A Revenue Function for Comparison-Based Hierarchical Clustering

Comparison-based learning addresses the problem of learning when, instead of explicit features or pairwise similarities, one only has access to comparisons of the form: \emph{Object $A$ is more similar to $B$ than to $C$.} Recently, it has been shown that, in Hierarchical Clustering, single and complete linkage can be directly implemented using only such comparisons while several algorithms have been proposed to emulate the behaviour of average linkage. Hence, finding hierarchies (or dendrograms) using only comparisons is a well understood problem. However, evaluating their meaningfulness when no ground-truth nor explicit similarities are available remains an open question. In this paper, we bridge this gap by proposing a new revenue function that allows one to measure the goodness of dendrograms using only comparisons. We show that this function is closely related to Dasgupta's cost for hierarchical clustering that uses pairwise similarities. On the theoretical side, we use the proposed revenue function to resolve the open problem of whether one can approximately recover a latent hierarchy using few triplet comparisons. On the practical side, we present principled algorithms for comparison-based hierarchical clustering based on the maximisation of the revenue and we empirically compare them with existing methods.Comment: 26 pages, 6 figures, 5 tables. Transactions on Machine Learning Research (2023

Théorie et algorithmes pour l'apprentissage de métriques à comportement contrôlé

Many Machine Learning algorithms make use of a notion of distance or similarity between examples to solve various problems such as classification, clustering or domain adaptation. Depending on the tasks considered these metrics should have different properties but manually choosing an adapted comparison function can be tedious and difficult. A natural trend is then to automatically tailor such metrics to the task at hand. This is known as Metric Learning and the goal is mainly to find the best parameters of a metric under some specific constraints. Standard approaches in this field usually focus on learning Mahalanobis distances or Bilinear similarities and one of the main limitations is that the control over the behaviour of the learned metrics is often limited. Furthermore if some theoretical works exist to justify the generalization ability of the learned models, most of the approaches do not come with such guarantees. In this thesis we propose new algorithms to learn metrics with a controlled behaviour and we put a particular emphasis on the theoretical properties of these algorithms. We propose four distinct contributions which can be separated in two parts, namely (i) controlling the metric with respect to a reference metric and (ii) controlling the underlying transformation corresponding to the learned metric. Our first contribution is a local metric learning method where the goal is to regress a distance proportional to the human perception of colors. Our approach is backed up by theoretical guarantees on the generalization ability of the learned metrics. In our second contribution we are interested in theoretically studying the interest of using a reference metric in a biased regularization term to help during the learning process. We propose to use three different theoretical frameworks allowing us to derive three different measures of goodness for the reference metric. These measures give us some insights on the impact of the reference metric on the learned one. In our third contribution we propose a metric learning algorithm where the underlying transformation is controlled. The idea is that instead of using similarity and dissimilarity constraints we associate each learning example to a so-called virtual point belonging to the output space associated with the learned metric. We theoretically show that metrics learned in this way generalize well but also that our approach is linked to a classic metric learning method based on pairs constraints. In our fourth contribution we also try to control the underlying transformation of a learned metric. However instead of considering a point-wise control we consider a global one by forcing the transformation to follow the geometrical transformation associated to an optimal transport problem. From a theoretical standpoint we propose a discussion on the link between the transformation associated with the learned metric and the transformation associated with the optimal transport problem. On a more practical side we show the interest of our approach for domain adaptation but also for a task of seamless copy in imagesDe nombreux algorithmes en Apprentissage Automatique utilisent une notion de distance ou de similarité entre les exemples pour résoudre divers problèmes tels que la classification, le partitionnement ou l'adaptation de domaine. En fonction des tâches considérées ces métriques devraient avoir des propriétés différentes mais les choisir manuellement peut-être fastidieux et difficile. Une solution naturelle est alors d'adapter automatiquement ces métriques à la tâche considérée. Il s'agit alors d'un problème connu sous le nom d'Apprentissage de Métriques et où le but est principalement de trouver les meilleurs paramètres d'une métrique respectant des contraintes spécifiques. Les approches classiques dans ce domaine se focalisent habituellement sur l'apprentissage de distances de Mahalanobis ou de similarités bilinéaires et l'une des principales limitations est le fait que le contrôle du comportement de ces métriques est souvent limité. De plus, si des travaux théoriques existent pour justifier de la capacité de généralisation des modèles appris, la plupart des approches ne présentent pas de telles garanties. Dans cette thèse nous proposons de nouveaux algorithmes pour apprendre des métriques à comportement contrôlé et nous mettons l'accent sur les propriétés théoriques de ceux-ci. Nous proposons quatre contributions distinctes qui peuvent être séparées en deux parties: (i) contrôler la métrique apprise en utilisant une métrique de référence et (ii) contrôler la transformation induite par la métrique apprise. Notre première contribution est une approche locale d'apprentissage de métriques où le but est de régresser une distance proportionnelle à la perception humaine des couleurs. Notre approche est justifiée théoriquement par des garanties en généralisation sur les métriques apprises. Dans notre deuxième contribution nous nous sommes intéressés à l'analyse théorique de l'intérêt d'utiliser une métrique de référence dans un terme de régularisation biaisé pour aider lors du processus d'apprentissage. Nous proposons d'utiliser trois cadres théoriques différents qui nous permettent de dériver trois mesures différentes de l'apport de la métrique de référence. Ces mesures nous donnent un aperçu de l'impact de la métrique de référence sur celle apprise. Dans notre troisième contribution nous proposons un algorithme d'apprentissage de métriques où la transformation induite est contrôlée. L'idée est que, plutôt que d'utiliser des contraintes de similarité et de dissimilarité, chaque exemple est associé à un point virtuel qui appartient déjà à l'espace induit par la métrique apprise. D'un point de vue théorique nous montrons que les métriques apprises de cette façon généralisent bien mais aussi que notre approche est liée à une méthode plus classique d'apprentissage de métriques basée sur des contraintes de paires. Dans notre quatrième contribution nous essayons aussi de contrôler la transformation induite par une métrique apprise. Cependant, plutôt que considérer un contrôle individuel pour chaque exemple, nous proposons une approche plus globale en forçant la transformation à suivre une transformation géométrique associée à un problème de transport optimal. D'un point de vue théorique nous proposons une discussion sur le lien entre la transformation associée à la métrique apprise et la transformation associée au problème de transport optimal. D'un point de vue plus pratique nous montrons l'intérêt de notre approche pour l'adaptation de domaine mais aussi pour l'édition d'image

FairGrad: Fairness Aware Gradient Descent

We tackle the problem of group fairness in classification, where the objective is to learn models that do not unjustly discriminate against subgroups of the population. Most existing approaches are limited to simple binary tasks or involve difficult to implement training mechanisms. This reduces their practical applicability. In this paper, we propose FairGrad, a method to enforce fairness based on a reweighting scheme that iteratively learns group specific weights based on whether they are advantaged or not. FairGrad is easy to implement and can accommodate various standard fairness definitions. Furthermore, we show that it is comparable to standard baselines over various datasets including ones used in natural language processing and computer vision

Regressive Virtual Metric Learning

International audienceWe are interested in supervised metric learning of Mahalanobis like distances. Existing approaches mainly focus on learning a new distance using similarity and dissimilarity constraints between examples. In this paper, instead of bringing closer examples of the same class and pushing far away examples of different classes we propose to move the examples with respect to virtual points. Hence, each example is brought closer to a a priori defined virtual point reducing the number of constraints to satisfy. We show that our approach admits a closed form solution which can be kernelized. We provide a theoretical analysis showing the consistency of the approach and establishing some links with other classical metric learning methods. Furthermore we propose an efficient solution to the difficult problem of selecting virtual points based in part on recent works in optimal transport. Lastly, we evaluate our approach on several state of the art datasets

A Theoretical Analysis of Metric Hypothesis Transfer Learning

International audienceWe consider the problem of transferring some a priori knowledge in the context of supervised metric learning approaches. While this setting has been successfully applied in some empirical contexts, no theoretical evidence exists to justify this approach. In this paper, we provide a theoretical justification based on the notion of algorithmic stability adapted to the regularized metric learning setting. We propose an on-average-replace-two-stability model allowing us to prove fast generalization rates when an auxiliary source metric is used to bias the regularizer. Moreover, we prove a consistency result from which we show the interest of considering biased weighted regularized formulations and we provide a solution to estimate the associated weight. We also present some experiments illustrating the interest of the approach in standard metric learning tasks and in a transfer learning problem where few labelled data are available

Bornes en Généralisation à Convergence Rapide pour le Transfert d'Hypothèses en Apprentissage de Métriques

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Entre ambition « tout-terrain » et impossible ubiquité : les ethnographes en mouvement

La pratique ethnographique est marquée par l’implication des chercheur·e·s et le partage d’un lieu avec des enqueté·e·s, pendant une durée prolongée. Cette manière de faire du terrain, en sciences sociales, est aujourd’hui questionnée par certaines transformations de la méthode, enrichie par les interrogations sur le rôle de l’ethnographe, la délimitation spatiale et temporelle de son travail, ou encore sur les conséquences sociales et politiques de ses observations. Dans un contexte..

A Revenue Function for Comparison-Based Hierarchical Clustering

25 pages, 5 figures, 7 tablesComparison-based learning addresses the problem of learning when, instead of explicit features or pairwise similarities, one only has access to comparisons of the form: \emph{Object $A$ is more similar to $B$ than to $C$.} Recently, it has been shown that, in Hierarchical Clustering, single and complete linkage can be directly implemented using only such comparisons while several algorithms have been proposed to emulate the behaviour of average linkage. Hence, finding hierarchies (or dendrograms) using only comparisons is a well understood problem. However, evaluating their meaningfulness when no ground-truth nor explicit similarities are available remains an open question. In this paper, we bridge this gap by proposing a new revenue function that allows one to measure the goodness of dendrograms using only comparisons. We show that this function is closely related to Dasgupta's cost for hierarchical clustering that uses pairwise similarities. On the theoretical side, we use the proposed revenue function to resolve the open problem of whether one can approximately recover a latent hierarchy using few triplet comparisons. On the practical side, we present principled algorithms for comparison-based hierarchical clustering based on the maximisation of the revenue and we empirically compare them with existing methods