Results 1  10
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25
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 58 (3 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.
Adaptive online gradient descent
 In Advances in Neural Information Processing Systems 21
, 2007
"... We study the rates of growth of the regret in online convex optimization. First, we show that a simple extension of the algorithm of Hazan et al eliminates the need for a priori knowledge of the lower bound on the second derivatives of the observed functions. We then provide an algorithm, Adaptive O ..."
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Cited by 31 (6 self)
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We study the rates of growth of the regret in online convex optimization. First, we show that a simple extension of the algorithm of Hazan et al eliminates the need for a priori knowledge of the lower bound on the second derivatives of the observed functions. We then provide an algorithm, Adaptive Online Gradient Descent, which interpolates between the results of Zinkevich for linear functions and of Hazan et al for strongly convex functions, achieving intermediate rates between √ T and log T. Furthermore, we show strong optimality of the algorithm. Finally, we provide an extension of our results to general norms. 1
Exponentiated gradient algorithms for loglinear structured prediction
 In Proc. ICML, 2007
"... Conditional loglinear models are a commonly used method for structured prediction. Efficient learning of parameters in these models is therefore an important problem. This paper describes an exponentiated gradient (EG) algorithm for training such models. EG is applied to the convex dual of the maxi ..."
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Cited by 29 (5 self)
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Conditional loglinear models are a commonly used method for structured prediction. Efficient learning of parameters in these models is therefore an important problem. This paper describes an exponentiated gradient (EG) algorithm for training such models. EG is applied to the convex dual of the maximum likelihood objective; this results in both sequential and parallel update algorithms, where in the sequential algorithm parameters are updated in an online fashion. We provide a convergence proof for both algorithms. Our analysis also simplifies previous results on EG for maxmargin models, and leads to a tighter bound on convergence rates. Experiments on a largescale parsing task show that the proposed algorithm converges much faster than conjugategradient and LBFGS approaches both in terms of optimization objective and test error. 1.
Mind the Duality Gap: Logarithmic regret algorithms for online optimization
"... We describe a primaldual framework for the design and analysis of online strongly convex optimization algorithms. Our framework yields the tightest known logarithmic regret bounds for FollowTheLeader and for the gradient descent algorithm proposed in Hazan et al. [2006]. We then show that one can ..."
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Cited by 23 (0 self)
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We describe a primaldual framework for the design and analysis of online strongly convex optimization algorithms. Our framework yields the tightest known logarithmic regret bounds for FollowTheLeader and for the gradient descent algorithm proposed in Hazan et al. [2006]. We then show that one can interpolate between these two extreme cases. In particular, we derive a new algorithm that shares the computational simplicity of gradient descent but achieves lower regret in many practical situations. Finally, we further extend our framework for generalized strongly convex functions. 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.
30 Regression and Classification on Large Datasets
 Journal of Computer and System Sciences
, 1997
"... We study boosting in the filtering setting, where the booster draws examples from an oracle instead of using a fixed training set and so may train efficiently on very large datasets. Our algorithm FilterBoost, which is based on a logistic regression technique proposed by Collins et al. (2002), requi ..."
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Cited by 17 (0 self)
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We study boosting in the filtering setting, where the booster draws examples from an oracle instead of using a fixed training set and so may train efficiently on very large datasets. Our algorithm FilterBoost, which is based on a logistic regression technique proposed by Collins et al. (2002), requires fewer assumptions to achieve bounds equivalent to or better than previous work. Our proofs demonstrate the algorithm’s strong theoretical properties for both classification and conditional probability estimation, and we validate these results through extensive experiments. Empirically, our algorithm proves more robust to noise and overfitting than batch boosters in conditional probability estimation and proves competitive in classification. We give several extensions of FilterBoost to the multiclass case, proving PAC bounds on each. In particular, we make use of the ideas of pseudoloss and ErrorCorrecting Output Codes used by Freund and Schapire (1997) and Schapire (1997) to create easily implementable boosters, and we show that the generalization of Output Codes by Allwein et al. (2000) extends to FilterBoost. This work represents one of the first studies of boostingbyfiltering
Noregret learning in convex games
, 2007
"... Quite a bit is known about minimizing different kinds of regret in experts problems, and how these regret types relate to types of equilibria in the multiagent setting of repeated matrix games. Much less is known about the possible kinds of regret in online convex programming problems (OCPs), or abo ..."
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Cited by 15 (4 self)
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Quite a bit is known about minimizing different kinds of regret in experts problems, and how these regret types relate to types of equilibria in the multiagent setting of repeated matrix games. Much less is known about the possible kinds of regret in online convex programming problems (OCPs), or about equilibria in the analogous multiagent setting of repeated convex games. This gap is unfortunate, since convex games are much more expressive than matrix games, and since many important machine learning problems can be expressed as OCPs. In this paper, we work to close this gap: we analyze a spectrum of regret types which lie between external and swap regret, along with their corresponding equilibria, which lie between coarse correlated and correlated equilibrium. We also analyze algorithms for minimizing these regret types. As examples of our framework, we derive algorithms for learning correlated equilibria in polyhedral convex games and extensiveform correlated equilibria in extensiveform games. The former is exponentially more efficient than previous algorithms, and the latter is the first of its type. 1.
Noregret algorithms for online convex programs
 In Neural Information Processing Systems 19
, 2007
"... Online convex programming has recently emerged as a powerful primitive for designing machine learning algorithms. For example, OCP can be used for learning a linear classifier, dynamically rebalancing a binary search tree, finding the shortest path in a graph with unknown edge lengths, solving a str ..."
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Cited by 10 (2 self)
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Online convex programming has recently emerged as a powerful primitive for designing machine learning algorithms. For example, OCP can be used for learning a linear classifier, dynamically rebalancing a binary search tree, finding the shortest path in a graph with unknown edge lengths, solving a structured classification problem, or finding a good strategy in an extensiveform game. Several researchers have designed noregret algorithms for OCP. But, compared to algorithms for special cases of OCP such as learning from expert advice, these algorithms are not very numerous or flexible. In learning from expert advice, one tool which has proved particularly valuable is the correspondence between noregret algorithms and convex potential functions: by reasoning about these potential functions, researchers have designed algorithms with a wide variety of useful guarantees such as good performance when the target hypothesis is sparse. Until now, there has been no such recipe for the more general OCP problem, and therefore no ability to tune OCP algorithms to take advantage of properties of the problem or data. In this paper we derive a new class of noregret learning algorithms for OCP. These Lagrangian Hedging algorithms are based on a general class of potential functions, and are a direct generalization of known learning rules like weighted majority and externalregret matching. In addition to proving regret bounds, we demonstrate our algorithms learning to play onecard poker. 1