#### Title

Calibrating Noise to Sensitivity in Private Data Analysis

#### Date of Original Version

2016

#### Type

Article

#### Abstract or Description

We continue a line of research initiated in~\cite{DiNi03,DwNi04,BDMN05} on privacy-preserving statistical databases.

Consider a trusted server that holds a database of sensitive information. Given a query function $f$ mapping databases to reals, the so-called {\em true answer} is the result of applying $f$ to the database. To protect privacy, the true answer is perturbed by the addition of random noise generated according to a carefully chosen distribution, and this response, the true answer plus noise, is returned to the user.

Previous work focused on the case of noisy sums, in which $f = \sum_i g(x_i)$, where $x_i$ denotes the $i$th row of the database and $g$ maps database rows to $[0,1]$. We extend the study to general functions $f$, proving that privacy can be preserved by calibrating the standard deviation of the noise according to the {\em sensitivity} of the function $f$. Roughly speaking, this is the amount that any single argument to $f$ can change its output. The new analysis shows that for several particular applications substantially less noise is needed than was previously understood to be the case.

The first step is a very clean definition of privacy---now known as differential privacy---and measure of its loss. We also provide a set of tools for designing and combining differentially private algorithms, permitting the construction of complex differentially private analytical tools from simple differentially private primitives.

Finally, we obtain separation results showing the increased value of interactive statistical release mechanisms over non-interactive ones.