In many domains, collecting sufficient labeled training data for supervised machine learning requires easily accessible but noisy sources, such as crowdsourcing services or tagged Web data. Noisy labels occur frequently in data sets harvested via these means, sometimes resulting in entire classes of data on which learned classifiers generalize poorly. For real world applications, we argue that it can be beneficial to avoid training on such classes entirely. In this work, we aim to explore the classes in a given data set, and guide supervised training to spend time on a class proportional to its learnability. By focusing the training process, we aim to improve model generalization on classes with a strong signal. To that end, we develop an online algorithm that works in conjunction with classifier and training algorithm, iteratively selecting training data for the classifier based on how well it appears to generalize on each class. Testing our approach on a variety of data sets, we show our algorithm learns to focus on classes for which the model has low generalization error relative to strong baselines, yielding a classifier with good performance on learnable classes.
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-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 expected per-round payoff of our algorithm approaches that of the best arm at the rate O(T sup -1 3 ), and we give an improved rate of convergence when the best arm has fairly low payoff. We also 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 payoff 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. Many applications require optimizing an unknown, noisy function that is expensive to evaluate. We formalize this task as a multi-armed bandit problem, where the payoff function is either sampled from a Gaussian process (GP) or has low RKHS norm. We resolve the important open problem of deriving regret bounds for this setting, which imply novel convergence rates for GP optimization. We analyze GP-UCB, an intuitive upper-confidence based algorithm, and bound its cumulative regret in terms of maximal information gain, establishing a novel connection between GP optimization and experimental design. Moreover, by bounding the latter in terms of operator spectra, we obtain explicit sublinear regret bounds for many commonly used covariance functions. In some important cases, our bounds have surprisingly weak dependence on the dimensionality. In our experiments on real sensor data, GP-UCB compares favorably with other heuristical GP optimization approaches. We consider the sequential Bayesian optimization problem with bandit feedback, adopting a formulation that allows for the reward function to vary with time. We model the reward function using a Gaussian process whose evolution obeys a simple Markov model. We introduce two natural extensions of the classical Gaussian process upper confidence bound (GP-UCB) algorithm. The first, R-GP-UCB, resets GP-UCB at regular intervals. The second, TV-GP-UCB, instead forgets about old data in a smooth fashion. Our main contribution comprises of novel regret bounds for these algorithms, providing an explicit characterization of the trade-off between the time horizon and the rate at which the function varies. We illustrate the performance of the algorithms on both synthetic and real data, and we find the gradual forgetting of TV-GP-UCB to perform favorably compared to the sharp resetting of R-GP-UCB. Moreover, both algorithms significantly outperform classical GP-UCB, since it treats stale and fresh data equally. In a multi-armed bandit problem, an online algorithm chooses from a set of strategies in a sequence of trials so as to maximize the total payoff of the chosen strategies. While the performance of bandit algorithms with a small finite strategy set is quite well understood, bandit problems with large strategy sets are still a topic of very active investigation, motivated by practical applications such as online auctions and web advertisement. The goal of such research is to identify broad and natural classes of strategy sets and payoff functions which enable the design of efficient solutions. In this work we study a very general setting for the multi-armed bandit problem in which the strategies form a metric space, and the payoff function satisfies a Lipschitz condition with respect to the metric. We refer to this problem as the "Lipschitz MAB problem". We present a solution for the multi-armed bandit problem in this setting. That is, for every metric space we define an isometry invariant which bounds from below the performance of Lipschitz MAB algorithms for this metric space, and we present an algorithm which comes arbitrarily close to meeting this bound. Furthermore, our technique gives even better results for benign payoff functions. We also address the full-feedback ("best expert") version of the problem, where after every round the payoffs from all arms are revealed. Until recently, statistical theory has been restricted to the design and analysis of sampling experiments in which the size and composition of the samples are completely determined before the experimentation begins. The reasons for this are partly historical, dating back to the time when the statistician was consulted, if at all, only after the experiment was over, and partly intrinsic in the mathematical difficulty of working with anything but a fixed number of independent random variables. A major advance now appears to be in the making with the creation of a theory of the sequential design of experiments, in which the size and composition of the samples are not fixed in advance but are functions of the observations themselves.