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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Dissertation•
01 Jun 2016
TL;DR: This document describes three new techniques for automated domain-independent deep representation learning that are computationally efficient relative to existing comparable methods, and often produce representations that yield improved performance on machine learning tasks.
Abstract: New Techniques in Deep Representation Learning Galen Andrew Chair of the Supervisory Committee: Associate Professor Emanuel Todorov CSE, joint with AMATH The choice of feature representation can have a large impact on the success of a machine learning algorithm at solving a given problem. Although human engineers employing taskspecific domain knowledge still play a key role in feature engineering, automated domainindependent algorithms, in particular methods from the area of deep learning, are proving more and more useful on a variety of difficult tasks, including speech recognition, image analysis, natural language processing, and game playing. This document describes three new techniques for automated domain-independent deep representation learning: • Sequential deep neural networks (SDNN) learn representations of data that is continuously extended in time such as audio. Unlike “sliding window” neural networks applied to such data or convolutional neural networks, SDNNs are capable of capturing temporal patterns of arbitrary span, and can encode that discovered features should exhibit greater or lesser degrees of continuity through time. • Deep canonical correlation analysis (DCCA) is a method to learn parametric nonlinear transformations of multiview data that capture latent shared aspects of the views so that the learned representation of each view is maximally predictive of (and predicted by) the other. DCCA may be able to learn to represent abstract properties when the two views are not superficially related. • The orthant-wise limited-memory quasi-Newton algorithm (OWL-QN) can be employed to train any parametric representation mapping to produce parameters that are sparse (mostly zero), resulting in more interpretable and more compact models. If the prior assumption that parameters should be sparse is reasonable for the data source, training with OWL-QN should also improve generalization. Experiments on many different tasks demonstrate that these new methods are computationally efficient relative to existing comparable methods, and often produce representations that yield improved performance on machine learning tasks.
1 citations
Posted Content•
TL;DR: In this article, a nonlinear primal-dual algorithm was proposed to solve a logistic regression problem regularized by an elastic net penalty in O(T(m,n) time.
Abstract: Logistic regression is a widely used statistical model to describe the relationship between a binary response variable and predictor variables in data sets. It is often used in machine learning to identify important predictor variables. This task, variable selection, typically amounts to fitting a logistic regression model regularized by a convex combination of $\ell_1$ and $\ell_{2}^{2}$ penalties. Since modern big data sets can contain hundreds of thousands to billions of predictor variables, variable selection methods depend on efficient and robust optimization algorithms to perform well. State-of-the-art algorithms for variable selection, however, were not traditionally designed to handle big data sets; they either scale poorly in size or are prone to produce unreliable numerical results. It therefore remains challenging to perform variable selection on big data sets without access to adequate and costly computational resources. In this paper, we propose a nonlinear primal-dual algorithm that addresses these shortcomings. Specifically, we propose an iterative algorithm that provably computes a solution to a logistic regression problem regularized by an elastic net penalty in $O(T(m,n)\log(1/\epsilon))$ operations, where $\epsilon \in (0,1)$ denotes the tolerance and $T(m,n)$ denotes the number of arithmetic operations required to perform matrix-vector multiplication on a data set with $m$ samples each comprising $n$ features. This result improves on the known complexity bound of $O(\min(m^2n,mn^2)\log(1/\epsilon))$ for first-order optimization methods such as the classic primal-dual hybrid gradient or forward-backward splitting methods.
Journal Article•
TL;DR: Numerical experiments validate the necessity and effectiveness of mentor identification towards improving the performance of a k-NN based product feature recommendation system for a real-world dataset and demonstrate that the proposed algorithm achieves a trade-off that is both quantitatively and qualitatively distinct from alternative approaches to mentor identification.
Abstract: A typical software tool for solving complex problems tends to expose a rich set of features to its users. This creates challenges such as new users facing a steep onboarding experience and current users tending to use only a small fraction of the software’s features. This paper describes and solves an unsupervised mentor pattern identification problem from product usage logs for softening both challenges. The problem is formulated as identifying a set of users (mentors) that satisfies three mentor qualification metrics: (a) the mentor set is small, (b) every user is close to some mentor as per usage pattern, and (c) every feature has been used by some mentor. The proposed solution models the task as a non-convex variant of an Open image in new window regularized logistic regression problem and develops an alternating minimization style algorithm to solve it. Numerical experiments validate the necessity and effectiveness of mentor identification towards improving the performance of a k-NN based product feature recommendation system for a real-world dataset. Further, t-SNE visuals demonstrate that the proposed algorithm achieves a trade-off that is both quantitatively and qualitatively distinct from alternative approaches to mentor identification such as Maximum Marginal Relevance and K-means.
19 Jul 2021
TL;DR: The results show that the Gradient Boosting Decision Tree model is the most accurate prediction model among the seven prediction models, and has the best prediction effect, which can accurately predict customers who will be lost.
Abstract: According to the traditional machine learning model as a learner, customer loss prediction model is implemented by Logistic Regression, Decision Tree, Random Forest, Support Vector Machine, Gradient Boosting Decision Tree, Multi-Layer Perceptron algorithm and Long Short-term Memory Neutral Network in deep learning, and experiments used the real sample data set of about 30,000 customers of an airline. The results show that the Gradient Boosting Decision Tree model is the most accurate prediction model among the seven prediction models, and has the best prediction effect, which can accurately predict customers who will be lost.
Posted Content•
TL;DR: An extension of the regularized least-squares in which the estimation parameters are stretchable is introduced and studied, and an input transformation is proposed to maintain the computation of power root terms within the real space.
Abstract: An extension of the regularized least-squares in which the estimation parameters are stretchable is introduced and studied in this paper. The solution of this ridge regression with stretchable parameters is given in primal and dual spaces and in closed-form. Essentially, the proposed solution stretches the covariance computation by a power term, thereby compressing or amplifying the estimation parameters. To maintain the computation of power root terms within the real space, an input transformation is proposed. The results of an empirical evaluation in both synthetic and real-world data illustrate that the proposed method is effective for compressive learning with high-dimensional data.
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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