of our approach.

Nicolas Goix, Anne Sabourin, Stéphan Clémençon

Extremes play a special role in Anomaly Detection. Beyond inference and simulation purposes, probabilistic tools borrowed from Extreme Value Theory (EVT), such as the angular measure, can also be used to design novel statistical learning methods for Anomaly Detection/ranking. This paper proposes a new algorithm based on multivariate EVT to learn how to rank observations in a high dimensional space with respect to their degree of 'abnormality'. The procedure relies on an original dimension-reduction technique in the extreme domain that possibly produces a sparse representation of multivariate extremes and allows to gain insight into the dependence structure thereof, escaping the curse of dimensionality. The representation output by the unsupervised methodology we propose here can be combined with any Anomaly Detection technique tailored to non-extreme data. As it performs linearly with the dimension and almost linearly in the data (in O(dn log n)), it fits to large scale problems. The approach in this paper is novel in that EVT has never been used in its multivariate version in the field of Anomaly Detection. Illustrative experimental results provide strong empirical evidence of the relevance of our approach.

Igor Colin, Aurélien Bellet, Joseph Salmon et al.

Efficient and robust algorithms for decentralized estimation in networks are essential to many distributed systems. Whereas distributed estimation of sample mean statistics has been the subject of a good deal of attention, computation of $U$-statistics, relying on more expensive averaging over pairs of observations, is a less investigated area. Yet, such data functionals are essential to describe global properties of a statistical population, with important examples including Area Under the Curve, empirical variance, Gini mean difference and within-cluster point scatter. This paper proposes new synchronous and asynchronous randomized gossip algorithms which simultaneously propagate data across the network and maintain local estimates of the $U$-statistic of interest. We establish convergence rate bounds of $O(1/t)$ and $O(\log t / t)$ for the synchronous and asynchronous cases respectively, where $t$ is the number of iterations, with explicit data and network dependent terms. Beyond favorable comparisons in terms of rate analysis, numerical experiments provide empirical evidence the proposed algorithms surpasses the previously introduced approach.

Charanpal Dhanjal, Romaric Gaudel, Stephan Clemencon

In recommendation systems, one is interested in the ranking of the predicted items as opposed to other losses such as the mean squared error. Although a variety of ways to evaluate rankings exist in the literature, here we focus on the Area Under the ROC Curve (AUC) as it widely used and has a strong theoretical underpinning. In practical recommendation, only items at the top of the ranked list are presented to the users. With this in mind, we propose a class of objective functions over matrix factorisations which primarily represent a smooth surrogate for the real AUC, and in a special case we show how to prioritise the top of the list. The objectives are differentiable and optimised through a carefully designed stochastic gradient-descent-based algorithm which scales linearly with the size of the data. In the special case of square loss we show how to improve computational complexity by leveraging previously computed measures. To understand theoretically the underlying matrix factorisation approaches we study both the consistency of the loss functions with respect to AUC, and generalisation using Rademacher theory. The resulting generalisation analysis gives strong motivation for the optimisation under study. Finally, we provide computation results as to the efficacy of the proposed method using synthetic and real data.

Nicolas Goix, Anne Sabourin, Stéphan Clémençon

Capturing the dependence structure of multivariate extreme events is a major concern in many fields involving the management of risks stemming from multiple sources, e.g. portfolio monitoring, insurance, environmental risk management and anomaly detection. One convenient (non-parametric) characterization of extremal dependence in the framework of multivariate Extreme Value Theory (EVT) is the angular measure, which provides direct information about the probable 'directions' of extremes, that is, the relative contribution of each feature/coordinate of the 'largest' observations. Modeling the angular measure in high dimensional problems is a major challenge for the multivariate analysis of rare events. The present paper proposes a novel methodology aiming at exhibiting a sparsity pattern within the dependence structure of extremes. This is done by estimating the amount of mass spread by the angular measure on representative sets of directions, corresponding to specific sub-cones of $R^d\_+$. This dimension reduction technique paves the way towards scaling up existing multivariate EVT methods. Beyond a non-asymptotic study providing a theoretical validity framework for our method, we propose as a direct application a --first-- anomaly detection algorithm based on multivariate EVT. This algorithm builds a sparse 'normal profile' of extreme behaviours, to be confronted with new (possibly abnormal) extreme observations. Illustrative experimental results provide strong empirical evidence of the relevance of our approach.

Nicolas Goix, Anne Sabourin, Stéphan Clémençon

Learning how to rank multivariate unlabeled observations depending on their degree of abnormality/novelty is a crucial problem in a wide range of applications. In practice, it generally consists in building a real valued "scoring" function on the feature space so as to quantify to which extent observations should be considered as abnormal. In the 1-d situation, measurements are generally considered as "abnormal" when they are remote from central measures such as the mean or the median. Anomaly detection then relies on tail analysis of the variable of interest. Extensions to the multivariate setting are far from straightforward and it is precisely the main purpose of this paper to introduce a novel and convenient (functional) criterion for measuring the performance of a scoring function regarding the anomaly ranking task, referred to as the Excess-Mass curve (EM curve). In addition, an adaptive algorithm for building a scoring function based on unlabeled data X1 , . . . , Xn with a nearly optimal EM is proposed and is analyzed from a statistical perspective.

Stéphan Clémençon, Aurélien Bellet, Igor Colin

In a wide range of statistical learning problems such as ranking, clustering or metric learning among others, the risk is accurately estimated by $U$-statistics of degree $d\geq 1$, i.e. functionals of the training data with low variance that take the form of averages over $k$-tuples. From a computational perspective, the calculation of such statistics is highly expensive even for a moderate sample size $n$, as it requires averaging $O(n^d)$ terms. This makes learning procedures relying on the optimization of such data functionals hardly feasible in practice. It is the major goal of this paper to show that, strikingly, such empirical risks can be replaced by drastically computationally simpler Monte-Carlo estimates based on $O(n)$ terms only, usually referred to as incomplete $U$-statistics, without damaging the $O_{\mathbb{P}}(1/\sqrt{n})$ learning rate of Empirical Risk Minimization (ERM) procedures. For this purpose, we establish uniform deviation results describing the error made when approximating a $U$-process by its incomplete version under appropriate complexity assumptions. Extensions to model selection, fast rate situations and various sampling techniques are also considered, as well as an application to stochastic gradient descent for ERM. Finally, numerical examples are displayed in order to provide strong empirical evidence that the approach we promote largely surpasses more naive subsampling techniques.

Stéphan Clémençon, Patrice Bertail, Emilie Chautru et al.

In certain situations that shall be undoubtedly more and more common in the Big Data era, the datasets available are so massive that computing statistics over the full sample is hardly feasible, if not unfeasible. A natural approach in this context consists in using survey schemes and substituting the "full data" statistics with their counterparts based on the resulting random samples, of manageable size. It is the main purpose of this paper to investigate the impact of survey sampling with unequal inclusion probabilities on stochastic gradient descent-based M-estimation methods in large-scale statistical and machine-learning problems. Precisely, we prove that, in presence of some a priori information, one may significantly increase asymptotic accuracy when choosing appropriate first order inclusion probabilities, without affecting complexity. These striking results are described here by limit theorems and are also illustrated by numerical experiments.