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
A game of prediction with expert advice
Journal of Computer and System Sciences - Special issue on the eighth annual workshop on computational learning theory, July 5–8, 1995
Path kernels and multiplicative updates
The Journal of Machine Learning Research
Efficient algorithms for online decision problems
Journal of Computer and System Sciences - Special issue: Learning theory 2003
Learning Permutations with Exponential Weights
The Journal of Machine Learning Research
Learning permutations with exponential weights
COLT'07 Proceedings of the 20th annual conference on Learning theory
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Consider the following setting for an on-line algorithm (introduced in [FS97]) that learns from a set of experts: In trial t the algorithm chooses an expert with probability p$^{t}_{i}$ . At the end of the trial a loss vector Lt∈[0,R]n for the n experts is received and an expected loss of ∑ip$^{t}_{i}$L$^{t}_{i}$ is incurred. A simple algorithm for this setting is the Hedge algorithm which uses the probabilities $p^{t}_{i} \sim exp^{-\eta L^{L*, R and n, then the total expected loss of the Hedge/WMR algorithm is at most $$L_{*} + \sqrt{\bf 2}\sqrt{L_{*}R{\rm log} n} + O({\rm log} n)$$ The factor of $\sqrt{\bf 2}$ is in some sense optimal [Vov97].