Minimax learning in iterated games via distributional majorization

We consider a framework for making a sequence of decisions based on a sequence of observations. Instead of pursuing a traditional model based approach, we take a game theoretic perspective and consider the case where we are given a pool of candidate algorithms and have reason to believe that some algorithm in the pool will do well. We exhibit the optimal method for combining these algorithms and show that it achieves a performance comparable to the best algorithm in the group. We address implementation issues and present several practical algorithms that achieve provably near optimal performance. This work is applicable to weather prediction, portfolio theory, supervised pattern recognition, gambling, and in general, iterated game playing. In the process of performing this analysis, we derive bounds on the expected maximum of N random variables with given marginal distributions in terms of an operator called the upper quantile expectation operator. Studying the properties of this operator leads to several partial orderings on the space of univariate distributions which we refer to as distributional majorization orderings. Distributional majorization is closely related to second order stochastic dominance, the Lorenz order, and classical majorization theory. We prove several new results about distributional majorization that shed light on all of these other areas, and we demonstrate a close relationship between distributional majorization and the statistical notion of uniform integrability.