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We consider prediction with expert advice under the log-loss with the goal ofderiving efficient and robust algorithms. We argue that existing algorithmssuch as exponentiated gradient, online gradient descent and online Newton stepdo not adequately satisfy both requirements. Our main contribution is ananalysis of the Prod algorithm that is robust to any data sequence and runs inlinear time relative to the number of experts in each round. Despite theunbounded nature of the log-loss, we derive a bound that is independent of thelargest loss and of the largest gradient, and depends only on the number ofexperts and the time horizon. Furthermore we give a Bayesian interpretation ofProd and adapt the algorithm to derive a tracking regret.

Soft-Bayes: Prod for Mixtures of Experts with Log-Loss