Min-max optimal universal prediction with side information

We consider the problem of sequential prediction of arbitrary real-valued sequences with side information. We first construct a universal algorithm that asymptotically achieves the performance of the best side-information dependent constant predictor, uniformly for all data and side-information sequences. We then extend these results to linear predictors of some fixed order. We derive matching upper and lower bounds, and show that the algorithms are not only universal but they are also optimal such that no sequential algorithm can give better performance for all sequences.