  Combinatorial dimensions play an important role in the theory of machine
learning. For example, VC dimension characterizes PAC learning, SQ dimension
characterizes weak learning with statistical queries, and Littlestone dimension
characterizes online learning.
  In this paper we aim to develop combinatorial dimensions that characterize
bounded memory learning. We propose a candidate solution for the case of
realizable strong learning under a known distribution, based on the SQ
dimension of neighboring distributions. We prove both upper and lower bounds
for our candidate solution, that match in some regime of parameters. In this
parameter regime there is an equivalence between bounded memory and SQ
learning. We conjecture that our characterization holds in a much wider regime
of parameters.
