Proceedings Article•
EfficientL 1 regularized logistic regression
16 Jul 2006-pp 401-408
TL;DR: Theoretical results show that the proposed efficient algorithm for L1 regularized logistic regression is guaranteed to converge to the global optimum, and experiments show that it significantly outperforms standard algorithms for solving convex optimization problems.
Abstract: L1 regularized logistic regression is now a workhorse of machine learning: it is widely used for many classification problems, particularly ones with many features. L1 regularized logistic regression requires solving a convex optimization problem. However, standard algorithms for solving convex optimization problems do not scale well enough to handle the large datasets encountered in many practical settings. In this paper, we propose an efficient algorithm for L1 regularized logistic regression. Our algorithm iteratively approximates the objective function by a quadratic approximation at the current point, while maintaining the L1 constraint. In each iteration, it uses the efficient LARS (Least Angle Regression) algorithm to solve the resulting L1 constrained quadratic optimization problem. Our theoretical results show that our algorithm is guaranteed to converge to the global optimum. Our experiments show that our algorithm significantly outperforms standard algorithms for solving convex optimization problems. Moreover, our algorithm outperforms four previously published algorithms that were specifically designed to solve the L1 regularized logistic regression problem.
Citations
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01 Jan 2013
TL;DR: This thesis applies an efficient block coordinate descent algorithm to real-world data and demonstrates that it produces a more accurate model of player effectiveness in basketball by showing that (1) the algorithm outperforms existing approaches and (2) it leads to a profitable betting strategy.
Abstract: This thesis examines two separate statistical problems for whichlow-dimensional models are effective.In the first part of this thesis, we examine the Robust Principal Components Analysis(RPCA) problem: given amatrix $\datam$ that is the sum of a low-rank matrix $\lowopt$ and a sparse noise matrix$\sparseopt$, recover $\lowopt$ and $\sparseopt$.This problem appears in various settings, including image processing,computer vision, and graphical models. Various polynomial-time heuristicsand algorithms have been proposed to solve this problem.We introduce a block coordinate descent algorithm for thisproblem and prove a convergence result. In addition, our iterativealgorithm has low complexity per iteration and empirically performs wellon synthetic datasets.In the second part of this thesis, we examine a variant of ridge regression:unlike in the classical setting where we know that the parameter ofinterest lies near a single point, we instead only know that it lies neara known low-dimensional subspace.We formulate this regression problem as a convex optimization problem, andintroduce an efficient block coordinate descent algorithm for solving it.We demonstrate that this ``subspace prior version of ridge regression isan appropriate model for understanding player effectiveness in basketball.In particular, we apply our algorithm to real-world data and demonstrateempirically that it produces a more accurate model of player effectivenessby showing that (1) the algorithm outperforms existing approaches and (2)it leads to a profitable betting strategy.
2 citations
Journal Article•
TL;DR: Experimental results show that the proposed variational image segmentation framework performs well in singleand multichannel segmentation tasks, and can be employed to the segmentation of various types of images, such as natural and texture images as well as medical images.
Abstract: In this paper, we propose a variational image segmentation framework for multichannel multiphase image segmentation based on the Chan-Vese active contour model. The core of our method lies in finding a variable u encoding the segmentation, by minimizing a multichannel energy functional that combines the information of multiple images. We create a decomposition of the input, either by multichannel filtering, or simply by using plain natural RGB, or medical images, which already consist of several channels. Subsequently we minimize the proposed functional for each of the channels simultaneously. Our model meets the necessary assumptions such that it can be solved efficiently by optimization techniques like the Chambolle–Pock method. We prove that the proposed energy functional has global minimizers, and show its stability and convergence with respect to noisy inputs. Experimental results show that the proposed method performs well in singleand multichannel segmentation tasks, and can be employed to the segmentation of various types of images, such as natural and texture images as well as medical images.
2 citations
Dissertation•
01 Jan 2013
TL;DR: This thesis proposes a technique for building visual descriptors based on a Bag-of-Words (BoWs) representation, which combines unsupervised and supervised information leading to more discriminative BoWs representations, and proposes an approach that exploits contextual information to improve the performance of object recognition techniques.
Abstract: Automatic visual recognition of generic objects is a highly relevant area of study. Nonetheless, intra-class and pose variations, as well as, background clutter and partial occlusion, are some of the main difficulties to achieve this goal. As an application case, robots with a robust visual system can achieve a high level of autonomy and a semantic understanding of their environments. Current state-of-the-art approaches to visual category-level object recognition are generally based on two main steps: generation of visual descriptors and training of visual classifiers using these descriptors and labeled images. Furthermore, these steps are usually complemented using techniques oriented to include contextual information in the models. In this thesis, we contribute to the area of visual recognition by proposing three techniques oriented to improve each of the previous steps, respectively. First, we introduce a technique for building visual descriptors based on a Bag-of-Words (BoWs) representation. In contrast to current approaches based on unsupervised clustering techniques, our proposal combines unsupervised and supervised information leading to more discriminative BoWs representations. Afterwards, we present a technique to improve the performance of current visual classifiers using a divide-and-conquer strategy based on a Mixture of Expert (MoE) approach. We innovate with respect to current MoE techniques by incorporating an embedded local feature selection scheme within each visual classifier. Finally, we propose an approach that exploits contextual information to improve the performance of object recognition techniques. We innovate with respect to state-of-theart techniques by considering scene dependent contextual relations among object classes. We test the performance of all these techniques by applying them to common benchmark datasets. Our results validate our main hypotheses indicating improvements with respect to alternative state-of-the-art methods. This also shows that the ideas presented in this thesis represent a relevant contribution to the state-of-the-art of category-level object recognition.
2 citations
Cites methods from "EfficientL 1 regularized logistic r..."
...Each of these problems is, in turn, solved using the strategy proposed in (Lee et al., 2006)....
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...To handle this case, we observe that each of these optimizations is equivalent to a regularized logistic regression (Lee et al., 2006)....
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...As shown in (Lee et al., 2006), this problem can be solved by using a coordinate ascent optimization strategy (Tseng, 2001) given by a sequential two-step approach that first models the problem as an unregularized logistic regression and afterwards incorporates the regularization constraints....
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...Optimization of the regularized likelihood Following the procedure in (Lee et al., 2006), we add the regularization term to the optimization problem given by Equation (3....
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01 Dec 2018
TL;DR: Preliminary results that have enormous potential in text mining research are reported, including a high-performance binary document segregator based on Logistic regression and the Bacterial Foraging Algorithm.
Abstract: In this paper, we report preliminary results that have enormous potential in text mining research. Logistic regression is a well-known and reliable prediction tool in Pattern Recognition. The simple regression tool is optimized for performance by an evolutionary approach and the result is a high-performance binary document segregator. The Bacterial Foraging Algorithm is the evolutionary algorithm used in our experiments. The theory behind our approach is presented, along with empirical results that prove the viability of our claim.
2 citations
Cites background from "EfficientL 1 regularized logistic r..."
...The problem with conventional gradient-based solutions to convex optimization problems like logistic regression, is that they do not scale sufficiently for large datasets [30]....
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...The problem statement incorporating the L1-regularized regression function [29, 30] is defined as...
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Posted Content•
TL;DR: This paper proposes an algorithm that leverages the unlabeled data (through Laplace smoothing) and learns classifiers with budget constraints, which is, to the authors' knowledge, the first algorithm for semi-supervised budgeted learning.
Abstract: As machine learning transitions increasingly towards real world applications controlling the test-time cost of algorithms becomes more and more crucial. Recent work, such as the Greedy Miser and Speedboost, incorporate test-time budget constraints into the training procedure and learn classifiers that provably stay within budget (in expectation). However, so far, these algorithms are limited to the supervised learning scenario where sufficient amounts of labeled data are available. In this paper we investigate the common scenario where labeled data is scarce but unlabeled data is available in abundance. We propose an algorithm that leverages the unlabeled data (through Laplace smoothing) and learns classifiers with budget constraints. Our model, based on gradient boosted regression trees (GBRT), is, to our knowledge, the first algorithm for semi-supervised budgeted learning.
2 citations
Cites methods from "EfficientL 1 regularized logistic r..."
...The first baseline we compare against is logistic regression with weighted l1 regularization [36] (LRL1), where the weight is the feature extraction cost....
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References
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TL;DR: A new method for estimation in linear models called the lasso, which minimizes the residual sum of squares subject to the sum of the absolute value of the coefficients being less than a constant, is proposed.
Abstract: SUMMARY We propose a new method for estimation in linear models. The 'lasso' minimizes the residual sum of squares subject to the sum of the absolute value of the coefficients being less than a constant. Because of the nature of this constraint it tends to produce some coefficients that are exactly 0 and hence gives interpretable models. Our simulation studies suggest that the lasso enjoys some of the favourable properties of both subset selection and ridge regression. It produces interpretable models like subset selection and exhibits the stability of ridge regression. There is also an interesting relationship with recent work in adaptive function estimation by Donoho and Johnstone. The lasso idea is quite general and can be applied in a variety of statistical models: extensions to generalized regression models and tree-based models are briefly described.
40,785 citations
"EfficientL 1 regularized logistic r..." refers methods in this paper
...(Tibshirani 1996) Several algorithms have been developed to solve L1 constrained least squares problems....
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...See, Tibshirani (1996) for details.)...
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...(Tibshirani 1996) Several algorithms have been developed to solve L1 constrained least squares problems....
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Book•
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TL;DR: In this article, the focus is on recognizing convex optimization problems and then finding the most appropriate technique for solving them, and a comprehensive introduction to the subject is given. But the focus of this book is not on the optimization problem itself, but on the problem of finding the appropriate technique to solve it.
Abstract: Convex optimization problems arise frequently in many different fields. A comprehensive introduction to the subject, this book shows in detail how such problems can be solved numerically with great efficiency. The focus is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. The text contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance, and economics.
33,341 citations
Book•
01 Jan 1983TL;DR: In this paper, a generalization of the analysis of variance is given for these models using log- likelihoods, illustrated by examples relating to four distributions; the Normal, Binomial (probit analysis, etc.), Poisson (contingency tables), and gamma (variance components).
Abstract: The technique of iterative weighted linear regression can be used to obtain maximum likelihood estimates of the parameters with observations distributed according to some exponential family and systematic effects that can be made linear by a suitable transformation. A generalization of the analysis of variance is given for these models using log- likelihoods. These generalized linear models are illustrated by examples relating to four distributions; the Normal, Binomial (probit analysis, etc.), Poisson (contingency tables) and gamma (variance components).
23,215 citations
01 Jan 1998
12,940 citations
"EfficientL 1 regularized logistic r..." refers methods in this paper
...We tested each algorithm’s performance on 12 different datasets, consisting of 9 UCI datasets (Newman et al. 1998), one artificial dataset called Madelon from the NIPS 2003 workshop on feature extraction,3 and two gene expression datasets (Microarray 1 and 2).4 Table 2 gives details on the number…...
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...We tested each algorithm’s performance on 12 different real datasets, consisting of 9 UCI datasets (Newman et al. 1998) and 3 gene expression datasets (Microarray 1, 2 and 3) 3....
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TL;DR: This is the rst book on generalized linear models written by authors not mostly associated with the biological sciences, and it is thoroughly enjoyable to read.
Abstract: This is the rst book on generalized linear models written by authors not mostly associated with the biological sciences. Subtitled “With Applications in Engineering and the Sciences,” this book’s authors all specialize primarily in engineering statistics. The rst author has produced several recent editions of Walpole, Myers, and Myers (1998), the last reported by Ziegel (1999). The second author has had several editions of Montgomery and Runger (1999), recently reported by Ziegel (2002). All of the authors are renowned experts in modeling. The rst two authors collaborated on a seminal volume in applied modeling (Myers and Montgomery 2002), which had its recent revised edition reported by Ziegel (2002). The last two authors collaborated on the most recent edition of a book on regression analysis (Montgomery, Peck, and Vining (2001), reported by Gray (2002), and the rst author has had multiple editions of his own regression analysis book (Myers 1990), the latest of which was reported by Ziegel (1991). A comparable book with similar objectives and a more speci c focus on logistic regression, Hosmer and Lemeshow (2000), reported by Conklin (2002), presumed a background in regression analysis and began with generalized linear models. The Preface here (p. xi) indicates an identical requirement but nonetheless begins with 100 pages of material on linear and nonlinear regression. Most of this will probably be a review for the readers of the book. Chapter 2, “Linear Regression Model,” begins with 50 pages of familiar material on estimation, inference, and diagnostic checking for multiple regression. The approach is very traditional, including the use of formal hypothesis tests. In industrial settings, use of p values as part of a risk-weighted decision is generally more appropriate. The pedagologic approach includes formulas and demonstrations for computations, although computing by Minitab is eventually illustrated. Less-familiar material on maximum likelihood estimation, scaled residuals, and weighted least squares provides more speci c background for subsequent estimation methods for generalized linear models. This review is not meant to be disparaging. The authors have packed a wealth of useful nuggets for any practitioner in this chapter. It is thoroughly enjoyable to read. Chapter 3, “Nonlinear Regression Models,” is arguably less of a review, because regression analysis courses often give short shrift to nonlinear models. The chapter begins with a great example on the pitfalls of linearizing a nonlinear model for parameter estimation. It continues with the effective balancing of explicit statements concerning the theoretical basis for computations versus the application and demonstration of their use. The details of maximum likelihood estimation are again provided, and weighted and generalized regression estimation are discussed. Chapter 4 is titled “Logistic and Poisson Regression Models.” Logistic regression provides the basic model for generalized linear models. The prior development for weighted regression is used to motivate maximum likelihood estimation for the parameters in the logistic model. The algebraic details are provided. As in the development for linear models, some of the details are pushed into an appendix. In addition to connecting to the foregoing material on regression on several occasions, the authors link their development forward to their following chapter on the entire family of generalized linear models. They discuss score functions, the variance-covariance matrix, Wald inference, likelihood inference, deviance, and overdispersion. Careful explanations are given for the values provided in standard computer software, here PROC LOGISTIC in SAS. The value in having the book begin with familiar regression concepts is clearly realized when the analogies are drawn between overdispersion and nonhomogenous variance, or analysis of deviance and analysis of variance. The authors rely on the similarity of Poisson regression methods to logistic regression methods and mostly present illustrations for Poisson regression. These use PROC GENMOD in SAS. The book does not give any of the SAS code that produces the results. Two of the examples illustrate designed experiments and modeling. They include discussion of subset selection and adjustment for overdispersion. The mathematic level of the presentation is elevated in Chapter 5, “The Family of Generalized Linear Models.” First, the authors unify the two preceding chapters under the exponential distribution. The material on the formal structure for generalized linear models (GLMs), likelihood equations, quasilikelihood, the gamma distribution family, and power functions as links is some of the most advanced material in the book. Most of the computational details are relegated to appendixes. A discussion of residuals returns one to a more practical perspective, and two long examples on gamma distribution applications provide excellent guidance on how to put this material into practice. One example is a contrast to the use of linear regression with a log transformation of the response, and the other is a comparison to the use of a different link function in the previous chapter. Chapter 6 considers generalized estimating equations (GEEs) for longitudinal and analogous studies. The rst half of the chapter presents the methodology, and the second half demonstrates its application through ve different examples. The basis for the general situation is rst established using the case with a normal distribution for the response and an identity link. The importance of the correlation structure is explained, the iterative estimation procedure is shown, and estimation for the scale parameters and the standard errors of the coef cients is discussed. The procedures are then generalized for the exponential family of distributions and quasi-likelihood estimation. Two of the examples are standard repeated-measures illustrations from biostatistical applications, but the last three illustrations are all interesting reworkings of industrial applications. The GEE computations in PROC GENMOD are applied to account for correlations that occur with multiple measurements on the subjects or restrictions to randomizations. The examples show that accounting for correlation structure can result in different conclusions. Chapter 7, “Further Advances and Applications in GLM,” discusses several additional topics. These are experimental designs for GLMs, asymptotic results, analysis of screening experiments, data transformation, modeling for both a process mean and variance, and generalized additive models. The material on experimental designs is more discursive than prescriptive and as a result is also somewhat theoretical. Similar comments apply for the discussion on the quality of the asymptotic results, which wallows a little too much in reports on various simulation studies. The examples on screening and data transformations experiments are again reworkings of analyses of familiar industrial examples and another obvious motivation for the enthusiasm that the authors have developed for using the GLM toolkit. One can hope that subsequent editions will similarly contain new examples that will have caused the authors to expand the material on generalized additive models and other topics in this chapter. Designating myself to review a book that I know I will love to read is one of the rewards of being editor. I read both of the editions of McCullagh and Nelder (1989), which was reviewed by Schuenemeyer (1992). That book was not fun to read. The obvious enthusiasm of Myers, Montgomery, and Vining and their reliance on their many examples as a major focus of their pedagogy make Generalized Linear Models a joy to read. Every statistician working in any area of applied science should buy it and experience the excitement of these new approaches to familiar activities.
10,520 citations
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...(Nelder & Wedderbum 1972; McCullagh & Nelder 1989)...
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...(Nelder & Wedderbum 1972; McCullagh & Nelder 1989 )...
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