Achievability of asymptotic minimax optimality in online and batch coding

The normalized maximum likelihood model achieves the minimax coding (log-loss) regret for data of fixed sample size n. However, it is a batch strategy, i.e., it requires that n be known in advance. Furthermore, it is computationally infeasible for most statistical models, and several computationally feasible alternative strategies have been devised. We characterize the achievability of asymptotic minimaxity by batch strategies (i.e., strategies that depend on n) as well as online strategies (i.e., strategies independent of n). On one hand, we conjecture that for a large class of models, no online strategy can be asymptotically minimax. We prove that this holds under a slightly stronger definition of asymptotic minimaxity. We also show that in the multinomial model, a Bayes mixture defined by the conjugate Dirichlet prior with a simple dependency on n achieves asymptotic minimaxity for all sequences, thus providing a simpler asymptotic minimax strategy compared to earlier work by Xie and Barron. The numerical results also demonstrate superior finite-sample behavior by a number of novel batch algorithms.