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143
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.
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
Stochastic Approximation Approach to Stochastic Programming
"... In this paper we consider optimization problems where the objective function is given in a form of the expectation. A basic difficulty of solving such stochastic optimization problems is that the involved multidimensional integrals (expectations) cannot be computed with high accuracy. The aim of th ..."
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Cited by 266 (18 self)
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In this paper we consider optimization problems where the objective function is given in a form of the expectation. A basic difficulty of solving such stochastic optimization problems is that the involved multidimensional integrals (expectations) cannot be computed with high accuracy. The aim of this paper is to compare two computational approaches based on Monte Carlo sampling techniques, namely, the Stochastic Approximation (SA) and the Sample Average Approximation (SAA) methods. Both approaches, the SA and SAA methods, have a long history. Current opinion is that the SAA method can efficiently use a specific (say linear) structure of the considered problem, while the SA approach is a crude subgradient method which often performs poorly in practice. We intend to demonstrate that a properly modified SA approach can be competitive and even significantly outperform the SAA method for a certain class of convex stochastic problems. We extend the analysis to the case of convexconcave stochastic saddle point problems, and present (in our opinion highly encouraging) results of numerical experiments.
A unified framework for highdimensional analysis of Mestimators with decomposable regularizers
"... ..."
Dual averaging methods for regularized stochastic learning and online optimization
 In Advances in Neural Information Processing Systems 23
, 2009
"... We consider regularized stochastic learning and online optimization problems, where the objective function is the sum of two convex terms: one is the loss function of the learning task, and the other is a simple regularization term such as ℓ1norm for promoting sparsity. We develop extensions of Nes ..."
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Cited by 131 (7 self)
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We consider regularized stochastic learning and online optimization problems, where the objective function is the sum of two convex terms: one is the loss function of the learning task, and the other is a simple regularization term such as ℓ1norm for promoting sparsity. We develop extensions of Nesterov’s dual averaging method, that can exploit the regularization structure in an online setting. At each iteration of these methods, the learning variables are adjusted by solving a simple minimization problem that involves the running average of all past subgradients of the loss function and the whole regularization term, not just its subgradient. In the case of ℓ1regularization, our method is particularly effective in obtaining sparse solutions. We show that these methods achieve the optimal convergence rates or regret bounds that are standard in the literature on stochastic and online convex optimization. For stochastic learning problems in which the loss functions have Lipschitz continuous gradients, we also present an accelerated version of the dual averaging method.
Dual Averaging for Distributed Optimization: Convergence Analysis and Network Scaling
 IEEE TRANSACTIONS ON AUTOMATIC CONTROL
, 2010
"... The goal of decentralized optimization over a network is to optimize a global objective formed by a sum of local (possibly nonsmooth) convex functions using only local computation and communication. It arises in various application domains, including distributed tracking and localization, multiagen ..."
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Cited by 93 (12 self)
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The goal of decentralized optimization over a network is to optimize a global objective formed by a sum of local (possibly nonsmooth) convex functions using only local computation and communication. It arises in various application domains, including distributed tracking and localization, multiagent coordination, estimation in sensor networks, and largescale machine learning. We develop and analyze distributed algorithms based on dual subgradient averaging, and we provide sharp bounds on their convergence rates as a function of the network size and topology. Our analysis allows us to clearly separate the convergence of the optimization algorithm itself and the effects of communication dependent on the network structure. We show that the number of iterations required by our algorithm scales inversely in the spectral gap of the network and confirm this prediction’s sharpness both by theoretical lower bounds and simulations for various networks. Our approach includes the cases of deterministic optimization and communication as well as problems with stochastic optimization and/or communication.
Approximate Primal Solutions and Rate Analysis for Dual Subgradient Methods
, 2007
"... We study primal solutions obtained as a byproduct of subgradient methods when solving the Lagrangian dual of a primal convex constrained optimization problem (possibly nonsmooth). The existing literature on the use of subgradient methods for generating primal optimal solutions is limited to the met ..."
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Cited by 82 (7 self)
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We study primal solutions obtained as a byproduct of subgradient methods when solving the Lagrangian dual of a primal convex constrained optimization problem (possibly nonsmooth). The existing literature on the use of subgradient methods for generating primal optimal solutions is limited to the methods producing such solutions only asymptotically (i.e., in the limit as the number of subgradient iterations increases to infinity). Furthermore, no convergence rate results are known for these algorithms. In this paper, we propose and analyze dual subgradient methods using averaging to generate approximate primal optimal solutions. These algorithms use a constant stepsize as opposed to a diminishing stepsize which is dominantly used in the existing primal recovery schemes. We provide estimates on the convergence rate of the primal sequences. In particular, we provide bounds on the amount of feasibility violation of the generated approximate primal solutions. We also provide upper and lower bounds on the primal function values at the approximate solutions. The feasibility violation and primal value estimates are given per iteration, thus providing practical stopping criteria. Our analysis relies on the Slater condition and the inherited boundedness properties of the dual problem under this condition.
A Stochastic Gradient Method with an Exponential Convergence Rate for StronglyConvex Optimization with Finite Training Sets. arXiv preprint arXiv:1202.6258
, 2012
"... We propose a new stochastic gradient method for optimizing the sum of a finite set of smooth functions, where the sum is strongly convex. While standard stochastic gradient methods converge at sublinear rates for this problem, the proposed method incorporates a memory of previous gradient values in ..."
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Cited by 76 (11 self)
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We propose a new stochastic gradient method for optimizing the sum of a finite set of smooth functions, where the sum is strongly convex. While standard stochastic gradient methods converge at sublinear rates for this problem, the proposed method incorporates a memory of previous gradient values in order to achieve a linear convergence rate. In a machine learning context, numerical experiments indicate that the new algorithm can dramatically outperform standard algorithms, both in terms of optimizing the training objective and reducing the testing objective quickly. 1
Optimal distributed online prediction using minibatches
, 2010
"... Online prediction methods are typically presented as serial algorithms running on a single processor. However, in the age of webscale prediction problems, it is increasingly common to encounter situations where a single processor cannot keep up with the high rate at which inputs arrive. In this wor ..."
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Cited by 69 (7 self)
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Online prediction methods are typically presented as serial algorithms running on a single processor. However, in the age of webscale prediction problems, it is increasingly common to encounter situations where a single processor cannot keep up with the high rate at which inputs arrive. In this work, we present the distributed minibatch algorithm, a method of converting many serial gradientbased online prediction algorithms into distributed algorithms. We prove a regret bound for this method that is asymptotically optimal for smooth convex loss functions and stochastic inputs. Moreover, our analysis explicitly takes into account communication latencies between nodes in the distributed environment. We show how our method can be used to solve the closelyrelated distributed stochastic optimization problem, achieving an asymptotically linear speedup over multiple processors. Finally, we demonstrate the merits of our approach on a webscale online prediction problem.
Composite Objective Mirror Descent
"... We present a new method for regularized convex optimization and analyze it under both online and stochastic optimization settings. In addition to unifying previously known firstorder algorithms, such as the projected gradient method, mirror descent, and forwardbackward splitting, our method yields n ..."
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Cited by 66 (9 self)
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We present a new method for regularized convex optimization and analyze it under both online and stochastic optimization settings. In addition to unifying previously known firstorder algorithms, such as the projected gradient method, mirror descent, and forwardbackward splitting, our method yields new analysis and algorithms. We also derive specific instantiations of our method for commonly used regularization functions, such as ℓ1, mixed norm, and tracenorm. 1