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
More filters
08 Dec 2014
TL;DR: The proposed coordinate solver is simpler, in the sense that there is no need for any line search, and can directly be used for small to large scale learning problems with elastic net regularization.
Abstract: In this work, a coordinate solver for elastic net regularized logistic regression is proposed. In particular, a method based on majorization maximization using a cubic function is derived. This to reliably and accurately optimize the objective function at each step without resorting to line search. Experiments show that the proposed solver is comparable to, or improves, state-of-the-art solvers. The proposed method is simpler, in the sense that there is no need for any line search, and can directly be used for small to large scale learning problems with elastic net regularization.
7 citations
Cites methods from "EfficientL 1 regularized logistic r..."
...Recent approaches to solve L1-regularized logistic regression has focused on stochastic methods [5], [6], [4], efficient coordinate decent methods [7], [8], methods based on quasi-Newton solvers [9], [10], [11] and interior point techniques [12]....
[...]
10 Aug 2015
TL;DR: A Maximum Likelihood definition of the "optimal" client clustering along with an efficient Expectation-Maximization clustering algorithm that can be applied in large marketplaces is proposed and shown to yield significant gains compared to "learning" the same hiring criteria for all clients.
Abstract: An important problem that online work marketplaces face is grouping clients into clusters, so that in each cluster clients are similar with respect to their hiring criteria. Such a separation allows the marketplace to "learn" more accurately the hiring criteria in each cluster and recommend the right contractor to each client, for a successful collaboration. We propose a Maximum Likelihood definition of the "optimal" client clustering along with an efficient Expectation-Maximization clustering algorithm that can be applied in large marketplaces. Our results on the job hirings at oDesk over a seven-month period show that our client-clustering approach yields significant gains compared to "learning" the same hiring criteria for all clients. In addition, we analyze the clustering results to find interesting differences between the hiring criteria in the different groups of clients.
7 citations
Proceedings Article•
01 Jan 2010TL;DR: This paper studies the combination of compression and l1-norm regularization in a machine learning context: learning compressible models and shows that use of a compression operation provides an opportunity to leverage auxiliary information from various sources, e.g., domain knowledge, coding theories, unlabeled data.
Abstract: In this paper, we study the combination of compression and l1-norm regularization in a machine learning context: learning compressible models. By including a compression operation into the l1 regularization, the assumption on model sparsity is relaxed to compressibility: model coefficients are compressed before being penalized, and sparsity is achieved in a compressed domain rather than the original space. We focus on the design of different compression operations, by which we can encode various compressibility assumptions and inductive biases, e.g., piecewise local smoothness, compacted energy in the frequency domain, and semantic correlation. We show that use of a compression operation provides an opportunity to leverage auxiliary information from various sources, e.g., domain knowledge, coding theories, unlabeled data. We conduct extensive experiments on brain-computer interfacing, handwritten character recognition and text classification. Empirical results show clear improvements in prediction performance by including compression in l1 regularization. We also analyze the learned model coefficients under appropriate compressibility assumptions, which further demonstrate the advantages of learning compressible models instead of sparse models.
7 citations
Cites methods from "EfficientL 1 regularized logistic r..."
...Lasso is implemented using the spgl1 Matlab solver(3), and l1regularized logistic regression is implemented as [15] using lasso....
[...]
14 Dec 2013
TL;DR: Empirical studies compare LRBC with several state-of-the-art algorithms on an actual ad clicking data set and demonstrates that LRBC method is able to exhibit much better classification performance, and the distributed process for bias correction also scales well.
Abstract: Logistic regression is a classical classification method, it has been used widely in many applications which have binary dependent variable. However, when the data sets are imbalanced, the probability of rare event is underestimated in the use of traditional logistic regression. With data explosion in recent years, some researchers propose large scale logistic regression which still fails to consider the rare event, therefore, there exists bias when applying their models for large scale data sets with rare events. To address the problems, this paper proposes LRBC method to correct bias of logistic regression for large scale data sets with rare events. Empirical studies compare LRBC with several state-of-the-art algorithms on an actual ad clicking data set. It demonstrates that LRBC method is able to exhibit much better classification performance, and the distributed process for bias correction also scales well.
7 citations
07 Jul 2010
TL;DR: A novel method that learns the structure of Bayesian networks for BOA, called L1BOA, which uses L1-regularized regression to find the candidate parents of each variable, which leads to a sparse but nearly optimized network structure.
Abstract: The Bayesian optimization algorithm (BOA) uses Bayesian networks to explore the dependencies between decision variables of an optimization problem in pursuit of both faster speed of convergence and better solution quality. In this paper, a novel method that learns the structure of Bayesian networks for BOA is proposed. The proposed method, called L1BOA, uses L1-regularized regression to find the candidate parents of each variable, which leads to a sparse but nearly optimized network structure. The proposed method improves the efficiency of the structure learning in BOA due to the reduction and automated control of network complexity introduced with L1-regularized learning. Experimental studies on different types of benchmark problems are carried out, which show that L1BOA outperforms the standard BOA when no a-priori knowledge about the problem structure is available, and nearly achieves the best performance of BOA that applies explicit complexity controls.
7 citations
Cites methods from "EfficientL 1 regularized logistic r..."
...In this paper, the optimization is formulated as a bound-constrained problem and is solved by using an e.cient Two-Metric Projection strategy [1] with IRLS-LARS [10] to scale the gradient....
[...]
...In this paper, the optimization is formulated as a bound-constrained problem and is solved by using an efficient Two-Metric Projection strategy [1] with IRLS-LARS [10] to scale the gradient....
[...]
References
More filters
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....
[...]
...See, Tibshirani (1996) for details.)...
[...]
...(Tibshirani 1996) Several algorithms have been developed to solve L1 constrained least squares problems....
[...]
Book•
[...]
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…...
[...]
...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....
[...]
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)...
[...]
...(Nelder & Wedderbum 1972; McCullagh & Nelder 1989 )...
[...]