Smooth Lower Bounds for Differentially Private Algorithms via Padding-and-Permuting Fingerprinting Codes
CoRR(2023)
摘要
Fingerprinting arguments, first introduced by Bun, Ullman, and Vadhan (STOC
2014), are the most widely used method for establishing lower bounds on the
sample complexity or error of approximately differentially private (DP)
algorithms. Still, there are many problems in differential privacy for which we
don't know suitable lower bounds, and even for problems that we do, the lower
bounds are not smooth, and usually become vacuous when the error is larger than
some threshold.
In this work, we present a new framework and tools to generate smooth lower
bounds on the sample complexity of differentially private algorithms satisfying
very weak accuracy. We illustrate the applicability of our method by providing
new lower bounds in various settings:
1. A tight lower bound for DP averaging in the low-accuracy regime, which in
particular implies a lower bound for the private 1-cluster problem introduced
by Nissim, Stemmer, and Vadhan (PODS 2016).
2. A lower bound on the additive error of DP algorithms for approximate
k-means clustering, as a function of the multiplicative error, which is tight
for a constant multiplication error.
3. A lower bound for estimating the top singular vector of a matrix under DP
in low-accuracy regimes, which is a special case of DP subspace estimation
studied by Singhal and Steinke (NeurIPS 2021).
Our main technique is to apply a padding-and-permuting transformation to a
fingerprinting code. However, rather than proving our results using a black-box
access to an existing fingerprinting code (e.g., Tardos' code), we develop a
new fingerprinting lemma that is stronger than those of Dwork et al. (FOCS
2015) and Bun et al. (SODA 2017), and prove our lower bounds directly from the
lemma. Our lemma, in particular, gives a simpler fingerprinting code
construction with optimal rate (up to polylogarithmic factors) that is of
independent interest.
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关键词
private algorithms,padding-and-permuting
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