This paper studies a difficult and fundamental problem that arises throughout electrical engineering, applied mathematics, and statistics. Suppose that one forms a short linear combination of elementary signals drawn from a large, fixed collection. Given an observation of the linear combination that has been contaminated with additive noise, the goal is to identify which elementary signals participated and to approximate their coefficients. Although many algorithms have been proposed, there is little theory which guarantees that these algorithms can accurately and efficiently solve the problem. This paper studies a method called convex relaxation, which attempts to recover the ideal sparse signal by solving a convex program. This approach is powerful because the optimization can be completed in polynomial time with standard scientific software. The paper provides general conditions which ensure that convex relaxation succeeds. As evidence of the broad impact of these results, the paper describes how convex relaxation can be used for several concrete signal recovery problems. It also describes applications to channel coding, linear regression, and numerical analysis.

Logistic regression with ℓ1 regularization has been proposed as a promising method for feature selection in classification problems. In this paper we describe an efficient interior-point method for solving large-scale ℓ1-regularized logistic regression problems. Small problems with up to a thousand or so features and examples can be solved in seconds on a PC; medium sized problems, with tens of thousands of features and examples, can be solved in tens of seconds (assuming some sparsity in the data). A variation on the basic method, that uses a preconditioned conjugate gradient method to compute the search step, can solve very large problems, with a million features and examples (e.g., the 20 Newsgroups data set), in a few minutes, on a PC. Using warm-start techniques, a good approximation of the entire regularization path can be computed much more efficiently than by solving a family of problems independently.

We develop fast algorithms for estimation of generalized linear models with convex penalties. The models include linear regression, twoclass logistic regression, and multinomial regression problems while the penalties include ℓ1 (the lasso), ℓ2 (ridge regression) and mixtures of the two (the elastic net). The algorithms use cyclical coordinate descent, computed along a regularization path. The methods can handle large problems and can also deal efficiently with sparse features. In comparative timings we find that the new algorithms are considerably faster than competing methods.

Extracting useful information from high-dimensional data is an important part of the focus of today’s statistical research and practice. Penalized loss function minimiza-tion has been shown to be effective for this task both theoretically and empirically. With the virtues of both regularization and sparsity, the L1-penalized L2 minimization method Lasso has been popular in regression models. In this paper, we combine different norms including L1 to form an intelligent penalty in order to add side information to the fitting of a regression or classification model to obtain reasonable estimates. Specifically, we introduce the Composite Absolute Penal-ties (CAP) family which allows the grouping and hierarchical relationships between the predictors to be expressed. CAP penalties are built by defining groups and com-bining the properties of norm penalties at the across group and within group levels. Grouped selection occurs for non-overlapping groups. In that case, we give a Bayesian 1 interpretation for CAP penalties. Hierarchical variable selection is reached by defining groups with particular overlapping patterns. In the computation aspect, we propose using the BLASSO and cross-validation to obtain CAP estimates. For a subfamily of CAP estimates involving only the L1 and L ∞ norms, we introduce the iCAP algorithm to trace the entire regularization path for the grouped selection problem. Within this subfamily, unbiased estimates of the degrees of freedom (df) are derived allowing the regularization parameter to be selected without cross-validation. CAP is shown to im-prove on the predictive performance of the LASSO in a series of simulated experiments including cases with p>> n and mis-specified groupings. When the complexity of a model is properly calculated, iCAP is seen to be parsimonious in the experiments. 1

Sparse coding—that is, modelling data vectors as sparse linear combinations of basis elements—is widely used in machine learning, neuroscience, signal processing, and statistics. This paper focuses on the large-scale matrix factorization problem that consists of learning the basis set, adapting it to specific data. Variations of this problem include dictionary learning in signal processing, non-negative matrix factorization and sparse principal component analysis. In this paper, we propose to address these tasks with a new online optimization algorithm, based on stochastic approximations, which scales up gracefully to large datasets with millions of training samples, and extends naturally to various matrix factorization formulations, making it suitable for a wide range of learning problems. A proof of convergence is presented, along with experiments with natural images and genomic data demonstrating that it leads to state-of-the-art performance in terms of speed and optimization for both small and large datasets.

We consider the task of reinforcement learning with linear value function approximation. Temporal difference algorithms, and in particular the Least-Squares Temporal Difference (LSTD) algorithm, provide a method for learning the parameters of the value function, but when the number of features is large this algorithm can over-fit to the data and is computationally expensive. In this paper, we propose a regularization framework for the LSTD algorithm that overcomes these difficulties. In particular, we focus on the case of l1 regularization, which is robust to irrelevant features and also serves as a method for feature selection. Although the l1 regularized LSTD solution cannot be expressed as a convex optimization problem, we present an algorithm similar to the Least Angle Regression (LARS) algorithm that can efficiently compute the optimal solution. Finally, we demonstrate the performance of the algorithm experimentally. 1.

We consider the problem of learning a sparse multi-task regression, where the structure in the outputs can be represented as a tree with leaf nodes as outputs and internal nodes as clusters of the outputs at multiple granularity. Our goal is to recover the common set of relevant inputs for each output cluster. Assuming that the tree structure is available as prior knowledge, we formulate this problem as a new multi-task regularized regression called tree-guided group lasso. Our structured regularization is based on a grouplasso penalty, where groups are defined with respect to the tree structure. We describe a systematic weighting scheme for the groups in the penalty such that each output variable is penalized in a balanced manner even if the groups overlap. We present an efficient optimization method that can handle a largescale problem. Using simulated and yeast datasets, we demonstrate that our method shows a superior performance in terms of both prediction errors and recovery of true sparsity patterns compared to other methods for multi-task learning. 1.

We study the problem of selecting a subset of k random variables to observe that will yield the best linear prediction of another variable of interest, given the pairwise correlations between the observation variables and the predictor variable. Under approximation preserving reductions, this problem is also equivalent to the“sparse approximation”problem of approximating signals concisely. We propose and analyze exact and approximation algorithms for several special cases of practical interest. We give an FPTAS when the covariance matrix has constant bandwidth, and exact algorithms when the associated covariance graph, consisting of edges for pairs of variables with nonzero correlation, forms a tree or has a large (known) independent set. Furthermore, we give an exact algorithm when the variables can be embedded into a line such that the covariance decreases exponentially in the distance, and a constant-factor approximation when the variables have no “conditional suppressor variables”. Much of our reasoning is based on perturbation results for the R 2 multiple correlation measure, frequently used as a measure for “goodness-of-fit statistics”. It lies at the core of our FPTAS, and also allows us to extend exact algorithms to approximation algorithms when the matrix “nearly ” falls into one of the above classes. We also use perturbation analysis to prove approximation guarantees for the widely used “Forward Regression ” heuristic when the observation variables are nearly independent.

The LASSO-Patternsearch algorithm is proposed to efficiently identify patterns of multiple dichotomous risk factors for outcomes of interest in demographic and genomic studies. The patterns considered are those that arise naturally from the log linear expansion of the multivariate Bernoulli density. The method is designed for the case where there is a possibly very large number of candidate patterns but it is believed that only a relatively small number are important. A LASSO is used to greatly reduce the number of candidate patterns, using a novel computational algorithm that can handle an extremely large number of unknowns simultaneously. The patterns surviving the LASSO are further pruned in the framework of (parametric) generalized linear models. A novel tuning procedure based on the GACV for Bernoulli outcomes, modified to act

This report reviews and extends the field of similarity-based classification, presenting new analyses, algorithms, data sets, and the most comprehensive set of experimental results to date. Specifically, the generalizability of using similarities as features is analyzed, design goals and methods for weighting nearest-neighbors for similarity-based learning are proposed, and different methods for consistently converting similarities into kernels are compared. Experiments on eight real data sets compare eight approaches and their variants to similarity-based learning. 1