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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TL;DR: The proposed discriminative learning method for emotion recognition using both articulatory and acoustic information is proposed, and it is shown that the proposed method is more effective at distinguishing happiness from other emotions.
Abstract: Speech emotion recognition methods combining articulatory information with acoustic features have been previously shown to improve recognition performance. Collection of articulatory data on a large scale may not be feasible in many scenarios, thus restricting the scope and applicability of such methods. In this paper, a discriminative learning method for emotion recognition using both articulatory and acoustic information is proposed. A traditional l1-regularized logistic regression cost function is extended to include additional constraints that enforce the model to reconstruct articulatory data. This leads to sparse and interpretable representations jointly optimized for both tasks simultaneously. Furthermore, the model only requires articulatory features during training; only speech features are required for inference on out-of-sample data. Experiments are conducted to evaluate emotion recognition performance over vowels /AA/, /AE/, /IY/, /UW/ and complete utterances. Incorporating articulatory information is shown to significantly improve the performance for valence-based classification. Results obtained for within-corpus and cross-corpus categorical emotion recognition indicate that the proposed method is more effective at distinguishing happiness from other emotions.
7 citations
Cites background from "EfficientL 1 regularized logistic r..."
...(4) is convex, and a number of generic or application-dependent techniques are available to solve this problem [58-61]....
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TL;DR: This paper extends the idea of sparse learning to a non-Gaussian model, especially the powerful conditional Gaussian mixture, and proposes a novel sparse inverse covariance learning algorithm based on the expectation-maximization lower-bound optimization technique.
Abstract: We consider the task of multiple-output regression where both input and output are high-dimensional. Due to the limited amount of training samples compared to data dimensions, properly imposing loose statistical dependency in learning a regression model is crucial for reliable prediction accuracy. The sparse inverse covariance learning of conditional Gaussian random fields has been recently emerging to achieve this goal, shown to exhibit superior performance to non-sparse approaches. However, one of its main drawbacks is the strong assumption of linear Gaussianity in modeling the input-output relationship. For certain application domains, the assumption might be too restricted and less powerful in representation, and consequently, prediction based on the wrong models can result in suboptimal performance. In this paper, we extend the idea of sparse learning to a non-Gaussian model, especially the powerful conditional Gaussian mixture. For this latent-variable model, we propose a novel sparse inverse covariance learning algorithm based on the expectation-maximization lower-bound optimization technique. It is shown that each M-step reduces to solving the regular sparse inverse covariance estimation of linear Gaussian models, in conjunction with estimating sparse logistic regression. We demonstrate the improved prediction performance of the proposed algorithm over exisitng methods on several datasets.
7 citations
Cites background from "EfficientL 1 regularized logistic r..."
...Note that (21) is very similar to the sparse logistic regression [16]....
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TL;DR: This work was the first to assess the impact of phenotyping errors on the accuracy of genomic predictions in animal genetics and found Lasso was the second best method, while SVML, RF and KNN were very sensitive to noise.
Abstract: Statistical and machine learning applications are increasingly popular in animal breeding and genetics, especially to compute genomic predictions for phenotypes of interest. Noise (errors) in the data may have a negative impact on the accuracy of predictions. The effects of noisy data have been investigated in genome-wide association studies for case–control experiments, and in genomic predictions for binary traits in plants. No studies have been published yet on the impact of noisy data in animal genomics. In this work, the susceptibility to noise of five classification models (Lasso-penalised logistic regression—Lasso, K-nearest neighbours—KNN, random forest—RF, support vector machines with linear—SVML—or radial—SVMR—kernel) was tested. As illustration, the identification of carriers of a recessive mutation in cattle (Bos taurus) was used. A population of 3116 Fleckvieh animals with SNP genotypes on the same chromosome as the mutation locus (BTA 19) was available. The carrier status (0/1 phenotype) was randomly sampled to generate noise. Increasing proportions of noise—up to 20%— were introduced in the data. SVMR and Lasso were relatively more robust to noise in the data, with total accuracy still above 0.975 and TPR (true positive rate; accuracy in the minority class) in the range 0.5–0.80 also with 17.5–20% mislabeled observations. The performance of SVML and RF decreased monotonically with increasing noise in the data, while KNN constantly failed to identify mutation carriers (observations in the minority class). The computation time increased with noise in the data, especially for the two support vector machines classifiers. This work was the first to assess the impact of phenotyping errors on the accuracy of genomic predictions in animal genetics. The choice of the classification method can influence results in terms of higher or lower susceptibility to noise. In the presented problem, SVM with radial kernel performed relatively well even when the proportion of errors in the data reached 12.5%. Lasso was the second best method, while SVML, RF and KNN were very sensitive to noise. Taking into account both accuracy and computation time, Lasso provided the best combination.
7 citations
Cites methods from "EfficientL 1 regularized logistic r..."
...1 was fitted by maximizing the corresponding Lasso-penalized log likelihood function [28]....
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Posted Content•
TL;DR: This work introduces a flexible ICA algorithm that uses an effective PDF estimator to accurately capture the underlying statistical properties of the data and discusses several techniques to accurately estimate the parameters of the multivariate generalized Gaussian distribution, and how to integrate them into the IVA model.
Abstract: Independent component analysis (ICA) is a widely used BSS method that can uniquely achieve source recovery, subject to only scaling and permutation ambiguities, through the assumption of statistical independence on the part of the latent sources. Independent vector analysis (IVA) extends the applicability of ICA by jointly decomposing multiple datasets through the exploitation of the dependencies across datasets. Though both ICA and IVA algorithms cast in the maximum likelihood (ML) framework enable the use of all available statistical information in reality, they often deviate from their theoretical optimality properties due to improper estimation of the probability density function (PDF). This motivates the development of flexible ICA and IVA algorithms that closely adhere to the underlying statistical description of the data. Although it is attractive minimize the assumptions, important prior information about the data, such as sparsity, is usually available. If incorporated into the ICA model, use of this additional information can relax the independence assumption, resulting in an improvement in the overall separation performance. Therefore, the development of a unified mathematical framework that can take into account both statistical independence and sparsity is of great interest. In this work, we first introduce a flexible ICA algorithm that uses an effective PDF estimator to accurately capture the underlying statistical properties of the data. We then discuss several techniques to accurately estimate the parameters of the multivariate generalized Gaussian distribution, and how to integrate them into the IVA model. Finally, we provide a mathematical framework that enables direct control over the influence of statistical independence and sparsity, and use this framework to develop an effective ICA algorithm that can jointly exploit these two forms of diversity.
7 citations
Cites background from "EfficientL 1 regularized logistic r..."
...The `1 norm is a non-differentiable function, so it is replaced by the the sum of multi-quadratic functions [56], given by f (yn) = lim n→0 V ∑...
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TL;DR: An iterative computation of sparse representations of functions defined on R^d exploits a formulation of the sparsification problem equivalent to Support Vector Machine and based on Tikhonov regularization, which exploits a smooth and strictly convex approximation of the l"1-norm.
7 citations
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
Additional excerpts
...(Nelder & Wedderbum 1972; McCullagh & Nelder 1989)...
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...(Nelder & Wedderbum 1972; McCullagh & Nelder 1989 )...
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