Abstract. In the multiarmed bandit problem, a gambler must decide which arm of K nonidentical 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 T plays, we prove that the per-round payoff of our algorithm approaches that of the best arm at the rate O(T −1/2 ). We show by a matching lower bound that this is the 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 N strategies, then our algorithm approaches the per-round payoff of the strategy at the rate O((log N ) 1/2 T −1/2 ). 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 O(T −1/2 ).

In the multi-armed bandit problem, a gambler must decide which arm of K 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-o between exploration (trying out each arm to nd the best one) and exploitation (playing the arm believed to give the best payo ). 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 payo s 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 payo s.In a sequence of T plays, we prove that the expected per-round payo of our algorithm approaches that of the best arm at the rate O(T ?1=2 ), and we give an improved rate of convergence when the best arm has fairly low payo . We also prove a general matching lower bound on the best possible performance of any algorithm in our setting. In addition, we consider a setting in which the player has a team of \experts" advising him on which arm to play; here, we give a strategy that will guarantee expected payo close to that of the best expert. Finally, we apply our result to the problem of learning to play an unknown repeated matrix game against an all-powerful adversary.

Abstract. Reinforcement learning policies face the exploration versus exploitation dilemma, i.e. the search for a balance between exploring the environment to find profitable actions while taking the empirically best action as often as possible. A popular measure of a policy's success in addressing this dilemma is the regret, that is the loss due to the fact that the globally optimal policy is not followed all the times. One of the simplest examples of the exploration/exploitation dilemma is the multi-armed bandit problem. Lai and Robbins were the first ones to show that the regret for this problem has to grow at least logarithmically in the number of plays. Since then, policies which asymptotically achieve this regret have been devised by Lai and Robbins and many others. In this work we show that the optimal logarithmic regret is also achievable uniformly over time, with simple and efficient policies, and for all reward distributions with bounded support.

This paper explores the power and the limitations of weakly supervised categorization. We present a complete framework that starts with the extraction of various local regions of either discontinuity or homogeneity. A variety of local descriptors can be applied to form a set of feature vectors for each local region. Boosting is used to learn a subset of such feature vectors (weak hypotheses) and to combine them into one final hypothesis for each visual category. This combination of individual extractors and descriptors leads to recognition rates that are superior to other approaches which use only one specific extractor/descriptor setting. To explore the limitation of our system, we had to set up new, highly complex image databases that show the objects of interest at varying scales and poses, in cluttered background, and under considerable occlusion. We obtain classification results up to 81 percent ROC-equal error rate on the most complex of our databases. Our approach outperforms all comparable solutions on common databases.

In this paper we describe the first stage of a new learning system for object detection and recognition. For our system we propose Boosting [5] as the underlying learning technique. This allows the use of very diverse sets of visual features in the learning process within a common framework: Boosting -together with a weak hypotheses findermay choose very inhomogeneous features as most relevant for combination into a final hypothesis. As another advantage the weak hypotheses finder may search the weak hypotheses space without explicit calculation of all available hypotheses, reducing computation time. This contrasts the related work of Agarwal and Roth [1] where Winnow was used as learning algorithm and all weak hypotheses were calculated explicitly. In our first empirical evaluation we use four types of local descriptors: two basic ones consisting of a set of grayvalues and intensity moments and two high level descriptors: moment invariants [8] and SIFTs [12]. The descriptors are calculated from local patches detected by an interest point operator. The weak hypotheses finder selects one of the local patches and one type of local descriptor and efficiently searches for the most discriminative similarity threshold. This differs from other work on Boosting for object recognition where simple rectangular hypotheses [22] or complex classifiers [20] have been used. In relatively simple images, where the objects are prominent, our approach yields results comparable to the state-of-the-art [3]. But we also obtain very good results on more complex images, where the objects are located in arbitrary positions, poses, and scales in the images. These results indicate that our flexible approach, which also allows the inclusion of features from segmented regions and even spatial relationships, leads us a significant step towards generic object recognition.

This paper reports calculations of the collision-free expansion of a semi-infinite plasma. It is shown that the ion front is accelerated to velocities comparable with the thermal velocity of the electrons.

ABSTRACT. In the stochastic multi-armed bandit problem we consider a modification of the UCB algorithm of Auer et al. [4]. For this modified algorithm we give an improved bound on the regret with respect to the optimal reward. While for the original UCB algorithm the regret in Karmed bandits after T trials is bounded by const ·, where ∆ measures the distance between a suboptimal arm and the optimal arm, for the modified UCB algorithm we show an upper bound on the regret of const · K log(T ∆ 2 ) ∆ .

We study on-line learning in the linear regression framework. Most of the performance bounds for on-line algorithms in this framework assume a constant learning rate. To achieve these bounds the learning rate must be optimized based on a posteriori information. This information depends on the whole sequence of examples and thus it is not available to any strictly on-line algorithm. We introduce new techniques for adaptively tuning the learning rate as the data sequence is progressively revealed. Our techniques allow us to prove essentially the same bounds as if we knew the optimal learning rate in advance. Moreover, such techniques apply to a wide class of on-line algorithms, including p-norm algorithms for generalized linear regression and Weighted Majority for linear regression with absolute loss. Our adaptive tunings are radically different from previous techniques, such as the so-called doubling trick. Whereas the doubling trick restarts the on-line algorithm several times using a constant learning rate for each run, our methods save information by changing the value of the learning rate very smoothly. In fact, for Weighted Majority over a finite set of experts our analysis provides a better leading constant than the doubling trick. © 2002 Elsevier Science (USA)