# 2 Search Results for "Thakurta, Abhradeep"

Document
##### Efficient Differentially Private F₀ Linear Sketching

Authors: Rasmus Pagh and Nina Mesing Stausholm

Published in: LIPIcs, Volume 186, 24th International Conference on Database Theory (ICDT 2021)

##### Abstract
A powerful feature of linear sketches is that from sketches of two data vectors, one can compute the sketch of the difference between the vectors. This allows us to answer fine-grained questions about the difference between two data sets. In this work we consider how to construct sketches for weighted F₀, i.e., the summed weights of the elements in the data set, that are small, differentially private, and computationally efficient. Let a weight vector w ∈ (0,1]^u be given. For x ∈ {0,1}^u we are interested in estimating ||x∘w||₁ where ∘ is the Hadamard product (entrywise product). Building on a technique of Kushilevitz et al. (STOC 1998), we introduce a sketch (depending on w) that is linear over GF(2), mapping a vector x ∈ {0,1}^u to Hx ∈ {0,1}^τ for a matrix H sampled from a suitable distribution ℋ. Differential privacy is achieved by using randomized response, flipping each bit of Hx with probability p < 1/2. That is, for a vector φ ∈ {0,1}^τ where Pr[(φ)_j = 1] = p independently for each entry j, we consider the noisy sketch Hx + φ, where the addition of noise happens over GF(2). We show that for every choice of 0 < β < 1 and ε = O(1) there exists p < 1/2 and a distribution ℋ of linear sketches of size τ = O(log²(u)ε^{-2}β^{-2}) such that: 1) For random H∼ℋ and noise vector φ, given Hx + φ we can compute an estimate of ||x∘w||₁ that is accurate within a factor 1±β, plus additive error O(log(u)ε^{-2}β^{-2}), w. p. 1-u^{-1}, and 2) For every H∼ℋ, Hx + φ is ε-differentially private over the randomness in φ. The special case w = (1,… ,1) is unweighted F₀. Previously, Mir et al. (PODS 2011) and Kenthapadi et al. (J. Priv. Confidentiality 2013) had described a differentially private way of sketching unweighted F₀, but the algorithms for calibrating noise to their sketches are not computationally efficient, either using quasipolynomial time in the sketch size or superlinear time in the universe size u. For fixed ε the size of our sketch is polynomially related to the lower bound of Ω(log(u)β^{-2}) bits by Jayram & Woodruff (Trans. Algorithms 2013). The additive error is comparable to the bound of Ω(1/ε) of Hardt & Talwar (STOC 2010). An application of our sketch is that two sketches can be added to form a noisy sketch of the form H(x₁+x₂) + (φ₁+φ₂), which allows us to estimate ||(x₁+x₂)∘w||₁. Since addition is over GF(2), this is the weight of the symmetric difference of the vectors x₁ and x₂. Recent work has shown how to privately and efficiently compute an estimate for the symmetric difference size of two sets using (non-linear) sketches such as FM-sketches and Bloom Filters, but these methods have an error bound no better than O(√{̄{m}}), where ̄{m} is an upper bound on ||x₁||₀ and ||x₂||₀. This improves previous work when β = o (1/√{̄{m}}) and log(u)/ε = ̄{m}^{o(1)}. In conclusion our results both improve the efficiency of existing methods for unweighted F₀ estimation and extend to a weighted generalization. We also give a distributed streaming implementation for estimating the size of the union between two input streams.

##### Cite as

Rasmus Pagh and Nina Mesing Stausholm. Efficient Differentially Private F₀ Linear Sketching. In 24th International Conference on Database Theory (ICDT 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 186, pp. 18:1-18:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

```@InProceedings{pagh_et_al:LIPIcs.ICDT.2021.18,
author =	{Pagh, Rasmus and Stausholm, Nina Mesing},
title =	{{Efficient Differentially Private F₀ Linear Sketching}},
booktitle =	{24th International Conference on Database Theory (ICDT 2021)},
pages =	{18:1--18:19},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-179-5},
ISSN =	{1868-8969},
year =	{2021},
volume =	{186},
editor =	{Yi, Ke and Wei, Zhewei},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address =	{Dagstuhl, Germany},
URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2021.18},
URN =		{urn:nbn:de:0030-drops-137264},
doi =		{10.4230/LIPIcs.ICDT.2021.18},
annote =	{Keywords: Differential Privacy, Linear Sketches, Weighted F0 Estimation}
}```
Document
##### Erasure-Resilient Property Testing

Authors: Kashyap Dixit, Sofya Raskhodnikova, Abhradeep Thakurta, and Nithin Varma

Published in: LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

##### Abstract
Property testers form an important class of sublinear algorithms. In the standard property testing model, an algorithm accesses the input function f:D -> R via an oracle. With very few exceptions, all property testers studied in this model rely on the oracle to provide function values at all queried domain points. However, in many realistic situations, the oracle may be unable to reveal the function values at some domain points due to privacy concerns, or when some of the values get erased by mistake or by an adversary. The testers do not learn anything useful about the property by querying those erased points. Moreover, the knowledge of a tester may enable an adversary to erase some of the values so as to increase the query complexity of the tester arbitrarily or, in some cases, make the tester entirely useless. In this work, we initiate a study of property testers that are resilient to the presence of adversarially erased function values. An alpha-erasure-resilient epsilon-tester is given parameters alpha, epsilon in (0,1), along with oracle access to a function f such that at most an alpha fraction of function values have been erased. The tester does not know whether a value is erased until it queries the corresponding domain point. The tester has to accept with high probability if there is a way to assign values to the erased points such that the resulting function satisfies the desired property P. It has to reject with high probability if, for every assignment of values to the erased points, the resulting function has to be changed in at least an epsilon-fraction of the non-erased domain points to satisfy P. We design erasure-resilient property testers for a large class of properties. For some properties, it is possible to obtain erasure-resilient testers by simply using standard testers as a black box. However, there are more challenging properties for which all known testers rely on querying a specific point. If this point is erased, all these testers break. We give efficient erasure-resilient testers for several important classes of such properties of functions including monotonicity, the Lipschitz property, and convexity. Finally, we show a separation between the standard testing and erasure-resilient testing. Specifically, we describe a property that can be epsilon-tested with O(1/epsilon) queries in the standard model, whereas testing it in the erasure-resilient model requires number of queries polynomial in the input size.

##### Cite as

Kashyap Dixit, Sofya Raskhodnikova, Abhradeep Thakurta, and Nithin Varma. Erasure-Resilient Property Testing. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 91:1-91:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

```@InProceedings{dixit_et_al:LIPIcs.ICALP.2016.91,
author =	{Dixit, Kashyap and Raskhodnikova, Sofya and Thakurta, Abhradeep and Varma, Nithin},
title =	{{Erasure-Resilient Property Testing}},
booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
pages =	{91:1--91:15},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-013-2},
ISSN =	{1868-8969},
year =	{2016},
volume =	{55},
editor =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address =	{Dagstuhl, Germany},
URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.91},
URN =		{urn:nbn:de:0030-drops-61947},
doi =		{10.4230/LIPIcs.ICALP.2016.91},
annote =	{Keywords: Randomized algorithms, property testing, error correction, monotoneand Lipschitz functions}
}```
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