DIFFERENTIALLY PRIVATE LINEAR QUERIES ON HISTOGRAMS

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First Claim
1. A method comprising:
 receiving a dataset by a computing device;
receiving a query by the computing device;
performing base decomposition using the dataset and the query to generate an orthonormal basis, by the computing device;
generating an answer to the query;
adding noise to the answer, by the computing device, using at least one of correlated noise and least squares estimation; and
providing the answer with noise by the computing device.
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Accused Products
Abstract
The privacy of linear queries on histograms is protected. A database containing private data is queried. Base decomposition is performed to recursively compute an orthonormal basis for the database space. Using correlated (or Gaussian) noise and/or least squares estimation, an answer having differential privacy is generated and provided in response to the query. In some implementations, the differential privacy is εdifferential privacy (pure differential privacy) or is (ε,δ)differential privacy (i.e., approximate differential privacy). In some implementations, the data in the database may be dense. Such implementations may use correlated noise without using least squares estimation. In other implementations, the data in the database may be sparse. Such implementations may use least squares estimation with or without using correlated noise.
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20 Claims
 1. A method comprising:
receiving a dataset by a computing device; receiving a query by the computing device; performing base decomposition using the dataset and the query to generate an orthonormal basis, by the computing device; generating an answer to the query; adding noise to the answer, by the computing device, using at least one of correlated noise and least squares estimation; and providing the answer with noise by the computing device.  View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9, 10, 11)
 12. A method comprising:
receiving a linear query at a computing device; determining an answer to the linear query, by the computing device; adding privacy to the answer, by the computing device, using at least one of correlated noise and least squares estimation; and providing the answer with privacy by the computing device.  View Dependent Claims (13, 14, 15, 16)
 17. A system comprising:
a dataset provider that provides a dataset; and a privacy protector that; receives the dataset from the dataset provider, and receives a query; determines whether the dataset is dense or sparse; generates an answer to the query using the dataset and whether the dataset is dense or sparse, wherein the answer has differential privacy; and provides the answer with differential privacy to a client device.  View Dependent Claims (18, 19, 20)
1 Specification
In recent years, there has been an abundance of rich and finegrained data about individuals in domains such as healthcare, finance, retail, web search, and social networks. It is desirable for data collectors to enable third parties to perform complex data mining applications over such data. However, privacy is an obstacle that arises when sharing data about individuals with third parties, since the data about each individual may contain private and sensitive information.
One solution to the privacy problem is to add noise to the data. The addition of the noise may prevent a malicious third party from determining the identity of a user whose personal information is part of the data or from establishing with certainty any previously unknown attributes of a given user. However, while such methods are effective in providing privacy protection, they may overly distort the data, reducing the value of the data to third parties for data mining applications.
A system is said to provide differential privacy if the presence or absence of a particular record or value cannot be determined based on an output of the system, or can only be determined with a very low probability. For example, in the case of medical data, a system may be provided that outputs answers to queries supplied such as the number of users with diabetes. While the output of such a system may be anonymous in that it does not reveal the identity of the patients associated with the data, a curious user may attempt to make inferences about the presence or absence of patients by varying the queries made to the system and observing the changes in output. For example, a user may have preexisting knowledge about a rare condition associated with a patient and may infer other information about the patient by restricting queries to users having the condition. Such a system may not provide differential privacy because the presence or absence of a patient in the medical data (i.e., a record) may be inferred from the answers returned to the queries (i.e., output).
Typically, systems provide differential privacy (for protecting the privacy of user data stored in a database) by introducing some amount of error or noise to the data or to the results of operations or queries performed on the data to hide specific information of any individual user. For example, noise may be added to each query using a distribution such as a Laplacian distribution. At the same time, one would like the noise to be as small as possible so that the answers are still meaningful. Existing methods may add more error or noise than is necessary or optimal to provide differential privacy protection (i.e., ensuring the privacy goal be met).
Techniques are provided for protecting the privacy of datasets responsive to linear queries on histograms. A database containing private data is queried. Base decomposition is performed to recursively compute an orthonormal basis for the database space. Using correlated (or Gaussian) noise and/or least squares estimation, an answer having differential privacy is generated and provided in response to the query.
In some implementations, the differential privacy is εdifferential privacy (pure differential privacy). In some implementations, the differential privacy is (ε, δ)differential privacy (i.e., approximate differential privacy).
In some implementations, the data in the database may be dense. Such implementations may use correlated noise without using least squares estimation. In other implementations, the data in the database may be sparse. Such implementations may use least squares estimation with or without using correlated noise, depending on the implementation.
This summary is provided to introduce a selection of concepts in a simplified form that is further described below in the detailed description. This summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used to limit the scope of the claimed subject matter.
The foregoing summary, as well as the following detailed description of illustrative embodiments, is better understood when read in conjunction with the appended drawings. For the purpose of illustrating the embodiments, there is shown in the drawings example constructions of the embodiments; however, the embodiments are not limited to the specific methods and instrumentalities disclosed. In the drawings:
Differential privacy is a privacy definition that has become the standard notion of privacy in statistical databases. Informally, a mechanism (a randomized function on databases) satisfies differential privacy if the distribution of the outcome of the mechanism does not change noticeably when one individual'"'"'s input to the database is changed. Privacy is measured by how small this change is: an εdifferentially private mechanism M satisfies Pr[M(x)∈S]≦exp(ε)Pr[M(x′)∈S] for any pair x, x′ of neighboring databases, and for any measurable subset S of the range. A relaxation of this definition is approximate differential privacy. A mechanism M is (ε, δ)differentially private if Pr[M(x)∈S]≦exp(ε)Pr[M(x′)∈S]+δ with x, x′, S as before. Here, δ is thought of as negligible in the size of the database. Both these definitions satisfy properties such as composability, and are resistant to postprocessing of the output of the mechanism.
In recent years, research has shown that this strong privacy definition still allows for very accurate analyses of statistical databases. At the same time, answering a large number of adversarially chosen queries accurately is inherently impossible with any semblance of privacy. Thus, there is an inherent tradeoff between privacy and accuracy when answering a large number of queries. This tradeoff is contemplated herein in the context of counting queries and more generally linear queries over histograms.
In some implementations, the client device 110 may include a desktop personal computer, workstation, laptop, PDA, smart phone, cell phone, or any WAPenabled device or any other computing device capable of interfacing directly or indirectly with the network 120, such as the computing device 600 described with respect to
The dataset provider 130 may generate a dataset 135. The dataset 135 may be in a database format, for example, and comprise a collection of data and may include data related to a variety of topics including but not limited to healthcare, finance, retail, and social networking. The dataset 135 may have a plurality of rows and each row may have a number of values or columns. The number of values associated with each row in the dataset 135 is referred to as the dimension of the dataset 135. Thus, for example, a row with twenty columns has a dimension of twenty.
In some implementations, depending on the type of dataset 135, each row of the dataset 135 may correspond to a user, and each value may correspond to an attribute of the user. For example, where the dataset 135 is healthcare data, there may be a row for each user associated with the dataset 135 and the values of the row may include height, weight, sex, and blood type.
As may be appreciated, publishing or providing the dataset 135 by the dataset provider 130 may raise privacy issues, as would publishing or providing a query answer based on the dataset. Even where personal information such as name or social security number have been removed from the dataset 135, malicious users may still be able to identify users based on the dataset 135 or answers obtained from the dataset 135, or through combination with other information such as information found on the internet or from other datasets.
Accordingly, the privacy protector 160 may receive the dataset 135 and a query 115 and may generate an answer 165 with privacy using the dataset 135 and the query 115. The answer 165 may then be published or provided to the client device 110 (e.g., that provided the query). The answer 165 generated by the privacy protector 160 may provide one or more privacy guarantees. The desired privacy guarantee(s) may be received from a user or administrator, for example.
As described further with respect to
In an implementation, the dataset 135 may comprise a database that contains n people in a universe of size N (i.e., the number of types of people is denoted N). A histogram of the database is denoted x. The histogram x is a vector in R^{N}, with x_{i }denoting the number of people of type i in the database, and R^{N }denoting the set of all possible databases. The mapping from people to types may be application specific, depending on the implementation.
More particularly, a database is given by a multiset of database rows, one for each individual (i.e., private data may be modeled as a database D of n rows, where each row of database D contains information about an individual). Formally, a database D is a multiset of size n of elements of the universe N={t_{1}, . . . , t_{N}} of possible user types (i.e., N denotes the size of the universe that the rows come from, and n denotes the number of individuals in the database). The database can be represented as its histogram x∈R^{N }with x_{i }denoting the number of occurrences of the ith element of the universe. The algorithms herein take as input a histogram x∈R^{N }of the database D, where the ith component x_{i }of x encodes the number of individuals in D of type t_{i}. Thus, x would be a vector of nonnegative integers with ∥x∥_{1}=n. Therefore, in this histogram representation, ∥x∥_{1}=n when D is a database of size n. Also, two neighboring databases D and D′ that differ in the presence or absence of a single individual correspond to two histograms x and x′ satisfying ∥x−x′∥_{1}=1. As described further herein, accurate answers may be obtained for a given set of d linear queries over this histogram x. This set of queries can be represented by a matrix A∈R^{dxN }with the vector Ax∈R^{d }giving the correct answers to the queries. When A∈{0,1}^{dxN}, such queries are referred to as counting queries.
In other words, a histogram vector is a Ndimensional vector which counts the number of users of each type. The queries are specified by a d*N matrix A, and the query result is the vector Ax. The matrix A is a d*N matrix corresponding to d linear questions about the vector x. The correct answer to this set of d queries is given by the vector Ax. The definition of differential privacy and its error metric is well known to those of skill in the art.
In an implementation, nearly minimal error (in terms of the mean squared error) is added to the query results while guaranteeing (ε, δ)differential privacy regardless of the number of people n in the database. This noise distribution is a correlated Gaussian and depends on the query matrix A, and can often add a lot less noise than the worst case bound of approximately √{square root over (n)} noise per query. This implementation has error close to the best possible. As described further herein, the matrix A is decomposed into smaller components via the minimum volume enclosing ellipsoid of the symmetric convex hull of the column vectors of A.
Approximate differential privacy (i.e., (ε, δ)differential privacy) may be defined as follows. A (randomized algorithm) M with input domain R^{N }and output range Y is (ε, δ)differentially private if for every n, every x,x′ with ∥x−x′∥_{1}=1, and every S⊂Y, M satisfies Pr[M(x)∈S]≦exp(ε)Pr[M(x′)∈S]+δ.
The (ε, δ)differential privacy guarantee provides that a malicious user or thirdparty researcher who knows all of the attribute values of the dataset 135 but one attribute for one user, cannot infer with confidence the value of the attribute from the information published by the algorithm (i.e., the answer 165).
In some implementations (e.g., when δ=0), the privacy protector 160 may guarantee a stricter form of privacy protection called εdifferential privacy (or pure differential privacy). In εdifferential privacy, the δ parameter is set to zero. A basic property of differential privacy is that the privacy guarantees degrade smoothly under composition and are not affected by postprocessing. Other privacy guarantees may also be supported, such as privacy guarantees related to comparing posterior probabilities with prior probability, or guarantees related to anonymity.
In accordance with the implementations herein, efficient nearly optimal algorithms for approximate privacy in the cases of dense databases (n>d/ε, using the correlated noise engine 220 for example) and sparse databases (n=o(d/ε), using the least squares estimation engine 230 for example) are provided. Implementations use the base decomposition engine 210 to recursively compute an orthonormal basis for Rd, based on the minimum volume enclosing ellipsoid (MEE) or approximate MEE of the columns of the query matrix A.
In an implementation, a query 115 and the dataset 135 are provided to the privacy protector 160. The answer generation engine 205, using known techniques for example, determines the correct answer to the query. At this point, the correct answer does not contain differential privacy and does not have noise added to it. Depending on the implementation, differential privacy is subsequently provided to the correct answer by the base decomposition engine 210, the correlated noise engine 220, and/or the least squares estimation engine 230.
In an implementation, the base decomposition engine 210 may use a base decomposition technique (an example is shown below as Algorithm 1) to compute the orthonormal basis for R^{d}, which may then be used by the correlated noise engine 220 and/or the least squares estimation engine 230.
Algorithm 1, given a matrix A∈R^{dXN}, computes a set of orthonormal matrices U_{1}, . . . , U_{k}, where k≧┌1+log d┐. For each i≈j, U_{i}^{T}U_{j}=0, and the union of columns U_{1}, . . . , U_{k }forms an orthonormal basis for R^{d}. Thus, Algorithm 1 computes a basis for R^{d}, and partitions (“decomposes”) it into k=O(log d) bases of mutually orthogonal subspaces. This set of bases also induces a decomposition of A into A=A_{1}+ . . . +A_{k}, where A_{i}=U_{i}U_{i}^{T}A.
The base decomposition of Algorithm 1 may be used in both the dense case and sparse case following techniques and implementations described further herein. Intuitively, for both cases it can be shown that the error of a mechanism applied to A_{i }can be matched by an error lower bound for A_{i+1}+ . . . +A_{k}. The error lower bounds are based on the spectral lower bound on discrepancy; the geometric properties of the minimum enclosing ellipsoid of a convex body together with the known restricted invertibility principle of Bourgain and Tzafriri may be used in deriving the lower bounds.
In an implementation, the correlated noise engine 220 may use a technique (an example is shown below as Algorithm 2) whose expected error matches the spectral lower bound up to polylogarithmic factors and is therefore nearly optimal. The technique adds correlated unbiased Gaussian noise to the exact answer Ax. The noise distribution is computed based on the decomposition algorithm above (Algorithm 1). An example of Algorithm 2 is given as:
The output of Algorithm 2 satisfies (ε, δ)differential privacy.
In an implementation, a sequence of minimum volume enclosing ellipsoids and projections is computed by the correlated noise engine 220, which is only dependent on the query, not on the data. For each projection, the correlated noise engine 220 (alone or in conjunction with the answer generation engine 205, depending on the implementation) determines the correct answer, and then adds Gaussian noise to the answer. Answers to the original query 115 may be constructed using the answers to projections.
More particularly, in an implementation, look at the convex body K=AB_{1}^{N}, i.e., the image of the unit I_{1 }ball under the linear map A. B_{1}^{N }is the I_{1 }ball of radius 1 in R^{N }(this is the set of points x in R^{N }such that x_{1 }(defined as Σx_{i}) is at most 1). This is a symmetric convex body in d dimensions. First compute E, the minimum volume ellipsoid that encloses K (also known as the John ellipsoid of K).
Next look at the axes of this ellipsoid E and look at their lengths σ_{1}≦σ_{2}≧ . . . ≧σ_{N }in decreasing order. Suppose that the corresponding axes are v_{1}, . . . v_{N}. Let V be the subspace spanned by the axes v_{1}, . . . v_{n}, and let W be the complementary subspace. Let K_{V}, K_{W}, and E_{V}, E_{W }denote the projections of K and E to V and W, respectively.
Let y=Ax denote the true answer to the query and let y_{V }and y_{W }denote its projection to V and W, respectively. Next compute y′_{V }and y′_{W}: the projection of y′ on V and W, respectively (as described further below). The noisy answer y′ is equal to y′, +y′_{W}.
y′_{V }is defined by adding multidimensional Gaussian noise to y_{V }according to the distribution defined by a scaled version of E_{V}. More precisely, add noise proportional to
along the ith axis v_{i }of E, for 1≦l≦n.
y′_{W }is defined by adding Gaussian noise to each coordinate of y_{W}, of magnitude
and then using least squares projection to find the closest vector in nK_{W }to the resulting noisy answer. This closest vector in nK_{W }is defined as y′_{W}.
This gives an (ε,δ) differentially private mechanism which has error at most a polylog(d,n,N) times the optimal. Polylog(d,n,N) denotes some function which is bounded by some polynomial of log d, log n, and log N.
When a (ε,0) differentially private mechanism is desired, change the procedure to get y′_{V }and y′_{W}. y′_{V }can be obtained using the well known generalized Knorm mechanism (e.g., Moritz Hardt and Kunal Talwar) according to the body K_{V}. y′_{W }can be obtained by first running the generalized Knorm mechanism according to the body K_{W}, and then using leastsquares projection to find the closest vector in nK_{W }to the noisy answer returned by generalized Knorm.
In another implementation, one can replace the exact minimum enclosing ellipsoid by an approximate one. As long as the approximation is good enough, the optimality can still be guaranteed.
In another implementation, which can be used alone or in conjunction with other implementations described herein, noise is reduced when the histogram x is known to be sparse (e.g., when number of people n in the database D is small). In this denoising implementation, the data analyst (or the curator, user, or administrator, for example) applies the least squares estimation (e.g., via the least squares estimation engine 230) on the noisy answer, to fit the noisy histogram to an answer that is consistent with the histogram having total weight n=x_{1 }(i.e., n=Σx_{i}) where x_{i }is the number of people of type i, and n is the number of people in the database. It can be shown that even amongst mechanisms whose error is a function of the sparsity n, the correlated Gaussian noise coupled with least squares estimation leads to a near optimal mechanism. This denoising implementation reduces error and can be used with any differentially private mechanism, including those described herein.
In an implementation, the least squares estimation engine 230 may use a technique (an example is shown below as Algorithm 3) with stronger accuracy guarantees than Algorithm 2. It may be used for any query matrix A and any database size bound n. The technique combines the noise distribution of Algorithm 2 with a least squares estimation step. Privacy is guaranteed by noise addition, while the least squares estimation step reduces the error significantly when n=o(d/ε). An example of Algorithm 3 is given as:
The output of Algorithm 3 satisfies (ε,δ)differential privacy.
More particularly, the least squares estimation engine may take an answer, such as the first noisy answer y′ and find a y″ such that y″=Ax″ for some x″ with Σ_{i}x″_{i} being at most n, and y′−y″_{2 }being as small as possible amongst all such y″. This operation is referred to as least squares estimation where the nearest point to y′ in the convex body nAB_{1}^{N }is determined. Various algorithms are known to solve this least squares estimation problem efficiently, such as Algorithm 3 above. It can be shown that y″ is much closer to y than y′, and this implies that the noise added to an average coordinate is only √{square root over (n)} polylog(d,N).
In an implementation, a least squares projection to the image of the I_{o }ball of radius n may be used, instead of the I_{1 }ball as above (an I_{o }ball of radius n is the set of points x in R^{N }that have at most n nonzero coordinates). This may give better utility guarantee, at the cost of a more computationally intensive projection step.
It is contemplated that when the query A is a counting query or has entries in [−1,1], one can use spherical Gaussian noise (which is independent of the query) along with least squares to achieve the best possible bounds.
It is noted that in some settings n itself may not be public. This may be handled, for example, by publishing a noisy version of n by adding Laplacian noise to the true n.
At 320, a query 115, in the form of a query matrix such as a linear query or histogram, is received at the privacy protector 160. The query 115 may be received from a client device 110, for example.
At 330, a base decomposition engine 210 of the privacy protector 160 performs base decomposition using the dataset 135 and the query 115, as described above. In this manner, a set of orthonormal matrices is determined, and an orthonormal basis for the database space is recursively computed.
Using the results of the base decomposition, an answer 165 having differential privacy is generated at 340. The answer 165 with differential privacy may be generated using a correlated noise engine 220 and/or a least squares estimation engine 230 and their associated techniques as described above, for example.
At 350, the answer 165 having differential privacy is provided, e.g., by the privacy protector 160 to the client device 110. Alternatively or additionally, the answer 165 having differential privacy may be published so that it may be downloaded by interested thirdparty researchers or users.
At 430, the results of the base decomposition technique (e.g., from 330) are received at the correlated noise engine 220 from the base decomposition engine 210. A sequence of John ellipsoids and projections are computed at 440, using techniques such as those detailed above (e.g., with respect to
At 450, for each projection, a correct answer is determined by the correlated noise engine 220. The correct answer is the actual answer that does not yet have any privacy or noise added to it. At 460, for each correct answer, the correlated noise engine 220 adds Gaussian noise to it to obtain an answer 165 having differential privacy. As detailed above, the correct answer y=Ax is obtained, and independent Gaussian noise is added to each coordinate of y of standard deviation (√{square root over (d)} log(1^{δ}))/ε to get a noisy answer y′. This process guarantees that releasing y′ does not compromise the privacy of any individual in the database: formally, this guarantees (ε,δ) differential privacy. Additional details of these operations are provided above (e.g., with respect to
At 470, similar to 350, the answer 165 having differential privacy is provided, e.g., by the privacy protector 160 to the client device 110 and/or is published.
At 510 and 520 respectively, a dataset 135 and a query 115 are received by the privacy protector 160. These operations are similar to those described above with respect to 310 and 320, and their descriptions are omitted for brevity.
At 530, the results of the base decomposition technique (e.g., from 330) are received at the least squares estimation engine 230 from the base decomposition engine 210. At 540, a noise distribution technique (e.g., of the method 400) may be performed to obtain a noisy answer.
At 550, a least squares estimation technique (such as described above with respect to
Numerous other general purpose or special purpose computing system environments or configurations may be used. Examples of well known computing systems, environments, and/or configurations that may be suitable for use include, but are not limited to, personal computers (PCs), server computers, handheld or laptop devices, multiprocessor systems, microprocessorbased systems, network personal computers, minicomputers, mainframe computers, embedded systems, distributed computing environments that include any of the above systems or devices, and the like.
Computerexecutable instructions, such as program modules, being executed by a computer may be used. Generally, program modules include routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Distributed computing environments may be used where tasks are performed by remote processing devices that are linked through a communications network or other data transmission medium. In a distributed computing environment, program modules and other data may be located in both local and remote computer storage media including memory storage devices.
With reference to
Computing device 600 may have additional features/functionality. For example, computing device 600 may include additional storage (removable and/or nonremovable) including, but not limited to, magnetic or optical disks or tape. Such additional storage is illustrated in
Computing device 600 typically includes a variety of computer readable media. Computer readable media can be any available media that can be accessed by device 600 and includes both volatile and nonvolatile media, removable and nonremovable media.
Computer storage media include volatile and nonvolatile, and removable and nonremovable media implemented in any method or technology for storage of information such as computer readable instructions, data structures, program modules or other data. Memory 604, removable storage 608, and nonremovable storage 610 are all examples of computer storage media. Computer storage media include, but are not limited to, RAM, ROM, electrically erasable program readonly memory (EEPROM), flash memory or other memory technology, CDROM, digital versatile disks (DVD) or other optical storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to store the desired information and which can be accessed by computing device 600. Any such computer storage media may be part of computing device 600.
Computing device 600 may contain communication connection(s) 612 that allow the device to communicate with other devices. Computing device 600 may also have input device(s) 614 such as a keyboard, mouse, pen, voice input device, touch input device, etc. Output device(s) 616 such as a display, speakers, printer, etc. may also be included. All these devices are well known in the art and need not be discussed at length here.
It should be understood that the various techniques described herein may be implemented in connection with hardware or software or, where appropriate, with a combination of both. Thus, the methods and apparatus of the presently disclosed subject matter, or certain aspects or portions thereof, may take the form of program code (i.e., instructions) embodied in tangible media, such as floppy diskettes, CDROMs, hard drives, or any other machinereadable storage medium where, when the program code is loaded into and executed by a machine, such as a computer, the machine becomes an apparatus for practicing the presently disclosed subject matter.
Although exemplary implementations may refer to utilizing aspects of the presently disclosed subject matter in the context of one or more standalone computer systems, the subject matter is not so limited, but rather may be implemented in connection with any computing environment, such as a network or distributed computing environment. Still further, aspects of the presently disclosed subject matter may be implemented in or across a plurality of processing chips or devices, and storage may similarly be effected across a plurality of devices. Such devices might include personal computers, network servers, and handheld devices, for example.
Although the subject matter has been described in language specific to structural features and/or methodological acts, it is to be understood that the subject matter defined in the appended claims is not necessarily limited to the specific features or acts described above. Rather, the specific features and acts described above are disclosed as example forms of implementing the claims.