Exploiting random projections and sparsity with random forests and gradient boosting methods - Application to multi-label and multi-output learning, random forest model compression and leveraging input sparsity

Within machine learning, the supervised learning field aims at modeling the input-output relationship of a system, from past observations of its behavior. Decision trees characterize the input-output relationship through a series of nested ``if-then-else'' questions, the testing nodes, leading to a set of predictions, the leaf nodes. Several of such trees are often combined together for state-of-the-art performance: random forest ensembles average the predictions of randomized decision trees trained independently in parallel, while tree boosting ensembles train decision trees sequentially to refine the predictions made by the previous ones. The emergence of new applications requires scalable supervised learning algorithms in terms of computational power and memory space with respect to the number of inputs, outputs, and observations without sacrificing accuracy. In this thesis, we identify three main areas where decision tree methods could be improved for which we provide and evaluate original algorithmic solutions: (i) learning over high dimensional output spaces, (ii) learning with large sample datasets and stringent memory constraints at prediction time and (iii) learning over high dimensional sparse input spaces. A first approach to solve learning tasks with a high dimensional output space, called binary relevance or single target, is to train one decision tree ensemble per output. However, it completely neglects the potential correlations existing between the outputs. An alternative approach called multi-output decision trees fits a single decision tree ensemble targeting simultaneously all the outputs, assuming that all outputs are correlated. Nevertheless, both approaches have (i) exactly the same computational complexity and (ii) target extreme output correlation structures. In our first contribution, we show how to combine random projection of the output space, a dimensionality reduction method, with the random forest algorithm decreasing the learning time complexity. The accuracy is preserved, and may even be improved by reaching a different bias-variance tradeoff. In our second contribution, we first formally adapt the gradient boosting ensemble method to multi-output supervised learning tasks such as multi-output regression and multi-label classification. We then propose to combine single random projections of the output space with gradient boosting on such tasks to adapt automatically to the output correlation structure. The random forest algorithm often generates large ensembles of complex models thanks to the availability of a large number of observations. However, the space complexity of such models, proportional to their total number of nodes, is often prohibitive, and therefore these modes are not well suited under stringent memory constraints at prediction time. In our third contribution, we propose to compress these ensembles by solving a L1-based regularization problem over the set of indicator functions defined by all their nodes. Some supervised learning tasks have a high dimensional but sparse input space, where each observation has only a few of the input variables that have non zero values. Standard decision tree implementations are not well adapted to treat sparse input spaces, unlike other supervised learning techniques such as support vector machines or linear models. In our fourth contribution, we show how to exploit algorithmically the input space sparsity within decision tree methods. Our implementation yields a significant speed up both on synthetic and real datasets, while leading to exactly the same model. It also reduces the required memory to grow such models by exploiting sparse instead of dense memory storage for the input matrix.Parmi les techniques d'apprentissage automatique, l'apprentissage supervisé vise à modéliser les relations entrée-sortie d'un système, à partir d'observations de son fonctionnement. Les arbres de décision caractérisent cette relation entrée-sortie à partir d'un ensemble hiérarchique de questions appelées les noeuds tests amenant à une prédiction, les noeuds feuilles. Plusieurs de ces arbres sont souvent combinés ensemble afin d'atteindre les performances de l'état de l'art: les ensembles de forêts aléatoires calculent la moyenne des prédictions d'arbres de décision randomisés, entraînés indépendamment et en parallèle alors que les ensembles d'arbres de boosting entraînent des arbres de décision séquentiellement, améliorant ainsi les prédictions faites par les précédents modèles de l'ensemble. L'apparition de nouvelles applications requiert des algorithmes d'apprentissage supervisé efficaces en terme de puissance de calcul et d'espace mémoire par rapport au nombre d'entrées, de sorties, et d'observations sans sacrifier la précision du modèle. Dans cette thèse, nous avons identifié trois domaines principaux où les méthodes d'arbres de décision peuvent être améliorées pour lequel nous fournissons et évaluons des solutions algorithmiques originales: (i) apprentissage sur des espaces de sortie de haute dimension, (ii) apprentissage avec de grands ensembles d'échantillons et des contraintes mémoires strictes au moment de la prédiction et (iii) apprentissage sur des espaces d'entrée creux de haute dimension. Une première approche pour résoudre des tâches d'apprentissage avec un espace de sortie de haute dimension, appelée "binary relevance" ou "single target", est l’apprentissage d’un ensemble d'arbres de décision par sortie. Toutefois, cette approche néglige complètement les corrélations potentiellement existantes entre les sorties. Une approche alternative, appelée "arbre de décision multi-sorties", est l’apprentissage d’un seul ensemble d'arbres de décision pour toutes les sorties, faisant l'hypothèse que toutes les sorties sont corrélées. Cependant, les deux approches ont (i) exactement la même complexité en temps de calcul et (ii) visent des structures de corrélation de sorties extrêmes. Dans notre première contribution, nous montrons comment combiner des projections aléatoires (une méthode de réduction de dimensionnalité) de l'espace de sortie avec l'algorithme des forêts aléatoires diminuant la complexité en temps de calcul de la phase d'apprentissage. La précision est préservée, et peut même être améliorée en atteignant un compromis biais-variance différent. Dans notre seconde contribution, nous adaptons d'abord formellement la méthode d'ensemble "gradient boosting" à la régression multi-sorties et à la classification multi-labels. Nous proposons ensuite de combiner une seule projection aléatoire de l'espace de sortie avec l’algorithme de "gradient boosting" sur de telles tâches afin de s'adapter automatiquement à la structure des corrélations existant entre les sorties. Les algorithmes de forêts aléatoires génèrent souvent de grands ensembles de modèles complexes grâce à la disponibilité d'un grand nombre d'observations. Toutefois, la complexité mémoire, proportionnelle au nombre total de noeuds, de tels modèles est souvent prohibitive, et donc ces modèles ne sont pas adaptés à des contraintes mémoires fortes lors de la phase de prédiction. Dans notre troisième contribution, nous proposons de compresser ces ensembles en résolvant un problème de régularisation basé sur la norme L1 sur l'ensemble des fonctions indicatrices défini par tous leurs noeuds. Certaines tâches d'apprentissage supervisé ont un espace d'entrée de haute dimension mais creux, où chaque observation possède seulement quelques variables d'entrée avec une valeur non-nulle. Les implémentations standards des arbres de décision ne sont pas adaptées pour traiter des espaces d'entrée creux, contrairement à d'autres techniques d'apprentissage supervisé telles que les machines à vecteurs de support ou les modèles linéaires. Dans notre quatrième contribution, nous montrons comment exploiter algorithmiquement le creux de l'espace d'entrée avec les méthodes d'arbres de décision. Notre implémentation diminue significativement le temps de calcul sur des ensembles de données synthétiques et réelles, tout en fournissant exactement le même modèle. Cela permet aussi de réduire la mémoire nécessaire pour apprendre de tels modèles en exploitant des méthodes de stockage appropriées pour la matrice des entrées

Boosted Feature Generation for Classification Problems Involving High Numbers of Inputs and Classes

Classification problems involving high numbers of inputs and classes play an important role in the field of machine learning. Image classification, in particular, is a very active field of research with numerous applications. In addition to their high number, inputs of image classification problems often show significant correlation. Also, in proportion to the number of inputs, the number of available training samples is usually low. Therefore techniques combining low susceptibility to overfitting with good classification performance have to be found. Since for many tasks data has to be processed in real time, computational efficiency is crucial as well. Boosting is a machine learning technique, which is used successfully in a number of application areas, in particular in the field of machine vision. Due to it's modular design and flexibility, Boosting can be adapted to new problems easily. In addition, techniques for optimizing classifiers produced by Boosting with respect to computational efficiency exist. Boosting builds linear ensembles of base classifiers in a stage-wise fashion. Sample-weights reflect whether training samples are hard-to-classify or not. Therefore Boosting is able to adapt to the given classification problem over the course of training. The present work deals with the design of techniques for adapting Boosting to problems involving high numbers of inputs and classes. In the first part, application of Boosting to multi-class problems is analyzed. After giving an overview of existing approaches, a new formulation for base-classifiers solving multi-class problems by splitting them into pair-wise binary subproblems is presented. Experimental evaluation shows the good performance and computational efficiency of the proposed technique compared to state-of-the-art techniques. In the second part of the work, techniques that use Boosting for feature generation are presented. These techniques use the distribution of sample weights, produced by Boosting, to learn features that are adapted to the problems solved in each Boosting stage. By using smoothing-spline base classifiers, gradient descent schemes can be incorporated to find features that minimize the cost function of the current base classifier. Experimental evaluation shows, that Boosting with linear projective features significantly outperforms state-of-the-art approaches like e.g. SVM and Random Forests. In order to be applicable to image classification problems, the presented feature generation scheme is extended to produce shift-invariant features. The utilized features are inspired by the features used in Convolutional Neural Networks and perform a combination of convolution and subsampling. Experimental evaluation for classification of handwritten digits and car side-views shows that the proposed system is competitive to the best published results. The presented scheme has the advantages of being very simple and involving a low number of design parameters only

Totally Corrective Multiclass Boosting with Binary Weak Learners

In this work, we propose a new optimization framework for multiclass boosting learning. In the literature, AdaBoost.MO and AdaBoost.ECC are the two successful multiclass boosting algorithms, which can use binary weak learners. We explicitly derive these two algorithms' Lagrange dual problems based on their regularized loss functions. We show that the Lagrange dual formulations enable us to design totally-corrective multiclass algorithms by using the primal-dual optimization technique. Experiments on benchmark data sets suggest that our multiclass boosting can achieve a comparable generalization capability with state-of-the-art, but the convergence speed is much faster than stage-wise gradient descent boosting. In other words, the new totally corrective algorithms can maximize the margin more aggressively.Comment: 11 page

Approximation and Relaxation Approaches for Parallel and Distributed Machine Learning

Large scale machine learning requires tradeoffs. Commonly this tradeoff has led practitioners to choose simpler, less powerful models, e.g. linear models, in order to process more training examples in a limited time. In this work, we introduce parallelism to the training of non-linear models by leveraging a different tradeoff--approximation. We demonstrate various techniques by which non-linear models can be made amenable to larger data sets and significantly more training parallelism by strategically introducing approximation in certain optimization steps. For gradient boosted regression tree ensembles, we replace precise selection of tree splits with a coarse-grained, approximate split selection, yielding both faster sequential training and a significant increase in parallelism, in the distributed setting in particular. For metric learning with nearest neighbor classification, rather than explicitly train a neighborhood structure we leverage the implicit neighborhood structure induced by task-specific random forest classifiers, yielding a highly parallel method for metric learning. For support vector machines, we follow existing work to learn a reduced basis set with extremely high parallelism, particularly on GPUs, via existing linear algebra libraries. We believe these optimization tradeoffs are widely applicable wherever machine learning is put in practice in large scale settings. By carefully introducing approximation, we also introduce significantly higher parallelism and consequently can process more training examples for more iterations than competing exact methods. While seemingly learning the model with less precision, this tradeoff often yields noticeably higher accuracy under a restricted training time budget