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

TL;DR

This paper introduces Soft-Bayes, an efficient and robust algorithm for prediction with expert advice under log-loss, providing theoretical guarantees and a Bayesian interpretation, suitable for dynamic environments.

Contribution

It presents a new analysis and adaptation of the Prod algorithm that is robust, efficient, and applicable to tracking regret under log-loss.

Findings

Linear-time complexity relative to experts per round

Loss bounds independent of maximum loss or gradient

Effective tracking regret adaptation

Abstract

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.