On robustness and local differential privacy

Abstract

It is of soaring demand to develop statistical analysis tools that are robust against contamination as well as preserving individual data owners' privacy. In spite of the fact that both topics host a rich body of literature, to the best of our knowledge, we are the first to systematically study the connections between the optimality under Huber's contamination model and the local differential privacy (LDP) constraints.In this paper, we start with a general minimax lower bound result, which disentangles the costs of being robust against Huber's contamination and preserving LDP. We further study three concrete examples: a two-point testing problem, a potentially-diverging mean estimation problem and a nonparametric density estimation problem. For each problem, we demonstrate procedures that are optimal in the presence of both contamination and LDP constraints, comment on the connections with the state-of-the-art methods that are only studied under either contamination or privacy constraints, and unveil the connections between robustness and LDP via partially answering whether LDP procedures are robust and whether robust procedures can be efficiently privatised. Overall, our work showcases a promising prospect of joint study for robustness and local differential privacy.

Robust Estimation of Discrete Distributions under Local Differential Privacy

“…In this case, we prove that the resulting estimation rate is not merely the sum of the two estimation rates established in Duchi et al (2014) and Qiao and Valiant (2017) but is always slower. This phenomenon stands in contrast with Li et al (2022), for which the resulting rate is exactly the sum of the rate with privacy but no contamination, plus the rate with contamination but no privacy. The reason is that in Li et al (2022), contamination occurs before privacy.…”

“…Conversely, in our work, contamination occurs after privacy: none of the raw data are outliers and the adversary is allowed to choose the contamination directly on the set of privatized data. As we will highlight below, this difference yields fundamentally novel phenomena compared to the results in Li et al (2022).…”

“…Closer to our setting, the paper by Li et al (2022) combines robustness with local differential privacy. Li et al (2022) give a general lower bound for estimation and testing under local differential privacy and contamination. However, the setting considered by Li et al (2022) is not the same as ours.…”

“…Li et al (2022) give a general lower bound for estimation and testing under local differential privacy and contamination. However, the setting considered by Li et al (2022) is not the same as ours. They consider three settings, namely a two-point testing problem, a density estimation problem and a mean estimation problem, but do not consider discrete distributions.…”

“…In this paper, we study how local differential privacy interacts with robustness. The very recent work of Li et al (2022) is the first to consider this interaction. We explain in more details the differences between their setting and ours in Section 1.1.…”