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We show a regret minimization algorithm for setting the reserve price in second-price auctions. We make the assumption that all bidders draw their bids from the same unknown and arbitrary distribution. Our algorithm is computationally efficient, and achieves a regret of e O( √ T ), even when the number of bidders is stochastic with a known distribution.

Kernel-based linear-threshold algorithms, such as support vector machines and Perceptron-like algorithms, are among the best available techniques for solving pattern classification problems. In this paper, we describe an extension of the classical Perceptron algorithm, called second-order Perceptron, and analyze its performance within the mistake bound model of on-line learning. The bound achieved by our algorithm depends on the sensitivity to second-order data information and is the best known mistake bound for (efficient) kernel-based linear-threshold classifiers to date. This mistake bound, which strictly generalizes the well-known Perceptron bound, is expressed in terms of the eigenvalues of the empirical data correlation matrix and depends on a parameter controlling the sensitivity of the algorithm to the distribution of these eigenvalues. Since the optimal setting of this parameter is not known a priori, we also analyze two variants of the second-order Perceptron algorithm: one that adaptively sets the value of the parameter in terms of the number of mistakes made so far, and one that is parameterless, based on pseudoinverses.

We investigate the PAC learnability of classes of f0; : : :; ng-valued functions. For n = 1 it is known that the niteness of the Vapnik-Chervonenkis dimension is necessary and su cient for learning. In this paper we present a general scheme for extending the VC-dimension to the case n > 1. Our scheme de nes a wide variety of notions of dimension in which several variants of the VC-dimension, previously introduced in the context of learning, appear as special cases. Our main result is a simple condition characterizing the set of notions of dimension whose niteness is necessary and su cient for learning. This provides a variety of new tools for determining the learnability of a class of multi-valued functions. Our characterization is also shown to hold in the \robust" variant of PAC model and for any \reasonable" loss function.

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We present and study a partial-information model of online learning, where a decision maker repeatedly chooses from a finite set of actions, and observes some subset of the associated losses. This naturally models several situations where the losses of different actions are related, and knowing the loss of one action provides information on the loss of other actions. Moreover, it generalizes and interpolates between the well studied full-information setting (where all losses are revealed) and the bandit setting (where only the loss of the action chosen by the player is revealed). We provide several algorithms addressing different variants of our setting, and provide tight regret bounds depending on combinatorial properties of the information feedback structure.

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