  A recent line of work has shown a qualitative equivalence between
differentially private PAC learning and online learning: A concept class is
privately learnable if and only if it is online learnable with a finite mistake
bound. However, both directions of this equivalence incur significant losses in
both sample and computational efficiency. Studying a special case of this
connection, Gonen, Hazan, and Moran (NeurIPS 2019) showed that uniform or
highly sample-efficient pure-private learners can be time-efficiently compiled
into online learners. We show that, assuming the existence of one-way
functions, such an efficient conversion is impossible even for general
pure-private learners with polynomial sample complexity. This resolves a
question of Neel, Roth, and Wu (FOCS 2019).
