A game of prediction with expert advice

Vovk, Vladimir

doi:10.1145/225298.225304

Cited by 169 publications

(283 citation statements)

References 27 publications

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“…We will call the triple ( , , λ) our game. Our assumptions about the game ( , , λ) are the same as in (Vovk, 1998): 1. is a compact topological space. 2.…”

Section: Protocolmentioning

confidence: 99%

“…(In (Vovk, 1998) we allowed P to range over only simple probability distributions, but it can be shown that, under mild regularity conditions, these two definitions are equivalent; e.g., it is sufficient to assume that the topology on is generated by a metric.) Under our assumptions, the infimum in (3) is attained.…”

Section: Formal Descriptionmentioning

confidence: 99%

“…A standard algorithm for predicting with expert advice is the Aggregating Algorithm (AA) proposed in Vovk (1990Vovk ( , 1998; this algorithm generalizes, e.g., the Bayesian merging scheme (DeSantis, Markowsky, & Wegman, 1988), the Weighted Majority Algorithm (Littlestone & Warmuth, 1994;Vovk, 1992), and Cover's (Cover & Ordentlich, 1996) universal portfolio algorithm. The AA works well in many situations; it was felt, however, that for the important problem of tracking the best expert (Littlestone & Warmuth, 1994;Herbster & Warmuth, 1995;Freund et al, 1997) the AA was not sufficient, and modifications of the AA or entirely different algorithms were used.…”

Section: Introductionmentioning

confidence: 99%

“…The AA works well in many situations; it was felt, however, that for the important problem of tracking the best expert (Littlestone & Warmuth, 1994;Herbster & Warmuth, 1995;Freund et al, 1997) the AA was not sufficient, and modifications of the AA or entirely different algorithms were used. In this paper we show that a general form of the AA (described in, e.g., Vovk (1998), Appendix A) can be applied to the problem of tracking the best expert (Section 3.3) and to several other related problems. This general form involves uncountably many elementary predictors; applications of the AA to uncountable pools of elementary predictors have been considered earlier by, e.g., Cover and Ordentlich (1996) and Freund (1996).…”

Section: Introductionmentioning

confidence: 99%

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Vovk

1999

Machine Learning

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Abstract. In this paper we continue study of the games of prediction with expert advice with uncountably many experts. A convenient interpretation of such games is to construe the pool of experts as one "stochastic predictor", who chooses one of the experts in the pool at random according to the prior distribution on the experts and then replicates the (deterministic) predictions of the chosen expert. We notice that if the stochastic predictor's total loss is at most L with probability at least p then the learner's loss can be bounded by cL + a ln 1 p for the usual constants c and a. This interpretation is used to revamp known results and obtain new results on tracking the best expert. It is also applied to merging overconfident experts and to fitting polynomials to data.

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“…We will call the triple ( , , λ) our game. Our assumptions about the game ( , , λ) are the same as in (Vovk, 1998): 1. is a compact topological space. 2.…”

Section: Protocolmentioning

confidence: 99%

Section: Formal Descriptionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Vovk

1999

Machine Learning

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“…One of the main metaphors of online learning has been the aggregation of many simple "experts" [LW94,Vov98,FS97]. In our application, each combined list serves as an expert.…”

Section: Introductionmentioning

confidence: 99%

Combining Initial Segments of Lists

Warmuth

Koolen

Helmbold

2011

Lecture Notes in Computer Science

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Abstract. We propose a new way to build a combined list from K base lists, each containing N items. A combined list consists of top segments of various sizes from each base list so that the total size of all top segments equals N . A sequence of item requests is processed and the goal is to minimize the total number of misses. That is, we seek to build a combined list that contains all the frequently requested items. We first consider the special case of disjoint base lists. There, we design an efficient algorithm that computes the best combined list for a given sequence of requests. In addition, we develop a randomized online algorithm whose expected number of misses is close to that of the best combined list chosen in hindsight. We prove lower bounds that show that the expected number of misses of our randomized algorithm is close to the optimum. In the presence of duplicate items, we show that computing the best combined list is NP-hard. We show that our algorithms still apply to a linearized notion of loss in this case. We expect that this new way of aggregating lists will find many ranking applications.

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Adaptive Game Playing Using Multiplicative Weights

Freund

Schapire

1999

Games and Economic Behavior

403

398

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We present a simple algorithm for playing a repeated game. We show that a player using this algorithm suffers average loss that is guaranteed to come close to the minimum loss achievable by any fixed strategy. Our bounds are nonasymptotic and hold for any opponent. The algorithm, which uses the multiplicative-weight methods of Littlestone and Warmuth, is analyzed using the Kullback-Liebler divergence. This analysis yields a new, simple proof of the min-max theorem, as well as a provable method of approximately solving a game. A variant of our game-playing algorithm is proved to be optimal in a very strong sense. Journal of Economic Literature Classification Numbers: C44, C70, D83.

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A game of prediction with expert advice

Cited by 169 publications

References 27 publications

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Combining Initial Segments of Lists

Adaptive Game Playing Using Multiplicative Weights

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