A bounded-noise mechanism for differential privacy

Answering multiple counting queries is one of the best-studied problems in differential privacy. Its goal is to output an approximation of the average $\frac{1}{n}\sum_{i=1}^n \vec{x}^{(i)}$ of vectors $\vec{x}^{(i)} \in [0,1]^k$, while preserving the privacy with respect to any $\vec{x}^{(i)}$. We present an $(\epsilon,\delta)$-private mechanism with optimal $\ell_\infty$ error for most values of $\delta$. This result settles the conjecture of Steinke and Ullman [2020] for the these values of $\delta$. Our algorithm adds independent noise of bounded magnitude to each of the $k$ coordinates, while prior solutions relied on unbounded noise such as the Laplace and Gaussian mechanisms.