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34
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 323 (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.
Tracking the best expert
 In Proceedings of the 12th International Conference on Machine Learning
, 1995
"... Abstract. We generalize the recent relative loss bounds for online algorithms where the additional loss of the algorithm on the whole sequence of examples over the loss of the best expert is bounded. The generalization allows the sequence to be partitioned into segments, and the goal is to bound th ..."
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Cited by 204 (18 self)
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Abstract. We generalize the recent relative loss bounds for online algorithms where the additional loss of the algorithm on the whole sequence of examples over the loss of the best expert is bounded. The generalization allows the sequence to be partitioned into segments, and the goal is to bound the additional loss of the algorithm over the sum of the losses of the best experts for each segment. This is to model situations in which the examples change and different experts are best for certain segments of the sequence of examples. In the single segment case, the additional loss is proportional to log n, where n is the number of experts and the constant of proportionality depends on the loss function. Our algorithms do not produce the best partition; however the loss bound shows that our predictions are close to those of the best partition. When the number of segments is k +1and the sequence is of length ℓ, we can bound the additional loss of our algorithm over the best partition by O(k log n + k log(ℓ/k)). For the case when the loss per trial is bounded by one, we obtain an algorithm whose additional loss over the loss of the best partition is independent of the length of the sequence. The additional loss becomes O(k log n + k log(L/k)), where L is the loss of the best partition with k +1segments. Our algorithms for tracking the predictions of the best expert are simple adaptations of Vovk’s original algorithm for the single best expert case. As in the original algorithms, we keep one weight per expert, and spend O(1) time per weight in each trial.
Universal prediction
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 1998
"... This paper consists of an overview on universal prediction from an informationtheoretic perspective. Special attention is given to the notion of probability assignment under the selfinformation loss function, which is directly related to the theory of universal data compression. Both the probabili ..."
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Cited by 137 (11 self)
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This paper consists of an overview on universal prediction from an informationtheoretic perspective. Special attention is given to the notion of probability assignment under the selfinformation loss function, which is directly related to the theory of universal data compression. Both the probabilistic setting and the deterministic setting of the universal prediction problem are described with emphasis on the analogy and the differences between results in the two settings.
Empirical Support for Winnow and WeightedMajority Algorithms: Results on a Calendar Scheduling Domain
 Machine Learning
, 1995
"... This paper describes experimental results on using Winnow and WeightedMajority based algorithms on a realworld calendar scheduling domain. These two algorithms have been highly studied in the theoretical machine learning literature. We show here that these algorithms can be quite competitive pract ..."
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Cited by 128 (4 self)
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This paper describes experimental results on using Winnow and WeightedMajority based algorithms on a realworld calendar scheduling domain. These two algorithms have been highly studied in the theoretical machine learning literature. We show here that these algorithms can be quite competitive practically, outperforming the decisiontree approach currently in use in the Calendar Apprentice system in terms of both accuracy and speed. One of the contributions of this paper is a new variant on the Winnow algorithm (used in the experiments) that is especially suited to conditions with stringvalued classifications, and we give a theoretical analysis of its performance. In addition we show how Winnow can be applied to achieve a good accuracy/coverage tradeoff and explore issues that arise such as concept drift. We also provide an analysis of a policy for discarding predictors in WeightedMajority that allows it to speed up as it learns. Keywords: Winnow, WeightedMajority, Multiplicative alg...
Relative Loss Bounds for Online Density Estimation with the Exponential Family of Distributions
 MACHINE LEARNING
, 2000
"... We consider online density estimation with a parameterized density from the exponential family. The online algorithm receives one example at a time and maintains a parameter that is essentially an average of the past examples. After receiving an example the algorithm incurs a loss, which is the n ..."
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Cited by 120 (10 self)
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We consider online density estimation with a parameterized density from the exponential family. The online algorithm receives one example at a time and maintains a parameter that is essentially an average of the past examples. After receiving an example the algorithm incurs a loss, which is the negative loglikelihood of the example with respect to the past parameter of the algorithm. An oline algorithm can choose the best parameter based on all the examples. We prove bounds on the additional total loss of the online algorithm over the total loss of the best oline parameter. These relative loss bounds hold for an arbitrary sequence of examples. The goal is to design algorithms with the best possible relative loss bounds. We use a Bregman divergence to derive and analyze each algorithm. These divergences are relative entropies between two exponential distributions. We also use our methods to prove relative loss bounds for linear regression.
Regret in the Online Decision Problem
, 1999
"... At each point in time a decision maker must choose a decision. The payoff in a period from the decision chosen depends on the decision as well as the state of the world that obtains at that time. The difficulty is that the decision must be made in advance of any knowledge, even probabilistic, about ..."
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Cited by 116 (2 self)
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At each point in time a decision maker must choose a decision. The payoff in a period from the decision chosen depends on the decision as well as the state of the world that obtains at that time. The difficulty is that the decision must be made in advance of any knowledge, even probabilistic, about which state of the world will obtain. A range of problems from a variety of disciplines can be framed in this way. In this
Bounds on the sample complexity of Bayesian learning using information theory and the VCdimension
 Machine Learning
, 1994
"... ..."
A Game of Prediction with Expert Advice
 Journal of Computer and System Sciences
, 1997
"... We consider the following problem. At each point of discrete time the learner must make a prediction; he is given the predictions made by a pool of experts. Each prediction and the outcome, which is disclosed after the learner has made his prediction, determine the incurred loss. It is known that, u ..."
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Cited by 109 (8 self)
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We consider the following problem. At each point of discrete time the learner must make a prediction; he is given the predictions made by a pool of experts. Each prediction and the outcome, which is disclosed after the learner has made his prediction, determine the incurred loss. It is known that, under weak regularity, the learner can ensure that his cumulative loss never exceeds cL+ a ln n, where c and a are some constants, n is the size of the pool, and L is the cumulative loss incurred by the best expert in the pool. We find the set of those pairs (c; a) for which this is true.
Using and combining predictors that specialize
 In 29th STOC
, 1997
"... Abstract. We study online learning algorithms that predict by combining the predictions of several subordinate prediction algorithms, sometimes called “experts. ” These simple algorithms belong to the multiplicative weights family of algorithms. The performance of these algorithms degrades only loga ..."
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Cited by 96 (13 self)
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Abstract. We study online learning algorithms that predict by combining the predictions of several subordinate prediction algorithms, sometimes called “experts. ” These simple algorithms belong to the multiplicative weights family of algorithms. The performance of these algorithms degrades only logarithmically with the number of experts, making them particularly useful in applications where the number of experts is very large. However, in applications such as text categorization, it is often natural for some of the experts to abstain from making predictions on some of the instances. We show how to transform algorithms that assume that all experts are always awake to algorithms that do not require this assumption. We also show how to derive corresponding loss bounds. Our method is very general, and can be applied to a large family of online learning algorithms. We also give applications to various prediction models including decision graphs and “switching ” experts. 1