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.

Summary. The group lasso is an extension of the lasso to do variable selection on (predefined) groups of variables in linear regression models. The estimates have the attractive property of being invariant under groupwise orthogonal reparameterizations. We extend the group lasso to logistic regression models and present an efficient algorithm, that is especially suitable for high dimensional problems, which can also be applied to generalized linear models to solve the corresponding convex optimization problem. The group lasso estimator for logistic regression is shown to be statistically consistent even if the number of predictors is much larger than sample size but with sparse true underlying structure. We further use a two-stage procedure which aims for sparser models than the group lasso, leading to improved prediction performance for some cases. Moreover, owing to the two-stage nature, the estimates can be constructed to be hierarchical. The methods are used on simulated and real data sets about splice site detection in DNA sequences.

It has been widely observed that there is no single “dominant ” SAT solver; instead, different solvers perform best on different instances. Rather than following the traditional approach of choosing the best solver for a given class of instances, we advocate making this decision online on a per-instance basis. Building on previous work, we describe SATzilla, an automated approach for constructing per-instance algorithm portfolios for SAT that use so-called empirical hardness models to choose among their constituent solvers. This approach takes as input a distribution of problem instances and a set of component solvers, and constructs a portfolio optimizing a given objective function (such as mean runtime, percent of instances solved, or score in a competition). The excellent performance of our SATzilla portfolios has been independently verified in the 2007 SAT Competition, where our SATzilla-07 solvers won three gold, one silver and one bronze medal. In this article, we go well beyond SATzilla-07 by making the portfolio construction scalable and completely automated, and improving it by integrating local search solvers as candidate solvers, by predicting performance score instead of runtime, and by using hierarchical hardness models that take into account different types of SAT instances. We demonstrate the effectiveness of these new techniques in extensive experimental results on data sets including instances from the most recent SAT competition. 1.

Abstract. It has been widely observed that there is no “dominant” SAT solver; instead, different solvers perform best on different instances. Rather than following the traditional approach of choosing the best solver for a given class of instances, we advocate making this decision online on a per-instance basis. Building on previous work, we describe a per-instance solver portfolio for SAT, SATzilla-07, which uses socalled empirical hardness models to choose among its constituent solvers. We leverage new model-building techniques such as censored sampling and hierarchical hardness models, and demonstrate the effectiveness of our techniques by building a portfolio of state-of-the-art SAT solvers and evaluating it on several widely-studied SAT data sets. Overall, we show that our portfolio significantly outperforms its constituent algorithms on every data set. Our approach has also proven itself to be effective in practice: in the 2007 SAT competition, SATzilla-07 won three gold medals, one silver, and one bronze; it is available online at

Within the field of pattern classification, the Fisher kernel is a powerful framework which combines the strengths of generative and discriminative approaches. The idea is to characterize a signal with a gradient vector derived from a generative probability model and to subsequently feed this representation to a discriminative classifier. We propose to apply this framework to image categorization where the input signals are images and where the underlying generative model is a visual vocabulary: a Gaussian mixture model which approximates the distribution of low-level features in images. We show that Fisher kernels can actually be understood as an extension of the popular bag-of-visterms. Our approach demonstrates excellent performance on two challenging databases: an in-house database of 19 object/scene categories and the recently released VOC 2006 database. It is also very practical: it has low computational needs both at training and test time and vocabularies trained on one set of categories can be applied to another set without any significant loss in performance.

Abstract. We present a unified and complete account of maximum entropy distribution estimation subject to constraints represented by convex potential functions or, alternatively, by convex regularization. We provide fully general performance guarantees and an algorithm with a complete convergence proof. As special cases, we can easily derive performance guarantees for many known regularization types, including ℓ1, ℓ2, ℓ 2 2 and ℓ1 + ℓ 2 2 style regularization. Furthermore, our general approach enables us to use information about the structure of the feature space or about sample selection bias to derive entirely new regularization functions with superior guarantees. We propose an algorithm solving a large and general subclass of generalized maxent problems, including all discussed in the paper, and prove its convergence. Our approach generalizes techniques based on information geometry and Bregman divergences as well as those based more directly on compactness. 1

Techniques from sparse signal representation are beginning to see significant impact in computer vision, often on non-traditional applications where the goal is not just to obtain a compact high-fidelity representation of the observed signal, but also to extract semantic information. The choice of dictionary plays a key role in bridging this gap: unconventional dictionaries consisting of, or learned from, the training samples themselves provide the key to obtaining state-of-theart results and to attaching semantic meaning to sparse signal representations. Understanding the good performance of such unconventional dictionaries in turn demands new algorithmic and analytical techniques. This review paper highlights a few representative examples of how the interaction between sparse signal representation and computer vision can enrich both fields, and raises a number of open questions for further study.

We present a unified and complete account of maximum entropy density estimation subject to constraints represented by convex potential functions or, alternatively, by convex regularization. We provide fully general performance guarantees and an algorithm with a complete convergence proof. As special cases, we easily derive performance guarantees for many known regularization types, including ℓ1, ℓ2, ℓ 2 2, and ℓ1+ ℓ 2 2 style regularization. We propose an algorithm solving a large and general subclass of generalized maximum entropy problems, including all discussed in the paper, and prove its convergence. Our approach generalizes and unifies techniques based on information geometry and Bregman divergences as well as those based more directly on compactness. Our work is motivated by a novel application of maximum entropy to species distribution modeling, an important problem in conservation biology and ecology. In a set of experiments on real-world data, we demonstrate the utility of maximum entropy in this setting. We explore effects of different feature types, sample sizes, and regularization levels on the performance of maxent, and discuss interpretability of the resulting models.

Classifiers favoring sparse solutions, such as support vector machines, relevance vector machines, LASSO-regression based classifiers, etc., provide competitive methods for classification problems in high dimensions. However, current algorithms for training sparse classifiers typically scale quite unfavorably with respect to the number of training examples. This paper proposes online and multi-pass algorithms for training sparse linear classifiers for high dimensional data. These algorithms have computational complexity and memory requirements that make learning on massive datasets feasible. The central idea that makes this possible is a straightforward quadratic approximation to the likelihood function.