The on-line shortest path problem is considered under various models of partial monitoring. Given a weighted directed acyclic graph whose edge weights can change in an arbitrary (adversarial) way, a decision maker has to choose in each round of a game a path between two distinguished vertices such that the loss of the chosen path (defined as the sum of the weights of its composing edges) be as small as possible. In a setting generalizing the multi-armed bandit problem, after choosing a path, the decision maker learns only the weights of those edges that belong to the chosen path. For this problem, an algorithm is given whose average cumulative loss in n rounds exceeds that of the best path, matched off-line to the entire sequence of the edge weights, by a quantity that is proportional to 1/ √ n and depends only polynomially on the number of edges of the graph. The algorithm can be implemented with linear complexity in the number of rounds n and in the number of edges. An extension to the so-called label efficient setting is also given, in which the decision maker is informed about the weights of the edges corresponding to the chosen path at a total of m ≪ n time instances. Another extension is shown where the decision maker competes against a time-varying path, a generalization of the problem of tracking the best expert. A version of the multi-armed bandit setting for shortest path is also discussed where the decision maker learns only the total weight of the chosen path but not the weights of the individual edges on the path. Applications to routing in packet switched networks along with simulation results are also presented.

Abstract. In this paper the sequential prediction problem with expert advice is considered when the loss is unbounded under partial monitoring scenarios. We deal with a wide class of the partial monitoring problems: the combination of the label efficient and multi-armed bandit problem, that is, where the algorithm is only informed about the performance of the chosen expert with probability ε ≤ 1. For bounded losses an algorithm is given whose expected regret scales with the square root of the loss of the best expert. For unbounded losses we prove that Hannan consistency can be achieved, depending on the growth rate of the average squared losses of the experts.

A simple on-line procedure is considered for the prediction of a real valued sequence. The algorithm is based on a combination of several simple predictors. If the sequence is a realization of an unbounded stationary and ergodic random process then the average of squared errors converges, almost surely, to that of the optimum, given by the Bayes predictor. An analog result is offered for the classification of binary processes.

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Time series prediction covers a vast field of every-day statistical applications in medical, environmental and economic domains. In this paper we develop nonparametric prediction strategies based on the combination of a set of "experts" and show the universal consistency of these strategies under a minimum of conditions. We perform an indepth analysis of real-world data sets and show that these nonparametric strategies are more flexible, faster and generally outperform ARMA methods in terms of normalized cumulative prediction error.