On Multiple and Hierarchical Universality

Universal coding, prediction and learning usually consider the case where the data generating mechanism is unknown or non-existent, and the goal of the universal scheme is to compete with the best hypothesis from a given hypothesis class, either on the average or in a worst-case scenario. Multiple universality considers the case where the hypothesis class is also unknown: there are several hypothesis classes with possibly different complexities. In hierarchical universality, the simpler classes are nested within more complex classes. The main challenge is to correctly define the universality problem. We propose several possible definitions and derive their min-max optimal solutions. Interestingly, the proposed solutions can be used to obtain Elias codes for universal representation of the integers. We also utilize this approach for variable-memory Markov models, presenting a new interpretation for the known bound over the regret of the celebrated context-tree weighting algorithm and proposing a 3-part code that (slightly) out-performs it.