LIBLINEAR is an open source library for large-scale linear classification. It supports logistic regression and linear support vector machines. We provide easy-to-use command-line tools and library calls for users and developers. Comprehensive documents are available for both beginners and advanced users. Experiments demonstrate that LIBLINEAR is very efficient on large sparse data sets.

We describe and analyze a simple and effective stochastic sub-gradient descent algorithm for solving the optimization problem cast by Support Vector Machines (SVM). We prove that the number of iterations required to obtain a solution of accuracy ɛ is Õ(1/ɛ), where each iteration operates on a single training example. In contrast, previous analyses of stochastic gradient descent methods for SVMs require Ω(1/ɛ2) iterations. As in previously devised SVM solvers, the number of iterations also scales linearly with 1/λ, where λ is the regularization parameter of SVM. For a linear kernel, the total run-time of our method is Õ(d/(λɛ)), where d is a bound on the number of non-zero features in each example. Since the run-time does not depend directly on the size of the training set, the resulting algorithm is especially suited for learning from large datasets. Our approach also extends to non-linear kernels while working solely on the primal objective function, though in this case the runtime does depend linearly on the training set size. Our algorithm is particularly well suited for large text classification problems, where we demonstrate an order-of-magnitude speedup over previous SVM learning methods.

Anomaly detection is an important problem that has been researched within diverse research areas and application domains. Many anomaly detection techniques have been specifically developed for certain application domains, while others are more generic. This survey tries to provide a structured and comprehensive overview of the research on anomaly detection. We have grouped existing techniques into different categories based on the underlying approach adopted by each technique. For each category we have identified key assumptions, which are used by the techniques to differentiate between normal and anomalous behavior. When applying a given technique to a particular domain, these assumptions can be used as guidelines to assess the effectiveness of the technique in that domain. For each category, we provide a basic anomaly detection technique, and then show how the different existing techniques in that category are variants of the basic technique. This template provides an easier and succinct understanding of the techniques belonging to each category. Further, for each category, we identify the advantages and disadvantages of the techniques in that category. We also provide a discussion on the computational complexity of the techniques since it is an important issue in real application domains. We hope that this survey will provide a better understanding of the di®erent directions in which research has been done on this topic, and how techniques developed in one area can be applied in domains for which they were not intended to begin with.

To accelerate the training of kernel machines, we propose to map the input data to a randomized low-dimensional feature space and then apply existing fast linear methods. Our randomized features are designed so that the inner products of the transformed data are approximately equal to those in the feature space of a user specified shift-invariant kernel. We explore two sets of random features, provide convergence bounds on their ability to approximate various radial basis kernels, and show that in large-scale classification and regression tasks linear machine learning algorithms that use these features outperform state-of-the-art large-scale kernel machines. 1

In many applications, data appear with a huge number of instances as well as features. Linear Support Vector Machines (SVM) is one of the most popular tools to deal with such large-scale sparse data. This paper presents a novel dual coordinate descent method for linear SVM with L1- and L2loss functions. The proposed method is simple and reaches an ɛ-accurate solution in O(log(1/ɛ)) iterations. Experiments indicate that our method is much faster than state of the art solvers such as Pegasos, TRON, SVM perf, and a recent primal coordinate descent implementation. 1.

A wide variety of machine learning problems can be described as minimizing a regularized risk functional, with different algorithms using different notions of risk and different regularizers. Examples include linear Support Vector Machines (SVMs), Logistic Regression, Conditional Random Fields (CRFs), and Lasso amongst others. This paper describes the theory and implementation of a highly scalable and modular convex solver which solves all these estimation problems. It can be parallelized on a cluster of workstations, allows for data-locality, and can deal with regularizers such as ℓ1 and ℓ2 penalties. At present, our solver implements 20 different estimation problems, can be easily extended, scales to millions of observations, and is up to 10 times faster than specialized solvers for many applications. The open source code is freely available as part of the ELEFANT toolbox.

Large-scale logistic regression arises in many applications such as document classification and natural language processing. In this paper, we apply a trust region Newton method to maximize the log-likelihood of the logistic regression model. The proposed method uses only approximate Newton steps in the beginning, but achieves fast convergence in the end. Experiments show that it is faster than the commonly used quasi Newton approach for logistic regression. We also compare it with existing linear SVM implementations. 1

Log-linear and maximum-margin models are two commonly-used methods in supervised machine learning, and are frequently used in structured prediction problems. Efficient learning of parameters in these models is therefore an important problem, and becomes a key factor when learning from very large data sets. This paper describes exponentiated gradient (EG) algorithms for training such models, where EG updates are applied to the convex dual of either the log-linear or maxmargin objective function; the dual in both the log-linear and max-margin cases corresponds to minimizing a convex function with simplex constraints. We study both batch and online variants of the algorithm, and provide rates of convergence for both cases. In the max-margin case, O ( 1 ε) EG updates are required to reach a given accuracy ε in the dual; in contrast, for log-linear models only O(log (1/ε)) updates are required. For both the max-margin and log-linear cases, our bounds suggest that the online EG algorithm requires a factor of n less computation to reach a desired accuracy than the batch EG algorithm, where n is the number of training examples. Our experiments confirm that the online algorithms are much faster than the batch algorithms in practice. We describe how the EG updates factor in a convenient way for structured prediction problems, allowing the algorithms to be

We discuss how the runtime of SVM optimization should decrease as the size of the training data increases. We present theoretical and empirical results demonstrating how a simple subgradient descent approach indeed displays such behavior, at least for linear kernels. 1.

We present a globally convergent method for regularized risk minimization problems. Our method applies to Support Vector estimation, regression, Gaussian Processes, and any other regularized risk minimization setting which leads to a convex optimization problem. SVMPerf can be shown to be a special case of our approach. In addition to the unified framework we present tight convergence bounds, which show that our algorithm converges in O(1/ɛ) steps to ɛ precision for general convex problems and in O(log(1/ɛ)) steps for continuously differentiable problems. We demonstrate in experiments the performance of our approach. 1