Differentially-private frugal estimation of quantiles

Fast and accurate estimation of quantiles on data streams coming from communication networks, Internet of Things (IoT), and alike, is at the heart of important data processing applications including statistical analysis, latency monitoring, query optimization for parallel database management systems, and more. Indeed, quantiles are more robust indicators for the underlying distribution, compared to moment-based indicators such as mean and variance. The streaming setting additionally constrains accurate tracking of quantiles, as stream items may arrive at a very high rate and must be processed as quickly as possible and discarded, being their storage usually unfeasible. Since an exact solution is only possible when data are fully stored, the goal in practical contexts is to provide an approximate solution with a provably guaranteed bound on the approximation error committed, while using a minimal amount of space. At the same time, with the increasing amount of personal and sensitive information exchanged, it is essential to design privacy protection techniques to ensure confidentiality and data integrity. In this paper we present the following differentially private streaming algorithms for frugal estimation of a quantile: DP-FRUGAL-1U-L, DP-FRUGAL-1U-G, DP-FRUGAL-1U-\r{ho} and DP-FRUGAL-2U-SA. Frugality refers to the ability of the algorithms to provide a good approximation to the sought quantile using a modest amount of space, either one or two units of memory. We provide a theoretical analysis and extensive experimental results, in which we also compare DP-FRUGAL-1U-L with LDPQ, a recent state of the art algorithm, and show that DP-FRUGAL-1U-L outperforms LDPQ in both accuracy and speed.