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.
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Posted Content•
TL;DR: A non-linear correlation test for non-binary finite support case that involves hashing a variable and then correlating with the output variable is proposed, motivated by insights from the boolean case.
Abstract: We consider support recovery in the quadratic logistic regression setting - where the target depends on both p linear terms $x_i$ and up to $p^2$ quadratic terms $x_i x_j$. Quadratic terms enable prediction/modeling of higher-order effects between features and the target, but when incorporated naively may involve solving a very large regression problem. We consider the sparse case, where at most $s$ terms (linear or quadratic) are non-zero, and provide a new faster algorithm. It involves (a) identifying the weak support (i.e. all relevant variables) and (b) standard logistic regression optimization only on these chosen variables. The first step relies on a novel insight about correlation tests in the presence of non-linearity, and takes $O(pn)$ time for $n$ samples - giving potentially huge computational gains over the naive approach. Motivated by insights from the boolean case, we propose a non-linear correlation test for non-binary finite support case that involves hashing a variable and then correlating with the output variable. We also provide experimental results to demonstrate the effectiveness of our methods.
1 citations
Cites background or methods from "EfficientL 1 regularized logistic r..."
...Several methods have been used for sparse logistic regression, including maximum likelihood estimation with regularization and greedy methods [15, 10, 11]....
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...Obtaining faster convex optimization based solvers for this particular convex optimization problem is the focus for many prior studies (see [11][10] and [6])....
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TL;DR: The proposed particle swarm optimization is used to optimize the regularization parameter of the logistic regression and improve the accuracy of the entity alias classification significantly to 4.98% compared to that of thelogistic regression.
Abstract: An improvement in detection of alias names of an entity is an important factor in many cases like terrorist and criminal network. In this paper, the social network properties are used to construct a feature set for classification. The proposed particle swarm optimization is used to optimize the regularization parameter of the logistic regression and improve the accuracy of the entity alias classification significantly to 4.98% compared to that of the logistic regression. The experimental results demonstrated its performance and the results are compared to the logistic regression with alias Detection Dataset from Auton Lab.
1 citations
TL;DR: A unified framework for likelihood-based regression modeling when the response variable has finite support is proposed, which includes models previously considered for interval-censored variables with log-concave distributions as special cases.
Abstract: We propose a unified framework for likelihood-based regression modeling when the response variable has finite support. Our work is motivated by the fact that, in practice, observed data are discrete and bounded. The proposed methods assume a model which includes models previously considered for interval-censored variables with log-concave distributions as special cases. The resulting log-likelihood is concave, which we use to establish asymptotic normality of its maximizer as the number of observations tends to infinity with the number of parameters fixed, and rates of convergence of -regularized estimators when the true parameter vector is sparse and and both tend to infinity with . We consider an inexact proximal Newton algorithm for computing estimates and give theoretical guarantees for its convergence. The range of possible applications is wide, including but not limited to survival analysis in discrete time, the modeling of outcomes on scored surveys and questionnaires, and, more generally, interval-censored regression. The applicability and usefulness of the proposed methods are illustrated in simulations and data examples. This article is protected by copyright. All rights reserved.
1 citations
26 Jun 2016
TL;DR: This paper develops a regularization framework for logistic regression using Jeffreys prior, which is free of any tuning parameters, and shows that the proposed regularization outperforms other well-known regularization approaches.
Abstract: Logistic regression is a statistical model widely used for solving classification problems. Maximum likelihood is used train the model parameters. When data from two classes is linearly separable, maximum likelihood is ill-posed. To address this problem as well as to handle over-fitting issues, regularization is commonly considered. A regularization coefficient is used to control the tradeoff between model complexity and data fit and cross-validation is applied to determine the coefficient. In this paper, we develop a regularization framework for logistic regression using Jeffreys prior, which is free of any tuning parameters. Our experiments show that the proposed regularization outperforms other well-known regularization approaches.
1 citations
Cites background from "EfficientL 1 regularized logistic r..."
...For some settings, regularization is critical [5]....
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...As discussed in Section 2, l2- [1] and l1-regularization [5] terms are routinely considered for training a LR model....
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DOI•
01 Jan 2014
TL;DR: This thesis examines components of test-time cost, and develops different strategies to trade-off accuracy and high classifier evaluation cost of nonparametric classifiers, and proposes a model compression strategy and develop Compressed Vector Machines (CVM).
Abstract: OF THE DISSERTATION Supervised Machine Learning Under Test-Time Resource Constraints: A Trade-off Between Accuracy and Cost by Zhixiang (Eddie) Xu Doctor of Philosophy in Computer Science Washington University in St. Louis, 2014 Research Advisor: Professor Kilian Q. Weinberger, Chair The past decade has witnessed how the field of machine learning has established itself as a necessary component in several multi-billion-dollar industries. The real-world industrial setting introduces an interesting new problem to machine learning research: computational resources must be budgeted and cost must be strictly accounted for during test-time. A typical problem is that if an application consumes x additional units of cost during test-time, but will improve accuracy by y percent, should the additional x resources be allocated? The core of this problem is a trade-off between accuracy and cost. In this thesis, we examine components of test-time cost, and develop different strategies to manage this trade-off. We first investigate test-time cost and discover that it typically consists of two parts: feature extraction cost and classifier evaluation cost. The former reflects the computational efforts of transforming data instances to feature vectors, and could be highly variable when features are heterogeneous. The latter reflects the effort of evaluating a classifier, which could be substantial, in particular nonparametric algorithms. We then propose three strategies ix to explicitly trade-off accuracy and the two components of test-time cost during classifier training. To budget the feature extraction cost, we first introduce two algorithms: GreedyMiser [132] and Anytime Representation Learning (AFR)[135]. GreedyMiser employs a strategy that incorporates the extraction cost information during classifier training to explicitly minimize the test-time cost. AFR extends GreedyMiser to learn a cost-sensitive feature representation rather than a classifier, and turns traditional Support Vector Machines (SVM) [110] into testtime cost-sensitive anytime classifiers. GreedyMiser and AFR are evaluated on two real-world data sets from two different application domains, and both achieve record performance. We then introduce Cost Sensitive Tree of Classifiers (CSTC)[134] and Cost Sensitive Cascade of Classifiers (CSCC)[137], which share a common strategy that trades-off the accuracy and the amortized test-time cost. CSTC introduces a tree structure and directs test inputs along different tree traversal paths, each is optimized for a specific sub-partition of the input space, extracting different, specialized subsets of features. CSCC extends CSTC and builds a linear cascade, instead of a tree, to cope with class-imbalanced binary classification tasks. Since both CSTC and CSCC extract different features for different inputs, the amortized test-time cost is greatly reduced while maintaining high accuracy. Both approaches out-perform the current state-of-the-art on real-world data sets. To trade-off accuracy and high classifier evaluation cost of nonparametric classifiers, we propose a model compression strategy and develop Compressed Vector Machines (CVM). CVM focuses on the nonparametric kernel Support Vector Machines (SVM), whose testtime evaluation cost is typically substantial when learned from large training sets. CVM is a post-processing algorithm which compresses the learned SVM model by reducing and
1 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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