  We investigate the trade-off between rate, privacy and storage in federated
learning (FL) with top $r$ sparsification, where the users and the servers in
the FL system only share the most significant $r$ and $r'$ fractions,
respectively, of updates and parameters in the FL process, to reduce the
communication cost. We present schemes that guarantee information theoretic
privacy of the values and indices of the sparse updates sent by the users at
the expense of a larger storage cost. To this end, we generalize the scheme to
reduce the storage cost by allowing a certain amount of information leakage.
Thus, we provide the general trade-off between the communication cost, storage
cost, and information leakage in private FL with top $r$ sparsification, along
the lines of two proposed schemes.
