Asymptotic minimax regret for data compression, gambling, and prediction

Abstract

For problems of data compression, gambling, and prediction of individual sequences x/sub 1/, /spl middot//spl middot//spl middot/, x/sub n/ the following questions arise. Given a target family of probability mass functions p(x/sub 1/, /spl middot//spl middot//spl middot/, x/sub n/|/spl theta/), how do we choose a probability mass function q(x/sub 1/, /spl middot//spl middot//spl middot/, x/sub n/) so that it approximately minimizes the maximum regret/belowdisplayskip10ptminus6pt max (log1/q(x/sub 1/, /spl middot//spl middot//spl middot/, x/sub n/)-log1/p(x/sub 1/, /spl middot//spl middot//spl middot/, x/sub n/|/spl theta//spl circ/)) and so that it achieves the best constant C in the asymptotics of the minimax regret, which is of the form (d/2)log(n/2/spl pi/)+C+o(1), where d is the parameter dimension? Are there easily implementable strategies q that achieve those asymptotics? And how does the solution to the worst case sequence problem relate to the solution to the corresponding expectation version min/sub q/ max/sub 0/ E/sub 0/(log1/q(x/sub 1/, /spl middot//spl middot//spl middot/, x/sub n/)-log1/p(x/sub 1/, /spl middot//spl middot//spl middot/, x/sub n/|/spl theta/))? In the discrete memoryless case, with a given alphabet of size m, the Bayes procedure with the Dirichlet(1/2, /spl middot//spl middot//spl middot/, 1/2) prior is asymptotically maximin. Simple modifications of it are shown to be asymptotically minimax. The best constant is C/sub m/=log(/spl Gamma/(1/2)/sup m//(/spl Gamma/(m/2)) which agrees with the logarithm of the integral of the square root of the determinant of the Fisher information. Moreover, our asymptotically optimal strategies for the worst case problem are also asymptotically optimal for the expectation version. Analogous conclusions are given for the case of prediction, gambling, and compression when, for each observation, one has access to side information from an alphabet of size k. In this setting the minimax regret is shown to be k(m-1)/2logn/2/spl pi/k+kC/sub m/+o(1).