Function estimation over the Besov spaces under pointwise r (1 ≤ r < ∞) risks is considered. Minimax rates of convergence are derived using a constrained risk inequality and wavelets. Adaptation under pointwise risks is also considered. Sharp lower bounds on the cost of adaptation are obtained and are shown to be attainable by a wavelet estimator. The results demonstrate important differences b...

We apply recent results on the minimax risk in density esti mation to the related problem of pattern classi cation The notion of loss we seek to minimize is an information theoretic measure of how well we can predict the classi cation of future examples given the classi cation of previously seen examples We give an asymptotic characterization of the minimax risk in terms of the metric entropy p...

Voting has been the most general scheme for preference aggregation in multi-agent settings involving agents of diverse preferences. Here, we study a specific type of voting protocols for multi-winner elections, namely approval voting, and we investigate the complexity of computing or approximating the minimax solution in approval voting, concentrating on elections for committees of fixed size. ...

We present a very simple selective minimax search algorithm for two-player gaines. It ahvays expands next the frontier node at the end of the principal variation, or current best line of play, which is the node that determines the minimax value of the root. The algorithm requires no information other than a static evaluation function, and its time overhead per node is similar to that of alpha-b...

The classic minimax search for two-player zero-sum games such as chess has been thoroughly studied since the early years of AI; however for more general nonzero-sum games, minimax is non-optimal, given a player’s knowledge about the opponent. Previously, a few opponent models and algorithms such as M were introduced to improve minimax by simulating the opponent’s search, given the opponent’s st...

Of those things that can be estimated well in an inverse problem, which are best to estimate? Backus-Gilbert resolution theory answers a version of this question for linear (or linearized) inverse problems in Hilbert spaces with additive zero-mean errors with known, finite covariance, and no constraints on the unknown other than the data. This paper extends Backus-Gilbert resolution: it defines...

On the one hand we state {\it Nash equilibrium} (NE) as a formal theorem on multilinear forms and give a pedagogically simple proof, free of game theory terminology. On the other hand, inspired by this formalism, we prove a {\it multilinear minimax theorem}, a generalization of von Neumann’s bilinear minimax theorem. Next, we relate the two theorems by proving that the solution of a multilinear...

When each voter rates or ranks several candidates for a single office, a strong Condorcet winner (SCW) is one who beats all others in two-way races. Among 21 electoral systems examined, 18 will sometimes make candidate X the winner even if thousands of voters would need to change their votes to make X a SCW while another candidate Y could become a SCW with only one such change. Analysis support...

Redundancy of universal codes for a class of sources determines by how much the actual code length exceeds the optimal code length. In the minimax scenario one designs the best code for the worst source within the class. Such minimax redundancy comes in two flavors: either on average or for individual sequences. The latter is also known as the maximal or the worst case minimax redundancy. We st...

In this paper, decision theory was used to derive Bayes and minimax decision rules to estimate allelic frequencies and to explore their admissibility. Decision rules with uniformly smallest risk usually do not exist and one approach to solve this problem is to use the Bayes principle and the minimax principle to find decision rules satisfying some general optimality criterion based on their ris...