Abstract
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 L t ∈ [0,R]n for the n experts is received and an expected loss of ∑ i p \(^{t}_{i}\) L \(^{t}_{i}\) is incurred. A simple algorithm for this setting is the Hedge algorithm which uses the probabilities \(p^{t}_{i} \sim exp^{-\eta L^{<t}_{i}}\). This algorithm and its analysis is a simple reformulation of the randomized version of the Weighted Majority algorithm (WMR) [LW94] which was designed for the absolute loss. The total expected loss of the algorithm is close to the total loss of the best expert \(L_{*} = min_{i}L^{\leq T}_{i}\). That is, when the learning rate is optimally tuned based on L *, R and n, then the total expected loss of the Hedge/WMR algorithm is at most
The factor of \(\sqrt{\bf 2}\) is in some sense optimal [Vov97].
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Kuzmin, D., Warmuth, M.K. (2005). Optimum Follow the Leader Algorithm. In: Auer, P., Meir, R. (eds) Learning Theory. COLT 2005. Lecture Notes in Computer Science(), vol 3559. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11503415_46
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DOI: https://doi.org/10.1007/11503415_46
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