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9939ea1a6d6d6aa610032f53dcfaef91c7fc1e24 | subsection | 43 | 52 | Proofs of Technical Estimates | Note that \epsilon ^{-2} ( {\mathcal {M}}_\beta ^\epsilon -1-\gamma _\beta \epsilon ^2 \partial _X^2) is a Fourier multiplier with symbol \epsilon ^{-2}(m_\beta (\epsilon Z) - 1 + \gamma _\beta \epsilon ^2 Z^2)
and so Lemma REF and (REF ) gives us
\Vert J_0\Vert _{r,q} \le c_* \epsilon ^{2} \Vert \sigma _\beta ^{\prim... | {
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e3a1e725e7cf239eccac22eb044a3f6271cdca64 | subsection | 44 | 52 | Proofs of Technical Estimates | The main estimates we need to apply the above to prove
Lemmas REF and REF are contained in the following:
Lemma 17
There exists \epsilon _0>0 and q_0>0 such for all q \in (0,q_0] there exists C_q>0 for which \epsilon \in (0,\epsilon _0] implies
\sup _{K \in {\mathbb {R}}} \left|{(1+K^2)^{1/4} l^{-1}_\epsilon (K + iq)... | {
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1256e5e68018c53072281f3e8be0542f1a3051c2 | subsection | 45 | 52 | Proofs of Technical Estimates | Estimates near Z =0:
We begin by estimating
B(Z):=\left|l_\epsilon ^{-1} (Z) + \gamma _\beta ^{-1}(1+Z^2)^{-1}
\right|
when |\Re (Z)| \le k_1/\epsilon and |\Im _\beta (Z)| \le b/\epsilon .
It is clear that
B(Z) = \left|l_\epsilon (Z) + \gamma _\beta (1+Z^2) \over l_\epsilon (Z) \gamma _\beta (1+Z^2)
\right|.
Recalli... | {
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27bcc441f49894d1adb62781adc70e193a1fbadd | subsection | 46 | 52 | Proofs of Technical Estimates | Also, it should be evident that
(1+\Re (Z)^2)^{1/4} \le C \epsilon ^{-1/2} \quad \text{and}\quad (1+\Re (Z)^2)^{1/4} |1+Z^2|^{-1} \le C
when
|\Re (Z)| \le k_1/\epsilon and |\Im Z| \le 1/2.
These, the triangle inequality and some naive estimates allow us to conclude that
|q|\le 1/2 \Rightarrow \sup _{|K| \le k_1/\epsi... | {
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ce1b927332f4d85559792be59546640584a9983f | subsection | 47 | 52 | Proofs of Technical Estimates | These, the triangle inequality and some naive estimates allow us to conclude that
|q|\le 1/2 \Rightarrow \sup _{k_1/\epsilon \le |K| \le k_2/\epsilon } \left|{(1+K^2)^{1/4} l^{-1}_\epsilon (K + iq)}
\right|\le C_q \epsilon ^{1/2}.
and
|q|\le 1/2 \Rightarrow \sup _{k_1/\epsilon \le |K| \le k_2/\epsilon } \left|l^{-1}... | {
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} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
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1025ba462af01b0414fd2a22325e1f086a7ae35c | subsection | 48 | 52 | Proofs of Technical Estimates | Thus the triangle inequality tells us that
\Vert {\mathcal {G}}_\epsilon \Vert = \epsilon ^{-1} \Vert {\mathcal {L}}_\epsilon ^{-1}{\mathcal {P}}_\epsilon +\gamma _\beta ^{-1}(1-\partial _X^2)^{-1}\Vert _{B(E^r_q,E^r_q)} \le C_q + C \epsilon ^{-1} \Vert (1-\partial _X^2)^{-1} \left({\mathcal {P}}_\epsilon - 1 \right)\... | {
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} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
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"J. Douglas Wright"
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912333bd465c6d708d68e687b5faaa6cede4f362 | subsection | 49 | 52 | Proofs of Technical Estimates | These are modeled on the proofs for the estimates found in Appendix E.4 of .
(Lemma REF ).
We only address the second estimate since it implies the first.
First:
\Vert J_2 - \widetilde{J}_2\Vert _{r,q} = 2 \Vert \sigma _\beta \left(a (\Phi ^a_\epsilon -\Phi ^0_\epsilon ) - \widetilde{a} (\Phi ^{\widetilde{a}}_\epsilon... | {
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"raw": "Timothy E. Faver and J. Douglas Wright. Exact diatomic Fermi-Pasta-Ulam-Tsingou solitary waves with optical band ripples at infinity. SIAM J. Math. Anal., 50(1):182–250, 2018.",
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beba482cae5ea8e19dd5c6c0bb91cb0f9151e823 | subsection | 50 | 52 | Proofs of Technical Estimates | Then we deploy (REF ) and (REF )
\left|\phi _\epsilon ^a (K_\epsilon ^a X) - \phi _\epsilon ^{{a}}(K_\epsilon ^{\widetilde{a}}X) \right|\le C|a-\widetilde{a}| |X|
for all X.
Thus we have
\left|\Phi _\epsilon ^a (X)- \Phi _\epsilon ^{\widetilde{a}}(X)
\right|\le C|a-\widetilde{a}|(1+|X|).
for any X.
The same sort of... | {
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} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
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4c455703c394c79b70bf2e8999c92d3126dad5c4 | subsection | 51 | 52 | Proofs of Technical Estimates | The last term on the right hand side of this is easily estimated by C_r \epsilon ^{-r}|\widetilde{a}|\Vert R- \widetilde{R}\Vert _{r,0}.
The first two terms on the right hand side above can be handled almost identically to how we dealt with the terms on the right hand side in (REF ), but with R replacing
\sigma _\beta ... | {
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} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
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47109dbea6f77313f8ce434dd099b27a5d6901bd | abstract | 0 | 120 | Abstract | LDP (Local Differential Privacy) has been widely studied to estimate
statistics of personal data (e.g., distribution underlying the data) while
protecting users' privacy. Although LDP does not require a trusted third party,
it regards all personal data equally sensitive, which causes excessive
obfuscation hence the los... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
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5302d3a9c4f735cc387671f60e7ef14906f5b2fd | subsection | 1 | 120 | Introduction | blackDP (Differential Privacy) ,
is becoming a gold standard for data privacy;
it enables
big data
analysis
while protecting users' privacy
against adversaries with arbitrary background knowledge.
According to the underlying architecture,
DP
can be categorized into
the one in the centralized model
and
the one in the l... | {
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"raw": "C. Dwork, F. Mcsherry, K. Nissim, and A. Smith. Calibrating noise to sensitivity in private data analysis. In Proc. 3rd International Conference on... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
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40628d7e25761ef193b0b7125cab3b217d02e8a0 | subsection | 2 | 120 | Introduction | In particular, LDP has been widely studied in the literature.
For example,
Erlingsson et al. proposed the RAPPOR as an obfuscation mechanism providing LDP, and implemented it in Google Chrome browser.
Kairouz et al. showed that under the l_1 and l_2 losses,
the randomized response (generalized to multiple alphabets) an... | {
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Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
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8f57f898187e4ab72e93da566fb3fa80f4d968e8 | subsection | 3 | 120 | Introduction | A possible solution to these problems would be to incorporate ULDP with Pufferfish privacy , , which is used to protect correlated data.
We leave this as future work (see Section for
discussions on the case of multiple
data
per user
and the correlation issue).blackWe focus on a scenario in which it is easy for users t... | {
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"raw": "D. Kifer and A. Machanavajjhala. Pufferfish: A framework for mathematical privacy definitions. ACM Transactions on Database Systems, 39(1):1–36, 2014.",
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8de35d4497e15bcb8b3e74efebfe3abeaa92809a | subsection | 4 | 120 | Notations | Let
\mathbb {R}_{\ge 0} be the set of non-negative real numbers.
Let
n be the number of users,
black[n] = \lbrace 1, 2, \ldots , n\rbrace ,
\mathcal {X}
(resp. \mathcal {Y})
be a finite set of personal (resp. obfuscated) data.
We assume continuous data are discretized into bins in advance (e.g., a location map is divid... | {
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"raw": "P. Golle and K. Partridge. On the anonymity of home/work location pairs. In Proc. 7th International Conference on Pervasive Computing (Pe... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
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3c2fa93d9593902b76ad5896ebbe53eaf54591fe | subsection | 5 | 120 | Notations | In Section , we consider a general setting
that can deal with the user-specific sensitive data \mathcal {X}_S^{(i)} and user-specific mechanisms \mathbf {Q}^{(i)}.
We call the former case a common-mechanism scenario
and the latter a personalized-mechanism scenario.We assume that each user's personal data X^{(i)} is ind... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
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"cs.CR",
"cs.IT",
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981fcc0e82deb5defa800fe98da7b852b3b73ad8 | subsection | 6 | 120 | Privacy Measures | LDP (Local Differential Privacy) is defined as follows:Definition 1 (\epsilon -LDP)
Let \epsilon \in \mathbb {R}_{\ge 0}.
An obfuscation mechanism
\mathbf {Q}
from \mathcal {X} to \mathcal {Y}
provides \epsilon -LDP
if for any x,x^{\prime } \in \mathcal {X} and any y \in \mathcal {Y},\mathbf {Q}(y|x)\le e^\epsilon \ma... | {
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Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
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ced7ddcb8b66e31e9316d1605e4e117ea48663e5 | subsection | 7 | 120 | Utility Measures | In this paper, we use
the l_1 loss (i.e., absolute error) and
the l_2 loss (i.e., squared error)
as utility
measures.
Let
l_1
(resp. l_2^2)
be the l_1 (resp. l_2) loss function, which maps
the estimate \hat{\mathbf {p}} and the true distribution \mathbf {p} to
the loss; i.e.,
l_1(\hat{\mathbf {p}},\mathbf {p}) = \sum _... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
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19197720aeb044e0074480769f32bb0d7bff240b | subsection | 8 | 120 | Obfuscation Mechanisms | blackWe describe the RR (Randomized Response) , and
a generalized version of the RAPPOR as follows.Randomized response. The RR
for |\mathcal {X}|-ary alphabets was studied in , .
Its output range
is identical to the input domain;
i.e., \mathcal {X}= \mathcal {Y}.Formally,
given \epsilon \in \mathbb {R}_{\ge 0},
the
\e... | {
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"raw": "P. Kairouz, K. Bonawitz, and D. Ramage. Discrete distribution estimation under local privacy. In Proc. 33rd International Conference on Machine Learning (ICML'16), pa... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.... | |
ff1f846bc2fac6540d4d2f9e99e640890a3043bf | subsection | 9 | 120 | Obfuscation Mechanisms | In this paper, we compute \psi from two parameters \theta and \epsilon .blackSpecifically, given \theta \in [0,1] and \epsilon \in \mathbb {R}_{\ge 0}, the (\theta ,\epsilon )-generalized RAPPOR
maps x_i to
y
with the probability:\mathbf {Q}_{\it RAP}(y | x_i) &= \textstyle {\prod _{1 \le j \le |\mathcal {X}|} \Pr (y_j... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 765,
"openalex_id": "",
"raw": "U. Erlingsson, V. Pihur, and A. Korolova. RAPPOR: Randomized aggregatable privacy-preserving ordinal response. In Proc. 2014 ACM SIGSAC Conference on Computer and Communications Security (CCS'14), p... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.0... | |
b70c82b5c7553f4efdb93c2df5cdfb6e1f550807 | subsection | 10 | 120 | Distribution Estimation Methods | Here we explain
the empirical estimation method , ,
and the EM reconstruction method , .
Both of them assume that the data collector knows the obfuscation mechanism \mathbf {Q} used to generate \mathbf {Y} from \mathbf {X}.Empirical estimation method. The empirical estimation method
, ,
computes an empirical estimat... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1145/1066157.1066187",
"end": 89,
"openalex_id": "https://openalex.org/W2118024521",
"raw": "R. Agrawal, R. Srikant, and D. Thomas. Privacy preserving OLAP. In Proc. 2005 ACM SIGMOD international conference on Management of data (SIGMOD'... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.041931841522455215,
-0.01042955368757248,
-0.... | |
57702bdba5f23d0e59732a44e8a41f22e54be6c6 | subsection | 11 | 120 | Utility-Optimized LDP (ULDP) | In this section, we focus on the common-mechanism scenario
(outlined in Section REF )
and introduce
ULDP (Utility-optimized Local Differential Privacy),
which provides a privacy guarantee equivalent to \epsilon -LDP only for sensitive data.
Section REF provides the definition of ULDP.
Section REF shows
some theoretical... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.04001383110880852,
-0.024691984057426453,
0.... | |
77f7742dd2dc32affa2c4b74b890c2fc9735b72d | subsection | 12 | 120 | Definition | Figure REF
shows an overview of
ULDP.
An obfuscation mechanism
providing ULDP,
which we call the utility-optimized mechanism,
divides obfuscated data into protected data and invertible data.
Let \mathcal {Y}_P be a set of protected data, and \mathcal {Y}_I= \mathcal {Y}\setminus \mathcal {Y}_P be a set of invertible d... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1354,
"openalex_id": "",
"raw": "N. S. Mangat. An improved ranomized response strategy. Journal of the Royal Statistical Society. Series B (Methodological), 56(1):93–95, 1994.",
"source_ref_id": "89e38817ddaec7af12c827bb70d8... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.03908415883779526,
0.015575693920254707,
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0.027734193950891495,
0.008054815232753754,
-0.... | |
37a8ffbc0b2748697004bdc7fc9d16de4d3e8551 | subsection | 13 | 120 | Definition | For any x,x^{\prime } \in \mathcal {X} and any y \in \mathcal {Y}_P,
\mathbf {Q}(y|x)\le e^\epsilon \mathbf {Q}(y|x^{\prime }).We refer to an obfuscation mechanism \mathbf {Q} providing
(\mathcal {X}_S, \mathcal {Y}_P, \epsilon )-ULDP
as the (\mathcal {X}_S,\mathcal {Y}_P,\epsilon )-utility-optimized mechanism.Example... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 473,
"openalex_id": "",
"raw": "N. S. Mangat. An improved ranomized response strategy. Journal of the Royal Statistical Society. Series B (Methodological), 56(1):93–95, 1994.",
"source_ref_id": "89e38817ddaec7af12c827bb70d8a... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.03296232223510742,
-0.007958264090120792,
-... | |
fed1b161bc2c3628c51ba5d792305362d0459409 | subsection | 14 | 120 | Definition | It should also be noted that
the data collector needs to know
\mathbf {Q} to estimate \mathbf {p} from \mathbf {Y} (as described in Section REF ), and
that the (\mathcal {X}_S,\mathcal {Y}_P,\epsilon )-utility-optimized mechanism \mathbf {Q} itself
includes
the information on what is sensitive for users
(i.e.,
the data... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.033694684505462646,
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0... | |
b6ba360dc5ec40afc1f3a9d6a64422147377e01b | subsection | 15 | 120 | Basic Properties of ULDP | Previous work showed some basic properties of differential privacy (or its variant), such as compositionality
and immunity to post-processing .
We briefly explain theoretical properties of ULDP including
the ones above.Sequential composition. ULDP
is preserved under adaptive sequential composition
when the composed o... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1561/9781601988195",
"end": 144,
"openalex_id": "https://openalex.org/W2027595342",
"raw": "C. Dwork and A. Roth. The Algorithmic Foundations of Differential Privacy. Now Publishers, 2014.",
"source_ref_id": "eca4837ecac0d5ebb4cf5f... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.008699179627001286,
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-0.0008661025785841048,
-0.03076762519776821,
... | |
1e0c09ec26020efc660929a577e6c19719e9a0ca | subsection | 16 | 120 | Basic Properties of ULDP | ULDP
is immune to the post-processing by a randomized algorithm
blackthat preserves data types: protected data or invertible data.
blackSpecifically, if a mechanism \mathbf {Q}_0 provides (\mathcal {X}_S,\mathcal {Y}_P,\varepsilon )-ULDP and a randomized algorithm \mathbf {Q}_1 maps protected data over \mathcal {Y}_P (... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1694,
"openalex_id": "",
"raw": "J. C. Duchi, M. I. Jordan, and M. J. Wainwright. Local privacy, data processing inequalities, and minimax rates. CoRR, abs/1302.3203, 2013.",
"source_ref_id": "b3ac648cf13c7cd653a9cc10b6273f0... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.010551140643656254,
0.01187098678201437,
0.04... | |
0e447c31080f166e93cce3991176aa98582f1d86 | subsection | 17 | 120 | Utility-Optimized Mechanisms | In this section, we focus on the common-mechanism scenario
and propose
the utility-optimized RR (Randomized Response)
and utility-optimized RAPPOR (Sections REF and REF ).
We then analyze the data utility of these mechanisms (Section REF ). | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.026337821036577225,
-0.00019503492512740195,
0.0292... | |
f52728aa916a1803374191385d509053e7dbdda3 | subsection | 18 | 120 | Utility-Optimized Randomized Response | We propose the utility-optimized RR, which is a generalization of Mangat's randomized response to |\mathcal {X}|-ary alphabets with |\mathcal {X}_S| sensitive symbols.
As with the RR, the output range of the utility-optimized RR is identical to the input domain; i.e., \mathcal {X}= \mathcal {Y}.
In addition, we divide ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 168,
"openalex_id": "",
"raw": "N. S. Mangat. An improved ranomized response strategy. Journal of the Royal Statistical Society. Series B (Methodological), 56(1):93–95, 1994.",
"source_ref_id": "89e38817ddaec7af12c827bb70d8a... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.030389610677957535,
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0.05400563403964043,
0.005488284397870302,
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0.04344860091805458,
-0.010991823859512806,
0... | |
2bc896aae09b9b9ae0c7c6a12beaf1f46579f210 | subsection | 19 | 120 | Utility-Optimized Randomized Response | Then the (\mathcal {X}_S,\epsilon )-utility-optimized RR
black(uRR)
is an obfuscation mechanism that
maps x \in \mathcal {X} to y \in \mathcal {Y} (=\mathcal {X})
with the probability \mathbf {Q}_{\it uRR}(y | x) defined as follows:{black}{\mathbf {Q}_{\it uRR}(y | x) =
{\left\lbrace \begin{array}{ll}
c_1 & \text{(if $... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.06065326929092407,
0.02831299602985382,
-0.015193827450275421,
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0.029365578666329384,
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0.04018126428127289,
0.0018572775879874825,
0.02... | |
d89d154d0d455fda034fcedce76e5428cfbd77c3 | subsection | 20 | 120 | Utility-Optimized RAPPOR | Next, we
propose the utility-optimized RAPPOR
with the input alphabet \mathcal {X}= \lbrace x_1, x_2, \cdots , x_{|\mathcal {X}|}\rbrace and the output alphabet \mathcal {Y} = \lbrace 0,1\rbrace ^{|\mathcal {X}|}.
Without loss of generality,
we assume
that x_1, \cdots , x_{|\mathcal {X}_S|}
are
sensitive
and x_{|\mathc... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.03030838631093502,
0.02070920541882515,
-0.025211205706000328,
0.049689881503582,
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0.021762214601039886,
0.009805205278098583,
0.056221600621938705,
0.007912838831543922,
-0.0529862642288208,
0.014871864579617977,
0.0091184601187706,
-0.013712... | |
ca678dd12f50a7f902b25197ad06d1e1a6cfa31a | subsection | 21 | 120 | Utility-Optimized RAPPOR | We
formally define the utility-optimized RAPPOR black(uRAP):
[Figure: Utility-optimized RAPPOR in the case where \mathcal {X}_S=\lbrace x_1, \cdots , x_4\rbrace and \mathcal {X}_N=\lbrace x_5, \cdots , x_{10}\rbrace .]Definition 4 ((\mathcal {X}_S,{black}{\theta ,}\epsilon )-utility-optimized RAPPOR)
Let \mathcal {X}_... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.0344480462372303,
0.029871245846152306,
-0.03098493441939354,
0.02822359837591648,
-0.032647836953401566,
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0.012616711668670177,
0.029581382870674133,
0.04149631783366203,
0.040977612137794495,
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0.009298531338572502,
-0.005873559508472681,
0.0... | |
0d1e0c2ed27a0f0200c692c8a3cf1d312574d621 | subsection | 22 | 120 | Utility-Optimized RAPPOR | \end{array}\right.}
}
if |\mathcal {X}_S|+1 \le j \le |\mathcal {X}|:
\Pr (y_j | x_i) &=
{\left\lbrace \begin{array}{ll}
d_2 & \text{(if $i = j$, $y_j = 0$)}\\
1 - d_2 & \text{(if $i = j$, $y_j = 1$)}\\
1 & \text{(if $i \ne j$, $y_j = 0$)}\\
0 & \text{(if $i \ne j$, $y_j = 1$)}.\\
\end{array}\right.}propULDPrestRAP... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1200,
"openalex_id": "https://openalex.org/W2742225091",
"raw": "T. Wang, J. Blocki, N. Li, and S. Jha. Locally differentially private protocols for frequency estimation. In Proc. 26th USENIX Security Symposium (USENIX'17), pages ... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.07051600515842438,
0.02193322591483593,
-0.05800018087029457,
0.0023314545396715403,
-0.022330068051815033,
0.02922903746366501,
-0.01548452116549015,
0.018361635506153107,
0.030282199382781982,
0.0221774373203516,
-0.03641800582408905,
-0.0017571764765307307,
-0.023337440565228462,
0.0... | |
cc9af85cdc8ff256e5b15df9434d30ad579ec35d | subsection | 23 | 120 | Utility-Optimized RAPPOR | We leave
finding the optimal
\theta for
our
blackuRAP
(with respect to
the estimation error
over all personal data)
as future work.We
refer to
the
(\mathcal {X}_S,\theta ,\epsilon )-blackuRAP
with \theta = \frac{e^{\epsilon /2}}{e^{\epsilon /2} + 1}
in shorthand
as the (\mathcal {X}_S,\epsilon )-blackuRAP. | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.014364030212163925,
0.040277257561683655,
-0.05870715528726578,
0.030452048406004906,
-0.03086397610604763,
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0.0004586496506817639,
0.0038236696273088455,
0.024364691227674484,
0.024379946291446686,
0.00048105770838446915,
0.039636485278606415,
-0.015393846668303013,... | |
6f5468ebcb10d2cf2befd49b0eeb1d7006cb3baf | subsection | 24 | 120 | Utility Analysis | We
evaluate
the l_1 loss
of
the blackuRR
and
blackuRAP
when the empirical estimation method is used for
distribution estimationblackWe note that we use the empirical estimation method in the same way as , and that it might be possible that
other mechanisms have better utility with a different estimation method.
However... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 312,
"openalex_id": "https://openalex.org/W2964074929",
"raw": "P. Kairouz, K. Bonawitz, and D. Ramage. Discrete distribution estimation under local privacy. In Proc. 33rd International Conference on Machine Learning (ICML'16), pa... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.04603911563754082,
0.03209618851542473,
-0.015697233378887177,
-0.022973792627453804,
-0.030646979808807373,
0.012623382732272148,
0.052568189799785614,
0.00015362103295046836,
0.027672285214066505,
0.015865037217736244,
-0.05696158483624458,
0.0019755021203309298,
-0.034597985446453094,
... | |
cabf1349a7065b7ce0ccd60deca5e1dfe1a75266 | subsection | 25 | 120 | Utility Analysis | Then the expected l_1 loss of the (\mathcal {X}_S,\epsilon )-blackuRR
mechanism is given by:\mathbb {E}\left[ l_1(\hat{\mathbf {p}},\mathbf {p}) \right]
\approx & {\textstyle \sqrt{\!\frac{2}{n\pi }}} \biggl ( \sum _{x \in \mathcal {X}_S} \sqrt{\bigl ( \mathbf {p}(x) + 1/u^{\prime } \bigr ) \bigl ( v - \mathbf {p}(x) -... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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... | |
4f46700fa8e2e64e4b0c96716cce027d389a367e | subsection | 26 | 120 | Utility Analysis | Symmetrically, let \mathbf {p}_{U_{\!S}} be the uniform distribution over \mathcal {X}_S.For 0 < \epsilon < \ln (|\mathcal {X}_N|+1),
the l_1 loss is maximized by \mathbf {p}_{U_{\!N}}:proppropLoneRestRRsmall
For any 0 < \epsilon < \ln (|\mathcal {X}_N|+1) and |\mathcal {X}_S| \le |\mathcal {X}_N|,
(REF )
is maximized... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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4cf92c4dd2a9133acd57a11c0179f4b3048d54f7 | subsection | 27 | 120 | Utility Analysis | When \epsilon is close to 0,
we have e^\epsilon - 1 \approx \epsilon .
Thus,
the right-hand side of (REF )
in Proposition REF
can be simplified
as follows:\mathbb {E}\left[ l_1(\hat{\mathbf {p}},\mathbf {p}_{U_{\!N}}) \right] &\approx \! {\textstyle \sqrt{\frac{2}{n\pi }} \cdot \frac{|\mathcal {X}_S|\sqrt{ |\mathcal {... | {
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"raw": "P. Kairouz, K. Bonawitz, and D. Ramage. Discrete distribution estimation under local privacy. In Proc. 33rd International Conference on Machine Learning (ICML'16), pa... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
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69d4efbe528d2e13ec7655bbc6dfa4d9937afeaf | subsection | 28 | 120 | Utility Analysis | The expected l_1-loss of the (\mathcal {X}_S,\epsilon )-blackuRAP
mechanism is:\mathbb {E}\left[ l_1(\hat{\mathbf {p}},\mathbf {p}) \right]
\approx & {\textstyle \sqrt{\frac{2}{n\pi }}} \biggl ( \sum _{j=1}^{|\mathcal {X}_S|} \sqrt{\bigl ( \mathbf {p}(x_j) + 1/u^{\prime } \bigr ) \bigl ( v_N - \mathbf {p}(x_j) \bigr )}... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0... | |
c9cfcd07444ef24a51ddd5019dc9c57e067dda58 | subsection | 29 | 120 | Utility Analysis | Below we instantiate the l_1 loss in the high and low privacy regimes based on this proposition.blackuRAP
in the high privacy regime. If \epsilon is close to 0, we have
e^{\epsilon /2} - 1 \approx \epsilon /2.
Thus, the right-hand side of
(REF )
in Proposition REF
can be simplified as follows:\mathbb {E}\left[ l_1(\h... | {
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"doi": "",
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"raw": "P. Kairouz, K. Bonawitz, and D. Ramage. Discrete distribution estimation under local privacy. In Proc. 33rd International Conference on Machine Learning (ICML'16), pa... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.0... | |
f0cc28d8960c860fb9fd26349688619e1a7a7545 | subsection | 30 | 120 | Utility Analysis | In summary,
the blackuRR
and
blackuRAP
provide much higher utility than
the RR and RAPPOR
when |\mathcal {X}_S| \ll |\mathcal {X}|.
Moreover,
when \epsilon = \ln |\mathcal {X}| and |\mathcal {X}_S| \ll |\mathcal {X}| (resp. |\mathcal {X}_S| \ll |\mathcal {X}|^{\frac{3}{4}}),
the
blackuRR
(resp. blackuRAP)
achieves
almo... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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ba7223b0f8c36bda46a9154bb163a73b5da8a122 | subsection | 31 | 120 | Utility Analysis | Thus, after computing \hat{\mathbf {r}},
the data collector can easily compute
the worst-case value of the second term in (REF ) to know the effect of the estimation error of \hat{\mathbf {\pi }}_k on the accuracy of \hat{\mathbf {p}}.Last but not least, the second term in (REF ) does not depend on \epsilon (while the ... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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74a1d6348bd07067e925e8679b4d7619d6944e41 | subsection | 32 | 120 | Personalized ULDP Mechanisms | We now consider the personalized-mechanism scenario (outlined in Section REF ),
and
propose a
PUM (Personalized ULDP Mechanism)
to keep secret what is sensitive for each user
while enabling the data collector to estimate a distribution.Sections REF describes the PUM.
Section REF explains its privacy properties.
Section... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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... | |
3e1e3991fcbc66b368f37fc6e4a74dead80f4b20 | subsection | 33 | 120 | PUM with | Figure REF shows the overview of the
PUM \mathbf {Q}^{(i)} for the i-th user (i = 1, 2, \ldots , n).
It first
deterministically
maps personal data x \in \mathcal {X} to intermediate data
using a pre-processor f_{pre}^{(i)},
and then
maps the intermediate data to obfuscated data y \in \mathcal {Y}
using a utility-optimi... | {
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{
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"doi": "",
"end": 966,
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"raw": "D. Yang, D. Zhang, and B. Qu. Participatory cultural mapping based on collective behavior data in location based social network. ACM Transactions on Intelligent Systems and Technology, 7(3):30:1–30:... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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... | |
3611e7e2085dc7bc798dba5f6514289b34e7089b | subsection | 34 | 120 | PUM with | Then, \mathcal {X}_S^{(i)} is expressed as \mathcal {X}_S^{(i)} = \bigcup _{1 \le k \le \kappa } \mathcal {X}_{S,k}^{(i)}, and
f_{pre}^{(i)} is given by:f_{pre}^{(i)}(x) =
{\left\lbrace \begin{array}{ll}
\bot _k & \text{(if $x \in \mathcal {X}_{S,k}^{(i)}$)}\\
x & \text{(otherwise)}.\\
\end{array}\right.}After mapping ... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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-... | |
8431be6f0767012b860333d2bab0c21b1dbb121e | subsection | 35 | 120 | PUM with | For example, if
we
prepare only tags named “home” and “workplace”, then
sightseeing places, restaurants,
and any other places are not associated with these tags.
One way to deal with such data is to
create another bot
associated with a tag named “others”
(e.g., if \bot _1 and \bot _2 are associated with
“home” and “wor... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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-... | |
707ece386c09c747e1c9c1224095611812642d0c | subsection | 36 | 120 | Privacy Properties | blackWe analyze the privacy properties of the PUM \mathbf {Q}^{(i)}. First, we show that it provides ULDP.proppropPUMULDP
The PUM \mathbf {Q}^{(i)} (= \mathbf {Q}_{cmn} \circ f_{pre}^{(i)}) provides (\mathcal {X}_S\cup \mathcal {X}_S^{(i)}, \mathcal {Y}_P,\epsilon )-ULDP.
blackWe also show that
our PUM
provides DP in... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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... | |
423e30a19d1dabc3be96cb8cbf82d5f631ec2ce9 | subsection | 37 | 120 | Privacy Properties | Furthermore, the data collector cannot infer where her home is,
since \mathcal {X}_S^{(i)} cannot be inferred from \mathbf {Q}_{cmn} and y \in \mathcal {Y}_P as explained above.blackWe need to take a little care when the i-th user obfuscates
non-sensitive data x \in \mathcal {X}_N\setminus \mathcal {X}_S^{(i)}
using \m... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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... | |
454a543816e474cdda8be9dfb5fd47c82096dae2 | subsection | 38 | 120 | Distribution Estimation | We now explain how to estimate a distribution \mathbf {p} from data \mathbf {Y} obfuscated using the PUM.
Let \mathbf {r}^{(i)} be a distribution of intermediate data for the i-th user:\mathbf {r}^{(i)}(z) =
{\left\lbrace \begin{array}{ll}
\sum _{x \in \mathcal {X}_{S,k}^{(i)}} \mathbf {p}(x) & \hspace{-5.69054pt} \tex... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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... | |
746371552cbf83c0ab3c8eccebbda33feb3fdd22 | subsection | 39 | 120 | Distribution Estimation | Our estimation method first estimates a distribution \mathbf {r} of intermediate data
from obfuscated data \mathbf {Y} using \mathbf {Q}_{cmn}.
This can be performed in the same way as the common-mechanism scenario.
Let \hat{\mathbf {r}} be the estimate of \mathbf {r}.blackAfter computing \hat{\mathbf {r}}, our method ... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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... | |
eaa6adc09df066561840d4933781e3323a65ba89 | subsection | 40 | 120 | Experimental Set-up | We conducted experiments using
two large-scale datasets:Foursquare dataset. The Foursquare dataset (global-scale check-in dataset) is one of the largest location datasets among publicly available datasets (e.g., see , , , ); it contains 33278683 check-ins all over the world, each of which is associated with a POI ID a... | {
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"raw": "D. Yang, D. Zhang, and B. Qu. Participatory cultural mapping based on collective behavior data in location based social network. ACM Transactions on Intelligent Systems and Technology, 7(3):30:1–30:... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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5b1ec9c69b9faf4f4641d80c113ac6a36a44f2ec | subsection | 41 | 120 | Experimental Results | Common-mechanism scenario. We first focused on the common-mechanism scenario, and
evaluated the RR, RAPPOR,
blackuRR,
and
blackuRAP.
As distribution estimation methods,
we used empirical estimation,
empirical estimation with the significance threshold,
and EM reconstruction (denoted by “emp”, “emp+thr”, and “EM”, resp... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 531,
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"raw": "T. Wang, J. Blocki, N. Li, and S. Jha. Locally differentially private protocols for frequency estimation. In Proc. 26th USENIX Security Symposium (USENIX'17), pages 7... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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77c823b96ae9aecf8ca67d55bc14755db008171a | subsection | 42 | 120 | Experimental Results | Overall, our theoretical results in Section REF hold for the two real datasets.We also evaluated the performance when the number of attributes was increased from 4 to 9 in the US Census dataset.
We added, one by one, five attributes as to whether or not a user has served in the military during five periods (“Sept80”, “... | {
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"arxiv_id": "",
"doi": "",
"end": 399,
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"raw": "D. Dua and E. K. Taniskidou. UCI machine learning repository. http://archive.ics.uci.edu/ml, 2017.",
"source_ref_id": "d01588e1e615b059b9d8e7e0844d52081053f0c7"... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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ca4854431dc85b75c04746b090bfcf6c89c3077b | subsection | 43 | 120 | Experimental Results | We used the PUM with \kappa =2 semantic tags
(described in Section REF ), which maps “home” and `workplace” to bots \bot _1 and \bot _2, respectively.
As the background knowledge about the bot distribution \mathbf {\pi }_k (1 \le k \le 2), we considered three cases:
(I) we do not have any background knowledge;
(II) we ... | {
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"end": 605,
"openalex_id": "",
"raw": "D. Yang, D. Zhang, and B. Qu. Participatory cultural mapping based on collective behavior data in location based social network. ACM Transactions on Intelligent Systems and Technology, 7(3):30:1–30:... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.016371889039874077,
0.006919526029378176,
-0.02958535961806774,
0.006385494023561478,
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0.056729432195425034,
-0.02108662575483322,
0.0... | |
e8c458f43a1394d2c1894ccf2295f9c423b36999 | subsection | 44 | 120 | Discussions | On the case of multiple
data
per user. We have so far assumed that each user sends only
a single
blackdatum.
Now we discuss
the case where
each user sends multiple
data
based on the
compositionality of ULDP described in
Section REF .
Specifically,
when
a user sends t (>1)
data,
we obtain (\mathcal {X}_S, (\mathcal {Y}... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1260,
"openalex_id": "",
"raw": "U. Erlingsson, V. Pihur, and A. Korolova. RAPPOR: Randomized aggregatable privacy-preserving ordinal response. In Proc. 2014 ACM SIGSAC Conference on Computer and Communications Security (CCS'14), ... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.06958030164241791,
-0.0005269067478366196,
-0.06719991564750671,
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-0.012626687996089458,
0.009285002015531063,
-0.0336915142834186... | |
15f17d9424b59fa31418e246713c9746e83460f2 | subsection | 45 | 120 | Discussions | If either t_I or |\mathcal {T}^{(i)}| is much smaller than |\mathcal {X}|,
her home is kept strongly secret.blackNote that \mathbf {p} can be estimated
even if \mathcal {X}_S^{(i)} changes over time. \mathcal {X}_S^{(i)} is also kept strongly secret if t_I or |\mathcal {T}^{(i)}| is small.On the correlation between \ma... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 976,
"openalex_id": "https://openalex.org/W3104360443",
"raw": "M. E. Andrés, N. E. Bordenabe, K. Chatzikokolakis, and C. Palamidessi. Geo-indistinguishability: Differential privacy for location-based systems. In Proc. 20th ACM Co... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.06400597095489502,
0.03755545988678932,
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0.022301413118839264,
-0.019098063930869102,
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-0.009137173183262348,
-0.039446961134672165,
0.025336967781186104,
0.0276403296738863,
-0.008915988728404045,
0.016931990161538124,
-0.03404702618718147,
-0.... | |
fff1f2245bf94c2afa10313d9387b8736a67f47b | subsection | 46 | 120 | Conclusion | In this paper, we introduced the notion of ULDP that guarantees privacy equivalent to LDP for only sensitive data.
We proposed
ULDP mechanisms
in both the
common and personalized mechanism scenarios.
We evaluated the utility of our mechanisms theoretically
and
demonstrated the effectiveness of our mechanisms through ex... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.03868769854307175,
0.027322422713041306,
-0.024027256295084953,
0.03392801061272621,
-0.049122393131256104,
0.014904526993632317,
0.08860336989164352,
0.012509428896009922,
0.020213406533002853,
-0.010564365424215794,
-0.015758829191327095,
0.011845818720757961,
-0.021021943539381027,
-... | |
5c6ff26e31cfcf952fef9cf9f0bbb865b48b3bb4 | subsection | 47 | 120 | Properties of ULDP | In this section we present basic properties of ULDP:
adaptive sequential composition,
post-processing,
and the compatibility with LDP.
We also prove that
the utility-optimized RR and
the utility-optimized RAPPOR provide ULDP. | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.024901308119297028,
0.039243973791599274,
-0.017638426274061203,
0.03726041316986084,
-0.05410541594028473,
0.0028113150037825108,
0.012938914820551872,
-0.01100875809788704,
0.024901308119297028,
-0.003614275250583887,
-0.03652802109718323,
-0.019591469317674637,
0.007789287716150284,
... | |
006a528b8dec63b56d724f0b7d70fbc1b083d66b | subsection | 48 | 120 | Sequential Composition | Below we prove that ULDP provides the compositionality.*Let \mathcal {Y}_{0I} = \mathcal {Y}\setminus \mathcal {Y}_{0P} and \mathcal {Y}_{1I} = \mathcal {Y}\setminus \mathcal {Y}_{1P}.
Let \mathbf {Q} be the sequential composition of \mathbf {Q}_0 and \mathbf {Q}_1;
i.e.,\mathbf {Q}((y_0,y_1) | x)
= \mathbf {Q}_0(y_0 |... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.028160875663161278,
0.01823969930410385,
-0.02991615980863571,
-0.00003499837839626707,
-0.013439376838505268,
0.02903088554739952,
0.011493300087749958,
0.027840344235301018,
0.012889896519482136,
0.018529703840613365,
0.016041778028011322,
-0.021521318703889847,
-0.012874633073806763,
... | |
822ecbb81a82a828b6c418b8cef590f77ad13199 | subsection | 49 | 120 | Sequential Composition | Hence we obtain:&\mathbf {Q}((y_0,y_1) | x) \\
&= \mathbf {Q}_0(y_0 | x) \mathbf {Q}_1(y_1 | (y_0, x)) \\
&\le e^{\epsilon _0} \mathbf {Q}_0(y_0 | x^{\prime }) \mathbf {Q}_1(y_1 | (y_0, x))
~~~~~\text{(by $y_0\in \mathcal {Y}_{0P}$)} \\
&\le e^{\epsilon _0} \mathbf {Q}_0(y_0 | x^{\prime }) e^{\epsilon _1} \mathbf {Q}_1... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.0021993990521878004,
0.03015446849167347,
-0.021379761397838593,
-0.03485465794801712,
-0.015077234245836735,
-0.006832825485616922,
-0.0036663010250777006,
0.02896416001021862,
0.017610453069210052,
0.025530578568577766,
0.012605056166648865,
-0.0061155883595347404,
-0.02824692241847515,... | |
b7bc5528342271fb78a6e365999641b114bb3612 | subsection | 50 | 120 | Post-processing | black
We first define a class of post-processing randomized algorithms that
preserve data types:Definition 5 (Preservation of data types) Let \mathcal {Y}_P and \mathcal {Z}_P
be
sets of protected data,
and
\mathcal {Y}_I and \mathcal {Z}_I be
sets of invertible data.
Given a randomized algorithm \mathbf {Q}_1 from \ma... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.06519423425197601,
-0.0006018486456014216,
-0.020968370139598846,
0.021929800510406494,
0.014162043109536171,
-0.0008379145292565227,
0.04276082292199135,
0.03305494040250778,
-0.00032262332388199866,
0.0064362515695393085,
0.008462125435471535,
-0.007882214151322842,
-0.00882075540721416... | |
5321c56fd9e73c6d8e21fad5bca4e7543e53d813 | subsection | 51 | 120 | Post-processing | Hence we obtain:&(\mathbf {Q}_1\circ \mathbf {Q}_0)(z|x)
= \mathbf {Q}_0(y|x) \mathbf {Q}_1(z|y) > 0and for any x^{\prime } \ne x,(\mathbf {Q}_1\circ \mathbf {Q}_0)(z|x^{\prime }) &=
\mathbf {Q}_0(y|x^{\prime }) \mathbf {Q}_1(z|y) +
\sum _{y^{\prime }\ne y} \mathbf {Q}_0(y^{\prime }|x^{\prime }) \mathbf {Q}_1(z|y^{\pri... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1545,
"openalex_id": "",
"raw": "S. Krishnan, J. Wang, M. J. Franklin, K. Goldberg, and T. Kraska. PrivateClean: Data cleaning and differential privacy. In Proc. 2016 ACM International Conference on Management of Data (SIGMOD'16),... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.08776417374610901,
0.037626296281814575,
-0.03198082372546196,
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0.00581330806016922,
0.015593703836202621,
0.022398782894015312,
-0.00019930797861889005,
0.02464171312749386,
0.020201627165079117,
0.012778597883880138,
0.008857284672558308,
-0.0048024640418589115,
... | |
1cef89617ff648978d3ea218f00cc4904706e98d | subsection | 52 | 120 | blackCompatibility with LDP | black
Assume that data collectors A and B adopt
a mechanism \mathbf {Q}_A providing (\mathcal {X}_S,\mathcal {Y}_P,\epsilon _A)-ULDP
and
a mechanism \mathbf {Q}_B providing \epsilon _B-LDP, respectively.
In this case,
all protected
data in the data collector A
can be
combined with
all obfuscated data in the data collec... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.040004514157772064,
0.01663040556013584,
-0.03902805224061012,
0.016050629317760468,
-0.019605569541454315,
0.003718955209478736,
0.01870539039373398,
-0.011328510008752346,
0.06139518320560455,
0.032314859330654144,
0.025418581441044807,
0.010535133071243763,
0.002315287943929434,
0.00... | |
7c5d3edab3b753fe7a69d3cf811ba3c2f0d697b9 | subsection | 53 | 120 | ULDP of the utility-optimized RR | Below we prove that the utility-optimized RR provides ULDP.*It follows from (REF )
that
(REF ) holds.
Since c_1 / c_2 = e^{\epsilon },
the inequality
(REF )
also
holds
black(note that
c_3 is
uniquely determined
from c_2 so that the sum of probabilities from x \in \mathcal {X}_N is 1; i.e.,
c_3 = 1 - |\mathcal {X}_S|c_2... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.055695246905088425,
0.03595013916492462,
-0.03845261037349701,
0.0072823441587388515,
-0.0399785079061985,
0.0032978453673422337,
0.022659573704004288,
0.016662796959280968,
0.016922200098633766,
0.011467117816209793,
-0.03659101575613022,
0.03826950117945671,
0.012191918678581715,
0.00... | |
39b088b61f3f72ee089338fb0d90f1c4a1016352 | subsection | 54 | 120 | blackULDP of the utility-optimized RAPPOR | Below we prove that the utility-optimized RAPPOR provides ULDP.*Let i, i^{\prime } \in \lbrace 1, 2, \ldots , |\mathcal {X}|\rbrace .By (REF ),
if y \in \mathcal {Y}_I, then only one of y_{|\mathcal {X}_S|+1}, \cdots , y_{|\mathcal {X}|} is 1.
In addition,
it follows from (REF ) that
for any j \in \lbrace |\mathcal {X}... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.04362000897526741,
0.03131849318742752,
-0.03812553733587265,
-0.002108122454956174,
-0.03180689364671707,
-0.0006281441310420632,
-0.007001637015491724,
0.008585114032030106,
0.032081615179777145,
0.011095782741904259,
-0.02826600894331932,
0.018238596618175507,
-0.02287837490439415,
0... | |
49bc5842cf9ca53b786cabfcb695b7d3b6cb9894 | subsection | 55 | 120 | blackULDP of the utility-optimized RAPPOR | Otherwise, since |\mathcal {X}_S|+1 \le j \le |\mathcal {X}|
and y \in \mathcal {Y}_P, we have y_j = 0, hence
\frac{\Pr (y_j | x_i)}{\Pr (y_j | x_{i^{\prime }})} = \frac{1}{1} = 1.Now we show that the (\mathcal {X}_S,\theta ,\epsilon )-utility-optimized RAPPOR satisfies (REF ) as follows.If x_i, x_{i^{\prime }} \in \ma... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.026528358459472656,
0.03304222226142883,
-0.02300446480512619,
0.0068342178128659725,
-0.022577326744794846,
-0.009763168171048164,
0.0009605851373635232,
0.005320163909345865,
0.05339232459664345,
0.01702452450990677,
-0.03636780008673668,
0.003735555801540613,
-0.009168225340545177,
0... | |
6731bea6ea2af23842c9f6eb9ff1993b055f77f3 | subsection | 56 | 120 | blackULDP of the utility-optimized RAPPOR | If y_i = 1 then we have:&
\frac{\mathbf {Q}_{\it uRAP}(y | x_i)}{\mathbf {Q}_{\it uRAP}(y | x_{i^{\prime }})}
\\
&= \frac{\Pr (y_i | x_i)}{\Pr (y_i | x_{i^{\prime }})} \cdot \frac{\Pr (y_{i^{\prime }} | x_i)}{\Pr (y_{i^{\prime }} | x_{i^{\prime }})} \cdot \hspace{-8.5pt}
\prod _{{j \ne i \\ 1 \le j \le |\mathcal {X}_S|... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.040427982807159424,
0.027735121548175812,
-0.02785716950893402,
-0.020854737609624863,
-0.00683461781591177,
0.0025534466840326786,
-0.02833010070025921,
0.026514654979109764,
0.02143445983529091,
0.019756317138671875,
-0.026789259165525436,
0.0014731422998011112,
-0.044333480298519135,
... | |
153ab67da2ebf3a36a4f0958a097e8e32f0e4420 | subsection | 57 | 120 | blackULDP of the utility-optimized RAPPOR | If y_i = 0 then we obtain:&
\frac{\mathbf {Q}_{\it uRAP}(y | x_i)}{\mathbf {Q}_{\it uRAP}(y | x_{i^{\prime }})}
\\
&= \frac{\Pr (y_i | x_i)}{\Pr (y_i | x_{i^{\prime }})} \cdot \frac{\Pr (y_{i^{\prime }} | x_i)}{\Pr (y_{i^{\prime }} | x_{i^{\prime }})} \cdot \hspace{-8.5pt}
\prod _{{j \ne i \\ 1 \le j \le |\mathcal {X}_... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.04911409690976143,
0.030768806114792824,
-0.03372969478368759,
-0.017948471009731293,
-0.005231155082583427,
0.007299197372049093,
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0.03345497325062752,
0.03971251845359802,
0.01965784840285778,
-0.028937330469489098,
-0.011400941759347916,
-0.03687372803688049,
0.... | |
e701d9d3a00f417ab717ccad85b2f97c7cb65751 | subsection | 58 | 120 | blackULDP of the utility-optimized RAPPOR | Then:&
\frac{\mathbf {Q}_{\it uRAP}(y | x_i)}{\mathbf {Q}_{\it uRAP}(y | x_{i^{\prime }})}
\\
&= \frac{\Pr (y_i | x_i)}{\Pr (y_i | x_{i^{\prime }})} \cdot \frac{\Pr (y_{i^{\prime }} | x_i)}{\Pr (y_{i^{\prime }} | x_{i^{\prime }})} \cdot \hspace{-4.25pt}
\prod _{{1 \le j \le |\mathcal {X}_S|}}\hspace{0.0pt} \frac{\Pr (y... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.04248702898621559,
0.044470977038145065,
-0.03384922072291374,
-0.00698578916490078,
-0.0564967580139637,
0.026294952258467674,
-0.026645958423614502,
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0.0361078679561615,
0.012605706229805946,
-0.03494802117347717,
0.027576889842748642,
-0.030491767451167107,
0.02... | |
aa142a0c6287f022f87c921197e2431e86f2f0a1 | subsection | 59 | 120 | Relationship between LDP, ULDP and OSLDP | Our main contributions lie in the proposal of local obfuscation mechanisms (i.e., uRR, uRAP, PUM) and ULDP is introduced to characterize the main features of these mechanisms, i.e., LDP for sensitive data and high utility in distribution estimation.
Nonetheless, it is worth making clearer the reasons for using ULDP as ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1109/icde48307.2020.00049",
"end": 885,
"openalex_id": "https://openalex.org/W3029016412",
"raw": "S. Doudalis, I. Kotsoginannis, S. Haney, A. Machanavajjhala, and S. Mehrotra. One-sided differential privacy. CoRR, abs/1712.05888, 2017."... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.07246213406324387,
0.025960002094507217,
-0.06959295272827148,
-0.018436027690768242,
-0.009073028340935707,
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0.06251156330108643,
-0.009981093928217888,
0.042610421776771545,
-0.0029397679027169943,
-0.013376803137362003,
0.03299560397863388,
-0.020786315202713013,
... | |
b61bd78615c967d673a058729659fc60c2fb1cf0 | subsection | 60 | 120 | Relationship between LDP, ULDP and OSLDP | As described in Section REF , for \epsilon \in [0,1], the lower bound on the l_1 and l_2 losses of any \epsilon -LDP mechanism can be expressed as \Theta (\frac{|\mathcal {X}|}{\sqrt{n \epsilon ^2}}) and \Theta (\frac{|\mathcal {X}|}{n \epsilon ^2}), respectively.
On the other hand,
the lower bound on the l_1 and l_2 l... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.023348284885287285,
0.04071452468633652,
-0.010575399734079838,
-0.03454935923218727,
0.012528720311820507,
0.010720373131334782,
0... | |
b651a17830eb60600f5d28a404db96235b9db743 | subsection | 61 | 120 | Relationship between LDP, ULDP and OSLDP | Thus, (i) and (ii) only allow us to mix non-sensitive data with sensitive data or other non-sensitive data, and
reduce the amount of output data y \in \mathcal {Y}_I that can be inverted to x \in \mathcal {X}_N.Then, each OSLDP mechanism can be decomposed into
a ULDP mechanism and a randomized post-processing that mixe... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1163,
"openalex_id": "https://openalex.org/W2147435839",
"raw": "P. Kairouz, S. Oh, and P. Viswanath. Extremal mechanisms for local differential privacy. Journal of Machine Learning Research, 17(1):492–542, 2016.",
"source_r... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.060735322535037994,
0.029284192249178886,
-0.04877138137817383,
0.00502439821138978,
-0.019685570150613785,
-0.0002529843768570572,
0.045810915529727936,
-0.010117467492818832,
0.030215060338377953,
-0.0070616258308291435,
-0.013863829895853996,
-0.008477003313601017,
0.020723259076476097... | |
0aa846d721c605d39c3c24c7f15ad4aaf3946b57 | subsection | 62 | 120 | Relationship between LDP, ULDP and OSLDP | In addition, if \mathcal {X}_S= \mathcal {X} (i.e., \mathcal {X}_N= \emptyset ), then
all of
(\mathcal {X}_S,\epsilon )-OSLDP,
(\mathcal {X}_S,\mathcal {Y}_P,\epsilon )-ULDP, and
\epsilon -LDP are equivalent, and hence
(REF ) holds.Assume that
\mathbf {Q}_O does not provide (\mathcal {X}_S,\mathcal {Y}_P,\epsilon )-ULD... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.020692799240350723,
0.03088659606873989,
-0.0612238273024559,
-0.028032943606376648,
-0.026766348630189896,
-0.007942921482026577,
0.03366394713521004,
-0.002771629486232996,
0.03436591848731041,
-0.008171824738383293,
0.01895313896238804,
0.008133674040436745,
-0.021135343238711357,
0.... | |
ef3bbdb3c81d1f448db273ef63978370ec281492 | subsection | 63 | 120 | Relationship between LDP, ULDP and OSLDP | Then,
from \mathbf {Q}_O^\dagger ,
we construct a mechanism
\mathbf {Q}_U
from \mathcal {X} to
\mathcal {Z} (= \mathcal {Y}_P\cup \mathcal {X}_N^{\prime })
such that:&\mathbf {Q}_U(z|x) \\
&=
{\left\lbrace \begin{array}{ll}
\min \lbrace \mathbf {Q}_O^\dagger (z|x), \mathbf {Q}_{max}^\dagger (z)\rbrace & \hspace{-5.6905... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.008782831951975822,
0.03623013570904732,
-0.05127770081162453,
-0.03387990966439247,
-0.013857188634574413,
0.044745899736881256,
0.028935274109244347,
-0.003809582209214568,
0.011400131508708,
0.017977718263864517,
0.015268851071596146,
0.013437504880130291,
-0.003733276156708598,
0.00... | |
c1912f0f3277313dcfa641630714ce311d0179be | subsection | 64 | 120 | Relationship between LDP, ULDP and OSLDP | Furthermore, by (REF ) and (REF ), for any x\in \mathcal {X}_N and any z \in \mathcal {Y}_P,
we obtain:e^{-\epsilon } \mathbf {Q}_{max}^\dagger (z) \le \mathbf {Q}_U(z|x) \le \mathbf {Q}_{max}^\dagger (z).Thus, \mathbf {Q}_U satisfies the second condition (REF )
for any x,x^{\prime }\in \mathcal {X}_N and any z \in \ma... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.053320690989494324,
0.03601512312889099,
-0.02490537241101265,
-0.032383088022470474,
-0.01135391928255558,
0.022265279665589333,
0.027743851765990257,
-0.005562504753470421,
0.013009699992835522,
0.02350139245390892,
0.01553533598780632,
0.019960923120379448,
-0.009644727222621441,
-0.... | |
99c7aefacd79159d415d8aeae7c9f2eea7966c96 | subsection | 65 | 120 | Relationship between LDP, ULDP and OSLDP | By (REF ) and (REF ), we obtain:\mathbf {Q}_O = \mathbf {Q}_{R_1} \circ \mathbf {Q}_O^\dagger(note that in (REF ), if w=x, then \mathbf {Q}_O(y^{\prime }|x) = \mathbf {Q}_O(y^{\prime }|w)).Next,
for any z \in \mathcal {X}_N^{\prime } and any w \in \mathcal {Y}_P, we define \beta (z,w) by:\beta (z,w) = \frac{\mathbf {Q}... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.06598453223705292,
0.044376276433467865,
-0.024080386385321617,
-0.044345758855342865,
-0.008576158434152603,
0.026644079014658928,
0.013512790203094482,
0.01703023537993431,
0.005978131666779518,
0.04285027086734772,
-0.0021173343993723392,
0.010559966787695885,
-0.008576158434152603,
... | |
e40f84758aad424deafc104c2ca8a27428138caa | subsection | 66 | 120 | Relationship between LDP, ULDP and OSLDP | \sum _{w^{\prime }\in \mathcal {Y}_P} \beta (z,w^{\prime }) \le 1, since
\sum _{w^{\prime }\in \mathcal {Y}_P} \mathbf {Q}_O^\dagger (w^{\prime }|z) - \sum _{w^{\prime }\in \mathcal {Y}_P} \min \lbrace \mathbf {Q}_O^\dagger (w^{\prime }|z), \mathbf {Q}_{max}^\dagger (w^{\prime })\rbrace \le \alpha (z).
Furthermore,
\su... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.055434953421354294,
0.03473842144012451,
-0.0536033995449543,
-0.053359195590019226,
-0.010210898704826832,
0.02046758495271206,
0.009485909715294838,
0.0034017248544842005,
0.01758289337158203,
-0.006219643168151379,
-0.021749671548604965,
0.009111967869102955,
0.02678643725812435,
0.0... | |
1c73a52db7f23cb67ea68ea829fb91b97265727a | subsection | 67 | 120 | L1 loss of the utility-optimized Mechanisms | In this section we show the detailed analyses on the l_1 loss of the utility-optimized RR and the utility-optimized RAPPOR.
Table REF summarizes the l_1 loss of each obfuscation mechanism. | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.051327649503946304,
0.024138031527400017,
-0.01124508772045374,
-0.003017253940925002,
-0.033018141984939575,
-0.003766799345612526,
0.015944527462124825,
0.014586572535336018,
0.039426468312740326,
0.03114141710102558,
-0.06310676038265228,
-0.01611236482858658,
0.006023064721375704,
0... | |
dc8e915cca0fc8b44a8a2f9a9ad785b522aad0fa | subsection | 68 | 120 | Body | We first present the l_1 loss of the (\mathcal {X}_S,\epsilon )-utility-optimized RR.
In the theoretical analysis of utility, we use the empirical estimation method described in Section REF .
Then it follows from (REF )
that
the distribution \mathbf {m} of the obfuscated data can be written as follows:\mathbf {m}(x) =
... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.0760580450296402,
0.005383121315389872,
-0.03421391174197197,
0.005039761308580637,
-0.02928479015827179,
-0.020952587947249413,
0.04626966267824173,
0.004173730965703726,
0.04810091480612755,
0.032504741102457047,
-0.0446520559489727,
0.017473207786679268,
-0.026278482750058174,
-0.014... | |
277040c5b4caf117ae7c2988c7ba22906dc59394 | subsection | 69 | 120 | Body | By (REF ) and (REF ),
the l_1 loss of \hat{\mathbf {p}} can be written as follows:\operatornamewithlimits{\displaystyle \mathbb {E}}_{Y^n\sim \mathbf {m}^n}\left[ l_1(\hat{\mathbf {p}},\mathbf {p}) \right]
&= \mathbb {E}\left[ \sum _{x \in \mathcal {X}} |\hat{\mathbf {p}}(x) - \mathbf {p}(x)| \right] \\
&= \mathbb {E}\... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.046968765556812286,
-0.0030652538407593966,
-0.020875006914138794,
-0.037294238805770874,
-0.012528056278824806,
0.018860751762986183,
-0.021287014707922935,
0.003948397934436798,
0.01519847009330988,
0.022400958463549614,
-0.04980703443288803,
0.026093758642673492,
-0.023270750418305397,... | |
91f769589cdc3403d96cf2afcc4faf59c10246b5 | subsection | 70 | 120 | Body | (See for details.)
Hence we obtain:\lim _{n\rightarrow \infty } \operatornamewithlimits{\displaystyle \mathbb {E}}_{Y^n\sim \mathbf {m}^n}\left[\, \left|\frac{\mathbf {t}(x) - \mathbb {E}\mathbf {t}(x)}{\sqrt{n}} \right| \,\right]
= \sqrt{\frac{2}{\pi } \mathbf {m}(x) (1 - \mathbf {m}(x))}.Then we have:\mathbb {E}\lef... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.48550/arxiv.1209.4340",
"end": 19,
"openalex_id": "https://openalex.org/W1850578340",
"raw": "A. Winkelbauer. Moments and absolute moments of the normal distribution. CoRR, abs/1209.4340, 2012.",
"source_ref_id": "66e92180154060948... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.03625251352787018,
0.00797219667583704,
-0.029706921428442,
-0.022016994655132294,
-0.0334755964577198,
-0.0033357348293066025,
-0.024992262944579124,
0.0041119749657809734,
0.03249909728765488,
0.02630443312227726,
-0.05532475560903549,
0.017165016382932663,
-0.01965203694999218,
-0.02... | |
b78802808898c7909ae899994729a6219300022f | subsection | 71 | 120 | Body | By using this approximation, we simplify the l_1 loss of the utility-optimized RR for both the cases where |\mathcal {X}_S|\le |\mathcal {X}_N| and where |\mathcal {X}_S|>|\mathcal {X}_N|.Case 1: |\mathcal {X}_S|\le |\mathcal {X}_N|. By Proposition REF ,
the expected l_1 loss of the (\mathcal {X}_S,\epsilon )-utility-... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.04373492673039436,
0.032717265188694,
-0.03955370932817459,
-0.009537449106574059,
-0.022706758230924606,
0.02479736879467964,
0.013909416273236275,
0.011353379115462303,
0.043399207293987274,
0.029588982462882996,
-0.02583504281938076,
0.011856956407427788,
-0.01292515080422163,
0.0246... | |
a4fdb2d7b796a38e886b10b8cf8668bc3f81c812 | subsection | 72 | 120 | Body | Thus,
for \epsilon \approx 0, the expected l_1 loss of the (\mathcal {X}_S,\epsilon )-utility-optimized RR mechanism is given by:\mathbb {E}\left[ l_1(\hat{\mathbf {p}},\mathbf {p}) \right] &\lesssim \left[ l_1(\hat{\mathbf {p}},\mathbf {p}^{\!*}) \right] \\
&=\!\sqrt{\frac{2}{n\pi }}\,F(w^*) \\
&\approx \!\sqrt{\frac{... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.05199366807937622,
0.05059008300304413,
-0.041588831692934036,
-0.01420366857200861,
-0.03203834965825081,
0.00014779595949221402,
-0.011335473507642746,
-0.017697375267744064,
0.039422426372766495,
0.012739058583974838,
-0.04122267663478851,
0.025035683065652847,
-0.006426740437746048,
... | |
59afb3ac5fd804122cd36015f4d0aeb568e29d36 | subsection | 73 | 120 | Body | It is shown in that
the expected l_1 loss of the \epsilon -RR is at most
\sqrt{\frac{2}{n\pi }} \frac{|\mathcal {X}|\sqrt{ |\mathcal {X}| - 1 }}{\epsilon } when \epsilon \approx 0.
Thus,
the expected l_1 loss of the (\mathcal {X}_S,\epsilon )-utility-optimized RR is much smaller than that of the \epsilon -RR when |\mat... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 181,
"openalex_id": "https://openalex.org/W2964074929",
"raw": "P. Kairouz, K. Bonawitz, and D. Ramage. Discrete distribution estimation under local privacy. In Proc. 33rd International Conference on Machine Learning (ICML'16), pa... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.058703865855932236,
0.034081194549798965,
-0.04903176426887512,
-0.006754453759640455,
-0.03127415105700493,
0.005991669371724129,
0.01771184802055359,
-0.01604897901415825,
0.03877994418144226,
-0.003188437782227993,
-0.044302504509687424,
0.008894063532352448,
-0.012113012373447418,
0... | |
9fd1c76bf42d0a00df57e82cc8b00095519e888f | subsection | 74 | 120 | Body | It follows from (REF ), (REF ) and (REF ) that
\mathbf {m}_j can be written as follows:\mathbf {m}_j =
{\left\lbrace \begin{array}{ll}
\frac{e^{\epsilon /2} - 1}{e^{\epsilon /2} + 1} \mathbf {p}(x_j) + \frac{1}{e^{\epsilon /2} + 1} & \text{(if $1 \le j \le |\mathcal {X}_S|$)}\\
\frac{e^{\epsilon /2} - 1}{e^{\epsilon /2... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.0220178235322237,
0.025817157700657845,
-0.02812116965651512,
-0.009414412081241608,
-0.00007158319931477308,
-0.017241954803466797,
-0.0045164767652750015,
0.02444390393793583,
0.03134068474173546,
0.035490963608026505,
-0.03237825632095337,
0.02523733861744404,
-0.028289012610912323,
... | |
854c117bf58eab54a0a4d9b7b20d85099f1b03a3 | subsection | 75 | 120 | Body | It follows from (REF ) that for 1\le j \le |\mathcal {X}_S|, \mathbf {m}_j = \mathbf {p}(x_j)/v_S + 1/u, and for |\mathcal {X}_S|+1 \le j \le |\mathcal {X}|, \mathbf {m}_j = \mathbf {p}(x_j)/v_N.
Therefore, we obtain:&\mathbb {E}\left[ l_1(\hat{\mathbf {p}},\mathbf {p}) \right] \\
&\approx \sqrt{\frac{2}{n\pi }} \biggl... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.04596581310033798,
0.03250570222735405,
-0.04074658825993538,
-0.0065240319818258286,
-0.00672242371365428,
0.01405528374016285,
-0.01369665190577507,
0.007851730100810528,
0.004444735124707222,
0.024829475209116936,
-0.021884121000766754,
0.031681615859270096,
-0.026721825823187828,
-0... | |
786ef131cc071362e856dcb69a8db2bea974d57b | subsection | 76 | 120 | Body | By using this approximation, we simplify the l_1 loss of the utility-optimized RAPPOR for both the cases where |\mathcal {X}_S|\le |\mathcal {X}_N| and where |\mathcal {X}_S|>|\mathcal {X}_N|.Case 1: |\mathcal {X}_S|\le |\mathcal {X}_N|. By Proposition REF ,
the expected l_1 loss of the (\mathcal {X}_S,\epsilon )-util... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.041942402720451355,
0.03803510591387749,
-0.02854159101843834,
-0.005601477809250355,
-0.031472064554691315,
0.0035104630514979362,
0.006715668365359306,
0.025336384773254395,
0.02880105935037136,
0.03351728990674019,
-0.03287624940276146,
0.005838052835315466,
-0.006353174801915884,
0.... | |
8cd98ba079b5dbe1d46d8251148229549a4b4c9a | subsection | 77 | 120 | Body | Let
F be the function defined in Lemma REF ,
w^* = \operatornamewithlimits{argmax}_{w\in [0,1]} F(w),
and
\mathbf {p}^{\!*} be the prior distribution over \mathcal {X} defined by:\mathbf {p}^{\!*}(x_j) =
{\left\lbrace \begin{array}{ll}
\frac{w^*}{|\mathcal {X}_S|}
~~~(\text{if $1 \le j \le |\mathcal {X}_S|$}) \\
\frac{... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.02070394530892372,
0.011160601861774921,
-0.020734459161758423,
-0.01539445947855711,
-0.01820177398622036,
0.02271788939833641,
0.017347373068332672,
0.027737488970160484,
0.002450678963214159,
0.014929116703569889,
-0.003167764749377966,
0.01711851730942726,
-0.030697375535964966,
0.0... | |
f75136db41ec4a4b6c71a9570b461014a3983333 | subsection | 78 | 120 | Body | Thus,
for \epsilon \approx 0, the expected l_1 loss of the (\mathcal {X}_S,\epsilon )-utility-optimized RAPPOR mechanism is given by:\mathbb {E}\left[ l_1(\hat{\mathbf {p}},\mathbf {p}) \right] &\lesssim \left[ l_1(\hat{\mathbf {p}},\mathbf {p}^{\!*}) \right] \\
&=\!\sqrt{\frac{2}{n\pi }}\,F(w^*) \\
&\approx \!\sqrt{\f... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1549,
"openalex_id": "https://openalex.org/W2964074929",
"raw": "P. Kairouz, K. Bonawitz, and D. Ramage. Discrete distribution estimation under local privacy. In Proc. 33rd International Conference on Machine Learning (ICML'16), p... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.044150423258543015,
0.04161795601248741,
-0.04247228056192398,
-0.006579084787517786,
-0.036888640373945236,
0.0022330747451633215,
0.0008338274783454835,
-0.004714057315140963,
0.012349609285593033,
0.016522083431482315,
-0.04582856968045235,
0.026560431346297264,
-0.010564674623310566,
... | |
8e04b89abd4ef87f11e7bcaf39b38c798cc084e7 | subsection | 79 | 120 | Body | Thus,
the expected l_1 loss of the (\mathcal {X}_S,\epsilon )-utility-optimized RAPPOR is much smaller than that of the \epsilon -RAPPOR when |\mathcal {X}_S| \ll |\mathcal {X}|.Note that the expected l_1 loss of the utility-optimized RAPPOR in the worst case can also be expressed as
\Theta (\frac{|\mathcal {X}_S|}{\sq... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.045131806284189224,
0.04061557352542877,
-0.04967855289578438,
0.011214292608201504,
-0.06035883352160454,
0.006625587120652199,
-0.00010036601452156901,
0.000554993050172925,
0.015013420954346657,
0.005683434195816517,
-0.028058618307113647,
0.01319777313619852,
-0.00733887730166316,
0... | |
a57fb7d6cd251793b3346b6dc1abae8f710eaf9b | subsection | 80 | 120 | Body | Then by Proposition REF ,
the expected l_1 loss of the (\mathcal {X}_S,\epsilon )-utility-optimized RAPPOR mechanism is given by:&\mathbb {E}\left[ l_1(\hat{\mathbf {p}},\mathbf {p}) \right] \\
&\lesssim \mathbb {E}\left[ l_1(\hat{\mathbf {p}},\mathbf {p}_{U_{\!N}}) \right] \\
&=\!\sqrt{\frac{2}{n\pi }} \!\biggl (\!\sq... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1556,
"openalex_id": "https://openalex.org/W2964074929",
"raw": "P. Kairouz, K. Bonawitz, and D. Ramage. Discrete distribution estimation under local privacy. In Proc. 33rd International Conference on Machine Learning (ICML'16), p... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.03473205864429474,
0.03403009474277496,
-0.020006032660603523,
-0.01625204086303711,
-0.01872418075799942,
0.030337141826748848,
0.01594683714210987,
0.0018150019459426403,
0.022493433207273483,
0.030657604336738586,
-0.040866632014513016,
0.016969265416264534,
-0.014939668588340282,
0.... | |
397ffc7c20e89f68f90d605f1192e690b8df3c16 | subsection | 81 | 120 | Body | Thus, when \epsilon =\ln |\mathcal {X}| and
|\mathcal {X}_S| \ll |\mathcal {X}|^{\frac{3}{4}},
the (\mathcal {X}_S,\epsilon )-utility-optimized RAPPOR achieves almost the same data utility as the non-private mechanism,
whereas the expected l_1 loss of the \epsilon -RAPPOR is \sqrt{|\mathcal {X}|} times larger than that... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 351,
"openalex_id": "https://openalex.org/W2964074929",
"raw": "P. Kairouz, K. Bonawitz, and D. Ramage. Discrete distribution estimation under local privacy. In Proc. 33rd International Conference on Machine Learning (ICML'16), pa... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.05738994479179382,
0.02672705426812172,
-0.03511739894747734,
0.01694849133491516,
-0.03070865385234356,
0.014934809878468513,
0.03022048808634281,
-0.005449908785521984,
0.00862679723650217,
0.026986392214894295,
-0.04991491138935089,
0.01826043613255024,
-0.017619719728827477,
-0.0035... | |
4d09c8e4a827c245773348bb76da36335756f3ee | subsection | 82 | 120 | Body | Since \mathbf {t}(x) follows the binomial distribution with parameters n and \mathbf {m}(x),
the mean is given by \mathbb {E}[\mathbf {t}(x)] = n\mathbf {m}(x), and the variance of \mathbf {t}(x) is given by \text{Var}(\mathbf {t}(x)) = n\mathbf {m}(x)(1 - \mathbf {m}(x)).Then,
by (REF ) and (REF ),
the l_2-loss of \ha... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.03351828455924988,
-0.005658881738781929,
-0.06212177500128746,
-0.03312143683433533,
-0.0272450540214777,
0.022666051983833313,
0.0002539914275985211,
0.04047836363315582,
-0.03507514297962189,
-0.01918601244688034,
-0.02985508367419243,
0.03016035072505474,
-0.05101006478071213,
0.040... | |
4c2935c9e16bf9a46aec1bd271fba003b6d6f792 | subsection | 83 | 120 | Body | Therefore, we obtain:&~~~\mathbb {E}\left[ l_2^2(\hat{\mathbf {p}},\mathbf {p}) \right] \\
&= \frac{v^2}{n} \Bigl ( 1 - \sum _{x \in \mathcal {X}_S} \bigl ( \mathbf {p}(x)/v + 1/u \bigr )^2 - \sum _{x \in \mathcal {X}_N} \bigl ( \mathbf {p}(x)/v \bigr )^2 \Bigr ) \\
&= \frac{1}{n} \Bigl ( v^2 - \sum _{x \in \mathcal {X... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.04955311119556427,
0.023372957482933998,
-0.030848640948534012,
-0.003228656714782119,
-0.023632317781448364,
-0.0002960714336950332,
0.00460364855825901,
0.021084481850266457,
-0.0354866161942482,
-0.004447269719094038,
-0.01904011145234108,
0.04213844984769821,
-0.025371558964252472,
... | |
2fb78cbb0dce3b76164d5af287a097f434b80587 | subsection | 84 | 120 | Body | In this case, e^\epsilon - 1 \approx \epsilon .
By using this approximation, we simplify the l_2 loss of the
blackuRR.By Proposition REF , the expected l_2 loss of the (\mathcal {X}_S,\epsilon )-blackuRR
mechanism is maximized by \mathbf {p}_{U_{\!N}}:\mathbb {E}\left[ l_2^2(\hat{\mathbf {p}},\mathbf {p}) \right]
&\le ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1092,
"openalex_id": "https://openalex.org/W2964074929",
"raw": "P. Kairouz, K. Bonawitz, and D. Ramage. Discrete distribution estimation under local privacy. In Proc. 33rd International Conference on Machine Learning (ICML'16), p... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.03890553489327431,
0.040797412395477295,
-0.05013474076986313,
-0.00989421084523201,
-0.025570852681994438,
0.018369514495134354,
0.00758657930418849,
0.015257071703672409,
0.015592727810144424,
0.0161724966019392,
-0.0322534516453743,
0.02431977353990078,
-0.04073638096451759,
0.018781... | |
750d5858c705c1b603021658504490005b49b53f | subsection | 85 | 120 | Body | By Proposition REF ,
the expected l_2^2 loss of the (\mathcal {X}_S,\epsilon )-blackuRR
is given by:\mathbb {E}\left[ l_2^2(\hat{\mathbf {p}},\mathbf {p}) \right]
&\le \mathbb {E}\left[ l_2^2(\hat{\mathbf {p}},\mathbf {p}^*) \right] \\
&= \frac{(|\mathcal {X}_S|+|\mathcal {X}|-1)^2}{n(|\mathcal {X}|-1)^2} \Bigl ( 1 - \... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 881,
"openalex_id": "https://openalex.org/W2964074929",
"raw": "P. Kairouz, K. Bonawitz, and D. Ramage. Discrete distribution estimation under local privacy. In Proc. 33rd International Conference on Machine Learning (ICML'16), pa... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.04957985132932663,
0.04619316756725311,
-0.04265392944216728,
-0.005819912068545818,
-0.023325413465499878,
0.03951133042573929,
0.016475766897201538,
0.0015159993199631572,
0.007181450724601746,
0.002606756053864956,
-0.027261290699243546,
0.0062928274273872375,
-0.02871054783463478,
0... | |
fde0c1b489c73be38f15f9913965363e03d88a9b | subsection | 86 | 120 | Body | \hat{\mathbf {m}}_j) is the true probability (resp. empirical probability) that the j-th coordinate in obfuscated data is 1.[l_2 loss of the blackuRAP]proppropLoneRestRAPl2
Then the expected l_2-loss of the (\mathcal {X}_S,\epsilon )-blackuRAP
mechanism is given by:&\mathbb {E}\left[ l_2^2(\hat{\mathbf {p}},\mathbf {p... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.04459204897284508,
-0.012727504596114159,
-0.044103700667619705,
-0.006367567460983992,
-0.01675635389983654,
-0.006466762628406286,
-0.0037522483617067337,
0.01354395691305399,
-0.008965717628598213,
0.015421034768223763,
-0.05280235782265663,
0.019243864342570305,
-0.024005232378840446,... | |
aa92b94eb0de38423f5acfb2ecb5f01d0ed6e006 | subsection | 87 | 120 | Body | Since \mathbf {t}_j follows the binomial distribution with parameters n and \mathbf {m}_j,
the mean is given by \mathbb {E}[\mathbf {t}_j] = n\mathbf {m}_j, and the variance of \mathbf {t}_j is given by \text{Var}(\mathbf {t}_j) = n\mathbf {m}_j(1 - \mathbf {m}_j).Then,
by (REF ) and (REF ),
the l_2-loss of \hat{\mathb... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.05432413890957832,
0.002449164167046547,
-0.057009827345609665,
-0.032045137137174606,
-0.013237694278359413,
0.023194575682282448,
-0.009811916388571262,
0.0415671207010746,
-0.041841793805360794,
-0.00782053917646408,
-0.04391709715127945,
0.030183468014001846,
-0.05444621667265892,
0... | |
443678de822f0a0f4f73d0d6ef9d7c8302efae77 | subsection | 88 | 120 | Body | It follows from (REF ) that for 1\le j \le |\mathcal {X}_S|, \mathbf {m}_j = \mathbf {p}(x_j)/v_S + 1/u, and for |\mathcal {X}_S|+1 \le j \le |\mathcal {X}|, \mathbf {m}_j = \mathbf {p}(x_j)/v_N. | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.012879140675067902,
0.01033077947795391,
-0.042574409395456314,
0.0029966128058731556,
-0.0007758575957268476,
0.011925412341952324,
0.014138060621917248,
0.04965488612651825,
0.022218042984604836,
0.019578125327825546,
0.008041832596063614,
-0.015267274342477322,
-0.02420179732143879,
... | |
f9f51dd4d73a833b9b15f89caae79aa810029ed2 | subsection | 89 | 120 | Body | Therefore, we obtain:&\mathbb {E}\left[ l_2^2(\hat{\mathbf {p}},\mathbf {p}) \right] \\
&= \frac{v_S^2}{n} \sum _{j = 1}^{|\mathcal {X}_S|} \bigl ( {\textstyle \frac{\mathbf {p}(x_j)}{v_S} + \frac{1}{u}} \bigr ) \bigl ( 1 - {\textstyle \frac{\mathbf {p}(x_j)}{v_S} - \frac{1}{u}} \bigr ) \\
&~~~ + \frac{v_N^2}{n} \hspac... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.04248461499810219,
0.03854746371507645,
-0.03271803632378578,
-0.008423673920333385,
-0.020662417635321617,
0.001985743874683976,
-0.02037247270345688,
0.03168033808469772,
-0.029971187934279442,
0.006844235118478537,
-0.021471211686730385,
0.040775466710329056,
-0.017762964591383934,
-... | |
05f7af41e058af46a256a35cc631f27ea23a6c15 | subsection | 90 | 120 | Body | In this case, e^{\epsilon /2} - 1 \approx \epsilon /2.
By using this approximation, we simplify the l_2 loss of the
blackuRAP.By Proposition REF ,
the expected l_2 loss of the (\mathcal {X}_S,\epsilon )-blackuRAP
mechanism is given by:\mathbb {E}\left[ l_2^2(\hat{\mathbf {p}},\mathbf {p}) \right]
&\le \mathbb {E}\left[... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1451,
"openalex_id": "https://openalex.org/W2964074929",
"raw": "P. Kairouz, K. Bonawitz, and D. Ramage. Discrete distribution estimation under local privacy. In Proc. 33rd International Conference on Machine Learning (ICML'16), p... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.0511077381670475,
0.04515788331627846,
-0.055989671498537064,
-0.006590609904378653,
-0.0228688046336174,
0.016384989023208618,
-0.012326881289482117,
0.016964716836810112,
0.004252621438354254,
0.026667559519410133,
-0.022243307903409004,
0.01922261156141758,
-0.04369330033659935,
0.00... |
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