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72
Pegasos: Primal Estimated subgradient solver for SVM
"... We describe and analyze a simple and effective stochastic subgradient 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 singl ..."
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Cited by 531 (21 self)
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We describe and analyze a simple and effective stochastic subgradient 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 runtime of our method is Õ(d/(λɛ)), where d is a bound on the number of nonzero features in each example. Since the runtime 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 nonlinear 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 orderofmagnitude speedup over previous SVM learning methods.
Nonprojective dependency parsing using spanning tree algorithms
 In Proceedings of Human Language Technology Conference and Conference on Empirical Methods in Natural Language Processing
, 2005
"... We formalize weighted dependency parsing as searching for maximum spanning trees (MSTs) in directed graphs. Using this representation, the parsing algorithm of Eisner (1996) is sufficient for searching over all projective trees in O(n 3) time. More surprisingly, the representation is extended natura ..."
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Cited by 377 (10 self)
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We formalize weighted dependency parsing as searching for maximum spanning trees (MSTs) in directed graphs. Using this representation, the parsing algorithm of Eisner (1996) is sufficient for searching over all projective trees in O(n 3) time. More surprisingly, the representation is extended naturally to nonprojective parsing using ChuLiuEdmonds (Chu and Liu, 1965; Edmonds, 1967) MST algorithm, yielding an O(n 2) parsing algorithm. We evaluate these methods on the Prague Dependency Treebank using online largemargin learning techniques (Crammer et al., 2003; McDonald et al., 2005) and show that MST parsing increases efficiency and accuracy for languages with nonprojective dependencies. 1
Online largemargin training of dependency parsers
 In Proc. ACL
, 2005
"... We present an effective training algorithm for linearlyscored dependency parsers that implements online largemargin multiclass training (Crammer and Singer, 2003; Crammer et al., 2003) on top of efficient parsing techniques for dependency trees (Eisner, 1996). The trained parsers achieve a competi ..."
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Cited by 293 (23 self)
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We present an effective training algorithm for linearlyscored dependency parsers that implements online largemargin multiclass training (Crammer and Singer, 2003; Crammer et al., 2003) on top of efficient parsing techniques for dependency trees (Eisner, 1996). The trained parsers achieve a competitive dependency accuracy for both English and Czech with no language specific enhancements. 1
Adaptive Subgradient Methods for Online Learning and Stochastic Optimization
, 2010
"... Stochastic subgradient methods are widely used, well analyzed, and constitute effective tools for optimization and online learning. Stochastic gradient methods ’ popularity and appeal are largely due to their simplicity, as they largely follow predetermined procedural schemes. However, most common s ..."
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Cited by 287 (3 self)
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Stochastic subgradient methods are widely used, well analyzed, and constitute effective tools for optimization and online learning. Stochastic gradient methods ’ popularity and appeal are largely due to their simplicity, as they largely follow predetermined procedural schemes. However, most common subgradient approaches are oblivious to the characteristics of the data being observed. We present a new family of subgradient methods that dynamically incorporate knowledge of the geometry of the data observed in earlier iterations to perform more informative gradientbased learning. The adaptation, in essence, allows us to find needles in haystacks in the form of very predictive but rarely seenfeatures. Ourparadigmstemsfromrecentadvancesinstochasticoptimizationandonlinelearning which employ proximal functions to control the gradient steps of the algorithm. We describe and analyze an apparatus for adaptively modifying the proximal function, which significantly simplifies setting a learning rate and results in regret guarantees that are provably as good as the best proximal function that can be chosen in hindsight. In a companion paper, we validate experimentally our theoretical analysis and show that the adaptive subgradient approach outperforms stateoftheart, but nonadaptive, subgradient algorithms. 1
Learning globallyconsistent local distance functions for shapebased image retrieval and classification
 In ICCV
, 2007
"... We address the problem of visual category recognition by learning an imagetoimage distance function that attempts to satisfy the following property: the distance between images from the same category should be less than the distance between images from different categories. We use patchbased feat ..."
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Cited by 150 (3 self)
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We address the problem of visual category recognition by learning an imagetoimage distance function that attempts to satisfy the following property: the distance between images from the same category should be less than the distance between images from different categories. We use patchbased feature vectors common in object recognition work as a basis for our imagetoimage distance functions. Our largemargin formulation for learning the distance functions is similar to formulations used in the machine learning literature on distance metric learning, however we differ in that we learn local distance functions— a different parameterized function for every image of our training set—whereas typically a single global distance function is learned. This was a novel approach first introduced in Frome, Singer, & Malik, NIPS 2006. In that work we learned the local distance functions independently, and the outputs of these functions could not be compared at test time without the use of additional heuristics or training. Here we introduce a different approach that has the advantage that it learns distance functions that are globally consistent in that they can be directly compared for purposes of retrieval and classification. The output of the learning algorithm are weights assigned to the image features, which is intuitively appealing in the computer vision setting: some features are more salient than others, and which are more salient depends on the category, or image, being considered. We train and test using the Caltech 101 object recognition benchmark. Using fifteen training images per category, we achieved a mean recognition rate of 63.2 % and
Online and batch learning of pseudometrics
 In ICML
, 2004
"... We describe and analyze an online algorithm for supervised learning of pseudometrics. The algorithm receives pairs of instances and predicts their similarity according to a pseudometric. The pseudometrics we use are quadratic forms parameterized by positive semidefinite matrices. The core of the ..."
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Cited by 96 (4 self)
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We describe and analyze an online algorithm for supervised learning of pseudometrics. The algorithm receives pairs of instances and predicts their similarity according to a pseudometric. The pseudometrics we use are quadratic forms parameterized by positive semidefinite matrices. The core of the algorithm is an update rule that is based on successive projections onto the positive semidefinite cone and onto halfspace constraints imposed by the examples. We describe an efficient procedure for performing these projections, derive a worst case mistake bound on the similarity predictions, and discuss a dual version of the algorithm in which it is simple to incorporate kernel operators. The online algorithm also serves as a building block for deriving a largemargin batch algorithm. We demonstrate the merits of the proposed approach by conducting experiments on MNIST dataset and on document filtering. 1.
Large Margin Hierarchical Classification
 In Proceedings of the TwentyFirst International Conference on Machine Learning
"... We present an algorithmic framework for supervised classification learning where the set of labels is organized in a predefined hierarchical structure. This structure is encoded by a rooted tree which induces a metric over the label set. ..."
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Cited by 90 (6 self)
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We present an algorithmic framework for supervised classification learning where the set of labels is organized in a predefined hierarchical structure. This structure is encoded by a rooted tree which induces a metric over the label set.
Adaptive Regularization of Weight Vectors
 Advances in Neural Information Processing Systems 22
, 2009
"... We present AROW, a new online learning algorithm that combines several useful properties: large margin training, confidence weighting, and the capacity to handle nonseparable data. AROW performs adaptive regularization of the prediction function upon seeing each new instance, allowing it to perform ..."
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Cited by 65 (14 self)
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We present AROW, a new online learning algorithm that combines several useful properties: large margin training, confidence weighting, and the capacity to handle nonseparable data. AROW performs adaptive regularization of the prediction function upon seeing each new instance, allowing it to perform especially well in the presence of label noise. We derive a mistake bound, similar in form to the second order perceptron bound, that does not assume separability. We also relate our algorithm to recent confidenceweighted online learning techniques and show empirically that AROW achieves stateoftheart performance and notable robustness in the case of nonseparable data. 1
The Forgetron: A kernelbased perceptron on a fixed budget
 In Advances in Neural Information Processing Systems 18
, 2005
"... The Perceptron algorithm, despite its simplicity, often performs well on online classification problems. The Perceptron becomes especially effective when it is used in conjunction with kernels. However, a common difficulty encountered when implementing kernelbased online algorithms is the amount of ..."
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Cited by 44 (5 self)
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The Perceptron algorithm, despite its simplicity, often performs well on online classification problems. The Perceptron becomes especially effective when it is used in conjunction with kernels. However, a common difficulty encountered when implementing kernelbased online algorithms is the amount of memory required to store the online hypothesis, which may grow unboundedly. In this paper we describe and analyze a new infrastructure for kernelbased learning with the Perceptron while adhering to a strict limit on the number of examples that can be stored. We first describe a template algorithm, called the Forgetron, for online learning on a fixed budget. We then provide specific algorithms and derive a unified mistake bound for all of them. To our knowledge, this is the first online learning paradigm which, on one hand, maintains a strict limit on the number of examples it can store and, on the other hand, entertains a relative mistake bound. We also present experiments with real datasets which underscore the merits of our approach. 1