  We present an asymptotically optimal $(\epsilon,\delta)$ differentially
private mechanism for answering multiple, adaptively asked, $\Delta$-sensitive
queries, settling the conjecture of Steinke and Ullman [2020]. Our algorithm
has a significant advantage that it adds independent bounded noise to each
query, thus providing an absolute error bound. Additionally, we apply our
algorithm in adaptive data analysis, obtaining an improved guarantee for
answering multiple queries regarding some underlying distribution using a
finite sample. Numerical computations show that the bounded-noise mechanism
outperforms the Gaussian mechanism in many standard settings.
