Abstract. One of the surprising recurring phenomena observed in experiments with boosting is that the test error of the generated classifier usually does not increase as its size becomes very large, and often is observed to decrease even after the training error reaches zero. In this paper, we show that this phenomenon is related to the distribution of margins of the training examples with respect to the generated voting classification rule, where the margin of an example is simply the difference between the number of correct votes and the maximum number of votes received by any incorrect label. We show that techniques used in the analysis of Vapnik’s support vector classifiers and of neural networks with small weights can be applied to voting methods to relate the margin distribution to the test error. We also show theoretically and experimentally that boosting is especially effective at increasing the margins of the training examples. Finally, we compare our explanation to those based on the bias-variance decomposition. 1

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The problem of combining preferences arises in several applications, such as combining the results of different search engines. This work describes an efficient algorithm for combining multiple preferences. We first give a formal framework for the problem. We then describe and analyze a new boosting algorithm for combining preferences called RankBoost. We also describe an efficient implementation of the algorithm for certain natural cases. We discuss two experiments we carried out to assess the performance of RankBoost. In the first experiment, we used the algorithm to combine different WWW search strategies, each of which is a query expansion for a given domain. For this task, we compare the performance of RankBoost to the individual search strategies. The second experiment is a collaborative-filtering task for making movie recommendations. Here, we present results comparing RankBoost to nearest-neighbor and regression algorithms.

In the multi-armed bandit problem, a gambler must decide which arm of £ non-identical slot machines to play in a sequence of trials so as to maximize his reward. This classical problem has received much attention because of the simple model it provides of the trade-off between exploration (trying out each arm to find the best one) and exploitation (playing the arm believed to give the best payoff). Past solutions for the bandit problem have almost always relied on assumptions about the statistics of the slot machines. In this work, we make no statistical assumptions whatsoever about the nature of the process generating the payoffs of the slot machines. We give a solution to the bandit problem in which an adversary, rather than a well-behaved stochastic process, has complete control over the payoffs. In a sequence of ¤ plays, we prove that the per-round payoff of our algorithm approaches that of the best arm at the rate ¥§¦¨¤�©������� �. We show by a matching lower bound that this is best possible. We also prove that our algorithm approaches the per-round payoff of any set of strategies at a similar rate: if the best strategy is chosen from a pool of � strategies then our algorithm approaches the per-round payoff of the strategy at the rate ¥ ¦��¨���� � �§ � ���� � ¤ ©����� � �. Finally, we apply our results to the problem of playing an unknown repeated matrix game. We show that our algorithm approaches the minimax payoff of the unknown game at the rate ¥ ¦ ¤ ©����� � �.

Convex programming involves a convex set F R and a convex function c : F ! R. The goal of convex programming is to nd a point in F which minimizes c. In this paper, we introduce online convex programming. In online convex programming, the convex set is known in advance, but in each step of some repeated optimization problem, one must select a point in F before seeing the cost function for that step. This can be used to model factory production, farm production, and many other industrial optimization problems where one is unaware of the value of the items produced until they have already been constructed. We introduce an algorithm for this domain, apply it to repeated games, and show that it is really a generalization of in nitesimal gradient ascent, and the results here imply that generalized in nitesimal gradient ascent (GIGA) is universally consistent.

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How does a search engine company decide what ads to display with each query so as to maximize its revenue? This turns out to be a generalization of the online bipartite matching problem. We introduce the notion of a tradeoff revealing LP and use it to derive two optimal algorithms achieving competitive ratios of 1 − 1/e for this problem. 1

PAC-Bayesian learning methods combine the informative priors of Bayesian methods with distribution-free PAC guarantees. Building on earlier methods for PAC-Bayesian model selection, this paper presents a method for PAC-Bayesian model averaging. The main result is a bound on generalization error of an arbitrary weighted mixture of concepts that depends on the empirical error of that mixture and the KLdivergence of the mixture from the prior. A simple characterization is also given for the error bound achieved by the optimal weighting. 1

Two minimal requirements for a satisfactory multiagent learning algorithm are that it 1. learns to play optimally against stationary opponents and 2. converges to a Nash equilibrium in self-play. The previous algorithm that has come closest, WoLF-IGA, has been proven to have these two properties in 2-player 2-action (repeated) games -- assuming that the opponent's mixed strategy is observable. Another algorithm, ReDVaLeR (which was introduced after the algorithm described in this paper), achieves the two properties in games with arbitrary numbers of actions and players, but still requires that the opponents' mixed strategies are observable. In this paper we present AWESOME, the first algorithm that is guaranteed to have the two properties in games with arbitrary numbers of actions and players. It is still the only algorithm that does so while only relying on observing the other players' actual actions (not their mixed strategies). It also learns to play optimally against opponents that eventually become stationary. The basic idea behind AWESOME (Adapt When Everybody is Stationary, Otherwise Move to Equilibrium) is to try to adapt to the others' strategies when they appear stationary, but otherwise to retreat to a precomputed equilibrium strategy. We provide experimental results that suggest that AWESOME converges fast in practice. The techniques used to prove the properties of AWESOME are fundamentally different from those used for previous algorithms, and may help in analyzing future multiagent learning algorithms as well.

We consider the AdaBoost procedure for boosting weak learners. In AdaBoost, a key step is choosing a new distribution on the training examples based on the old distribution and the mistakes made by the present weak hypothesis. We show how AdaBoost 's choice of the new distribution can be seen as an approximate solution to the following problem: Find a new distribution that is closest to the old distribution subject to the constraint that the new distribution is orthogonal to the vector of mistakes of the current weak hypothesis. The distance (or divergence) between distributions is measured by the relative entropy. Alternatively, we could say that AdaBoost approximately projects the distribution vector onto a hyperplane dened by the mistake vector. We show that this new view of AdaBoost as an entropy projection is dual to the usual view of AdaBoost as minimizing the normalization factors of the updated distributions.