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: This paper proposes a variation on Hodrick-Prescott (H-P) filtering, a widely used method for trend estimation that substitutes a sum of absolute values for the sum of squares used in H-P filtering to penalize variations in the estimated trend.
Abstract: The problem of estimating underlying trends in time series data arises in a variety of disciplines. In this paper we propose a variation on Hodrick-Prescott (H-P) filtering, a widely used method for trend estimation. The proposed $\ell_1$ trend filtering method substitutes a sum of absolute values (i.e., $\ell_1$ norm) for the sum of squares used in H-P filtering to penalize variations in the estimated trend. The $\ell_1$ trend filtering method produces trend estimates that are piecewise linear, and therefore it is well suited to analyzing time series with an underlying piecewise linear trend. The kinks, knots, or changes in slope of the estimated trend can be interpreted as abrupt changes or events in the underlying dynamics of the time series. Using specialized interior-point methods, $\ell_1$ trend filtering can be carried out with not much more effort than H-P filtering; in particular, the number of arithmetic operations required grows linearly with the number of data points. We describe the method and some of its basic properties and give some illustrative examples. We show how the method is related to $\ell_1$ regularization-based methods in sparse signal recovery and feature selection, and we list some extensions of the basic method.
577 citations
Cites background from "EfficientL 1 regularized logistic r..."
...From (9), we can see that the regularization path of ℓ1 trend filtering is piecewise linear....
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17 Sep 2007
TL;DR: Two new techniques are proposed based on a smooth (differentiable) convex approximation for the L1 regularizer that does not depend on any assumptions about the loss function used and a new strategy that addresses the non-differentiability of the L 1-regularizer.
Abstract: L1 regularization is effective for feature selection, but the resulting optimization is challenging due to the non-differentiability of the 1-norm. In this paper we compare state-of-the-art optimization techniques to solve this problem across several loss functions. Furthermore, we propose two new techniques. The first is based on a smooth (differentiable) convex approximation for the L1 regularizer that does not depend on any assumptions about the loss function used. The other technique is a new strategy that addresses the non-differentiability of the L1-regularizer by casting the problem as a constrained optimization problem that is then solved using a specialized gradient projection method. Extensive comparisons show that our newly proposed approaches consistently rank among the best in terms of convergence speed and efficiency by measuring the number of function evaluations required.
396 citations
Cites background or methods or result from "EfficientL 1 regularized logistic r..."
...We can extend the IRLS-LARS algorithm to a general algorithm by observing that the algorithm is an IRLS reformulation of a Sequential Quadratic Programming (SQP) update (where a unit step length is assumed)....
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...As in SQP, this algorithm achieves a superlinear rate of convergence [7]....
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...Efficient algorithms have been proposed for the special cases where L(x) has a specific functional form, such as a Gaussian [3] or Logistic [11] negative log-likelihood....
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...We examined optimizing two more complicated objectives than those described above: Multinomial Logistic Regression (using the Softmax function) and (2- dimensional) Conditional Random Fields (CRFs)....
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...6 do this is to replace λ with a variable t ∝ 1/λ and solve the constrained problem: min x L(x) s.t.||x||1 ≤ t (6) Recently, [11] presented an algorithm for L1-regularized Logistic Regression, where the Logistic Regression IRLS update is computed subject to the constraint ||x||1 ≤ t....
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14 May 2007
TL;DR: It is found that the discriminatively trained CRF performs as well as or better than an HMM even when the model features do not violate the independence assumptions of the HMM, and it is confirmed that CRFs are robust against any degradation in performance.
Abstract: Activity recognition is a key component for creating intelligent, multi-agent systems. Intrinsically, activity recognition is a temporal classification problem. In this paper, we compare two models for temporal classification: hidden Markov models (HMMs), which have long been applied to the activity recognition problem, and conditional random fields (CRFs). CRFs are discriminative models for labeling sequences. They condition on the entire observation sequence, which avoids the need for independence assumptions between observations. Conditioning on the observations vastly expands the set of features that can be incorporated into the model without violating its assumptions. Using data from a simulated robot tag domain, chosen because it is multi-agent and produces complex interactions between observations, we explore the differences in performance between the discriminatively trained CRF and the generative HMM. Additionally, we examine the effect of incorporating features which violate independence assumptions between observations; such features are typically necessary for high classification accuracy. We find that the discriminatively trained CRF performs as well as or better than an HMM even when the model features do not violate the independence assumptions of the HMM. In cases where features depend on observations from many time steps, we confirm that CRFs are robust against any degradation in performance.
377 citations
Cites background from "EfficientL 1 regularized logistic r..."
...[58] consider the specialized case of logistic regression....
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TL;DR: It is shown that comparative evaluations of predictors that do not address two types of circularity may erroneously conclude that circularity confounded tools are most accurate among all tools, and may even outperform optimized combinations of tools.
Abstract: Prioritizing missense variants for further experimental investigation is a key challenge in current sequencing studies for exploring complex and Mendelian diseases A large number of in silico tools have been employed for the task of pathogenicity prediction, including PolyPhen-2, SIFT, FatHMM, MutationTaster-2, MutationAssessor, Combined Annotation Dependent Depletion, LRT, phyloP, and GERP++, as well as optimized methods of combining tool scores, such as Condel and Logit Due to the wealth of these methods, an important practical question to answer is which of these tools generalize best, that is, correctly predict the pathogenic character of new variants We here demonstrate in a study of 10 tools on five datasets that such a comparative evaluation of these tools is hindered by two types of circularity: they arise due to (1) the same variants or (2) different variants from the same protein occurring both in the datasets used for training and for evaluation of these tools, which may lead to overly optimistic results We show that comparative evaluations of predictors that do not address these types of circularity may erroneously conclude that circularity confounded tools are most accurate among all tools, and may even outperform optimized combinations of tools
282 citations
Cites methods from "EfficientL 1 regularized logistic r..."
...Text S1), we compared the original FatHMM-W method with an L1-regularized logistic regression [Lee et al., 2006] over the log-transformed features ln(Wn) and ln(Wd)....
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TL;DR: Recent mathematical advances that provide ways to combat dimensionality in specific situations are reviewed, shed light on two dual questions in neuroscience, and how do brains themselves process information in their intrinsically high-dimensional patterns of neural activity as well as learn meaningful, generalizable models of the external world from limited experience.
Abstract: The curse of dimensionality poses severe challenges to both technical and conceptual progress in neuroscience. In particular, it plagues our ability to acquire, process, and model high-dimensional data sets. Moreover, neural systems must cope with the challenge of processing data in high dimensions to learn and operate successfully within a complex world. We review recent mathematical advances that provide ways to combat dimensionality in specific situations. These advances shed light on two dual questions in neuroscience. First, how can we as neuroscientists rapidly acquire high-dimensional data from the brain and subsequently extract meaningful models from limited amounts of these data? And second, how do brains themselves process information in their intrinsically high-dimensional patterns of neural activity as well as learn meaningful, generalizable models of the external world from limited experience?
274 citations
Cites methods from "EfficientL 1 regularized logistic r..."
...Indeed it has been used successfully in learning logistic regression (Lee et al. 2006b) and in various graphical models (Lee et al. 2006a, Wainwright et al. 2007), as well as in point process models of neuronal spike trains (Kelly et al. 2010)....
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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
Additional excerpts
...(Nelder & Wedderbum 1972; McCullagh & Nelder 1989)...
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...(Nelder & Wedderbum 1972; McCullagh & Nelder 1989 )...
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