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Register a new userA note on differentially private clustering with large additive error
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Added by
Huy Nguyen
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Authors
Huy L. Nguyen
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In this note, we describe a simple approach to obtain a differentially private algorithm for k-clustering with nearly the same multiplicative factor as any non-private counterpart at the cost of a large polynomial additive error. The approach is the combination of a simple geometric observation independent of privacy consideration and any existing private algorithm with a constant approximation.
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Given a data set of size $n$ in $d$-dimensional Euclidean space, the $k$-means problem asks for a set of $k$ points (called centers) so that the sum of the $ell_2^2$-distances between points of a given data set of size $n$ and the set of $k$ centers is minimized. Recent work on this problem in the locally private setting achieves constant multiplicative approximation with additive error $tilde{O} (n^{1/2 + a} cdot k cdot max {sqrt{d}, sqrt{k} })$ and proves a lower bound of $Omega(sqrt{n})$ on the additive error for any solution with a constant number of rounds. In this work we bridge the gap between the exponents of $n$ in the upper and lower bounds on the additive error with two new algorithms. Given any $alpha>0$, our first algorithm achieves a multiplicative approximation guarantee which is at most a $(1+alpha)$ factor greater than that of any non-private $k$-means clustering algorithm with $k^{tilde{O}(1/alpha^2)} sqrt{d n} mbox{poly}log n$ additive error. Given any $c>sqrt{2}$, our second algorithm achieves $O(k^{1 + tilde{O}(1/(2c^2-1))} sqrt{d n} mbox{poly} log n)$ additive error with constant multiplicative approximation. Both algorithms go beyond the $Omega(n^{1/2 + a})$ factor that occurs in the additive error for arbitrarily small parameters $a$ in previous work, and the second algorithm in particular shows for the first time that it is possible to solve the locally private $k$-means problem in a constant number of rounds with constant factor multiplicative approximation and polynomial dependence on $k$ in the additive error arbitrarily close to linear.
Correlation clustering is a widely used technique in unsupervised machine learning. Motivated by applications where individual privacy is a concern, we initiate the study of differentially private correlation clustering. We propose an algorithm that achieves subquadratic additive error compared to the optimal cost. In contrast, straightforward adaptations of existing non-private algorithms all lead to a trivial quadratic error. Finally, we give a lower bound showing that any pure differentially private algorithm for correlation clustering requires additive error of $Omega(n)$.
We introduce a new $(epsilon_p, delta_p)$-differentially private algorithm for the $k$-means clustering problem. Given a dataset in Euclidean space, the $k$-means clustering problem requires one to find $k$ points in that space such that the sum of squares of Euclidean distances between each data point and its closest respective point among the $k$ returned is minimised. Although there exist privacy-preserving methods with good theoretical guarantees to solve this problem [Balcan et al., 2017; Kaplan and Stemmer, 2018], in practice it is seen that it is the additive error which dictates the practical performance of these methods. By reducing the problem to a sequence of instances of maximum coverage on a grid, we are able to derive a new method that achieves lower additive error then previous works. For input datasets with cardinality $n$ and diameter $Delta$, our algorithm has an $O(Delta^2 (k log^2 n log(1/delta_p)/epsilon_p + ksqrt{d log(1/delta_p)}/epsilon_p))$ additive error whilst maintaining constant multiplicative error. We conclude with some experiments and find an improvement over previously implemented work for this problem.
Densest subgraph detection is a fundamental graph mining problem, with a large number of applications. There has been a lot of work on efficient algorithms for finding the densest subgraph in massive networks. However, in many domains, the network is private, and returning a densest subgraph can reveal information about the network. Differential privacy is a powerful framework to handle such settings. We study the densest subgraph problem in the edge privacy model, in which the edges of the graph are private. We present the first sequential and parallel differentially private algorithms for this problem. We show that our algorithms have an additive approximation guarantee. We evaluate our algorithms on a large number of real-world networks, and observe a good privacy-accuracy tradeoff when the network has high density.
Large data collections required for the training of neural networks often contain sensitive information such as the medical histories of patients, and the privacy of the training data must be preserved. In this paper, we introduce a dropout technique that provides an elegant Bayesian interpretation to dropout, and show that the intrinsic noise added, with the primary goal of regularization, can be exploited to obtain a degree of differential privacy. The iterative nature of training neural networks presents a challenge for privacy-preserving estimation since multiple iterations increase the amount of noise added. We overcome this by using a relaxed notion of differential privacy, called concentrated differential privacy, which provides tighter estimates on the overall privacy loss. We demonstrate the accuracy of our privacy-preserving dropout algorithm on benchmark datasets.
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