This paper develops a theory for group Lasso using a concept called strong group sparsity. Our result shows that group Lasso is superior to standard Lasso for strongly group-sparse signals. This provides a convincing theoretical justification for using group sparse regularization when the underlying group structure is consistent with the data. Moreover, the theory predicts some limitations of the group Lasso formulation that are confirmed by simulation studies.
Finally, we shall mention that independent of the authors, results similar to those presented in this paper have also been obtained in @cite_1 with a similar technical analysis. However, while our paper studies the general group Lasso formulation, only the special case of multi-task learning is considered in @cite_1 .
@cite_1
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The Group-Lasso is a well-known tool for joint regularization in machine learning methods. While the l1,2 and the l1,∞ version have been studied in detail and efficient algorithms exist, there are still open questions regarding other l1,p variants. We characterize conditions for solutions of the l1,p Group-Lasso for all p-norms with 1 ≤ p ≤ ∞, and we present a unified active set algorithm. For all p-norms, a highly efficient projected gradient algorithm is presented. This new algorithm enables us to compare the prediction performance of many variants of the Group-Lasso in a multi-task learning setting, where the aim is to solve many learning problems in parallel which are coupled via the Group-Lasso constraint. We conduct large-scale experiments on synthetic data and on two real-world data sets. In accordance with theoretical characterizations of the different norms we observe that the weak-coupling norms with p between 1.5 and 2 consistently outperform the strong-coupling norms with p ≫ 2. We propose a novel group regularization which we call exclusive lasso. Unlike the group lasso regularizer that assumes covarying variables in groups, the proposed exclusive lasso regularizer models the scenario when variables in the same group compete with each other. Analysis is presented to illustrate the properties of the proposed regularizer. We present a framework of kernel based multi-task feature selection algorithm based on the proposed exclusive lasso regularizer. An efficient algorithm is derived to solve the related optimization problem. Experiments with document categorization show that our approach outperforms state-of-theart algorithms for multi-task feature selection. We consider a special type of multi-label learning where class assignments of training examples are incomplete. As an example, an instance whose true class assignment is (c 1 , c 2 , c 3 ) is only assigned to class c 1 when it is used as a training sample. We refer to this problem as multi-label learning with incomplete class assignment. Incompletely labeled data is frequently encountered when the number of classes is very large (hundreds as in MIR Flickr dataset) or when there is a large ambiguity between classes (e.g., jet vs plane). In both cases, it is difficult for users to provide complete class assignments for objects. We propose a ranking based multi-label learning framework that explicitly addresses the challenge of learning from incompletely labeled data by exploiting the group lasso technique to combine the ranking errors. We present a learning algorithm that is empirically shown to be efficient for solving the related optimization problem. Our empirical study shows that the proposed framework is more effective than the state-of-the-art algorithms for multi-label learning in dealing with incompletely labeled data. We consider the problem of learning a structured multi-task regression, where the output consists of multiple responses that are related by a graph and the correlated response variables are dependent on the common inputs in a sparse but synergistic manner. Previous methods such as l1 l2-regularized multi-task regression assume that all of the output variables are equally related to the inputs, although in many real-world problems, outputs are related in a complex manner. In this paper, we propose graph-guided fused lasso (GFlasso) for structured multi-task regression that exploits the graph structure over the output variables. We introduce a novel penalty function based on fusion penalty to encourage highly correlated outputs to share a common set of relevant inputs. In addition, we propose a simple yet efficient proximal-gradient method for optimizing GFlasso that can also be applied to any optimization problems with a convex smooth loss and the general class of fusion penalty defined on arbitrary graph structures. By exploiting the structure of the non-smooth ''fusion penalty'', our method achieves a faster convergence rate than the standard first-order method, sub-gradient method, and is significantly more scalable than the widely adopted second-order cone-programming and quadratic-programming formulations. In addition, we provide an analysis of the consistency property of the GFlasso model. Experimental results not only demonstrate the superiority of GFlasso over the standard lasso but also show the efficiency and scalability of our proximal-gradient method. The Group-Lasso method for finding important explanatory factors suffers from the potential non-uniqueness of solutions and also from high computational costs. We formulate conditions for the uniqueness of Group-Lasso solutions which lead to an easily implementable test procedure that allows us to identify all potentially active groups. These results are used to derive an efficient algorithm that can deal with input dimensions in the millions and can approximate the solution path efficiently. The derived methods are applied to large-scale learning problems where they exhibit excellent performance and where the testing procedure helps to avoid misinterpretations of the solutions. In multi-task learning (MTL), multiple tasks are learnt jointly. A major assumption for this paradigm is that all those tasks are indeed related so that the joint training is appropriate and beneficial. In this paper, we study the problem of multi-task learning of shared feature representations among tasks, while simultaneously determining "with whom" each task should share. We formulate the problem as a mixed integer programming and provide an alternating minimization technique to solve the optimization problem of jointly identifying grouping structures and parameters. The algorithm mono-tonically decreases the objective function and converges to a local optimum. Compared to the standard MTL paradigm where all tasks are in a single group, our algorithm improves its performance with statistical significance for three out of the four datasets we have studied. We also demonstrate its advantage over other task grouping techniques investigated in literature. We develop a novel online learning algorithm for the group lasso in order to efficiently find the important explanatory factors in a grouped manner. Different from traditional batch-mode group lasso algorithms, which suffer from the inefficiency and poor scalability, our proposed algorithm performs in an online mode and scales well: at each iteration one can update the weight vector according to a closed-form solution based on the average of previous subgradients. Therefore, the proposed online algorithm can be very efficient and scalable. This is guaranteed by its low worst-case time complexity and memory cost both in the order of O(d), where d is the number of dimensions. Moreover, in order to achieve more sparsity in both the group level and the individual feature level, we successively extend our online system to efficiently solve a number of variants of sparse group lasso models. We also show that the online system is applicable to other group lasso models, such as the group lasso with overlap and graph lasso. Finally, we demonstrate the merits of our algorithm by experimenting with both synthetic and real-world datasets. We consider the problem of learning a coefficient vector xο ∈ RN from noisy linear observation y = Axo + ∈ Rn. In many contexts (ranging from model selection to image processing), it is desirable to construct a sparse estimator x. In this case, a popular approach consists in solving an l1-penalized least-squares problem known as the LASSO or basis pursuit denoising. For sequences of matrices A of increasing dimensions, with independent Gaussian entries, we prove that the normalized risk of the LASSO converges to a limit, and we obtain an explicit expression for this limit. Our result is the first rigorous derivation of an explicit formula for the asymptotic mean square error of the LASSO for random instances. The proof technique is based on the analysis of AMP, a recently developed efficient algorithm, that is inspired from graphical model ideas. Simulations on real data matrices suggest that our results can be relevant in a broad array of practical applications. We study the problem of estimating multiple linear regression equations for the purpose of both prediction and variable selection. Following recent work on multi-task learning [2008], we assume that the regression vectors share the same sparsity pattern. This means that the set of relevant predictor variables is the same across the different equations. This assumption leads us to consider the Group Lasso as a candidate estimation method. We show that this estimator enjoys nice sparsity oracle inequalities and variable selection properties. The results hold under a certain restricted eigenvalue condition and a coherence condition on the design matrix, which naturally extend recent work in [2007], Lounici [2008]. In particular, in the multi-task learning scenario, in which the number of tasks can grow, we are able to remove completely the effect of the number of predictor variables in the bounds. Finally, we show how our results can be extended to more general noise distributions, of which we only require the variance to be finite. In multi-instance learning, the training set comprises labeled bags that are composed of unlabeled instances, and the task is to predict the labels of unseen bags. In this paper, a web mining problem, i.e. web index recommendation, is investigated from a multi-instance view. In detail, each web index page is regarded as a bag, while each of its linked pages is regarded as an instance. A user favoring an index page means that he or she is interested in at least one page linked by the index. Based on the browsing history of the user, recommendation could be provided for unseen index pages. An algorithm named Fretcit-kNN, which employs the Minimal Hausdorff distance between frequent term sets and utilizes both the references and citers of an unseen bag in determining its label, is proposed to solve the problem. Experiments show that in average the recommendation accuracy of Fretcit-kNN is 81.0 with 71.7 recall and 70.9 precision, which is significantly better than the best algorithm that does not consider the specific characteristics of multi-instance learning, whose performance is 76.3 accuracy with 63.4 recall and 66.1 precision. This paper suggests that it may be easier to learn several hard tasks at one time than to learn these same tasks separately. In effect, the information provided by the training signal for each task serves as a domain-specific inductive bias for the other tasks. Frequently the world gives us clusters of related tasks to learn. When it does not, it is often straightforward to create additional tasks. For many domains, acquiring inductive bias by collecting additional teaching signal may be more practical than the traditional approach of codifying domain-specific biases acquired from human expertise. We call this approach Multitask Learning (MTL). Since much of the power of an inductive learner follows directly from its inductive bias, multitask learning may yield more powerful learning. An empirical example of multitask connectionist learning is presented where learning improves by training one network on several related tasks at the same time. Multitask decision tree induction is also outlined. In this paper, we address the challenging task of simultaneously optimizing (i) the weights of a neural network, (ii) the number of neurons for each hidden layer, and (iii) the subset of active input features (i.e., feature selection). While these problems are traditionally dealt with separately, we propose an efficient regularized formulation enabling their simultaneous parallel execution, using standard optimization routines. Specifically, we extend the group Lasso penalty, originally proposed in the linear regression literature, to impose group-level sparsity on the networks connections, where each group is defined as the set of outgoing weights from a unit. Depending on the specific case, the weights can be related to an input variable, to a hidden neuron, or to a bias unit, thus performing simultaneously all the aforementioned tasks in order to obtain a compact network. We carry out an extensive experimental evaluation, in comparison with classical weight decay and Lasso penalties, both on a toy dataset for handwritten digit recognition, and multiple realistic mid-scale classification benchmarks. Comparative results demonstrate the potential of our proposed sparse group Lasso penalty in producing extremely compact networks, with a significantly lower number of input features, with a classification accuracy which is equal or only slightly inferior to standard regularization terms. This paper develops a theory for group Lasso using a concept called strong group sparsity. Our result shows that group Lasso is superior to standard Lasso for strongly group-sparse signals. This provides a convincing theoretical justification for using group sparse regularization when the underlying group structure is consistent with the data. Moreover, the theory predicts some limitations of the group Lasso formulation that are confirmed by simulation studies. We consider the problem of using a large unlabeled sample to boost performance of a learning algorit,hrn when only a small set of labeled examples is available. In particular, we consider a problem setting motivated by the task of learning to classify web pages, in which the description of each example can be partitioned into two distinct views. For example, the description of a web page can be partitioned into the words occurring on that page, and the words occurring in hyperlinks t,hat point to that page. We assume that either view of the example would be sufficient for learning if we had enough labeled data, but our goal is to use both views together to allow inexpensive unlabeled data to augment, a much smaller set of labeled examples. Specifically, the presence of two distinct views of each example suggests strategies in which two learning algorithms are trained separately on each view, and then each algorithm’s predictions on new unlabeled examples are used to enlarge the training set of the other. Our goal in this paper is to provide a PAC-style analysis for this setting, and, more broadly, a PAC-style framework for the general problem of learning from both labeled and unlabeled data. We also provide empirical results on real web-page data indicating that this use of unlabeled examples can lead to significant improvement of hypotheses in practice. *This research was supported in part by the DARPA HPKB program under contract F30602-97-1-0215 and by NSF National Young investigator grant CCR-9357793. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. TO copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and or a fee. COLT 98 Madison WI USA Copyright ACM 1998 l-58113-057--0 98 7... 5.00 92 Tom Mitchell School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213-3891 mitchell+@cs.cmu.edu We investigate the relation of two fundamental tools in machine learning and signal processing, that is the support vector machine (SVM) for classification, and the Lasso technique used in regression. We show that the resulting optimization problems are equivalent, in the following sense. Given any instance of an L2-loss soft-margin (or hard-margin) SVM, we construct a Lasso instance having the same optimal solutions, and vice versa. As a consequence, many existing optimization algorithms for both SVMs and Lasso can also be applied to the respective other problem instances. Also, the equivalence allows for many known theoretical insights for SVM and Lasso to be translated between the two settings. One such implication gives a simple kernelized version of the Lasso, analogous to the kernels used in the SVM setting. Another consequence is that the sparsity of a Lasso solution is equal to the number of support vectors for the corresponding SVM instance, and that one can use screening rules to prune the set of support vectors. Furthermore, we can relate sublinear time algorithms for the two problems, and give a new such algorithm variant for the Lasso. We also study the regularization paths for both methods. Visual attributes, from simple objects (e.g., backpacks, hats) to soft-biometrics (e.g., gender, height, clothing) have proven to be a powerful representational approach for many applications such as image description and human identification. In this paper, we introduce a novel method to combine the advantages of both multi-task and curriculum learning in a visual attribute classification framework. Individual tasks are grouped based on their correlation so that two groups of strongly and weakly correlated tasks are formed. The two groups of tasks are learned in a curriculum learning setup by transferring the acquired knowledge from the strongly to the weakly correlated. The learning process within each group though, is performed in a multi-task classification setup. The proposed method learns better and converges faster than learning all the tasks in a typical multi-task learning paradigm. We demonstrate the effectiveness of our approach on the publicly available, SoBiR, VIPeR and PETA datasets and report state-of-the-art results across the board. Multi-task learning (MTL) aims to improve the performance of multiple related tasks by exploiting the intrinsic relationships among them. Recently, multi-task feature learning algorithms have received increasing attention and they have been successfully applied to many applications involving high dimensional data. However, they assume that all tasks share a common set of features, which is too restrictive and may not hold in real-world applications, since outlier tasks often exist. In this paper, we propose a Robust Multi-Task Feature Learning algorithm (rMTFL) which simultaneously captures a common set of features among relevant tasks and identifies outlier tasks. Specifically, we decompose the weight (model) matrix for all tasks into two components. We impose the well-known group Lasso penalty on row groups of the first component for capturing the shared features among relevant tasks. To simultaneously identify the outlier tasks, we impose the same group Lasso penalty but on column groups of the second component. We propose to employ the accelerated gradient descent to efficiently solve the optimization problem in rMTFL, and show that the proposed algorithm is scalable to large-size problems. In addition, we provide a detailed theoretical analysis on the proposed rMTFL formulation. Specifically, we present a theoretical bound to measure how well our proposed rMTFL approximates the true evaluation, and provide bounds to measure the error between the estimated weights of rMTFL and the underlying true weights. Moreover, by assuming that the underlying true weights are above the noise level, we present a sound theoretical result to show how to obtain the underlying true shared features and outlier tasks (sparsity patterns). Empirical studies on both synthetic and real-world data demonstrate that our proposed rMTFL is capable of simultaneously capturing shared features among tasks and identifying outlier tasks. We consider the least-square linear regression problem with regularization by the l1-norm, a problem usually referred to as the Lasso. In this paper, we present a detailed asymptotic analysis of model consistency of the Lasso. For various decays of the regularization parameter, we compute asymptotic equivalents of the probability of correct model selection (i.e., variable selection). For a specific rate decay, we show that the Lasso selects all the variables that should enter the model with probability tending to one exponentially fast, while it selects all other variables with strictly positive probability. We show that this property implies that if we run the Lasso for several bootstrapped replications of a given sample, then intersecting the supports of the Lasso bootstrap estimates leads to consistent model selection. This novel variable selection algorithm, referred to as the Bolasso, is compared favorably to other linear regression methods on synthetic data and datasets from the UCI machine learning repository. Learning multiple related tasks from data simultaneously can improve predictive performance relative to learning these tasks independently. In this paper we propose a novel multi-task learning algorithm called MT-Adaboost: it extends Adaboost algorithm Freund1999Short to the multi-task setting, it uses as multi-task weak classifier a multi-task decision stump. This allows to learn different dependencies between tasks for different regions of the learning space. Thus, we relax the conventional hypothesis that tasks behave similarly in the whole learning space. Moreover, MT-Adaboost can learn multiple tasks without imposing the constraint of sharing the same label set and or examples between tasks. A theoretical analysis is derived from the analysis of the original Adaboost. Experiments for multiple tasks over large scale textual data sets with social context (Enron and Tobacco) give rise to very promising results. The Lasso is one of the most popular methods in high dimensional statistical learning. Most existing theoretical results for the Lasso, however, require the samples to be iid. Recent work has provided guarantees for the Lasso assuming that the time series is generated by a sparse Vector Auto-Regressive (VAR) model with Gaussian innovations. Proofs of these results rely critically on the fact that the true data generating mechanism (DGM) is a finite-order Gaussian VAR. This assumption is quite brittle: linear transformations, including selecting a subset of variables, can lead to the violation of this assumption. In order to break free from such assumptions, we derive nonasymptotic inequalities for estimation error and prediction error of the Lasso estimate of the best linear predictor without assuming any special parametric form of the DGM. Instead, we rely only on (strict) stationarity and mixing conditions to establish consistency of the Lasso in the following two scenarios: (a) alpha-mixing Gaussian processes, and (b) beta-mixing sub-Gaussian random vectors. Our work provides an alternative proof of the consistency of the Lasso for sparse Gaussian VAR models. But the applicability of our results extends to non-Gaussian and non-linear times series models as the examples we provide demonstrate. In order to prove our results, we derive a novel Hanson-Wright type concentration inequality for beta-mixing sub-Gaussian random vectors that may be of independent interest.