A hedged mixture of Cover and Robbins betting strategies on bounded data yields O(ln n) worst-case regret and almost-sure O(ln ln n) regret on typical paths, witnessing a sharp game-theoretic LIL.
Estimating means of bounded random variables by betting
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A sub-Gaussian mixture achieves almost sure ln ln V_T regret on unbounded data via a pathwise bound that holds on the probability-one Ville event.
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Cover meets Robbins while Betting on Bounded Data: $\ln n$ Regret and Almost Sure $\ln\ln n$ Regret
A hedged mixture of Cover and Robbins betting strategies on bounded data yields O(ln n) worst-case regret and almost-sure O(ln ln n) regret on typical paths, witnessing a sharp game-theoretic LIL.
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Eventually LIL Regret: Almost Sure $\ln\ln T$ Regret for a sub-Gaussian Mixture on Unbounded Data
A sub-Gaussian mixture achieves almost sure ln ln V_T regret on unbounded data via a pathwise bound that holds on the probability-one Ville event.