Differentially Private Sparse Vectors with Low Error, Optimal Space, and Fast Access

Abstract

Representing a sparse histogram, or more generally a sparse vector, is a fundamental task in differential privacy. An ideal solution would use space close to information-theoretical lower bounds, have an error distribution that depends optimally on the desired privacy level, and allow fast random access to entries in the vector. However, existing approaches have only achieved two of these three goals.In this paper we introduce the Approximate Laplace Projection (ALP) mechanism for approximating 𝑘-sparse vectors. This mechanism is shown to simultaneously have information-theoretically optimal space (up to constant factors), fast access to vector entries, and error of the same magnitude as the Laplace-mechanism applied to dense vectors. A key new technique is a unary representation of small integers, which is shown to be robust against "randomized response" noise. This representation is combined with hashing, in the spirit of Bloom filters, to obtain a space-efficient, differentially private representation. Our theoretical performance bounds are complemented by simulations which show that the constant factors on the main performance parameters are quite small, suggesting practicality of the technique. CCS CONCEPTS• Security and privacy → Privacy-preserving protocols.

Bounded Space Differentially Private Quantiles

“…(3) We extend our results to the continual release setting. (4) We empirically validate our results by evaluating DPExpGK, analyzing and comparing various aspects of performance on real-world and synthetic datasets.…”

“…Differentially private statistical estimation: Estimation of global data statistics (or more generally, inference) is in general an important use-case of differential privacy [34]. The related private histogram release problem was studied by [4,5,10,38] and in the continual observation privacy setting [11,12,21]. Perrier et al [47] make improvements over prior work for private release of some statistics such as the moving average of a stream when the data distribution is light-tailed.…”

“…Using Theorem 4.11, we set the first checkpoint value to log | X |/𝛽 12𝛼 * 𝜖 * . 4 As a result, there are at most 𝑘 ≤ log 1+𝛼/2 𝑛 = 𝑂 ( 1 𝛼 log 𝑛) checkpoints using the fact that for all 𝑥 > −1, 𝑥 1+𝑥 ≤ log(1 + 𝑥) ≤ 𝑥. Since 𝑣 cp(𝑠) is the output of DPExpGK given a GK sketch with accuracy parameter 𝛼 * and privacy parameter 𝜖 * it follows that with probability 1 − 𝛽 * where 𝛽 * = 𝑂 (𝛼𝛽/log 𝑛), rank 𝑋 [1:𝑠 ′ ] ∈ (1 − 𝛼/2, 1 + 𝛼/2)𝑞𝑛.…”

Locally Differentially Private Sparse Vector Aggregation

“…Sparse vector data releasing under central-DP. Previous works [5,17,30] discussed a related setting that a single entity wishes to differentially privately release a k-sparse vector v ∈ [0, u] d (u can be large). The neighboring notion is also defined by L 1 distance -a neighboring input pair v ∼ v iff.…”

“…v −v 1 ≤ 1. For example, the newest work on this line -the ALP mechanism [5] showed how to privately encode the k-sparse vector with O(k log(d + u)) bits with L ∞ decoding error of O( log d ). However, the encoding-decoding processes of these works are biased.…”