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36
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 284 (15 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.
A Dual Coordinate Descent Method for Largescale Linear SVM
"... 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 largescale sparse data. This paper presents a novel dual coordinate descent method for linear SVM with L1 and L2loss functi ..."
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Cited by 99 (11 self)
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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 largescale 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 ɛaccurate 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.
SVM Optimization: Inverse Dependence on Training Set Size
"... 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. ..."
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Cited by 53 (13 self)
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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.
Learning Graph Matching
"... As a fundamental problem in pattern recognition, graph matching has found a variety of applications in the field of computer vision. In graph matching, patterns are modeled as graphs and pattern recognition amounts to finding a correspondence between the nodes of different graphs. There are many way ..."
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Cited by 41 (9 self)
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As a fundamental problem in pattern recognition, graph matching has found a variety of applications in the field of computer vision. In graph matching, patterns are modeled as graphs and pattern recognition amounts to finding a correspondence between the nodes of different graphs. There are many ways in which the problem has been formulated, but most can be cast in general as a quadratic assignment problem, where a linear term in the objective function encodes node compatibility functions and a quadratic term encodes edge compatibility functions. The main research focus in this theme is about designing efficient algorithms for solving approximately the quadratic assignment problem, since it is NPhard. In this paper, we turn our attention to the complementary problem: how to estimate compatibility functions such that the solution of the resulting graph matching problem best matches the expected solution that a human would manually provide. We present a method for learning graph matching: the training examples are pairs of graphs and the “labels” are matchings between pairs of graphs. We present experimental results with real image data which give evidence that learning can improve the performance of standard graph matching algorithms. In particular, it turns out that linear assignment with such a learning scheme may improve over stateoftheart quadratic assignment relaxations. This finding suggests that for a range of problems where quadratic assignment was thought to be essential for securing good results, linear assignment, which is far more efficient, could be just sufficient if learning is performed. This enables speedups of graph matching by up to 4 orders of magnitude while retaining stateoftheart accuracy. 1.
Bundle Methods for Regularized Risk Minimization
"... 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), Gaussian Processes, Logistic Regression, Conditional ..."
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Cited by 35 (2 self)
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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), Gaussian Processes, Logistic Regression, Conditional Random Fields (CRFs), and Lasso amongst others. This paper describes the theory and implementation of a scalable and modular convex solver which solves all these estimation problems. It can be parallelized on a cluster of workstations, allows for datalocality, and can deal with regularizers such as L1 and L2 penalties. 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 the performance of our general purpose solver on a variety of publicly available datasets.
A quasiNewton approach to nonsmooth convex optimization
 In ICML
, 2008
"... We extend the wellknown BFGS quasiNewton method and its limitedmemory variant LBFGS to the optimization of nonsmooth convex objectives. This is done in a rigorous fashion by generalizing three components of BFGS to subdifferentials: The local quadratic model, the identification of a descent direc ..."
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Cited by 24 (2 self)
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We extend the wellknown BFGS quasiNewton method and its limitedmemory variant LBFGS to the optimization of nonsmooth convex objectives. This is done in a rigorous fashion by generalizing three components of BFGS to subdifferentials: The local quadratic model, the identification of a descent direction, and the Wolfe line search conditions. We apply the resulting subLBFGS algorithm to L2regularized risk minimization with binary hinge loss, and its directionfinding component to L1regularized risk minimization with logistic loss. In both settings our generic algorithms perform comparable to or better than their counterparts in specialized stateoftheart solvers. 1.
On the equivalence of weak learnability and linear separability: New relaxations and efficient boosting algorithms
 IN: PROCEEDINGS OF THE 21ST ANNUAL CONFERENCE ON COMPUTATIONAL LEARNING THEORY
"... Boosting algorithms build highly accurate prediction mechanisms from a collection of lowaccuracy predictors. To do so, they employ the notion of weaklearnability. The starting point of this paper is a proof which shows that weak learnability is equivalent to linear separability with ℓ1 margin. Whil ..."
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Cited by 20 (6 self)
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Boosting algorithms build highly accurate prediction mechanisms from a collection of lowaccuracy predictors. To do so, they employ the notion of weaklearnability. The starting point of this paper is a proof which shows that weak learnability is equivalent to linear separability with ℓ1 margin. While this equivalence is a direct consequence of von Neumann’s minimax theorem, we derive the equivalence directly using Fenchel duality. We then use our derivation to describe a family of relaxations to the weaklearnability assumption that readily translates to a family of relaxations of linear separability with margin. This alternative perspective sheds new light on known softmargin boosting algorithms and also enables us to derive several new relaxations of the notion of linear separability. Last, we describe and analyze an efficient boosting framework that can be used for minimizing the loss functions derived from our family of relaxations. In particular, we obtain efficient boosting algorithms for maximizing hard and soft versions of the ℓ1 margin.
Improving maximum margin matrix factorization
, 2008
"... Collaborative filtering is a popular method for personalizing product recommendations. Maximum Margin Matrix Factorization (MMMF) has been proposed as one successful learning approach to this task and has been recently extended to structured ranking losses. In this paper we discuss a number of exte ..."
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Cited by 16 (3 self)
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Collaborative filtering is a popular method for personalizing product recommendations. Maximum Margin Matrix Factorization (MMMF) has been proposed as one successful learning approach to this task and has been recently extended to structured ranking losses. In this paper we discuss a number of extensions to MMMF by introducing offset terms, item dependent regularization and a graph kernel on the recommender graph. We show equivalence between graph kernels and the recent MMMF extensions by Mnih and Salakhutdinov (Advances in Neural Information Processing Systems 20, 2008). Experimental evaluation of the introduced extensions show improved performance over the original MMMF formulation.
Estimating labels from label proportions
 Proceedings of the 25th Annual International Conference on Machine Learning
, 2008
"... Consider the following problem: given sets of unlabeled observations, each set with known label proportions, predict the labels of another set of observations, also with known label proportions. This problem appears in areas like ecommerce, spam filtering and improper content detection. We present ..."
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Cited by 15 (2 self)
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Consider the following problem: given sets of unlabeled observations, each set with known label proportions, predict the labels of another set of observations, also with known label proportions. This problem appears in areas like ecommerce, spam filtering and improper content detection. We present consistent estimators which can reconstruct the correct labels with high probability in a uniform convergence sense. Experiments show that our method works well in practice. 1
Entropy Regularized LPBoost
, 2008
"... In this paper we discuss boosting algorithms that maximize the soft margin of the produced linear combination of base hypotheses. LPBoost is the most straightforward boosting algorithm for doing this. It maximizes the soft margin by solving a linear programming problem. While it performs well on nat ..."
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Cited by 15 (3 self)
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In this paper we discuss boosting algorithms that maximize the soft margin of the produced linear combination of base hypotheses. LPBoost is the most straightforward boosting algorithm for doing this. It maximizes the soft margin by solving a linear programming problem. While it performs well on natural data, there are cases where the number of iterations is linear in the number of examples instead of logarithmic. By simply adding a relative entropy regularization to the linear objective of LPBoost, we arrive at the Entropy Regularized LPBoost algorithm for which we prove a logarithmic iteration bound. A previous algorithm, called SoftBoost, has the same iteration bound, but the generalization error of this algorithm often decreases slowly in early iterations. Entropy Regularized LPBoost does not suffer from this problem and has a simpler, more natural motivation.