Learning with ordinal-bounded memory from positive data

A bounded example memory learner operates incrementally and maintains a memory of finitely many data items. The paradigm is well-studied and known to coincide with set-driven learning. A hierarchy of stronger and stronger learning criteria had earlier been obtained when one considers, for each [email protected]?N, iterative learners that can maintain a memory of at most k previously processed data items. We investigate an extension of the paradigm into the constructive transfinite. For this purpose we use [email protected]?s universal ordinal notation system O. To each ordinal notation in O one can associate a learning criterion in which the number of times a learner can extend its example memory is bounded by an algorithmic count-down from the notation. We prove a general hierarchy result: if b is larger than a in [email protected]?s system, then learners that extend their example memory ''at most b times'' can learn strictly more than learners that can extend their example memory ''at most a times''. For notations for ordinals below @w^2 the result only depends on the ordinals and is notation-independent. For higher ordinals it is notation-dependent. In the setting of learners with ordinal-bounded memory, we also study the impact of requiring that a learner cannot discard an element from memory without replacing it with a new one. A learner satisfying this condition is called cumulative.