A generalization of the Mixing Past Posteriors algorithm for an uncountable set of experts

A paper [2, Herbster, Warmuth, 1998] examines an on-line learning problem of aggregating the predictions of a finite set of experts with the goal of predicting almost as well as the best sequence of such experts. The regret in this problem is 𝑅𝑇 = Θ(𝑘 ln𝑁), that is, it gets arbitrary large with a large number of experts 𝑁. This paper considers a similar setting, except with an uncountable set of experts 𝒩 = [𝑐, 𝑑]. We propose a generalization of the Mixing Past Posteriors algorithm for an uncountable set of experts. We provide an upper bound on the regret of this generalization and a lower bound on regret of any algorithm that solves this problem. We also propose a generalization of a standard approach of estimating the regret of algorithms similar to MPP by taking into account additional information regarding the losses of individual experts.