The weighted majority algorithm
Information and Computation
A decision-theoretic generalization of on-line learning and an application to boosting
Journal of Computer and System Sciences - Special issue: 26th annual ACM symposium on the theory of computing & STOC'94, May 23–25, 1994, and second annual Europe an conference on computational learning theory (EuroCOLT'95), March 13–15, 1995
Prediction, Learning, and Games
Prediction, Learning, and Games
Finite-memory universal prediction of individual sequences
IEEE Transactions on Information Theory
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We study the online decision problem in which there are T steps to play and n actions to choose. For this problem, several algorithms achieve an optimal regret of O(√T ln n), but they all require about Tn states, which one may not be able to afford when n and T are very large. We are interested in such large scale problems, and we would like to understand what an online algorithm can achieve with only a bounded number of states. We provide two algorithms, both with mn-1 states, for a parameter m, which achieve regret of O(m + (T/m) ln (mn)) and O(n√m+T/√m), respectively. We also show that any online algorithm with mn-1 states must suffer a regret of Ω(T/m), which is close to what our algorithms achieve.