Prediction with expert advice under additive noise

Prediction with expert advice serves as a fundamental model in online learning and sequential decision-making. However, in many real-world settings, this classical model proves insufficient as the feedback available to the decision-maker is often subject to noise, errors, or communication constraints. This paper provides fundamental limits on performance, quantified by the regret, in the case when the feedback is corrupted by an additive noise. Our general analysis achieves sharp regret bounds for canonical examples of such additive noise as the Gaussian distribution, the uniform distribution, and a general noise with a log-concave density. This analysis demonstrates how different noise characteristics affect regret bounds and identifies how the regret fundamentally scales as a function of the properties of the noise distribution.