  We consider the task of constructing confidence intervals with differential
privacy. We propose two private variants of the non-parametric bootstrap, which
privately compute the median of the results of multiple "little" bootstraps run
on partitions of the data and give asymptotic bounds on the coverage error of
the resulting confidence intervals. For a fixed differential privacy parameter
$\epsilon$, our methods enjoy the same error rates as that of the non-private
bootstrap to within logarithmic factors in the sample size $n$. We empirically
validate the performance of our methods for mean estimation, median estimation,
and logistic regression with both real and synthetic data. Our methods achieve
similar coverage accuracy to existing methods (and non-private baselines) while
providing notably shorter ($\gtrsim 10$ times) confidence intervals than
previous approaches.
