Soft-Bayes: Prod for Mixtures of Experts with Log-Loss
This work addresses the need for efficient and robust algorithms in online learning for log-loss scenarios, though it appears incremental as it builds on existing Prod algorithm analysis.
The paper tackled the problem of prediction with expert advice under log-loss by analyzing the Prod algorithm, achieving a robust and efficient method with a bound independent of the largest loss or gradient, depending only on the number of experts and time horizon.
We consider prediction with expert advice under the log-loss with the goal of deriving efficient and robust algorithms. We argue that existing algorithms such as exponentiated gradient, online gradient descent and online Newton step do not adequately satisfy both requirements. Our main contribution is an analysis of the Prod algorithm that is robust to any data sequence and runs in linear time relative to the number of experts in each round. Despite the unbounded nature of the log-loss, we derive a bound that is independent of the largest loss and of the largest gradient, and depends only on the number of experts and the time horizon. Furthermore we give a Bayesian interpretation of Prod and adapt the algorithm to derive a tracking regret.