Worst-case separation caused by retroactivity

Determine whether imposing retroactivity in addition to continual differential privacy yields a worst-case separation for the CountDistinct problem, rather than only the instance-adaptive separation established in the paper.

Background

The paper studies the CountDistinct problem under continual differential privacy and retroactivity. It proves that any private and retroactive algorithm cannot retain instance-adaptive error guarantees: even insertion-only sequences with maximum flippancy one must incur error of order T^{1/4}, matching the worst-case lower bound.

The authors explicitly note that this result is only an instance-adaptive separation, because it shows that retroactivity destroys data-dependent improvements on easy instances. They leave unresolved whether retroactivity also creates a worst-case separation—namely, whether there is a task or setting for which private algorithms achieve substantially smaller worst-case error without retroactivity than any private retroactive algorithm can achieve.