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72
The Weighted Majority Algorithm
, 1994
"... We study the construction of prediction algorithms in a situation in which a learner faces a sequence of trials, with a prediction to be made in each, and the goal of the learner is to make few mistakes. We are interested in the case that the learner has reason to believe that one of some pool of kn ..."
Abstract

Cited by 678 (39 self)
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We study the construction of prediction algorithms in a situation in which a learner faces a sequence of trials, with a prediction to be made in each, and the goal of the learner is to make few mistakes. We are interested in the case that the learner has reason to believe that one of some pool of known algorithms will perform well, but the learner does not know which one. A simple and effective method, based on weighted voting, is introduced for constructing a compound algorithm in such a circumstance. We call this method the Weighted Majority Algorithm. We show that this algorithm is robust in the presence of errors in the data. We discuss various versions of the Weighted Majority Algorithm and prove mistake bounds for them that are closely related to the mistake bounds of the best algorithms of the pool. For example, given a sequence of trials, if there is an algorithm in the pool A that makes at most m mistakes then the Weighted Majority Algorithm will make at most c(log jAj + m) mi...
Large Margin Classification Using the Perceptron Algorithm
 Machine Learning
, 1998
"... We introduce and analyze a new algorithm for linear classification which combines Rosenblatt 's perceptron algorithm with Helmbold and Warmuth's leaveoneout method. Like Vapnik 's maximalmargin classifier, our algorithm takes advantage of data that are linearly separable with large margins. Compa ..."
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Cited by 415 (1 self)
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We introduce and analyze a new algorithm for linear classification which combines Rosenblatt 's perceptron algorithm with Helmbold and Warmuth's leaveoneout method. Like Vapnik 's maximalmargin classifier, our algorithm takes advantage of data that are linearly separable with large margins. Compared to Vapnik's algorithm, however, ours is much simpler to implement, and much more efficient in terms of computation time. We also show that our algorithm can be efficiently used in very high dimensional spaces using kernel functions. We performed some experiments using our algorithm, and some variants of it, for classifying images of handwritten digits. The performance of our algorithm is close to, but not as good as, the performance of maximalmargin classifiers on the same problem, while saving significantly on computation time and programming effort. 1 Introduction One of the most influential developments in the theory of machine learning in the last few years is Vapnik's work on supp...
How to Use Expert Advice
 JOURNAL OF THE ASSOCIATION FOR COMPUTING MACHINERY
, 1997
"... We analyze algorithms that predict a binary value by combining the predictions of several prediction strategies, called experts. Our analysis is for worstcase situations, i.e., we make no assumptions about the way the sequence of bits to be predicted is generated. We measure the performance of the ..."
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Cited by 317 (66 self)
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We analyze algorithms that predict a binary value by combining the predictions of several prediction strategies, called experts. Our analysis is for worstcase situations, i.e., we make no assumptions about the way the sequence of bits to be predicted is generated. We measure the performance of the algorithm by the difference between the expected number of mistakes it makes on the bit sequence and the expected number of mistakes made by the best expert on this sequence, where the expectation is taken with respect to the randomization in the predictions. We show that the minimum achievable difference is on the order of the square root of the number of mistakes of the best expert, and we give efficient algorithms that achieve this. Our upper and lower bounds have matching leading constants in most cases. We then show howthis leads to certain kinds of pattern recognition/learning algorithms with performance bounds that improve on the best results currently known in this context. We also compare our analysis to the case in which log loss is used instead of the expected number of mistakes.
Exponentiated Gradient Versus Gradient Descent for Linear Predictors
 Information and Computation
, 1995
"... this paper, we concentrate on linear predictors . To any vector u 2 R ..."
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Cited by 247 (12 self)
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this paper, we concentrate on linear predictors . To any vector u 2 R
On the Generalization Ability of Online Learning Algorithms
 IEEE Transactions on Information Theory
, 2001
"... In this paper we show that online algorithms for classification and regression can be naturally used to obtain hypotheses with good datadependent tail bounds on their risk. Our results are proven without requiring complicated concentrationofmeasure arguments and they hold for arbitrary onlin ..."
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Cited by 133 (8 self)
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In this paper we show that online algorithms for classification and regression can be naturally used to obtain hypotheses with good datadependent tail bounds on their risk. Our results are proven without requiring complicated concentrationofmeasure arguments and they hold for arbitrary online learning algorithms. Furthermore, when applied to concrete online algorithms, our results yield tail bounds that in many cases are comparable or better than the best known bounds.
Noisetolerant learning, the parity problem, and the statistical query model
 J. ACM
"... We describe a slightly subexponential time algorithm for learning parity functions in the presence of random classification noise. This results in a polynomialtime algorithm for the case of parity functions that depend on only the first O(log n log log n) bits of input. This is the first known ins ..."
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Cited by 116 (2 self)
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We describe a slightly subexponential time algorithm for learning parity functions in the presence of random classification noise. This results in a polynomialtime algorithm for the case of parity functions that depend on only the first O(log n log log n) bits of input. This is the first known instance of an efficient noisetolerant algorithm for a concept class that is provably not learnable in the Statistical Query model of Kearns [7]. Thus, we demonstrate that the set of problems learnable in the statistical query model is a strict subset of those problems learnable in the presence of noise in the PAC model. In codingtheory terms, what we give is a poly(n)time algorithm for decoding linear k × n codes in the presence of random noise for the case of k = clog n log log n for some c> 0. (The case of k O(log n) is trivial since one can just individually check each of the 2 k possible messages and choose the one that yields the closest codeword.) A natural extension of the statistical query model is to allow queries about statistical properties that involve ttuples of examples (as opposed to single examples). The second result of this paper is to show that any class of functions learnable (strongly or weakly) with twise queries for t = O(log n) is also weakly learnable with standard unary queries. Hence this natural extension to the statistical query model does not increase the set of weakly learnable functions. 1.
The Relaxed Online Maximum Margin Algorithm
 Machine Learning
, 2000
"... We describe a new incremental algorithm for training linear threshold functions: the Relaxed Online Maximum Margin Algorithm, or ROMMA. ROMMA can be viewed as an approximation to the algorithm that repeatedly chooses the hyperplane that classifies previously seen examples correctly with the maximum ..."
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Cited by 73 (1 self)
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We describe a new incremental algorithm for training linear threshold functions: the Relaxed Online Maximum Margin Algorithm, or ROMMA. ROMMA can be viewed as an approximation to the algorithm that repeatedly chooses the hyperplane that classifies previously seen examples correctly with the maximum margin. It is known that such a maximummargin hypothesis can be computed by minimizing the length of the weight vector subject to a number of linear constraints. ROMMA works by maintaining a relatively simple relaxation of these constraints that can be eciently updated. We prove a mistake bound for ROMMA that is the same as that proved for the perceptron algorithm. Our analysis implies that the more computationally intensive maximummargin algorithm also satis es this mistake bound; this is the rst worstcase performance guarantee for this algorithm. We describe some experiments using ROMMA and a variant that updates its hypothesis more aggressively as batch algorithms to recognize handwr...
Knows What It Knows: A Framework For SelfAware Learning
"... We introduce a learning framework that combines elements of the wellknown PAC and mistakebound models. The KWIK (knows what it knows) framework was designed particularly for its utility in learning settings where active exploration can impact the training examples the learner is exposed to, as is ..."
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Cited by 44 (17 self)
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We introduce a learning framework that combines elements of the wellknown PAC and mistakebound models. The KWIK (knows what it knows) framework was designed particularly for its utility in learning settings where active exploration can impact the training examples the learner is exposed to, as is true in reinforcementlearning and activelearning problems. We catalog several KWIKlearnable classes and open problems. 1.
Analysis of two gradientbased algorithms for online regression
 Journal of Computer and System Sciences
, 1999
"... In this paper we present a new analysis of two algorithms, Gradient Descent and Exponentiated Gradient, for solving regression problems in the online framework. Both these algorithms compute a prediction that depends linearly on the current instance, and then update the coefficients of this linear ..."
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Cited by 40 (5 self)
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In this paper we present a new analysis of two algorithms, Gradient Descent and Exponentiated Gradient, for solving regression problems in the online framework. Both these algorithms compute a prediction that depends linearly on the current instance, and then update the coefficients of this linear combination according to the gradient of the loss function. However, the two algorithms have distinctive ways of using the gradient information for updating the coefficients. For each algorithm, we show general regression bounds for any convex loss function. Furthermore, we show special bounds for the absolute and the square loss functions, thus extending previous results by Kivinen and Warmuth. In the nonlinear regression case, we show general bounds for pairs of transfer and loss functions satisfying a certain condition. We apply this result to the Hellinger loss and the entropic loss in case of logistic regression (similar results, but only for the entropic loss, were also obtained by Helmbold et al. using a different analysis.) Finally, we describe the connection between our approach and a general family of gradientbased algorithms proposed by Warmuth et al. in recent works. 1999 Academic Press 1.
Probably Approximately Correct Learning
 Proceedings of the Eighth National Conference on Artificial Intelligence
, 1990
"... This paper surveys some recent theoretical results on the efficiency of machine learning algorithms. The main tool described is the notion of Probably Approximately Correct (PAC) learning, introduced by Valiant. We define this learning model and then look at some of the results obtained in it. We th ..."
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Cited by 40 (1 self)
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This paper surveys some recent theoretical results on the efficiency of machine learning algorithms. The main tool described is the notion of Probably Approximately Correct (PAC) learning, introduced by Valiant. We define this learning model and then look at some of the results obtained in it. We then consider some criticisms of the PAC model and the extensions proposed to address these criticisms. Finally, we look briefly at other models recently proposed in computational learning theory. 2 Introduction It's a dangerous thing to try to formalize an enterprise as complex and varied as machine learning so that it can be subjected to rigorous mathematical analysis. To be tractable, a formal model must be simple. Thus, inevitably, most people will feel that important aspects of the activity have been left out of the theory. Of course, they will be right. Therefore, it is not advisable to present a theory of machine learning as having reduced the entire field to its bare essentials. All ...