Results 1  10
of
107
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 ..."
Abstract

Cited by 133 (7 self)
 Add to MetaCart
(Show Context)
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.
Efficient Online and Batch Learning using Forward Backward Splitting
"... We describe, analyze, and experiment with a framework for empirical loss minimization with regularization. Our algorithmic framework alternates between two phases. On each iteration we first perform an unconstrained gradient descent step. We then cast and solve an instantaneous optimization problem ..."
Abstract

Cited by 130 (1 self)
 Add to MetaCart
(Show Context)
We describe, analyze, and experiment with a framework for empirical loss minimization with regularization. Our algorithmic framework alternates between two phases. On each iteration we first perform an unconstrained gradient descent step. We then cast and solve an instantaneous optimization problem that trades off minimization of a regularization term while keeping close proximity to the result of the first phase. This view yields a simple yet effective algorithm that can be used for batch penalized risk minimization and online learning. Furthermore, the two phase approach enables sparse solutions when used in conjunction with regularization functions that promote sparsity, such as ℓ1. We derive concrete and very simple algorithms for minimization of loss functions with ℓ1, ℓ2, ℓ 2 2, and ℓ ∞ regularization. We also show how to construct efficient algorithms for mixednorm ℓ1/ℓq regularization. We further extend the algorithms and give efficient implementations for very highdimensional data with sparsity. We demonstrate the potential of the proposed framework in a series of experiments with synthetic and natural datasets.
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 ..."
Abstract

Cited by 67 (8 self)
 Add to MetaCart
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
A comparison of optimization methods and software for largescale l1regularized linear classification
 The Journal of Machine Learning Research
"... Largescale linear classification is widely used in many areas. The L1regularized form can be applied for feature selection; however, its nondifferentiability causes more difficulties in training. Although various optimization methods have been proposed in recent years, these have not yet been com ..."
Abstract

Cited by 54 (7 self)
 Add to MetaCart
Largescale linear classification is widely used in many areas. The L1regularized form can be applied for feature selection; however, its nondifferentiability causes more difficulties in training. Although various optimization methods have been proposed in recent years, these have not yet been compared suitably. In this paper, we first broadly review existing methods. Then, we discuss stateoftheart software packages in detail and propose two efficient implementations. Extensive comparisons indicate that carefully implemented coordinate descent methods are very suitable for training large document data.
Convergence Rates of Inexact ProximalGradient Methods for Convex Optimization
 NIPS'11 25 TH ANNUAL CONFERENCE ON NEURAL INFORMATION PROCESSING SYSTEMS
, 2011
"... We consider the problem of optimizing the sum of a smooth convex function and a nonsmooth convex function using proximalgradient methods, where an error is present in the calculation of the gradient of the smooth term or in the proximity operator with respect to the nonsmooth term. We show that b ..."
Abstract

Cited by 49 (6 self)
 Add to MetaCart
(Show Context)
We consider the problem of optimizing the sum of a smooth convex function and a nonsmooth convex function using proximalgradient methods, where an error is present in the calculation of the gradient of the smooth term or in the proximity operator with respect to the nonsmooth term. We show that both the basic proximalgradient method and the accelerated proximalgradient method achieve the same convergence rate as in the errorfree case, provided that the errors decrease at appropriate rates. Using these rates, we perform as well as or better than a carefully chosen fixed error level on a set of structured sparsity problems.
Stochastic Gradient Descent Training for L1regularized Loglinear Models with Cumulative Penalty
"... Stochastic gradient descent (SGD) uses approximate gradients estimated from subsets of the training data and updates the parameters in an online fashion. This learning framework is attractive because it often requires much less training time in practice than batch training algorithms. However, L1re ..."
Abstract

Cited by 42 (0 self)
 Add to MetaCart
(Show Context)
Stochastic gradient descent (SGD) uses approximate gradients estimated from subsets of the training data and updates the parameters in an online fashion. This learning framework is attractive because it often requires much less training time in practice than batch training algorithms. However, L1regularization, which is becoming popular in natural language processing because of its ability to produce compact models, cannot be efficiently applied in SGD training, due to the large dimensions of feature vectors and the fluctuations of approximate gradients. We present a simple method to solve these problems by penalizing the weights according to cumulative values for L1 penalty. We evaluate the effectiveness of our method in three applications: text chunking, named entity recognition, and partofspeech tagging. Experimental results demonstrate that our method can produce compact and accurate models much more quickly than a stateoftheart quasiNewton method for L1regularized loglinear models. 1
Accelerated proximal stochastic dual coordinate ascent for regularized loss minimization.
 Mathematical Programming,
, 2015
"... Abstract We introduce a proximal version of the stochastic dual coordinate ascent method and show how to accelerate the method using an innerouter iteration procedure. We analyze the runtime of the framework and obtain rates that improve stateoftheart results for various key machine learning op ..."
Abstract

Cited by 36 (2 self)
 Add to MetaCart
(Show Context)
Abstract We introduce a proximal version of the stochastic dual coordinate ascent method and show how to accelerate the method using an innerouter iteration procedure. We analyze the runtime of the framework and obtain rates that improve stateoftheart results for various key machine learning optimization problems including SVM, logistic regression, ridge regression, Lasso, and multiclass SVM. Experiments validate our theoretical findings.
Recent Advances of Largescale Linear Classification
"... Linear classification is a useful tool in machine learning and data mining. For some data in a rich dimensional space, the performance (i.e., testing accuracy) of linear classifiers has shown to be close to that of nonlinear classifiers such as kernel methods, but training and testing speed is much ..."
Abstract

Cited by 32 (6 self)
 Add to MetaCart
(Show Context)
Linear classification is a useful tool in machine learning and data mining. For some data in a rich dimensional space, the performance (i.e., testing accuracy) of linear classifiers has shown to be close to that of nonlinear classifiers such as kernel methods, but training and testing speed is much faster. Recently, many research works have developed efficient optimization methods to construct linear classifiers and applied them to some largescale applications. In this paper, we give a comprehensive survey on the recent development of this active research area.
Predicting bounce rates in sponsored search advertisements
 In SIGKDD Conference on Knowledge Discovery and Data Mining (KDD
, 2009
"... This paper explores an important and relatively unstudied quality measure of a sponsored search advertisement: bounce rate. The bounce rate of an ad can be informally defined as the fraction of users who click on the ad but almost immediately move on to other tasks. A high bounce rate can lead to po ..."
Abstract

Cited by 32 (3 self)
 Add to MetaCart
(Show Context)
This paper explores an important and relatively unstudied quality measure of a sponsored search advertisement: bounce rate. The bounce rate of an ad can be informally defined as the fraction of users who click on the ad but almost immediately move on to other tasks. A high bounce rate can lead to poor advertiser return on investment, and suggests search engine users may be having a poor experience following the click. In this paper, we first provide quantitative analysis showing that bounce rate is an effective measure of user satisfaction. We then address the question, can we predict bounce rate by analyzing the features of the advertisement? An affirmative answer would allow advertisers and search engines to predict the effectiveness and quality of advertisements before they are shown. We propose solutions to this problem involving largescale learning methods that leverage features drawn from ad creatives in addition
Stochastic Alternating Direction Method of Multipliers
"... The Alternating Direction Method of Multipliers (ADMM) has received lots of attention recently due to the tremendous demand from largescale and datadistributed machine learning applications. In this paper, we present a stochastic setting for optimization problems with nonsmooth composite objectiv ..."
Abstract

Cited by 31 (0 self)
 Add to MetaCart
(Show Context)
The Alternating Direction Method of Multipliers (ADMM) has received lots of attention recently due to the tremendous demand from largescale and datadistributed machine learning applications. In this paper, we present a stochastic setting for optimization problems with nonsmooth composite objective functions. To solve this problem, we propose a stochastic ADMM algorithm. Our algorithm applies to a more general class of convex and nonsmooth objective functions, beyond the smooth and separable least squares loss used in lasso. We also demonstrate the rates of convergence for our algorithm under various structural assumptions of the stochastic function: O(1 / √ t) for convex functions and O(log t/t) for strongly convex functions. Compared to previous literature, we establish the convergence rate of ADMM for convex problems in terms of both the objective value and the feasibility violation. A novel application named GraphGuided SVM is proposed to demonstrate the usefulness of our algorithm.