  We study the problem of multi-task learning under user-level differential
privacy, in which $n$ users contribute data to $m$ tasks, each involving a
subset of users. One important aspect of the problem, that can significantly
impact quality, is the distribution skew among tasks. Certain tasks may have
much fewer data samples than others, making them more susceptible to the noise
added for privacy. It is natural to ask whether algorithms can adapt to this
skew to improve the overall utility.
  We give a systematic analysis of the problem, by studying how to optimally
allocate a user's privacy budget among tasks. We propose a generic algorithm,
based on an adaptive reweighting of the empirical loss, and show that when
there is task distribution skew, this gives a quantifiable improvement of
excess empirical risk.
  Experimental studies on recommendation problems that exhibit a long tail of
small tasks, demonstrate that our methods significantly improve utility,
achieving the state of the art on two standard benchmarks.
