  We propose and analyze algorithms to solve a range of learning tasks under
user-level differential privacy constraints. Rather than guaranteeing only the
privacy of individual samples, user-level DP protects a user's entire
contribution ($m \ge 1$ samples), providing more stringent but more realistic
protection against information leaks. We show that for high-dimensional mean
estimation, empirical risk minimization with smooth losses, stochastic convex
optimization, and learning hypothesis classes with finite metric entropy, the
privacy cost decreases as $O(1/\sqrt{m})$ as users provide more samples. In
contrast, when increasing the number of users $n$, the privacy cost decreases
at a faster $O(1/n)$ rate. We complement these results with lower bounds
showing the minimax optimality of our algorithms for mean estimation and
stochastic convex optimization. Our algorithms rely on novel techniques for
private mean estimation in arbitrary dimension with error scaling as the
concentration radius $\tau$ of the distribution rather than the entire range.
