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49
Boosting a Weak Learning Algorithm By Majority
, 1995
"... We present an algorithm for improving the accuracy of algorithms for learning binary concepts. The improvement is achieved by combining a large number of hypotheses, each of which is generated by training the given learning algorithm on a different set of examples. Our algorithm is based on ideas pr ..."
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Cited by 516 (15 self)
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We present an algorithm for improving the accuracy of algorithms for learning binary concepts. The improvement is achieved by combining a large number of hypotheses, each of which is generated by training the given learning algorithm on a different set of examples. Our algorithm is based on ideas presented by Schapire in his paper "The strength of weak learnability", and represents an improvement over his results. The analysis of our algorithm provides general upper bounds on the resources required for learning in Valiant's polynomial PAC learning framework, which are the best general upper bounds known today. We show that the number of hypotheses that are combined by our algorithm is the smallest number possible. Other outcomes of our analysis are results regarding the representational power of threshold circuits, the relation between learnability and compression, and a method for parallelizing PAC learning algorithms. We provide extensions of our algorithms to cases in which the conc...
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 376 (72 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.
Schapire R., Bounds on the Sample Complexity of Bayesian Learning Using Information Theory and the VC Dimension
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Learning conjunctions of Horn clauses
 In Proceedings of the 31st Annual Symposium on Foundations of Computer Science
, 1990
"... Abstract. An algorithm is presented for learning the class of Boolean formulas that are expressible as conjunctions of Horn clauses. (A Horn clause is a disjunction of literals, all but at most one of which is a negated variable.) The algorithm uses equivalence queries and membership queries to prod ..."
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Cited by 120 (14 self)
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Abstract. An algorithm is presented for learning the class of Boolean formulas that are expressible as conjunctions of Horn clauses. (A Horn clause is a disjunction of literals, all but at most one of which is a negated variable.) The algorithm uses equivalence queries and membership queries to produce a formula that is logically equivalent to the unknown formula to be learned. The amount of time used by the algorithm is polynomial in the number of variables and the number of clauses in the unknown formula.
Sphere Packing Numbers for Subsets of the Boolean nCube with Bounded VapnikChervonenkis Dimension
, 1992
"... : Let V ` f0; 1g n have VapnikChervonenkis dimension d. Let M(k=n;V ) denote the cardinality of the largest W ` V such that any two distinct vectors in W differ on at least k indices. We show that M(k=n;V ) (cn=(k + d)) d for some constant c. This improves on the previous best result of ((cn ..."
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Cited by 114 (4 self)
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: Let V ` f0; 1g n have VapnikChervonenkis dimension d. Let M(k=n;V ) denote the cardinality of the largest W ` V such that any two distinct vectors in W differ on at least k indices. We show that M(k=n;V ) (cn=(k + d)) d for some constant c. This improves on the previous best result of ((cn=k) log(n=k)) d . This new result has applications in the theory of empirical processes. 1 The author gratefully acknowledges the support of the Mathematical Sciences Research Institute at UC Berkeley and ONR grant N0001491J1162. 1 1 Statement of Results Let n be natural number greater than zero. Let V ` f0; 1g n . For a sequence of indices I = (i 1 ; . . . ; i k ), with 1 i j n, let V j I denote the projection of V onto I, i.e. V j I = f(v i 1 ; . . . ; v i k ) : (v 1 ; . . . ; v n ) 2 V g: If V j I = f0; 1g k then we say that V shatters the index sequence I. The VapnikChervonenkis dimension of V is the size of the longest index sequence I that is shattered by V [VC71] (t...
Online portfolio selection using multiplicative updates
 Mathematical Finance
, 1998
"... We present an online investment algorithm which achieves almost the same wealth as the best constantrebalanced portfolio determined in hindsight from the actual market outcomes. The algorithm employs a multiplicative update rule derived using a framework introduced by Kivinen and Warmuth. Our algo ..."
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Cited by 94 (10 self)
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We present an online investment algorithm which achieves almost the same wealth as the best constantrebalanced portfolio determined in hindsight from the actual market outcomes. The algorithm employs a multiplicative update rule derived using a framework introduced by Kivinen and Warmuth. Our algorithm is very simple to implement and requires only constant storage and computing time per stock ineach trading period. We tested the performance of our algorithm on real stock data from the New York Stock Exchange accumulated during a 22year period. On this data, our algorithm clearly outperforms the best single stock aswell as Cover's universal portfolio selection algorithm. We also present results for the situation in which the We present an online investment algorithm which achieves almost the same wealth as the best constantrebalanced portfolio investment strategy. The algorithm employsamultiplicative update rule derived using a framework introduced by Kivinen and Warmuth [20]. Our algorithm is very simple to implement and its time and storage requirements grow linearly in the number of stocks.
Sample compression, learnability, and the VapnikChervonenkis dimension
 MACHINE LEARNING
, 1995
"... Within the framework of paclearning, we explore the learnability of concepts from samples using the paradigm of sample compression schemes. A sample compression scheme of size k for a concept class C ` 2 X consists of a compression function and a reconstruction function. The compression function r ..."
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Cited by 83 (5 self)
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Within the framework of paclearning, we explore the learnability of concepts from samples using the paradigm of sample compression schemes. A sample compression scheme of size k for a concept class C ` 2 X consists of a compression function and a reconstruction function. The compression function receives a finite sample set consistent with some concept in C and chooses a subset of k examples as the compression set. The reconstruction function forms a hypothesis on X from a compression set of k examples. For any sample set of a concept in C the compression set produced by the compression function must lead to a hypothesis consistent with the whole original sample set when it is fed to the reconstruction function. We demonstrate that the existence of a sample compression scheme of fixedsize for a class C is sufficient to ensure that the class C is paclearnable. Previous work has shown that a class is paclearnable if and only if the VapnikChervonenkis (VC) dimension of the class i...
Tracking drifting concepts by minimizing disagreements
 Machine Learning
, 1994
"... Abstract. In this paper we consider the problem of tracking a subset of a domain (called the target) which changes gradually over time. A single (unknown) probability distribution over the domain is used to generate random examples for the learning algorithm and measure the speed at which the target ..."
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Cited by 75 (3 self)
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Abstract. In this paper we consider the problem of tracking a subset of a domain (called the target) which changes gradually over time. A single (unknown) probability distribution over the domain is used to generate random examples for the learning algorithm and measure the speed at which the target changes. Clearly, the more rapidly the target moves, the harder it is for the algorithm to maintain a good approximation of the target. Therefore we evaluate algorithms based on how much movement of the target can be tolerated between examples while predicting with accuracy e. Furthermore, the complexity of the class 7/of possible targets, as measured by d, its VCdimension, also effects the difficulty of tracking the target concept. We show that if the problem of minimizing the number of disagreements with a sample from among concepts in a class 7 { can be approximated to within a factor k, then there is a simple tracking algorithm for 7t which can achieve a probability e of making a mistake if the target movement rate is at most a constant times e2/(k(d + k) In 1), where d is the VapnikChervonenkis dimension of 7t. Also, we show that if 7 / is properly PAClearnable, then there is an efficient (randomized) algorithm that with high probability approximately minimizes disagreements to within a factor of 7d + 1, yielding an efficient tracking algorithm for 7I which tolerates drift rates up to a constant times e2/(d 2 In ). In addition, we prove complementary results for the classes of halfspaces and axisaligned hyperrectangles showing that the maximum rate of drift that any algorithm (even with unlimited computational power) can tolerate is a constant times e2/d.
Risk bounds for Statistical Learning
"... We propose a general theorem providing upper bounds for the risk of an empirical risk minimizer (ERM).We essentially focus on the binary classi…cation framework. We extend Tsybakov’s analysis of the risk of an ERM under margin type conditions by using concentration inequalities for conveniently weig ..."
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Cited by 62 (2 self)
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We propose a general theorem providing upper bounds for the risk of an empirical risk minimizer (ERM).We essentially focus on the binary classi…cation framework. We extend Tsybakov’s analysis of the risk of an ERM under margin type conditions by using concentration inequalities for conveniently weighted empirical processes. This allows us to deal with other ways of measuring the ”size”of a class of classi…ers than entropy with bracketing as in Tsybakov’s work. In particular we derive new risk bounds for the ERM when the classi…cation rules belong to some VCclass under margin conditions and discuss the optimality of those bounds in a minimax sense.
Improved Bounds on the Sample Complexity of Learning
 Journal of Computer and System Sciences
, 2000
"... We present two improved bounds on the sample complexity of learning. First, we present a new general upper bound on the number of examples required to estimate all of the expectations of a set of random variables uniformly well. The quality of the estimates is measured using a variant of the relativ ..."
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Cited by 53 (1 self)
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We present two improved bounds on the sample complexity of learning. First, we present a new general upper bound on the number of examples required to estimate all of the expectations of a set of random variables uniformly well. The quality of the estimates is measured using a variant of the relative error proposed by Haussler and Pollard. We also show that our bound is within a constant factor of the best possible. Our upper bound implies improved bounds on the sample complexity of learning according to Haussler's decision theoretic model. Next, we prove a lower bound on the sample complexity for learning according to the prediction model that is optimal to within a factor of 1+o(1). 1 Introduction Many important applied problems can be modeled as learning from random examples. Examples include text categorization [21], handwritten character recognition [15, 4, 26], speech recognition [2, 1], and virtual circuit holding times in IPoverATM networks [20, 17, 14]. In this paper, we p...