This manuscript investigates the relationship between Blackwell Approachability, a stochastic vector-valued repeated game, and minimax theory, a single-play scalar-valued scenario. First, it is established in a general setting — one not permitting invocation of minimax theory — that Blackwell’s Approachability Theorem (Blackwell [1]) and its generalization due to Hou [6] are still valid. Second...

For the best-choice (or secretary) problem with an unknown number N of objects, minimax-optimal strategies for the observer and minimax distributions for N are derived under the assumption that N is a random variable with expected value at most M, where M is known. The solution is derived as a special case of the situation where N is constrained by Ef(N) <_ M, where f is increasing with f(i) f(...

We consider the problem of identifying a small sub-matrix of activation in a large noisy matrix. We establish the minimax rate for the problem by showing tight (up to constants) upper and lower bounds on the signal strength needed to identify the sub-matrix. We consider several natural computationally tractable procedures and show that under most parameter scalings they are unable to identify t...

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