In this paper, we propose a trust region filter method for minimax problems. Based on the filter technique, the minimax problem is transformed to a constrained optimization problem and solved by the traditional filter idea. In the presented algorithm, the acceptable criterion of the trial points is relaxed, so compared to the existed SQP and Newton-type methods for minimax method, this method i...

It is well known that a near minimax polynomial approximation p is obtained by truncating the Chebyshev series of a function fafter n + 1 terms. It is shown that if /'E C' " + " [-1, I], then 1I.f-pII may be expressed in terms off' " ' I) in the same manner as the error of minimax approximation. The result is extended to other types of near minimax approximation.

We consider the random design regression with square loss. We propose a method that aggregates empirical minimizers (ERM) over appropriately chosen random subsets and reduces to ERM in the extreme case, and we establish exact oracle inequalities for its risk. We show that, under the −p growth of the empirical -entropy, the excess risk of the proposed method attains the rate n− 2 2+p for p ∈ (0,...

IMP-minimax is the analog to minimax for games with imperfect information, like card games such as bridge or poker. It computes an optimal strategy for the game if the game has a single player and a certain natural property called perfect recall. IMP-minimax is described fully in a companion paper in this proceedings. Here we introduce an algorithm IMP-alpha-beta that is to IMP-minimax as alpha...

Cai et al. (2010) [4] have studied the minimax optimal estimation of a collection of large bandable covariance matrices whose off-diagonal entries decay to zero at a polynomial rate. They have shown that the minimax optimal procedures are fundamentally different under Frobenius and spectral norms, regardless of the rate of polynomial decay. To gain more insight into this interesting problem, we...

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: average minimax or worst case minimax. We study the worst case minimax redundancy of universal block codes for Markovian sources o...

This paper describes comparisons of the minimax back-up rule and the product back-up rule on a wide variety of games, including Pgames, G-games, three-hole kalah, othello, and Ballard's incremental game. In three-hole kalah, the product rule plays better than a minimax search to the same depth. This is a remarkable result, since it is the first widely known game in which product has been found ...

Let Xn = (X1; ; Xn) be a memoryless source with unknown distribution on a finite alphabet of size k. We identify the asymptotic minimax coding redundancy for this class of sources, and provide a sequence of asymptotically minimax codes. Equivalently, we determine the limiting behavior of the minimax relative entropy minQ maxP D(PX kQX ), where the maximum is over all independent and identically...

We attempt to recover an unknown function from noisy, sampled data. Using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coe cients. The shrinkage can be tuned to be nearly minimax over any member of a wide range of Triebeland Besov-type smoothness constraints, and asymptoti...

IMP-minimax is the analog to minimax for games with imperfect information, like card games such as bridge or poker. It computes an optimal strategy for the game if the game has a single player and a certain natural property called perfect recall. IMP-minimax is described fully in a companion paper in this proceedings. Here we introduce an algorithm IMP-alpha-beta that is to IMP-minimax s alpha-...