Lower Bounds in Differential Privacy

By Anindya De. 2011

136 citations recorded by the source corpus.

This paper is about private data analysis, in which a trusted curator holding a confidential database responds to real vector-valued queries. A common approach to ensuring privacy for the database elements is to add appropriately generated random noise to the answers, releasing only these noisy responses. A line of study initiated in [7] examines the amount of distortion needed to prevent privacy violations of various kinds. The results in the literature vary according to several parameters, including the size of the database, the size of the universe from which data elements are drawn, the "amount" of privacy desired, and for the purposes of the current work, the arity of the query. In this paper we sharpen and unify these bounds. Our foremost result combines the techniques of Hardt and Talwar [11] and McGregor et al. [13] to obtain linear lower bounds on distortion when providing differential privacy for a (contrived) class of low-sensitivity queries. (A query has low sensitivity if the data of a single individual has small effect on the answer.) Several structural results follow as immediate corollaries: — We separate so-called counting queries from arbitrary low-sensitivity queries, proving the latter requires more noise, or distortion, than does the former; — We separate (e,0)-differential privacy from its well-studied relaxation (e,δ)-differential privacy, even when δ∈2−o(n) is negligible in the size n of the database, proving the latter requires less distortion than the former; — We demonstrate that (e,δ)-differential privacy is much weaker than (e,0)-differential privacy in terms of mutual information of the transcript of the mechanism with the database, even when δ∈2−o(n) is negligible in the size n of the database. We also simplify the lower bounds on noise for counting queries in [11] and also make them unconditional. Further, we use a characterization of (e,δ) differential privacy from [13] to obtain lower bounds on the distortion needed to ensure (e,δ)-differential privacy for e,δ>0. We next revisit the LP decoding argument of [10] and combine it with a recent result of Rudelson [15] to improve on a result of Kasiviswanathan et al. [12] on noise lower bounds for privately releasing l-way marginals.

Subfields: Artificial Intelligence, Sociology and Political Science, Privacy & Security

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Citations in the corpus

Pure Differentially Private Summation from Anonymous Messages (2020, background): “In the central and local settings, several prior works either obtained pure protocols in regimes where approximate protocols were previously known, or proved separations between pure and approximate protocols (e.g., [HT10, De12, NTZ13, SU15, BNS18]).”
Rényi Differential Privacy (2017, background): “Extending this argument to domains of exponential sizes (for instance, to a boolean hypercube) cannot be done without diluting the guarantee exponentially [4].”
Extremal Mechanisms for Local Differential Privacy (2014, neutral): “These results follow from our main theorems and are of independent interest in other applications where lower bounds in statistical analysis are studied [3, 21, 6, 9].”
Concentrated Differential Privacy: Simplifications, Extensions, and Lower Bounds (2016, methodology): “These lower bounds reinforce the connection between group privacy and lower bounds [HT10, De12, SU15a].”
Local Privacy and Minimax Bounds: Sharp Rates for Probability Estimation (2013, methodology): “In this paper, we bring together the formal treatment of disclosure risk provided by diﬀerential privacy with the tools of minimax decision theory to provide a theoretical treatment of probability estimation under privacy constraints.”
The geometry of differential privacy: the sparse and approximate cases (2012, background): “Lower bounds on the amount of noise needed for general low sensitivity queries have been shown in [DN03, DMT07, DY08, DMNS06, RHS07, HT10, De12].”
Fingerprinting codes and the price of approximate differential privacy (2013, background): “Their result was improved by De [11], who proved that the same sample complexity lower bound applies even for algorithms that can introduce arbitrarily large error on a constant fraction of the queries.”
Private Learning and Sanitization: Pure vs. Approximate Differential Privacy (2013, background): “Another related work is the work of De [10], who proved a separation between pure ε-differential privacy and approximate (ε,δ )-differential privacy.”
The Power of Linear Reconstruction Attacks (2012, methodology): “Definition 2.”
The optimal mechanism in differential privacy (2012, background): “Anindya De [20] studies lower bound on the additive noise for Lipschitz query functions in ( , δ)-differential privacy which uses a different metric for the noise, and the lower bound depends on the size of the database.”
Unconditional differentially private mechanisms for linear queries (2012, background): “Finally, recent work by De [De11] has shown that the lower bounds on err(F ) for certain speci.c F s shown in[HT10] canbe extended to hold under the weaker de.nition as well.”
Exposed! A Survey of Attacks on Private Data (2017, background): “A more sophisticated argument, based on restricted isometry properties, was used by Dwork et al. (2007) (and later generalized by De 2012) to handle releases where most statistics are answered α-accurately, but the error on some fraction (bounded by a parameter η) of statistics is arbitrarily high.”
Accuracy-Privacy Tradeoffs for Two-Party Differentially Private Protocols (2013, background): “The optimality of differentially private mechanisms has since been studied in different models such as answering multiple linear queries [17], contingency tables [20], or certain classes of low-sensitivity queries [8].”