P RIVACY I MPLICATIONS OF S HUFFLING
A BSTRACT
LDP deployments are vulnerable to inference attacks as an adversary can link the noisy responses to their identity and subsequently, auxiliary information using the order of the data.
An alternative model, shufﬂe DP , prevents this by shufﬂing the noisy responses uniformly at random.
However, this limits the data learnability – only symmetric functions (input order agnostic) can be learned.
In this paper, we strike a balance and show that systematic shufﬂing of the noisy responses can thwart speciﬁc inference attacks while retaining some meaningful data learnability.
To this end, we propose a novel privacy guarantee, d σ -privacy, that captures the privacy of the order of a data sequence.
d σ -privacy allows tuning the granularity at which the ordinal information is maintained, which formalizes the degree the resistance to inference attacks trading it off with data learnability.
Additionally, we propose a novel shufﬂing mechanism that can achieve d σ -privacy and demonstrate the practicality of our mechanism via evaluation on real-world datasets.
I NTRODUCTION
Differential Privacy ( DP ) and its local variant ( LDP ) are the most commonly accepted notions of data privacy.
LDP has the signiﬁcant advantage of not requiring a trusted centralized aggregator, and has become a popular model for commercial deployments, such as those of Microsoft (Ding et al., 2017), Apple (Greenberg, 2016), and Google (Erlingsson et al., 2014; Fanti et al., 2015; Bittau et al., 2017b).
Its formal guarantee asserts that an adversary cannot infer the value of an individual’s private input by observing the noisy output.
However in practice, a vast amount of public auxiliary information , such as address, social media connections, court records, property records, income and birth dates (La Corte, 2019), is available for every individual.
An adversary, with access to such auxiliary information, can learn about an individual’s private data from several other participants’ noisy responses.
We illustrate this as follows.
Problem.
An analyst runs a medical survey in Alice’s community to investigate how the prevalence of a highly contagious disease changes from neighborhood to neighborhood.
Community members report a binary value indicating whether they have the disease.
Next, consider the following two data reporting strategies.
Strategy 1 .
Each data owner passes their data through an appropriate randomizer (that ﬂips the input bit with some probability) in their local devices and reports the noisy output to the untrusted data analyst.
Strategy 2 .
The noisy responses from the local devices of each of the data owners are collected by an intermediary trusted shufﬂer which dissociates the device IDs (metadata) from the responses and uniformly randomly shufﬂes them before sending them to the analyst.
Strategy 1 corresponds to the standard LDP deployment model (for example, Apple and Microsoft’s deployments).
Here the order of the noisy responses is informative of the identity of the data owners – the noisy response at index 1 corresponds to the ﬁrst data owner and so on.
Thus, the noisy responses can be directly linked with its associated device/account ID and subsequently, auxiliary information.
This puts Alice’s data under the threat of inference attacks.
For instance, an adversary 1 may know the home addresses of the participants and use this to identify the responses of all the individuals from Alice’s household.
Being highly infectious, all or most of them will have the same true value (or 1 ).
Hence, the adversary can reliably infer Alice’s value by taking a simple majority vote of her and her household’s noisy responses.
Note that this does not violate the LDP guarantee since the inputs are appropriately randomized when observed in isolation.
Additionally, on account of being
[Figure]
public, the auxiliary information is known to the adversary (and analyst) a priori – no mechanism can prevent their disclosure.
For instance, any attempts to include Alice’s address as an additional feature of the data and then report via LDP is futile – the adversary would simply discard the reported noisy address and use the auxiliary information about the exact addresses to identify the responses of her household members.
We call such threats inference attacks – recovering an individual’s private input using all or a subset of other participants’ noisy responses.
It is well known that protecting against inference attacks that rely on underlying data correlations is beyond the purview of DP (Kifer & Machanavajjhala, 2014; Tschantz et al., 2020).
Strategy 2 corresponds to the recently introduced shufﬂe DP model, such as Google’s Prochlo (Bittau et al., 2017b).
Here, the noisy responses are completely anonymized – the adversary cannot identify which LDP responses correspond to Alice and her household.
Under such a model, only information that is completely order agnostic (i.e., symmetric functions that can be computed over just the bag of values, such as aggregate statistics) can be extracted.
Consequently, the analyst also fails to accomplish their original goal as all the underlying data correlation is destroyed.
Thus, we see that the two models of deployment for LDP present a trade-off between vulnerability to inference attacks and scope of data learnability.
In fact, as demonstrated in Kifer & Machanavajjhala (2011), it is impossible to defend against all inference attacks while simultaneously maintaining utility for learning.
In the extreme case that the adversary knows everyone in Alice’s community has the same true value (but not which one), no mechanism can prevent revelation of Alice’s datapoint short of destroying all utility of the dataset.
This then begs the question: Can we formally suppress speciﬁc inference attacks targeting each data owner while maintaining some meaningful learnability of the private data?
Referring back to our example, can we thwart attacks inferring Alice’s data using speciﬁcally her households’ responses and still allow the medical analyst to learn its target trends?
Can we offer this to every data owner participating?
In this paper, we strike a balance and propose a generalized shufﬂe framework that meets the utility requirements of the above analyst while formally protecting data owners against inference attacks.
Our solution is based on the key insight: the order of the data acts as the proxy for the identity of data owners as illustrated above.
The granularity at which the ordering is maintained formalizes resistance to inference attacks while retaining some meaningful learnability of the private data.
Speciﬁcally, we guarantee each data owner that their data is shufﬂed together with a carefully chosen group of other data owners.
Revisiting our example, consider uniformly shufﬂing the responses from Alice’s household and her immediate neighbors.
Now an adversary cannot use her household’s responses to predict her value any better than they could with a random sample of responses from this group.
In the same way that LDP prevents reconstruction of her datapoint using speciﬁcally her noisy response, this scheme prevents reconstruction of her datapoint using speciﬁcally her households’ responses.
The real challenge is offering such guarantees equally to every data owner.
Bob, Alice’s neighbor, needs his households’ responses shufﬂed in with his neighbors, as does Luis who is a neighbor of Bob but not of Alice.
Thus, we have n data owners with n distinct, overlapping groups.
Our scheme supports arbitrary groupings (overlapping or not), introducing a diverse and tunable class of privacy/utility trade-offs which is not attainable with either LDP or uniform shufﬂing alone.
For the above example, our scheme can formally protect each data owner from inference attacks using speciﬁcally their household, while still learning how disease prevalence changes across the neighborhoods of Alice’s community.
This work offers two key contributions to the machine learning privacy literature:
• Novel privacy guarantee.
We propose a novel privacy deﬁnition, d σ -privacy that captures the privacy of the order of a data sequence (Sec. 4.2) and formalizes the degree of resistance against inference attacks (Sec. 4.3).
d σ -privacy allows assigning an arbitrary group, G i , to each data owner, DO i , i ∈ [ n ] .
For instance, the groups can represent individuals in the same age bracket, ‘friends’ on social media, or individuals living in each other’s vicinity (as in case of Alice in our example).
Recall
that the order is informative of the data owner’s identity.
Intuitively, d σ -privacy protects
DO i from inference attacks that arise from knowing the identity of the members of their
group G i (Sec. 4.3).
Additionally, this grouping determines a threshold of learnability – any learning that is order agnostic within a group (disease prevalence in a neighborhood – the data analyst’s goal in our example) is utilitarian and allowed; whereas analysis that involves identifying the values of individuals within a group (disease prevalence within speciﬁc households – the adversary’s goal) is regarded as a privacy threat and protected against.
See Fig.
1 for a toy demonstration of how our guarantee allows tuning the granularity at which trends can be learned .
• Novel shufﬂe framework.
We propose a novel mechanism that shufﬂes the data systematically and achieves d σ -privacy.
This provides a generalized shufﬂe framework that interpolates between no shufﬂing ( LDP ) and
uniform random shufﬂing (shufﬂe model) in terms of protection against inference attacks and data learnability.
R ELATED W ORK
The shufﬂe model of DP (Bittau et al., 2017a; Cheu et al., 2019; Erlingsson et al., 2019) differs from our scheme as follows.
These works (1) study DP beneﬁts of shufﬂing whereas we study the inferential privacy beneﬁts, and (2) only study uniformly random shufﬂing where ours generalizes this to tunable, non-uniform shufﬂing (see App. A.15).
A steady line of work has studied inferential privacy (Kasiviswanathan & Smith, 2014; Kifer &
Machanavajjhala, 2011; Ghosh & Kleinberg, 2016; Dalenius, 1977; Dwork & Naor, 2010; Tschantz et al., 2020).
Our work departs from those in that we focus on local inferential privacy and do so via the new angle of shufﬂing.
Older works such as k -anonymity (Sweeney, 2002), l -diversity Machanavajjhala et al.
Anatomy (Xiao & Tao, 2006) and others (Wong et al., 2010; Tassa et al., 2012; Xue et al., 2012;
Choromanski et al., 2013; Doka et al., 2015) have studied the privacy risk of non-sensitive auxiliary information or ‘quasi identiﬁers’.
These works (1) focus on the setting of dataset release, whereas we focus on dataset collection, and (2) do not offer each data owner formal inferential guarantees, whereas we do.
The De Finetti attack (Kifer, 2009) shows how shufﬂing schemes are vulnerable to inference attacks that correlate records to recover the original permutation of sensitive attributes.
A strict instance of our privacy guarantee can thwart such attacks (at the cost of no utility, App. A.3).
B ACKGROUND
Notations.
Boldface (such as x = (cid:104) x 1 , · · · , x n (cid:105) ) denotes a data sequence (ordered list); normal font (such as x 1 ) denotes individual values and {·} represents a multiset or bag of values.
3. L OCAL D IFFERENTIAL P RIVACY
The local model consists of a set of data owners and an untrusted data aggregator (analyst); each individual perturbs their data using a LDP algorithm (randomizers) and sends it to the analyst.
The
LDP guarantee is formally deﬁned as
Deﬁnition 3.1.
[Local Differential Privacy, LDP Warner (1965); Evﬁmievski et al.
; Kasiviswanathan et al.
A randomized algorithm M : X → Y is (cid:15) -locally differentially private (or (cid:15) - LDP ), if for any pair of private values x, x (cid:48) ∈ X and any subset of output,
[Equation]
The shufﬂe model is an extension of the local model where the data owners ﬁrst randomize their inputs.
Additionally, an intermediate trusted shufﬂer applies a uniformly random permutation to all the noisy responses before the analyst can view them.
The anonymity provided by the shufﬂer requires less noise than the local model for achieving the same privacy.
3. M ALLOWS M ODEL
A permutation of a set S is a bijection S (cid:55)→ S .
The set of permutations of [ n ] , n ∈ N forms a symmetric group S n .
As a shorthand, we use σ ( x ) to denote applying permutation σ ∈ S n to a data sequence x of length n .
Additionally, σ ( i ) , i ∈ [ n ] , σ ∈ S n denotes the value at index i in σ and σ − 1 denotes its inverse.
For example, if σ = (1 3 5 4 2) and x = (cid:104) 21 , 33 , 45 , 65 , 67 (cid:105) , then σ ( x ) = (cid:104) 21 , 45 , 67 , 65 , 33 (cid:105) , σ (2) = 3 , σ (3)
= 5 and σ − 1 = (1 5 2 4 3) .
Mallows model is a popular probabilistic model for permutations (MALLOWS, 1957).
The mode of the distribution is given by the reference permutation σ 0 – the probability of a permutation increases as we move ‘closer’ to σ 0 as measured by rank distance metrics, such as the Kendall’s tau distance (Def. A.2).
The dispersion parameter θ controls how fast this increase happens.
Deﬁnition 3.2.
For a dispersion parameter θ , a reference permutation σ o ∈ S n , and a rank distance measure d : S n × S n (cid:55)→ R , P Θ , d ( σ : σ 0 ) = 1 ψ ( θ, d ) e − θ d ( σ,σ 0 ) is the Mallows model where ψ ( θ, d )
=
(cid:80) σ ∈ S n e − θ d
( σ,σ 0 ) is a normalization term and σ ∈ S n .
D ATA P RIVACY AND S HUFFLING
In this section, we present d σ -privacy and a shufﬂing mechanism capable of achieving the d σ -privacy guarantee.
4. P ROBLEM S ETTING
In our problem setting, we have n data owners DO i , i ∈ [ n ] each with a private input x
i ∈ X (Fig. 2).
The data owners ﬁrst randomize their inputs via a (cid:15) - LDP mechanism to generate y i
=
M ( x i ) .
Additionally, just like in the shufﬂe model, we have a trusted shufﬂer.
It mediates upon the noisy responses y
=
(cid:104)
y 1 , · · · , y n (cid:105) to obtain the ﬁnal output sequence z
= A ( y )
( A corresponds to Alg. 1) which is sent to the untrusted data analyst.
The shufﬂer can be implemented via trusted execution environments (TEE) just like Google’s Prochlo.
Next, we formally discuss the notion of order and its implications.
Deﬁnition 4.1.
(Order)
The order of a sequence x = (cid:104) x 1 , · · · , x n (cid:105) refers to the indices of its set of values { x i } and is represented by permutations from S n .
When the noisy response sequence y =
(cid:104)
y 1 , · · · , y n (cid:105) is represented by the identity permutation σ I =
(1 2 · · · n ) , the value at index 1 corresponds to DO 1 and so on.
Standard LDP releases the identity permutation w.p. 1.
The output of the shufﬂer, z , is some permutation of the sequence y ,
i.e., z
= σ ( y ) =
(cid:104)
y σ (1) , · · · , y σ ( n ) (cid:105) where σ is determined via A ( · ) .
For example, for σ = (4 5 2 3 1) , we have z = (cid:104) y 4 , y 5 , y 2 , y 3 , y 1 (cid:105) which means that the value at index 1 ( DO 1 ) now corresponds to that of DO 4 and so on.
4. D EFINITION OF d σ - PRIVACY
Inferential risk captures the threat of an adversary who infers
DO i ’s private x i using all or a subset of other data owners’ released y j ’s. Since we cannot prevent all such attacks and maintain utility, our aim is to formally limit which data owners can be leveraged in inferring
DO i ’s private x i .
To make this precise, each DO i may choose a corresponding group, G i ⊆
[ n ] , of data owners.
d σ -privacy guarantees that y j values originating from a data owner’s group G i are shufﬂed together.
In doing so, the LDP values corresponding to subsets of DO
i ’s group
I ⊂ G i cannot be reliably identiﬁed, and thus cannot be singled out to make inferences about DO i ’s x i .
If Alice’s group includes her whole neighborhood, LDP data originating from her household cannot be singled out to recover her private x i .
Any choice of grouping G = { G 1 , G 2 , . .
G n } can be accommodated under d σ -privacy.
Each data owner may choose a group large enough to hide anyone they feel sufﬁcient risk from.
We outline two systematic approaches to assigning groups as follows:
• Let t = (cid:104) t 1 , · · · , t n (cid:105) , t i ∈ T denote some public auxiliary information about each individual.
DO i ’s group, G i , could consist of all those DO j ’s who are similar to DO i w.r
t. the public auxiliary information t i , t j according to some distance measure d : T × T → R .
Here, we deﬁne
‘similar’ as being under a threshold 2 r ∈ R such that G i = { j ∈ [ n ] (cid:12)(cid:12) d ( t i , t j ) ≤ r } , ∀ i ∈ [ n ] .
For example, d ( · ) can be Euclidean distance if T corresponds to geographical locations, thwarting inference attacks leveraging one’s household or immediate neighbors.
If T represents a social media connectivity graph, d ( · ) can measure the path length between two nodes, thwarting inference attacks using speciﬁcally one’s close friends.
For the example social media connectivity graph depicted in Fig.
3, assuming distance metric path length and r = 2 , the groups are deﬁned as G 1 = { 1 , 7 , 8 , 2 , 5 , 6 } , G 2 = { 2 , 1 , 7 , 5 , 6 , 3 } and so on.
• Alternatively, the data owners might opt for a group of a speciﬁc size r < n .
Collecting private data from a social media network, we may set r = 50 , where each G i is encouraged to include the 50 data owners DO i interacts with most frequently.
Intuitively, d σ -privacy protects DO i against inference attacks that leverages correlations at a ﬁner granularity than G i .
In other words, under d σ -privacy, one subset of k data owners ⊂ G i (e.g household) is no more useful for targeting x
i than any other subset of k data owners ⊂ G i (e.g. some combination of neighbors).
This leads to the following key insight for the formal privacy deﬁnition.
Key Insight.
Formally, our privacy goal is to prevent the leakage of ordinal information from within a group.
We achieve this by systematically bounding the dependence of the mechanism’s output on the relative ordering (of data values corresponding to the data owners) within each group .
First, we introduce the notion of neighboring permutations.
Deﬁnition 4.2.
(Neighboring Permutations)
Given a group assignment G , two permutations σ,σ (cid:48)
∈ S n are deﬁned to be neighboring w.r.t.
a group G i ∈ G (denoted as σ ≈ G i
[Equation]
Neighboring permutations differ only in the indices of its corresponding group G i .
For example, σ = (1 2 4 5 7 6 10 3 8 9) and σ (cid:48)
= (7 3 4 5 6 2 1 10 8 9) are neighboring w.r.t G 1 (Fig. 3) since they differ only in σ (1) ,σ (2) ,σ (5) ,σ (6) ,σ (7) and σ (8) .
We denote the set of all neighboring permutations as
[Equation]
Now, we formally deﬁne d σ -privacy as follows.
Deﬁnition 4.3 ( d σ -privacy) .
For a given group assignment G on a set of n entities and a privacy parameter α ∈ R ≥ 0 , a randomized mechanism A : Y n (cid:55)→ V is ( α, G ) - d
σ private if for all y ∈ Y n and neighboring permutations σ, σ (cid:48) ∈ N G and any subset of output O ⊆ V , we have
[Equation]
σ ( y ) and σ (cid:48)
( y ) are deﬁned to be neighboring sequences .
d σ -privacy states that, for any group G i , the mechanism is (almost) agnostic of the order of the data within the group.
Even after observing the output, an adversary cannot learn about the relative ordering of the data within any group.
Thus, two neighboring sequences are indistinguishable to an adversary.
An important property of d σ -privacy is that post-processing computations does not degrade privacy.
Additionally, when applied multiple times, the privacy guarantee degrades gracefully.
Both the properties are analogous to DP and are presented in App.
A.4.
Note.
Any data sequence x = (cid:104) x 1 , · · · , x n (cid:105) can be viewed as a two-tuple, (cid:0) { x } , σ (cid:1) , where { x } denotes the bag of values and σ ∈ S n denotes the corresponding indices of the values which represents the order of the data.
The (cid:15) -LDP protects the bag of data values, { x } , while d σ -privacy protects the order, σ .
Thus, the two privacy guarantees cater to orthogonal parts of a data sequence (see Thm. 4.2 ).
Also, α = ∞ (0) , r = 0 ( n ) represents the standard LDP (shufﬂe DP ) setting.
4. P RIVACY I MPLICATIONS
The group assignment G delineates a threshold of learnability which determines the privacy/utility tradeoff as follows.
• Learning allowed (Analyst’s goal) .
d σ -privacy can answer queries that are order agnostic within groups, such as aggregate statistics of a group.
In Alice’s case, the analyst can estimate the disease prevalence in her neighborhood.
• Learning disallowed (Adversary’s goal) .
Adversaries cannot identify (noisy) values of individuals within any group.
While they may learn the disease prevalence in Alice’s neighborhood, they cannot determine the prevalence within her household and use that to recover her value x i .
To make this precise, we ﬁrst formalize the privacy implications of the d σ guarantee in the standard
Bayesian framework, typically used for studying inferential privacy.
Next, we formalize the privacy provided by the combination of LDP and d σ guarantees by way of a decision theoretic adversary.
Bayesian Adversary.
Consider a Bayesian adversary with any prior P on the joint distribution of noisy responses, Pr P [ y ] , which models their beliefs on the correlation between the participants (such as the correlation between Alice and her households’ disease status).
Their goal is to infer DO i ’s private input x i .
As with early DP works (Dwork et al., 2006), we consider an informed adversary.
Here, the adversary knows (1) the sequence (assignment) of noisy values outside G i , y G i , and (2) the (unordered) bag of noisy values in G i , { y G i } .
d σ -privacy bounds the prior-posterior odds gap on x i for such as informed adversary as follows:
Theorem 4.1.
For a given group assignment G on a set of n data owners, if a shufﬂing mechanism
A : Y n (cid:55)→ Y n is ( α, G ) - d σ private, then for each data owner DO i , i ∈ [ n ] ,
[Equation]
for a prior distribution P , where z =
A ( y ) and y G i is the noisy sequence for data owners outside G i .
See App A.5 for the proof and further discussion on the semantic meaning of the above guarantee.
Decision Theoretic Adversary.
Here, we analyse the privacy provided by the combination of
LDP and d σ guarantees.
Consider a decision theoretic adversary who aims to identify the noisy responses, { z I } , that originated from a speciﬁc subset of data owners, I ⊂ G i (such as the members of Alice’s household).
We denote the adversary by a (possibly randomized) function mapping from the output z sequence to a set of k indices, D Adv : Y n → [ n ] k , where k
= | I | .
These k indices,
H ∈ [ n ] k , represent the elements of z that D Adv believes originated from the data owners in I .
D Adv wins if > k/ 2 of the chosen indices indeed originated from I , i.e, | σ ( H )
∩ I | > k/ 2 , where z i =
y σ ( i ) and σ ( H )
=
{ σ ( i ) : i ∈ H } .
D Adv loses if most of H did not originate from I , i.e., | σ ( H ) ∩ I | ≤ k/ 2 .
We choose the above adversary because this re-identiﬁcation is a key step in carrying out inference attacks – in failing to reliably re-identify the noisy values originating from I , one cannot make inferences on x i speciﬁcally from the subset I ⊂ G i .
Theorem 4.2.
For A ( M ( x ))
=
z where M ( · ) is (cid:15) - LDP and A ( · ) is α - d σ private, we have
[Equation]
for any input subgroup I ⊂ G i , r = | G i | and k < r/ 2 .
The adversary’s ability to re-identify the { z I } values comes partially from the bag of values (quantiﬁed by (cid:15) ) and partially from the order (quantiﬁed by α ).
We highlight two implications of this fact.
• When (cid:15) is small ( (cid:28) 1 ), an adversary’s ability to re-identify the noisy values { z I } originating from I may very well be dominated by α .
For instance, if (cid:15) = 0 .
2 and k = 5 , the adversary’s advantage is dominated by α for any α > 2 .
When using LDP alone (no shufﬂing), α = ∞ and the adversary can exactly recover which values came from Alice’s household.
As such, even a moderate α value (obtained via d σ -privacy) signiﬁcantly reduces the ability to re-identify the values.
• When the loss is dominated by (cid:15) ( 2 k(cid:15) (cid:29) α ), the above expression allows us to disentangle the source of privacy loss .
In this regime, adversaries get most of their advantage from the bag of values released, not from the order of the release.
That is, even if α = 0 (uniform random shufﬂing), participants still suffer a large risk of re-identiﬁcation simply due to the noisy values being reported.
Thus, no shufﬂing mechanism can prevent re-identiﬁcation in this regime.
Discussion.
In spirit, DP does not guarantee protection against recovering
DO i ’s private x i value.
It guarantees that – had a user not participated (or equivalently submitted a false value x (cid:48) i ) – the adversary would have about the same ability to learn their true value, potentially from the responses of other data owners.
In other words, the choice to participate is unlikely to be responsible for the disclosure of x i .
Similarly, d σ -privacy does not prevent disclosure of x i .
By requiring indistinguishability of neighboring permutations, it guarantees that – had the data owners of any group G i completely swapped identities – the adversary would have about the same ability to learn x i .
So most likely,
Alice’s household is not uniquely responsible for a disclosure of her x i : had her household swapped identities with any of her neighbors, the adversary would probably draw the same conclusion on x i .
Or, as detailed in Thm.4.2, an adversary cannot reliably resolve which { z } values originated from
Alice’s household, so they cannot draw conclusions based on her household’s responses.
In a nutshell,
• Inference attacks can recover a data owner
DO i ’s private data x i from the responses of other data owners.
The order of the data acts as the proxy for the data owner’s identity which can aid an adversary in corralling the subset of other data owners who correlate with DO i (required to make a reliable inference of x i ).
• DP alleviates concerns that DO i ’s choice to share data ( y i ) will result in disclosure of x i , and
d σ -privacy alleviates concerns that DO i ’s group
’s ( G i ) choice to share their identity will result in disclosure of x i .
4. d σ - PRIVATE S HUFFLING M ECHANISM
Input
LDP y =
y 1 , ··· y n t = t 1 , · · · t n threshold r ; Priv param α ; 7 ˆ σ ∼ P θ, ; 8 σ ∗ = σ − 1 0 ˆ σ ; y σ ∗
· · · y σ ∗
We now describe our novel shufﬂing mechanism that can achieve d σ -privacy.
In a nutshell, our mechanism samples a permutation from a suitable
Mallows model and shufﬂes the data sequence accordingly.
We can characterize the d σ -privacy guarantee of our mechanism in the same way as that of the DP guarantee of classic mechanisms (Dwork & Roth, 2014) – with variance and sensitivity.
Intuitively, a larger dispersion parameter θ ∈ R (Def. 3.2) reduces randomness over permutations, increasing utility and increasing (worsening)
the privacy parameter α .
The maximum value of θ for a given α guarantee depends on the sensitivity of the rank distance measure d ( · ) over all neighboring permutations
N G .
Formally, we deﬁne the sensitivity as
[Equation]
the maximum change in distance d ( · ) from the reference permutation σ 0 for any pair of neighboring permutations ( σ, σ (cid:48) )
∈ N G permuted by σ 0 .
The privacy parameter of the mechanism is then proportional to its sensitivity
α = θ · ∆( σ 0 : d , G ) .
Given G and a reference permutation σ 0 , the sensitivity of a rank distance measure d ( · ) depends on the width , ω σ G , which measures how ‘spread apart’ the members of any group of G are in σ 0 :
[Equation]
For example, for σ = (1 3 7 8 6 4 5 2 9 10) and G 1 = { 1 , 7 , 8 , 2 , 5 , 6 } , ω σG 1 = | σ − 1 (1) − σ − 1 (2) | = 7 .
The sensitivity is an increasing function of the width.
For instance, for Kendall’s τ distance d τ ( · ) we have ∆( σ 0 : d τ , G ) =
ω σ 0 G ( ω σ 0 G + 1) / 2 .
If a reference permutation clusters the members of each group closely together (low width), then the groups are more likely to permute within themselves.
This has two beneﬁts.
First, for the same θ ( θ is an indicator of utility as it determines the dispersion of the sampled permutation), a lower value of width gives lower α (better privacy).
Second, if a group is likely to shufﬂe within itself, it will have better ( η,δ ) -preservation – a novel utility metric, we propose, for a shufﬂing mechanism.
Intuitively, a mechanism is ( η, δ ) -preserving w.r.t a subset of indices S ⊂ [ n ] if at least η % of its indices are shufﬂed within itself with probability (1 − δ ) .
The rationale behind this metric is that it captures the utility of the learning allowed by d σ -privacy – if S is equal to some group G ∈ G , high ( η,δ ) -preservation allows overall statistics of G to be captured since η % of the correct data values remain preserved.
We present the formal discussion in App.
A.7.
Unfortunately, minimizing ω σ G is an NP-hard problem (Thm. A.3 in App. A.9).
Instead, we estimate the optimal σ 0 using the following heuristic 3 approach based on a graph breadth ﬁrst search.
Algorithm Description.
Alg.
1 above proceeds as follows.
We ﬁrst compute the group assignment,
G , based on the public auxiliary information and desired threshold r following discussion in Sec. 4.
(Step 1).
Then we construct σ 0 with a breadth ﬁrst search (BFS) graph traversal.
We translate G into an undirected graph ( V,E ) , where the vertices are indices V =
[ n ] and two indices i,j are connected by an edge if they are both in some group (Step 2).
Next, σ 0 is computed via a breadth
ﬁrst search traversal (Step 4) – if the k -th node in the traversal is i , then σ 0 ( k )
=
i .
The rationale is that neighbors of i (members of G i ) would be traversed in close succession.
Hence, a neighboring node j is likely to be traversed at some step h near k which means | σ − 1
( i ) − σ − 1 0 ( j ) | = | h − k | would be small (resulting in low width).
Additionally, starting from the node with the highest degree (Steps 3-4) which corresponds to the largest group in G (lower bound for ω σ G for any σ ) helps to curtail the maximum width in σ 0 .
See App.
This is followed by the computation of the dispersion parameter, θ , for our Mallows model (Steps 5-6).
Next, we sample a permutation from the Mallows model (Step 7) ˆ σ ∼ P θ ( σ : σ 0 ) and we apply the inverse reference permutation to it, σ ∗ =
σ − 1 0 ˆ σ to obtain the desired permutation for shufﬂing.
Recall that ˆ σ is (most likely) close to σ 0 , which is unrelated to the original order of the data.
σ − 1therefore brings σ ∗ back to a shufﬂed version of the original sequence (identity permutation σ I ).
Note that since Alg. 1 is publicly known, the adversary/analyst knows σ 0 .
Hence, even in the absence of this step from our algorithm, the adversary/analyst could perform this anyway.
Finally, we permute y according to σ ∗ and output the result z
= ˆ σ ( y ) (Steps 9-10).
Theorem 4.3.
Alg. 1 is ( α, G ) - d σ private where α = θ · ∆( σ 0 : d , G ) .
The proof is in App.
A.11.
Note that Alg.
1 provides the same level of privacy ( α ) for any two group assignment G , G (cid:48) as long as they have the same sensitivity, i.e, ∆( σ 0 :
d τ , G ) =
∆( σ 0 :
d τ , G (cid:48) ) .
This leads to the following theorem which generalizes the privacy guarantee for any group assignment.
Theorem 4.4.
Alg.
1 satisﬁes ( α (cid:48) , G (cid:48) ) -
d σ privacy for any group assignment G (cid:48) with α (cid:48)
=
α ∆( σ 0 : d , G (cid:48) )
∆( σ 0 : d , G ) (proof in App. A.12.)
Note.
Producing σ ∗ is completely data ( y ) independent.
It only requires access to the public auxiliary information t .
Hence, Steps 1 − 6 can be performed in a pre-processing phase and do not contribute to the actual running time.
See App.
A.10 for an illustration of Alg.
1 and runtime analysis.
E VALUATION
[Figure]
The previous sections describe how our shufﬂing framework interpolates between standard LDP and uniform random shufﬂing.
We now experimentally evaluate this asking the following two questions –
Q
Does the Alg.
1 mechanism protect against realistic inference attacks?
Q
How well can Alg.
1 tune a model’s ability to learn trends within the shufﬂed data, i.e., tune data
learnability
We evaluate on four datasets.
We are not aware of any prior work that provides comparable local inferential privacy.
Hence, we baseline our mechanism with the two extremes: standard LDP and uniform random shufﬂing.
For concreteness, we detail our procedure with the PUDF dataset (PUD) (license), which comprises n ≈ 29 k psychiatric patient records from Texas.
Each data owner’s sensitive value x i is their medical payment method, which is reﬂective of socioeconomic class (such as medicaid or charity).
Public auxiliary information t ∈ T is the hospital’s geolocation.
Such information is used for understanding how payment methods (and payment amounts) vary from town to town for insurances in practice (Eric Lopez, 2020).
Uniform shufﬂing across Texas precludes such analyses.
Standard LDP risks inference attacks, since patients attending hospitals in the same neighborhood have similar socioeconomic standing and use similar payment methods, allowing an adversary to correlate their noisy y i ’s. To trade these off, we apply Alg.
1 with d ( · ) being distance (km) between hospitals, α = 4 and Kendall’s τ rank distance measure for permutations.
Our inference attack predicts DO i ’s x
i by taking a majority vote of the z j values of the 25 data owners within r ∗ of t i and who are most similar to DO i w.r.t some additional privileged auxiliary information t pj ∈ T p .
For PUDF, this includes the 25 data owners who attended hospitals that are within r ∗ km of
DO i ’s hospital, and are most similar in payment amount t pj .
Using an (cid:15) = 2 .
5 randomized response mechanism, we resample the LDP sequence y 50 times, and apply Alg.
1’s chosen permutation to each, producing 50 z ’s. We then mount the majority vote attack on each x
i for each z .
If the attack on a given x i is successful across ≥ 90% of these LDP trials, we mark that data owner as vulnerable – although they randomize with LDP , there is a ≥ 90% chance that a simple inference attack can recover their true value.
We record the fraction of vulnerable data owners as ρ .
We report 1-standard deviation error bars over 10 trials.
Additionally, we evaluate data learnability – how well the underlying statistics of the dataset are preserved across T .
For PUDF , this means training a model on the shufﬂed z to predict the distribution of payment methods used near, for instance, t i = Houston for DO i .
For this, we train a calibrated model, Cal : T → D x , on the shufﬂed outputs where D x is the set of all distributions on the domain of sensitive attributes X .
We implement Cal as a gradient boosted decision tree (GBDT) model (Friedman, 2001) calibrated with Platt scaling (Niculescu-Mizil & Caruana, 2005).
For each location t i , we treat the empirical distribution of x i values within r ∗ as the ground truth distribution at t i , denoted by E ( t i )
∈
D x .
Then, for each t i , we measure the Total Variation error between the predicted and ground truth distributions TV (cid:0) E ( t i ) , Cal r ( t i ) (cid:1) .
We then report λ ( r ) – the average TV error for distributions predicted at each t i ∈ t normalized by the TV error of naively guessing the uniform distribution at each t i .
With standard LDP , this task can be performed relatively well at the risk of inference attacks.
With uniformly shufﬂed data, it is impossible to make geographically localized predictions unless the distribution of payment methods is identical in every Texas locale.
We additionally perform the above experiments on the following three datasets
• Twitch
(Rozemberczki et al., 2019).
This dataset, gathered from the Twitch social media platform, includes a graph of ≈ 9 K edges (mutual friendships) along with node features.
The user’s history of explicit language is private X
= { 0 , 1 } .
T is a user’s mutual friendships, i.e. t i is the i ’th row of the graph’s adjacency matrix.
We do not have any T P here and select the 25 neighbors randomly.
• Syn .
This is a synthetic dataset of size 20 K which can be classiﬁed at three granularities – 8-way, 4-way and 2-way (Fig.
1a shows a scaled down version of the dataset).
The eight color labels are private X = [8] ; the 2D-positions are public T
= R 2 .
For learnability, we measure the accuracy of 8 -way, 4 -way and 2 -way GBDT models trained on z on an equal sized test set at each r .
• Adult (Dua & Graff, 2017).
This dataset is derived from the 1994 Census and has ≈ 33 K records.
Whether DO i ’s annual income is ≥ 50 k is considered private, X = {≥ 50 k, < 50 k } .
T =
[17 , 90] is age and T P is the individual’s marriage status.
Due to lack of space ﬁgures are in App.
A.14.2.
Experimental Results
Q1.
Our formal guarantee on the inferential privacy loss (Thm. 4.1) is described w.r.t to a ‘strong’ adversary (with access to { y G i } , y G i ).
Here, we test how well does our proposed scheme (Alg. 1) protect against inference attacks on real-world datasets without any such assumptions.
Additionally, to make our attack more realistic, the adversary has access to extra privileged auxiliary information
T P which is not used by Alg.
Fig.
4a → 4b show that our scheme signiﬁcantly reduces the attack efﬁcacy.
For instance, ρ is reduced by 2 .
7 X at the attack distance threshold r ∗ for PUDF .
Additionally, ρ for our scheme varies from that of LDP 4 (minimum privacy) to uniform shufﬂe (maximum privacy) with increasing r (equivalently group size as in Fig. 4b) thereby spanning the entire privacy spectrum.
As expected, ρ decreases with decreasing privacy parameter α (Fig. 8b).
Q2.
Fig.4c → 4d show that λ varies from that of LDP (maximum learnability) to that of uniform shufﬂe (minimum learnability) with increasing r (equivalently, group size), thereby providing tunability.
Interestingly, for Adult our scheme reduces ρ by 1 .
7 X at the same λ as that of LDP for r = 1 (Fig. 8c).
Fig.
5 shows that the distance threshold r deﬁnes the granularity at which the data can be classiﬁed.
LDP allows 8-way classiﬁcation while uniform shufﬂing allows none.
The granularity of classiﬁcation can be tuned by our scheme – r 8 , r 4 and r 2 mark the thresholds for 8 -way, 4 -way and 2 -way classiﬁcations, respectively.
C ONCLUSION
We have proposed a new privacy deﬁnition, d σ -privacy that casts new light on the inferential privacy beneﬁts of shufﬂing and a novel shufﬂing mechanism to achieve the same.
E THIS S TATEMENT
It is the aim of this paper to formalize the enhanced privacy guarantees offered by shufﬂing, to provide intuition of what those formal guarantees semantically offer to data owners, and to demonstrate an algorithm + experiments which offer these guarantees while meeting analyst utility requirements.
We feel that all of these aims as well as the public datasets used are ethical.
R EPRODUCIBILITY S TATEMENT
The majority of this paper formalizes a novel perspective on the privacy guarantees achieved by shufﬂing (i e. randomizing the order of the data as opposed to the values).
Detailed proofs as well as intuitive discussions are provided in the Appendix.
All datasets are public.
A .zip ﬁle demonstrating code of each experiment has been uploaded as supplementary material.
A A PPENDIX A. B ACKGROUND C NTD . A. L OCAL I NFERENTIAL P RIVACY
Local inferential privacy captures what information a Bayesian adversary Kifer & Machanavajjhala (2014), with some prior, can learn in the LDP setting.
Speciﬁcally, it measures the largest possible ratio between the adversary’s posterior and prior beliefs about an individual’s data after observing a mechanism’s output .
Deﬁnition A.1.
(Local Inferential Privacy Loss Kifer & Machanavajjhala (2014))
Let x = (cid:104) x 1 , · · · , x n (cid:105) and let y =
(cid:104)
y 1 , · · · , y n (cid:105) denote the input (private) and output sequences (observable to the adversary) in the LDP setting.
Additionally, the adversary’s auxiliary knowledge is modeled by a prior distribution P on x .
The inferential privacy loss for the input sequence x is given by
[Equation]
Bounding L P ( x ) would imply that the adversary’s belief about the value of any x i does not change by much even after observing the output sequence y .
This means that an informed adversary does not learn much about the individual i ’s private input upon observation of the entire private dataset y .
Here we deﬁne two rank distance measures
Deﬁnition A.2 (Kendall’s τ Distance) .
For any two permutations, σ, π ∈ S n , the Kendall’s τ distance d τ ( σ, π ) counts the number of pairwise disagreements between σ and π , i.e., the number of item pairs that have a relative order in one permutation and a different order in the other.
Formally,
[Equation]
For example, if σ = (1 2 3 4 5 6 7 8 9 10) and π = (1 2 3 6 5 4 7 8 9 10) , then d τ ( σ, π ) = 3 .
Next, Hamming distance measure is deﬁned as follows.
Deﬁnition A.3 (Hamming Distance) .
For any two permutations, σ, π ∈ S n , the Hamming distance d H ( σ, π ) counts the number of positions in which the two permutations disagree.
Formally,
[Equation]
Repeating the above example, if σ = (1 2 3 4 5 6 7 8 9 10) and π = (1 2 3 6 5 4 7 8 9 10) , then d H ( σ, π ) = 2 .
A. d σ - PRIVACY AND THE D E F INETTI ATTACK
We now show that a strict instance of d σ privacy is sufﬁcient for thwarting any de Finetti attack
Kifer (2009) on individuals.
The de Finetti attack involves a Bayesian adversary, who, assuming some degree of correlation between data owners, attempts to recover the true permutation from the shufﬂed data.
As written, the de Finetti attack assumes the sequence of sensitive attributes and side information ( x 1 , t 1 ) , . . .
, ( x n , t n ) are exchangeable : any ordering of them is equally likely.
By the de Finetti theorem, this implies that they are i.i.d.
conditioned on some latent measure θ .
To balance privacy with utility, the x sequence is non-uniformly randomly shufﬂed w.r.t.
the t sequence producing a shufﬂed sequence z , which the adversary observes.
Conditioning on z the adversary updates their posterior on θ (i.e. posterior on a model predicting x i | t i ), and thereby their posterior predictive on the true x .
The deﬁnition of privacy in Kifer (2009) holds that the adversary’s posterior beliefs are close to their prior beliefs by some metric on distributions in X , δ ( [Equation]
[Equation]
We now translate the de Finetti attack to our setting.
First, to align notation with the rest of the paper we provide privacy to the sequence of LDP values y since we shufﬂe those instead of the x values as in Kifer (2009).
We use max divergence (multiplicative bound on events used in DP ) for δ :
[Equation]
which, for compactness, we write as
[Equation]
We restrict ourselves to shufﬂing mechanisms, where we only randomize the order of sensitive values.
By learning the unordered values { y } alone, an adversary may have arbitrarily large updates to its posterior (e.g. if all values are identical), breaking the privacy requirement above.
With this in mind, we assume the adversary already knows the unordered sequence of values { y } (which they will learn anyway), and has a prior on permutations σ allocating values from that sequence to individuals.
We then generalize the de Finetti problem to an adversary with an arbitrary prior on the true permutation σ , and observes a randomize permutation σ (cid:48) from the shufﬂing mechanism.
We require that the adversary’s prior belief that σ ( i ) =
j is close to their posterior belief for all i, j ∈ [ n ] :
[Equation]
where Σ i,j = { σ ∈ S n : σ ( i ) = j } , the set of permutations assigning element j to DO i .
Conditioning on any unordered sequence { y } with all unique values, the above condition is necessary to satisfy Eq.
equation 6 for events of the form O = { y i = a } , since { y i = a } = { Σ i,j } for some j ∈ [ n ] .
For any { y } with repeat values, it is sufﬁcient since Pr[ y i
=
a ] is the sum of probabilities of disjoint events of the form Pr[ σ ∈ Σ i,k ] for various k ∈ [ n ] values.
We now show that a strict instance of d σ -privacy satisﬁes Eq.
equation 7.
Let (cid:98) G be any group assignment such that at least one G i ∈ (cid:98) G includes all data owners, G i = { 1 , 2 , . . . , n } .
Property 1.
A ( (cid:98) G , α ) - d σ -private shufﬂing mechanism σ (cid:48) ∼ A satisﬁes
[Equation]
for all i, j ∈ [ n ] and all priors on permutations Pr[ σ ] .
Proof.
Lemma 1.
For any prior Pr[ σ ] , Eq. equation 7 is equivalent to the condition (cid:80)
[Equation]
where the set Σ i,j is the complement of Σ i,j .
Under grouping ˆ G , every permutation σ a ∈ Σ i,j neighbors every permutation σ b ∈ Σ i,j , σ a ≈ ˆ G σ b , for any i, j .
By the deﬁnition of d σ -privacy, we have that for any observed permutation σ (cid:48) output by the mechanism:
[Equation]
This implies Eq. 8.
Thus, ( (cid:98) G , α ) - d σ -privacy implies Eq. 8, which implies Eq.
7, thus proving the property.
Using Lemma 1, we may also show that this strict instance of d σ -privacy is necessary to block all de
Finetti attacks:
Property 2.
A (
[Equation]
G , α ) - d σ -private shufﬂing mechanism σ (cid:48) ∼ A is necessary to satisfy
[Equation]
for all i, j ∈ [ n ] and all priors on permutations Pr[ σ ] .
Proof.
If our mechanism A is not ( (cid:98) G , α ) - d σ -private, then for some pair of true (input) permutations σ a (cid:54) =
σ b and some released permutation σ (cid:48) ∼ A , we have that
Pr[ σ (cid:48) | σ b ] ≥ e α Pr[ σ (cid:48) | σ a ] .
Under ˆ G , all permutations neighbor each other, so σ a ≈ ˆ G σ b .
Since σ a (cid:54) = σ b , then for some i, j ∈ [ n ] , σ a ∈ Σ i,j and σ b ∈ Σ
i,j : one of the two permutations assigns some j to some DO i and the other does not.
Given this, we may construct a bimodal prior on the true σ that assigns half its probability mass to σ a and the rest to σ b ,
[Equation]
Therefore, for released permutation σ (cid:48) , the RHS of Eq.
8 is 1, and the LHS is
[Equation]
violating Eq. 8, thus violating Eq. 7, and failing to prevent de Finetti attacks against this bimodal prior.
Ultimately, unless we satisfy d σ -privacy shufﬂing the entire dataset, there exists some prior on the true permutation Pr[ σ ] such that after observing the shufﬂed z permuted by σ (cid:48) , the adversary’s posterior belief on one permutation is larger than their prior belief by a factor ≥ e α .
If we suppose that the set of values { y } are all distinct, this means that for some a ∈ { y } , the adversary’s belief that
y i
=
a is signﬁcantly larger after observing z than it was before.
Now to prove Lemma 1:
Proof.
[Equation]
[Table]
As such
[Table]
a strict instance of d σ -privacy can defend against any de Finetti attack (i [Table] e. for any prior
Pr[ σ ] on permutations), wherein at least one group G i ∈ G includes all data owners.
Furthermore, it is necessary.
This makes sense.
In order to defend against any prior, we need to signiﬁcantly shufﬂe the entire dataset.
Without a restriction of priors as in Pufferﬁsh Kifer & Machanavajjhala (2014), the de Finetti attack (i.e. uninformed Bayesian adversaries) is an indelicate metric for evaluating the privacy of shufﬂing mechanisms: to achieve signiﬁcant privacy, we must sacriﬁce all utility.
This in many regards is reminiscent of the no free lunch for privacy theorem established in Kifer &
Machanavajjhala (2011).
As such, there is a need for more ﬂexible privacy deﬁnitions for shufﬂing mechanisms.
A. A DDITIONAL P ROPERTIES OF d σ - PRIVACY
Lemma 2 (Convexity) .
Let A 1 , . . .
A k : Y n (cid:55)→ V be a collection of k ( α, G ) - d σ private mechanisms for a given group assignment G on a set of n entities.
Let A : Y n (cid:55)→ V be a convex combination of these k mechanisms, where the probability of releasing the output of mechanism
A
i is p i , and (cid:80)
ki
=1 p i = 1 .
A is also ( α, G ) - d σ private w.r.t.
G .
Proof.
For any ( σ, σ (cid:48) )
∈ N G and
y ∈ Y :
[Equation]
Theorem A.1 (Post-processing) .
Let A : Y n (cid:55)→ V be ( α, G ) -
d σ private for a given group assignment
G on a set of n entities.
Let f : V (cid:55)→ V (cid:48) be an arbitrary randomized mapping.
Then f ◦ A : Y n (cid:55)→ V (cid:48) is also ( α, G ) - d σ private.
Proof.
Let g : V → V (cid:48) be a deterministic, measurable function.
For any output event Z ⊂ V (cid:48) , let W be its preimage:
[Equation]
Then, for any ( σ, σ
[Equation]
G ,
[Equation]
This concludes our proof because any randomized mapping can be decomposed into a convex combination of measurable, deterministic functions Dwork & Roth (2014), and as Lemma 2 shows, a convex combination of ( α, G ) - d σ private mechanisms is also ( α, G ) - d σ private.
Theorem A.2 (Sequential Composition) .
If A 1 and A 2 are ( α 1 , G ) - and ( α 2 , G ) - d σ private mechanisms, respectively, that use independent randomness, then releasing the outputs (cid:0)
A 1 ( y ) , A 2 ( y ) (cid:1) satisﬁes ( α 1 + α 2 , G ) -
d σ privacy.
Proof.
We have that
A 1 : Y n → V (cid:48) and A 1 : Y n → V (cid:48)(cid:48) each satisfy d σ -privacy for different α values.
Let A : Y n → ( V (cid:48)
× V (cid:48)(cid:48) ) output (cid:0)
A 1 ( y ) , A 2 ( y ) (cid:1) .
Then, we may write any event
Z ∈ ( V (cid:48)
× V (cid:48)(cid:48) ) as
Z (cid:48)
×
Z (cid:48)(cid:48) , where Z (cid:48) ∈ V (cid:48) and Z (cid:48)(cid:48) ∈ V (cid:48)(cid:48) .
We have for any ( σ, σ (cid:48) )
∈ N G ,
Pr (cid:2)
A (cid:0) σ ( y ) (cid:1)
{+∈ Z (cid:3) = Pr (cid:2)(cid:0) A 1 (cid:0) σ ( y ) (cid:1) , A 2 (cid:0) σ ( y ) (cid:1)(cid:
+}
) ∈
Z (cid:3) =
Pr (cid:2) {A 1 (cid:0) σ ( y ) (cid:1) ∈
Z (cid:48) } ∩ {A 2 (cid:0) σ ( y ) (cid:1)
∈
Z (cid:48)(cid:48) } (cid:3)
[Equation]
A. P ROOF FOR T HM . 4.
Theorem 4.1 For a given group assignment G on a set of n data owners, if a shufﬂing mechanism
A : Y n (cid:55)→ Y n is ( α, G ) - d σ private, then for each data owner DO i , i ∈ [ n ] ,
[Equation]
α for a prior distribution P , where z =
A ( y ) and y G i is the noisy sequence for data owners outside G i .
Proof.
We prove the above by bounding the following equivalent expression for any i ∈ [ n ] and a, b ∈ X .
[Equation]
The second line simply marginalizes out the full noisy sequence y .
The third line reduces this to a sum over permutations of of y G i , where r = | G i | and y ∗ G i is any ﬁxed permutation of values { y G i } .
This is possible since we are given the values outside the group, y G i , and the unordered set of values inside the group, { y G i } .
Note that the permutations σ written here are possible permutations of the
LDP input, not permutations output by the mechanism applied to the input as sometimes written in other parts of this document.
The fourth line uses the fact that the numerator and denominator are both convex combinations of
Pr A [ z | σ ( y ∗ G i ) , y G i ] over all σ ∈ S r .
The ﬁfth line uses the fact that for any y G i , ( σ ( y ∗ G
[Equation]
, y G
[Equation]
This allows a further upper bound over all neighboring sequences w.r.t.
G i , and thus over any permutation of y G i , as long as it is the same in the numerator and denominator.
Discussion
The above Bayesian analysis measures what can be learned about DO
i ’s x
i from observing the private release z relative to some other known information (the conditioned information).
Under d σ -privacy, we condition on the bag of LDP values in Alice’s group { y G i } as well as the sequence (order and value) of LDP values outside her group y G i .
This implies that releasing the shufﬂed sequence z cannot provide much more information about Alice’s x
i than would releasing the
LDP values outside her neighborhood (her group) and the unordered bag of LDP values inside her neighborhood, regardless of the adversary’s prior knowledge P .
This is a communicable guarantee: if
Alice feels comfortable with the data collection knowing that her entire neighborhood’s responses will be uniformly shufﬂed together (including those of her household), then she ought to be comfortable with d σ -privacy.
Now, we have to provide this guarantee to Bob, a neighbor of Alice, as well as Luis, a neighbor of Bob but not of Alice.
Thus, Bob, Alice and Luis have distinct and overlapping groups (neighborhoods).
Hence, the trivial solution of uniformly shufﬂing the noisy responses of every group separately does not work in this case.
d σ -privacy, however, offers the above guarantee to each user (knowing that their entire neighborhood is nearly uniformly shufﬂed) while still maintaining utility (estimate disease prevalence within neighborhoods).
Semantically, this is very powerful, since it implies that the noisy responses speciﬁc to one’s household cannot be leveraged to infer one’s disease state x i .
A
P ROOF OF T HEOREM
Theorem 4.
For A ( M ( x ))
=
z where M ( · ) is (cid:15) - LDP and A ( · ) is α - d σ private, we have [Equation] loses ]
[Equation]
wins ] for any input subgroup I ⊂ G i , r = | G i | and k < r/ 2 .
Proof.
We ﬁrst focus on deterministic adversaries and then expand to randomized adversaries afterwards using the fact that randomized adversaries are mixtures of deterministic ones.
Our adversary D Adv is then deﬁned by a deterministic decision function η : Y n → [ n ] k .
Upon observing z , η ( z ) selects k elements in z which it believes originated from I ⊂ G i .
In the following, let Pr z be the probability of events conditioned on the shufﬂed output sequence z , where randomness is over the (cid:15) - LDP mechanism M and the α - d σ -private shufﬂing mechanism A .
The adversary wins if it reidentiﬁes > k 2 of the LDP values originating from I .
Let H =
η ( z ) be the indices of elements in z selected by η .
Let W = { σ ∈ S n : | σ ( H ) ∩ I | > k 2 } be the set of permutations where the adversary wins and let L = { σ ∈ S n : σ ( H ) ∩ I | ≤ k 2 } be the set of permutations where the adversary loses.
[Equation]
where σ is the shufﬂing permutation produced by A ,
z = σ ( y ) i.e. z i
=
y
σ ( i ) .
Concretely, this is equivalent to DO i releasing DO σ ( i ) ’s
LDP response.
Since the permutation and LDP outputs are randomized, many subgroups of size k in G i could have produced the LDP values ( z H 1 , . .
. , z H k ) and then been mapped to H by a permutation.
Concretely, there is a reasonable probability that
Alice’s household output the LDP values of another k -member household in her neighborhood and they output her household’s LDP values.
In the worst case, this is e − 2 k(cid:15) less likely than without swapping values, by group DP guarantees.
Since both households are part of the same group G i , the permutation that maps her household to elements H in the output is close in probability to that which maps the other household to elements H in the output.
As such, we have in the worst case a e − (2 k(cid:15) + α ) reduction in probability of the other household having swapped LDP values with Alice’s and permuting to subset H .
The above provides intuition on how we could get the same output z many different ways, and how
Alice’s household could or could not contribute to elements H .
It does not, however, explain why an adversary who is given output z has limited advantage in choosing a subset H such that they recover most of Alice’s household’s values.
We formalize this fact as follows.
We may rewrite the probabilities of winning or losing by marginalizing out all possible LDP sequences y .
Conditioning on the output sequence z , the only possible LDP sequences y are permutations of z .
Note that the probability of any sequence y is determined by the input x and the
LDP mechanism M :
[Equation]
Note that Pr z [ σ | y ] =
Pr z [ σ ] for the mallows mechanism, which chooses its permutations independently of y .
Now consider when η ( z ) loses.
By similar arguments as above:
[Equation]
The odds of losing versus winning is given by
[Equation]
We now show that for each σ in the denominator, we may construct m = (cid:98) r − kk (cid:99) distinct permutations σ (cid:48) in the numerator that are close in probability to it.
Lemma 3.
For every σ ∈ W there exists a set of m = (cid:98) r − kk (cid:99) permutations, E ( σ ) , such that
E ( σ ) ⊆
L 2.
σ − 1 ≈ G i σ (cid:48) −3.
E ( σ a ) ∩ E ( σ b ) = ∅ for any pair σ a , σ b ∈ W 4.
Pr (cid:2)
A ( x ) =
y
=
σ − 1 ( z ) (cid:3) ≤ e 2 k(cid:15)
Pr (cid:104) A ( x ) =
y
= σ (cid:48) − 1 ( z ) (cid:105) for any x ∈ X n and any z ∈ Y n
Proof.
Given σ ∈ W , we construct E ( σ ) by ﬁrst taking the inverse σ − 1 .
Recall that, since σ ∈ W , we have that | σ − 1 ( I ) ∩ H | > k 2 .
( σ − 1 ( i ) =
j could be interpreted as data owner i ’s LDP value will be output at position j ).
We then divide the remainder of the group G i \ I into m disjoint subsets of size k each, J 1 , J 2 , . . .
, J m .
These represent the other distinct subsets of size k that Alice’s household could swap LDP values with.
We then produce m permutations, σ (cid:48)
− 1 1 , . . .
, σ (cid:48) − 1 m , by making σ (cid:48)
− 1 i ( I ) =
σ − 1 ( J i ) and σ (cid:48) − 1 i ( J i ) =
σ − 1 ( I ) (preserving order within those subsets) and σ (cid:48) − 1 =
σ − 1 everywhere else.
On the ﬁrst point, we know that every σ (cid:48)
∈ E ( σ ) is also in L .
We know this because σ (cid:48) − 1 i ( I ) =
σ − 1 ( J i ) .
Since σ ∈ W , we have that | σ − 1 ( J i ) ∩ H | < k 2 since | σ − 1 ( I ) ∩ H | ≥ k
and I ∩ J i = ∅ by deﬁnition.
Thus, | σ (cid:48) − [-
-]1 i (
[Equation]
2 , so | σ (cid:48)
i ( H ) ∩ I | < k 2 and σ (cid:48)
i ∈ L .
On the second point, we know that the inverse permutations are neighboring σ − 1 ≈ G i σ (cid:48) − 1 simply by construction – they only differ on elements in G i .
On the third point, we know that the sets E ( σ a ) and E ( σ b ) are distinct since we can map any permutation σ (cid:48)
∈ E ( σ a ) uniquely back to σ a for any σ a ∈ W .
We do so by taking its inverse σ (cid:48) − 1 , ﬁnding which subset J i has majority elements from H i.e. | σ (cid:48) − 1 ( J i ) ∩ H | > k 2 .
Swap elements back: σ (cid:48) − 1 ( J i ) with σ (cid:48) − 1 ( I ) .
Invert back to σ a .
On the fourth point, we know that σ − 1 ( z ) and σ (cid:48) − 1 ( z ) differ on at most 2 k indices.
As such, by group DP guarantees, we know that their probabilities must be close to a factor of e − 2 k(cid:15) regardless of z and x .
Using the above Lemma we may bound the odds of losing vs. winning.
[Equation]
where the last line follows from the fourth point of the above Lemma (for the 2 k(cid:15) term) and the fact that the inverse permutations σ (cid:48)− 1 , σ − 1 are neighboring (second point of the Lemma) so the probabilities of the mechanism to produce σ vs. σ (cid:48) to reach z from these neighboring permutations must be close by a factor of e α .
Since the above holds for any z and x , the bound holds on average across all outcomes z , thus
[Equation]
for any deterministic adversary with decision function η .
Finally, we may write any probabilistic adversary as mixture of decision functions.
By convexity (same argument used in Lemma 2), the above bound still holds.
As such,
[Equation]
A. U TILITY OF S HUFFLING M ECHANISM
We now introduce a novel metric, ( η, δ ) -preservation, for assessing the utility of any shufﬂing mechanism.
Let S ⊆
[ n ] correspond to a set of indices in y .
The metric is deﬁned as follows.
Deﬁnition A.4.
( ( η, δ ) -preservation)
A shufﬂing mechanism A : Y n (cid:55)→ Y n is deﬁned to be ( η, δ ) - preserving ( η, δ ∈ [0 , 1]) w.r.t to a given subset S ⊆ [ n ] , if
[Equation]
where z =
A ( y ) =
σ ( y ) and S σ = { σ ( i ) | i ∈ S } .
For example, consider S = { 1 , 4 , 5 , 7 , 8 } .
If A ( · ) permutes the output according to σ = (5 3 2 6 7 9 8 1 4 10) , then S σ = { 5 , 6 , 7 , 8 , 1 } which preserves 4 or 80% of its original indices.
This means that for any data sequence y , at least η fraction of its data values corresponding to the subset S overlaps with that of shufﬂed sequence z with high probability (1 − δ ) .
Assuming, { y S } =
{ y i | i ∈ S } and { z S } =
{ z i | i ∈ S } = { y σ ( i ) | i ∈ S } denotes the set of data values corresponding to S in data sequences y and z respectively, we have Pr (cid:2) |{ y S } ∩ { z S }| ≥ η · | S | (cid:3) ≥ 1 − δ, ∀ y .
For example, let S be the set of individuals from Nevada.
Then, for a shufﬂing mechanism that provides ( η = 0 .
8 ,δ = 0 . 1) -preservation to S , with probability ≥ 0 .
9 , ≥ 80% of the values that are reported to be from Nevada in z are genuinely from Nevada.
The rationale behind this metric is that it captures the utility of the learning allowed by d σ -privacy – if S is equal to some group G ∈ G , ( η,δ ) preservation allows overall statistics of G to be captured.
Note that this utility metric is agnostic of both the data distribution and the analyst’s query .
Hence, it is a conservative analysis of utility which serves as a lower bound for learning from { z S } .
We suspect that with the knowledge of the data distribution and/or the query, a tighter utility analysis is possible.
A formal utility analysis of Alg.
10 is presented in App.
A.13.
Empirical evaluation of ( η, δ ) - preservation is presented in App.
A.14.
A. D ISCUSSION ON P ROPERTIES OF M ALLOWS M ECHANISM
Property 3.
For group assignment G , a mechanism A ( · ) that shufﬂes according to a permutation sampled from the Mallows model P θ, d ( · ) , satisﬁes ( α, G ) - d σ privacy where
[Equation]
and
[Equation]
We refer to ∆( σ 0 : d , G ) as the sensitivity of the rank-distance measure d ( · )
Proof.
Consider two permutations of the initial sequence y , σ 1 ( y ) , σ 2 ( y ) that are neighboring w.r.t.
some group G i ∈ G , σ
≈ G i σ 2 .
Additionally consider any ﬁxed released shufﬂed sequence z .
Let Σ 1 , Σ 2 be the set of permutations that turn σ 1 ( y ) , σ 2 ( y ) into z , respectively:
[Equation]
In the case that { y } consists entirely of unique values, Σ 1 , Σ 2 will each contain exactly one permutation, since only one permutation can map σ i ( y ) to z .
Lemma 4.
For each permutation σ (cid:48) 1 ∈ Σ 1 there exists a permutation in σ (cid:48) 2 ∈ Σ 2 such that
[Equation]
Proof follows from the fact that — since only the elements j ∈ G i differ in σ 1 ( y ) and σ 2 ( y ) — only those elements need to differ to achieve the same output permutation.
In other words, we may deﬁne σ (cid:48) 1 , σ (cid:48) 2 at all inputs i / ∈ G i identically, and then deﬁne all inputs i ∈ G i differently as needed.
As such, they are neighboring w.r.t.
G i .
Recalling that Alg.
1 applies σ − 1to the sampled permutation, we must sample σ 0 σ (cid:48) 1 (for some σ (cid:48) 1 ∈ Σ 1 ) for the mechanism to produce z from σ 1 ( y ) .
Formally, since σ (cid:48) 1 σ 1 ( y ) =
z we must sample σ 0 σ (cid:48) 1 to get z since we are going to apply σ − 1 0 to the sampled permutation.
[Equation]
Taking the odds, we have P
[Equation]
[Figure]
[Equation]
[Figure]
[Equation]
[Figure]
[Equation]
[Figure]
[Equation]
[Figure]
[Equation]
where
[Equation]
Therefore, setting α = θ · ∆ , we achieve ( α, G ) - d σ privacy.
Property 4.
The sensitivity of a rank-distance is an increasing function of the width ω
σ 0 G .
For instance, for Kendall’s τ distance d τ ( · ) , we have ∆( σ 0 : d τ , G ) = ω σ [Equation]
G ( ω σ [Equation] G + [Equation]
To show the sensitivity of Kendall’s τ , we make use of its triangle inequality.
Proof.
Recall from the proof of the previous property that the expression d ( σ, σ 0 ) =
d (cid:0) σ 0 σ, σ 0 (cid:1) , where d is the actual rank distance measure e.g. Kendall’s τ .
As such, we require that
[Equation]
for any pair of permutations ( σ a , σ b ) ∈ N G .
For any group G i ∈ G , let W i ⊆ n represent the smallest contiguous subsequence of indices in σthat contains all of G i .
For instance, if σ 0 = [2 , 4 , 6 , 8 , 1 , 3 , 5 , 7] and G i = { 2 , 6 , 8 } , then W i = { 2 , 4 , 6 , 8 } .
Then the group width width is ω i =
| W i |− 1 = 3 .
Now consider two permutations neighboring w.r.t.
G i , σ a ≈ G i σ b , so only the elements of G i are shufﬂed between them.
We want to bound (cid:12)(cid:12)
[Equation]
For this, we use a pair of triangle inequalities:
[Equation]
so,
[Equation]
Since σ 0 σ a and σ 0 σ b only differ in the contiguous subset W i , the largest number of discordant pairs between them is given by the maximum Kendall’s τ distance between two permutations of size
ω i + 1 :
[Equation]
Since ω σ 0 G ≥ ω i for all G i ∈ G , we have that
[Equation]
A. H ARDNESS OF C OMPUTING T HE O PTIMUM R EFERENCE P ERMUTATION
Theorem A.3.
The problem of ﬁnding the optimum reference permutation, i.e., σ ∗= arg min σ ∈ S n ω σ G is NP-hard.
Proof.
We start with the formal representation of the problem as follows.
Optimum Reference Permutation Problem.
Given n subsets G = { G i ∈ 2 [ n ] , i ∈ [ n ] } , ﬁnd the permutation σ ∗ 0 = arg min σ ∈ S n ω σ G .
Now, consider the following job-shop scheduling problem.
Job Shop Scheduling.
There is one job J with n operations o i , i ∈ [ n ] and n machines such that
o i needs to run on machine M i .
Additionally, each machine has a sequence dependent processing time p i .
Let S be the sequence till There are n subsets S i ⊆
[ n ] , each corresponding to a set of operations that need to occur in contiguous machines, else the processing times incur penalty as follows.
Let p i denote the processing time for the machine running the i -th operation scheduled.
Let S i be the preﬁx sequence with i schedulings.
For instance, if the ﬁnal scheduling is 1 3 4 5 9 8 10 6 7 2 then
S 4 = 1345 .
Additionally, let P j S
i be the shortest subsequence such of S
i such that it contains all the elements in S j ∩ { S i } .
For example for S 1 = { 3 , 5 , 7 } , P 1 S 4 = 345 .
[Equation]
The objective is to ﬁnd a scheduling for J such that it minimizes the makespan, i.e., the completion time of the job.
Note that p n = max i p i , hence the problem reduces to minimizing p n .
Lemma 5.
The aforementioned job shop scheduling problem with sequence-dependent processing time is NP-hard.
Proof.
Consider the following instantiation of the sequence-dependent job shop scheduling problem where the processing time is given by p i
= p
i − 1 + w kl , p 1 = 0 where S i
[ i − 1]
= k , S i [ i ] =
l and w ij , j ∈ S
i represents some associated weight.
This problem is equivalent to the travelling salesman problem (TSP) Balas (2008) and is therefore, NP-hard.
Thus, our aforementioned job shop scheduling problem is also clearly NP-hard.
Reduction: Let the n subsets S
i correspond to the groups in G .
Clearly, minimizing ω σ G minimizes p n .
Hence, the optimal reference permutation gives the solution to the scheduling problem as well.
A. I LLUSTRATION OF A LG .
We now provide a small-scale step-by-step example of how Alg. 1 operates.
Fig.
6a is an example of a grouping G on a dataset of n = 8 elements.
The group of DO i includes i and its neighbors.
For instance, G 8 = { 8 , 3 , 5 } .
To build a reference permutation, Alg.
1 starts at the index with the largest group, i = 5 (highlighted in purple), with G 5 = { 5 , 2 , 3 , 8 , 4 } .
As shown in Figure 6b, the σ 0 is then constructed by following a BFS traversal from i = 5 .
Each j ∈ G 5 is visited, queuing up the neighbors of each j ∈ G 5 that haven’t been visited along the way, and so on.
The algorithm completes after the entire graph has been visited.
The goal is to produce a reference permutation in which the width of each group in the reference permutation ω
i is small.
In this case, the width of the largest group G 5 is as small as it can be ω 5 = 5 − 1 = 4 .
However, the width of G 4 = { 4 , 5 , 7 } is the maximum possible since σ − 1 (5) =and σ − 1 (7) = 8 , so ω 4 = 7 .
This is difﬁcult to avoid when the maximum group size is large as compared to the full dataset size n .
Realistically, we expect n to be signiﬁcantly larger, leading to relatively smaller groups.
With the reference permutation in place, we compute the sensitivity:
[Equation]
[Figure]
Which lets us set θ
= α 28 for any given α privacy value.
To reiterate, lower θ results in more randomness in the mechanism.
We then sample the permutation ˆ σ = P θ, d ( σ 0 ) .
Suppose
[Equation]
Then, the released z is given as
[Equation]
One can think of the above operation as follows.
What was 5 in the reference permutation ( σ 0 (1) = 5 ) is 3 in the sampled permutation (ˆ σ (1) = 3) .
So, index 5 corresponding to DO 5 now holds DO 3 ’s noisy data y 3 .
As such, we shufﬂe mostly between members of the same group, and minimally between groups.
The runtime of this mechanism is dominated by the Repeated Insertion Model sampler Doignon et al.
(2004), which takes O ( n 2 ) time.
It is very possible that there are more efﬁcient samplers available, but RIM is a standard and simple to implement for this ﬁrst proposed mechanism.
Additionally, the majority of this is spent computing sampling parameters which can be stored in advanced with
O ( n 2 ) memory.
Furthermore, sampling from a Mallows model with some reference permutation σis equivalent to sampling from a Mallows model with the identity permutation and applying it to σ 0 .
As such, permutations may be sampled in advanced, and the runtime is dominated by computation of σ 0 which takes O ( | V | + | E | ) time (the number of vertices and edges in the graph).
A future direction could be exploring even better heuristics for computing σ 0 .
One possibility could be based on ranked enumeration of the permutations Deep & Koutris (2021); Deep et al.
A. P ROOF OF T HM . 4.
Theorem 4.3 Alg. 1 is ( α, G ) - d σ private.
Proof.
The proof follows from Prop. 3.
Having computed the sensitivity of the reference permutation σ 0 , ∆ , and set θ = α/ ∆ , we are guaranteed by Property 3 that shufﬂing according to the permutation ˆ σ guarantees ( α, G ) - d σ privacy.
A. P ROOF OF T HM . 4.
Theorem 4.4 Alg.
1 satisﬁes ( α (cid:48) , G (cid:48) ) -
d σ privacy for any group assignment G (cid:48) where α (cid:48) =
α ∆( σ 0 : d , G (cid:48) )
∆( σ 0 : d , G )
Proof.
Recall from Property 3 that we satisfy ( α, G ) d σ -privacy by setting θ = α/ ∆( σ 0 : d , G ) .
Given alternative grouping G (cid:48) with sensitivity ∆( σ 0 : d , G (cid:48) ) , this same mechanism provides
[Equation]
A. F ORMAL U TILITY A NALYSIS OF A LG .
Theorem A.4.
For a given set S ⊂ [ n ] and Hamming distance metric, d H ( · ) , Alg. 1 is ( η, δ ) - preserving for δ = 1 ψ ( θ, d H ) (cid:80) nh =2 k +1 ( e − θ · h · c h ) where k = (cid:100) (1 − η ) · | S |(cid:101) and c h is the number of permutations with hamming distance h from the reference permutation that do not preserve η % of S and is given by
[Equation]
h/ 2 (cid:99) )
(cid:88)
j = k + (cid:18)
l s j (cid:19) · (cid:18) n − l s j (cid:19) · (cid:34) min( l s − j,h − 2 j ) (cid:88) i
= (cid:18) l s − j i (cid:19) · (cid:18) i + j j (cid:19) ·
f ( i, j ) · (cid:18)
n − l s − j h − 2 j − i (cid:19) · f ( h − 2 j − i, j )!
(cid:35) f
( i, 0)
[Equation]
i, f (0 , q ) = q !
f ( i, [Equation] =
min( i,j ) (cid:88)
q = (cid:34)(cid:18)
i q (cid:19) · (cid:18)
j j − q (cid:19) · j ! · f ( i
[Equation]
q, q ) (cid:35)
[Equation]
Proof.
Let l s = | S | denote the size of the set S and k = (cid:100) (1 − η ) · l S (cid:101) denote the maximum number of correct values that can be missing from S .
Now, for a given permutation σ ∈ S n , let h denote
its
Hamming distance from the reference permutation σ 0 , i.e, h = d H ( σ, σ 0 ) .
This means that σ and σdiffer in h indices.
Now, h can be analysed in the the following two cases,
Case I. h ≤ 2 k +
For (1 − η ) fraction of indices to be removed from S , we need at least k + 1 indices from S to be replaced by k + 1 values from outside S .
This is clearly not possible for h ≤ 2 k + 1 .
Hence, here
Case II. h > 2 k
For the following analysis we consider we treat the permutations as strings (multi-digit numbers are treated as a single string character).
Now, Let S σ 0 denote the non-contiguous substring of σ 0 such that it consists of all the elements of S , i.e.,
[Equation]
Let S σ denote the substring corresponding to the positions occupied by S σ 0 in σ .
Formally,
[Equation]
S
[Equation]
For example, for σ 0 = (1 2 3 5 4 7 8 10 9 6) , σ = (1 3 2 7 8 5 4 6 10 9) and S = { 2 , 4 , 5 , 8 } , we have S σ 0 = 2548 and S σ = 3784 where h =
d H ( σ, σ 0 )
= 9 .
Let { S σ } denote the set of the elements of string S σ .
Let A be the set of characters in S σ such that they do not belong to S , i.e,
A = { S σ
[ i ] | S σ [ i ] (cid:54)∈ S, i ∈
[ l S ] } .
Let B be the set of characters in S σ that belong to S but differ from S σ 0 in position, i.e., B = { S σ
[ i ] | S σ [ i ] ∈ S, S σ
[ i ] (cid:54) =
S σ
[ i ] , i ∈ [ l S ] } .
Additionally, let C = S −{ S σ } .
For instance, in the above example, A = { 3 , 7 } , B = { 4 , 8 } , C = { 2 , 5 } .
Now consider an initial arrangement of p + m distinct objects that are subdivided into two types – p objects of Type A and m objects of Type B. Let f ( p, m ) denote the number of permutations of these p + m objects such that the m Type B objects can occupy any position but no object of Type A can occupy its original position.
For example, for f ( p, 0) this becomes the number of derangements der denoted as !
p = (cid:98) p !
e + 12 (cid:99) .
Therefore, f ( | B | , | A | ) denotes the number of permutations of S σ such that d H ( S σ 0 , S σ ) =
| A | + | B | .
This is because if elements of B are allowed to occupy their original position then this will reduce the Hamming distance.
Now, let ¯ S σ ( ¯ S σ 0 ) denote the substring left out after extracting from S σ ( S σ 0 ) from σ ( σ 0 ).
For example, ¯ S σ = 1256109 and ¯ S σ 0 = 1371096 in the above example.
Let D be the set of elements outside of S and A that occupy different positions in ¯ S σ and ¯ S σ 0 (thereby contributing to the hamming distance), i.e., D = { ¯ S σ
[ i ] | ¯ S σ
[ i ] (cid:54)∈ S, ¯ S σ
[ i ] (cid:54) =
¯ S σ
[ i ] , i ∈ [ n − l S ] } .
For instance, in the above example D = { 9 , 6 , 10 } .
Hence, h =
d H ( σ, σ 0 ) =
| A | + | B | + | C | + | D | and clearly f ( | D | , | C | ) represents the number of permutations of ¯ S σ such that d H (¯ S σ , ¯ S σ 0 ) =
| C | + | D | .
Finally, we have
[Equation]
of selecting set A · (cid:34) min( l s − j,h − 2 j ) (cid:88)
i = (cid:18) l s − j i (cid:19) (cid:124) (cid:123)(cid:122) (cid:125)
# ways of selecting set B · f ( i, j ) · (cid:18) n − l s − j h − 2 j − i (cid:19) (cid:124)
(cid:123)(cid:122) (cid:125) # ways of selecting set D · f ( h − 2 j − i, j ) (cid:35)
Now, for f ( i, j ) let E be the set of original positions of Type A that are occupied by Type B objects in the resulting permutation.
Additionally, let F be the set of the original positions of Type B objects that are still occupied by some Type B object.
Clearly, Type B objects can occupy these | E | + | F | = m in any way they like.
However, the type A objects can only result in f ( p − q, q ) permutations.
Therefore, f ( p, m ) is given by the following recursive function
[Equation]
(cid:32) (cid:18) p
[Equation]
(cid:19) (cid:124)(cid:123)(cid:122)(cid:125) # ways of selecting set E · (cid:18) m
[Equation]
(cid:19) (cid:124) (cid:123)(cid:122) (cid:125) # ways of selecting set F
[Equation]
(cid:33)
Thus, the total probability of failure is given by
[Equation]
A. A DDITIONAL E XPERIMENTAL D ETAILS A.14. E VALUATION OF ( η, δ ) - PRESERVATION
[Figure]
In this section, we evaluate the characteristics of the ( η, δ ) -preservation for Kendall’s τ distance d τ ( · , · ) .
Each sweep of Fig.
7 ﬁxes δ = 0 . 01 , and observes η .
We consider a dataset of size n = 10 K and a subset S of size l S corresponding to the indices in the middle of the reference permutation σ(the actual value of the reference permutation is not signiﬁcant for measuring preservation).
For the rest of the discussion, we denote the width of a permutation by ω for notational brevity.
For each value of the independent axis, we generate 50 trials of the permutation σ from a Mallows model with the appropriate θ (given the ω and α parameters).
We then report the largest η (fraction of subset preserved) that at least 99% of trials satisfy.
In Fig.
7a, we see that preservation is highest for higher α and increases gradually with declining width ω and increasing subset size l s .
Fig.
7b demonstrates that preservation declines with increasing width.
∆ increases quadratically with width ω for d τ , resulting in declining θ and increasing randomness.
We also see that larger subset sizes result in a more gradual decline in η .
This is due to the fact that the worst-case preservation (uniform random shufﬂing) is better for larger subsets.
i.e. we cannot do worse than 80% preservation for a subset that is 80% of indices.
Finally, Fig.
7c demonstrates how preservation grows rapidly with increasing subset size.
For large widths, we are nearly uniformly randomly permuting, so preservation will equal the size of the subset relative to the dataset size.
For smaller widths, we see that preservation offers diminishing returns as we grow subset size past some critical l s .
For ω = 30 , we see that subset sizes much larger than a quarter of the dataset gain little in preservation.
A.14. A DULT D ATASET A. A DDITIONAL R ELATED W ORK
In this section, we discuss the relevant existing work.
The anonymization of noisy responses to improve differential privacy was ﬁrst proposed by Bittau et al.
Bittau et al.
(2017a) who proposed a principled system architecture for shufﬂing.
This model was formally studied later in Erlingsson et al.
; Cheu et al.
Erlingsson et al. Erlingsson et al.
(2019) showed that for arbitrary (cid:15) - LDP randomizers, random shufﬂing results in privacy ampliﬁcation.
Cheu et al. Cheu et al.
(2019) formally deﬁned the shufﬂe DP model and analyzed the privacy guarantees of the binary randomized response in this model.
The shufﬂe DP model
[Figure]
differs from our approach in two ways.
First, it focuses completely on the DP guarantee.
The privacy ampliﬁcation is manifested in the from of a lower (cid:15) (roughly a factor of √ n ) when viewed in an alternative DP model known as the central DP model.
Erlingsson et al.
; Cheu et al.
Balle et al.
; Feldman et al.
; Bittau et al.
(2017a); Balcer & Cheu (2020).
However, our result caters to local inferential privacy.
Second, the shufﬂe model involves an uniform random shufﬂing of the entire dataset.
In contrast, our approach the granularity at which the data is shufﬂed is tunable which delineates a threshold for the learnability of the data.
A steady line of work has sudied the inferential privacy setting Kasiviswanathan & Smith (2014);
Kifer & Machanavajjhala (2011); Ghosh & Kleinberg (2016); Dalenius (1977); Dwork & Naor (2010); Tschantz et al.
Kifer et al.
Kifer & Machanavajjhala (2011) formally studied privacy degradation in the face of data correlations and later proposed a privacy framework, Pufferﬁsh Kifer & Machanavajjhala (2014); Song et al.
; He et al.
(2014), for analyzing inferential privacy.
Subsequently, several other privacy deﬁnitions have also been proposed for the inferential privacy setting Liu et al.
; Yang et al.
; Chen et al.
(2014); Zhu et al.
; Bassily et al.
For instance, Gehrke et al.
proposed a zero-knowledge privacy Gehrke et al.
(2011; 2012) which is based on simulation semantics.
Bhaskar et al.
proposed noiseless privacy Bhaskar et al.
Grining & Klonowski (2017) by restricting the set of prior distributions that the adversary may have access to.
A recent work by Zhang et al.
proposes attribute privacy Zhang et al.
(2020) which focuses on the sensitive properties of a whole dataset.
In another recent work, Ligett et al.
study a relaxation of DP that accounts for mechanisms that leak some additional, bounded information about the database Ligett et al.
Some early work in local inferential privacy include proﬁle-based privacy Geumlek & Chaudhuri (2019) by Gehmke et al.
where the problem setting comes with a graph of data generating distributions, whose edges encode sensitive pairs of distributions that should be made indistinguishable.
In another work by Kawamoto et al., the authors propose distribution privacy Kawamoto & Murakami (2018) – local differential privacy for probability distributions.
The major difference between our work and prior research is that we provide local inferential privacy through a new angle – data shufﬂing.
Finally, older works such as k -anonymity Sweeney (2002), l -diversity Machanavajjhala et al.
(2007), and Anatomy Xiao & Tao (2006) and other Wong et al.
; Tassa et al.
; Xue et al.
Choromanski et al.
; Doka et al. (2015) have studied the privacy risk of non-sensitive auxiliary information, or ‘quasi identiﬁers’ (QIs).
In practice, these works focus on the setting of dataset release, where we focus on dataset collection.
As such, QIs can be manipulated and controlled, whereas we place no restriction on the amount or type of auxiliary information accessible to the adversary, nor do we control it.
Additionally, our work offers each individual formal inferential guarantees against informed adversaries, whereas those works do not.
We emphasize this last point since formalized guarantees are critical for providing meaningful privacy deﬁnitions.
As established by Kifer and Lin in An Axiomatic View of Statistical Privacy and Utility (2012), privacy deﬁnitions ought to at least satisfy post-processing and convexity properties which our formal deﬁnition does.
[Figure]
[Figure]
[Figure]
Algorithm 10 is designed to ﬁnd a reference permutation σ 0 with low width ω
σ G w.r.t.
the given grouping G .
A low width is desirable, since it leads to low sensitivity ∆( σ 0 : d , G ) , which in turn leads to higher dispersion parameter θ = α/ ∆ , and thus less randomness over permutations (higher utility).
Theorem A.3 proves that computing the optimal reference permutation (minimum width) is
NP-hard.
As such, we propose a BFS-based heuristic.
Comparison with optimal reference permutation
To demonstrate the value of the heuristic used in Alg. 10, we provide two evaluations of its performance.
For our ﬁrst evaluation, we compare the performance of our heuristic BFS reference permutation selection ( σ 0 ) with that of the optimal reference permutation and that of a random reference permutation.
As identiﬁed by Theorem A.3, ﬁnding the optimal reference permutation for a given grouping G is NP-hard.
For these experiments, we ﬁrst create an optimal reference permutation, where each group G i ∈ G is equally sized w and maximally compact.
The optimal width, ω σ G , is then min( n, w ) .
We then generate a graph from this optimal reference permutation.
Finally, we run the
BFS reference permutation computation described in Alg.
10 attempting to approximate the optimal σ ∗ 0 , and compute its width.
To compare with a naive approach, we also plot the performance of a randomly chosen reference permutation.
We expect the maximum width across groups ω σ G to be large for this technique.
If one of the n groups has a single entry low (near 0) in σ 0 and a single entry high (near n ) in σ 0 , the width will be near n .
The random baseline is averaged over 10 trials with a 1 standard deviation envelope plotted (but difﬁcult to see, since the variance is low).
Figure 9 depicts our ﬁndings.
Each plot has a different group size w , listed at the top, used in the optimal reference permutation.
We ﬁnd that the random baseline (orange) consistently chooses a reference permutation such that ω σ G is near n , as expected.
Our method (blue), on the other hand, closely tracks the optimal solution (green).
We ﬁnd that in the worst case, our algorithm’s solution has a width ≤ 2 .
5 × larger than the optimal.
Note that for r = 0 (upper left), all methods trivially have a width of one, since the corresponding graph has no edges.
While there may be room for improvement, we ﬁnd this to be sufﬁcient for the present work.
Performance on randomly generated graphs
For our second evaluation, we observe how well our BFS heuristic (in Algorithm 10) performs on randomly generated graphs.
Here, we sample n points uniformly on the unit interval.
We then say that the i th point’s group, G i , consists of all other points within r of it.
As r increases, so does the groups size.
Since computing the optimal reference permutation is NP-hard (Theorem A.3), we do not show the optimal width.
Instead, we show a loose lower bound of the optimal width (green) by plotting the average group size for a given r (recall that the width is greater than or equal to the largest group size, so we expect this to be a loose lower bound, solely for reference).
For comparison, we evaluate the performance of a random σ 0 choice as well.
For both of these methods, we runtrials of generating a random graph (and picking a random σ 0 ) at each value of n and plot the mean along with a 1 standard deviation envelope, which is difﬁcult to see due to low variance.
Figure 10 depicts our ﬁndings.
We ﬁnd that — across values of n and r — our heuristic (blue) signiﬁcantly outperforms the random baseline (orange).
Additionally, we observe the trends we expect.
For a low r values, our heuristic BFS algorithm chooses a σ 0 with width close to the lower bound (green) of the optimal width ω σ G .
As r increases, the graph become signiﬁcantly more connected.
Both the lower bound and our heuristic move closer to the width of the random baseline.
Note that for r = 0 (upper left), all methods trivially have a width of one, since the corresponding graph has no edges.
Ultimately, these ﬁndings indicate that our heuristic for computing σ 0 signiﬁcantly outperforms a naive random choice, and follows the same trend as the lower bound of the optimal.