Differential Privacy for Functions and Functional Data

Introduction

Suppose we have database $D$ which consists of measurements of a set of individuals. We want to release a summary of $D$ without compromising the privacy of those individuals in the database. One framework for defining privacy rigorously in such problems is differential privacy . The basic idea is to produce an output via random noise addition. An algorithm which does this may be thought of as inducing a distribution $P_{D}$ on the output space (where the randomness is due to internal “coin flips” of the algorithm), for every input data set $D$. Differential privacy, defined in Section 2, requires that $P_{D}$ not depend too strongly on any single element of the database $D$.

The literature on differential privacy is vast. Algorithms that preserve differential privacy have been developed for boosting, parameter estimation, clustering, logistic regression, SVM learning and many other learning tasks. See, for example, , , , , , , , and references therein. In all these cases, the data (both the input and output) are assumed to be real numbers or vectors. In this paper we are concerned with a setting in which the output, and possibly the input data set, consist of functions.

where $W$ is a kernel (see, for instance ) and $h>0$ is the bandwidth parameter. The density estimator is useful for may tasks such as clustering and classification. We may then want to release a “version” of the density estimator $\widehat{f}$ in a way which fulfills the criteria of differential privacy. The utility of such a procedure goes beyond merely estimating the underlying density. In fact, suppose the goal is to release a privatized database. With a differentially private density estimator in hand, a large sample of data may be drawn from that density. The release of such a sample would inherit the differential privacy properties of the density estimator: see, in particular, and . This is a very attractive proposition, since a differentially private sample of data could be used as the basis for any number of statistical analyses which may have been brought to bear against the original data (for instance, exploratory data analysis, model fitting, etc).

Histograms are an example of a density estimator that has been “privatized” in previous literature . However, as density estimators, histograms are suboptimal because they are not smooth. Specifically, they do converge at the minimax rate under the assumption that the true density is smooth. The preferred method for density estimation in statistics is kernel density estimation. The methods developed in this paper lead to a private kernel density estimator.

In addition to kernel density estimation, there are a myriad of other scenarios in which the result of a statistical analysis is a function. For example, the regression function or classification function from a supervised learning task. We demonstrate how the theory we develop may be applied in these contexts as well.

Outline. We introduce some notation and review the definition of differential privacy in Section 2. We also give a demonstration of a technique to achieve the differential privacy for a vector valued output. The theory for demonstrating differentially privacy of functions is established in Section 3. In Section 4 we apply the theory to the problems of kernel density estimation and kernel SVM learning. We also demonstrate how the theory may apply to a broad class of functions (a Sobolev space). Section 5 discusses possible algorithms for outputting functions.

Differential Privacy

Here we recall the definition of differential privacy and introduce some notation. Let $D=(d_{1},\ldots,d_{n})\in\mathcal{D}$ be an input database in which $d_{i}$ represents a row or an individual, and where $\mathcal{D}$ is the space of all such databases of $n$ elements. For two databases $D,D^{\prime}$, we say they are “adjacent” or “neighboring” and write $D\sim D^{\prime}$ whenever both have the same number of elements, but differ in one element. In other words, there exists a permutation of $D$ having Hamming distance of 2 to $D^{\prime}$. In some other works databases are called “adjacent” whenever one database contains the other together with exactly one additional element.

A set of distributions $\{P_{D}:D\in\mathcal{D}\}$ is called $(\alpha,\beta)$-differentially private, or said to “achieve $(\alpha,\beta)$-DP” whenever for all $D\sim D^{\prime}\in{\cal D}$ we have:

where $\alpha,\beta\geq 0$ are parameters, and $\mathcal{A}$ is the finest $\sigma$-field on which all $P_{D}$ are defined.

Typically the above definition is called “approximate differential privacy” whenever $\beta>0$, and “$(\alpha,0)$-differential privacy” is shortened to “$\alpha$-differential privacy.” It is important to note that the relation $D\sim D^{\prime}$ is symmetric, and so the inequality (1) is required to hold when $D$ and $D^{\prime}$ are swapped. Throughout this paper we take $\alpha\leq 1$, since this simplifies some proofs.

The $\sigma$-field $\mathcal{A}$ is rarely mentioned in the literature on differential privacy but is actually quite important. For example if we were to take $\mathcal{A}=\{\Omega,\emptyset\}$ then the condition (1) is trivially satisfied by any randomized algorithm. To make the definition as strong as possible we insist that $\mathcal{A}$ be the finest available $\sigma$-field on which the $P_{D}$ are defined. Therefore when $\Omega$ is discrete the typical $\sigma$-field is $\mathcal{A}=2^{\Omega}$ (the class of all subsets of $\Omega$), and when $\Omega$ is a space with a topology it is typical to use the completion of the Borel $\sigma$-field (the smallest $\sigma$-field containing all open sets). We raise this point since when $\Omega$ is a space of functions, the choice of $\sigma$-field is more delicate.

give a technique to achieve approximate differential privacy for general vector valued outputs in which the “sensitivity” may be bounded. We review this below, since the result is important in the demonstration of the privacy of our methods which output functions. What follows in this section is a mild alteration to the technique developed by and , in that the “sensitivity” of the class of vectors is measured in the Mahalanobis distance rather than the usual Euclidean distance.

In demonstrating the differential privacy, we make use of the following lemma which is simply an explicit statement of an argument used in a proof by .

Suppose that, for all $D\sim D^{\prime}$, there exists a set $A^{\star}_{D,D^{\prime}}\in\mathcal{A}$ such that, for all $S\in\mathcal{A}$,

Then the family $\{P_{D}\}$ achieves the $(\alpha,\beta)$-DP.

The first inequality is due to (3), the second is due to (2) and the third is due to the subadditivity of measures. ∎

The above result shows that, so long as there is a large enough (in terms of the measure $P_{D}$) set on which the $(\alpha,0)$-DP condition holds, then the approximate $(\alpha,\beta)$-DP is achieved.

If $(\Omega,\mathcal{A})$ has a $\sigma$-finite dominating measure $\lambda$, then for (2) to hold a sufficient condition is that the ratio of the densities be bounded on some set $A^{\star}_{D,D^{\prime}}$:

Then the randomized algorithm which, for input database $D$ outputs

This ratio exceeds $e^{\alpha}$ only when

We consider the probability of this set under $P_{D}$, in which case we have $x=v_{D}+\frac{c(\beta)\Delta}{\alpha}M^{1/2}z$, where $z$ is an isotropic normal with unit variance. We have

Multiplying by $\frac{\alpha}{c(\beta)\Delta}$ and using (5) gives

Note that the left side is a normal random variable with mean zero and variance smaller than $\Delta^{2}$. The probability of this set is increasing with the variance of said variable, and so we examine the probability when the variance equals $\Delta^{2}$. We also restrict to $\alpha\leq 1$, and let $y\sim\mathcal{N}(0,1)$, yielding

where $c(\beta)$ is as defined in (6) and the final inequality is proved in . Thus lemma 2.2 gives the differential privacy. ∎

The quantity (5) is a mild modification of the usual notion of “sensitivity” or “global sensitivity” . It is nothing more than the sensitivity measured in the Mahalanobis distance corresponding to the matrix $M$. The case $M=I$ corresponds to the usual Euclidean distance, a setting that has been studied previously by , among others.

2 The Implications of Approximate Differential Privacy

The above definitions provide a strong privacy guarantee in the sense that they aim to protect against an adversary having almost complete knowledge of the private database. Specifically, an adversary knowing all but one of the data elements and having observed the output of a private procedure, will remain unable to determine the identity of the data element which is unknown to him. To see this, we provide an analog of theorem 2.4 of , who consider the case of $\alpha$-differential privacy.

Let the adversary’s database be denoted by $D_{A}=(d_{1},\ldots,d_{n-1})$, and the private database by $D=(d_{1},\ldots,d_{n})$. First note that before observing the output of the private algorithm, the adversary could determine that the private database $D$ lay in the set $\left\{(d_{1},\ldots,d_{n-1},d)\in\mathcal{D}\right\}.$ Thus, the private database comprises his data with one more element. Since all other databases may be excluded from consideration by the adversary we concentrate on those in the above set. In particular, we obtain the following analog of theorem 2.4 of .

Let $X\sim P_{D}$ where the family $P_{D}$ achieves the $(\alpha,\beta)$-approximate DP. Any level $\gamma$ test of: $H:D=D_{0}$ vs $V:D\neq D_{0}$ has power bounded above by $\gamma e^{\alpha}+\beta$.

The above result follows immediately from noting that the rejection region of the test is a measurable set in the space and so obeys the constraint of the differential privacy. The implication of the above proposition is that the power of the test will be bounded close to its size. When $\alpha,\beta$ are small, this means that the test is close to being “trivial” in the sense that it is no more likely to correctly reject a false hypothesis than it is to incorrectly reject the true one.

Approximate Differential Privacy for Functions

The goal of the release a function raises a number of questions. First what does it mean for a computer program to output a function? Second, how can the differential privacy be demonstrated? In this section we continue to treat randomized algorithms as measures, however now they are measures over function spaces. In section 5 we demonstrate concrete algorithms, which in essence output the function on any arbitrary countable set of points.

We cannot expect the techniques for finite dimensional vectors to apply directly when dealing with functions. The reason is that $\sigma$-finite dominating measures of the space of functions do not exist, and, therefore, neither do densities. However, there exist probability measures on the spaces of functions. Below, we demonstrate the approximate differential privacy of measures on function spaces, by considering random variables which correspond to evaluating the random function on a finite set of points.

This is a field (see page 508) although not a $\sigma$-field, since it does not have the requisite closure under countable intersections (namely it does not contain cylinder sets for which $S$ is countably infinite). We focus on the creation of algorithms for which the differential privacy holds over the field of cylinder sets, in the sense that, for all $D\sim D^{\prime}\in\mathcal{D}$,

Let $x_{1},\ldots,x_{n}$ be any finite set of points in $T$ chosen a-priori. Then whenever (7) holds, the release of the vector

The claimed privacy guarantee follows from (7). ∎

We now give a limiting argument to extend (7) to the generated $\sigma$-field (or, equivalently, the $\sigma$-field generated by the cylinders of dimension 1)

where the union extends over all the countable subsets $S$ of $T$. The second equality above is due to theorem 36.3 part ii.

Note that, for countable $S$, the cylinder sets take the form

Define $C_{S,B,n}=\bigcap_{i=1}^{n}C_{\{t_{i}\},B_{i}}.$ Then, the sets $C_{S,B,n}$ form a sequence of sets which decreases towards $C_{S,B}$ and $C_{S,B}=\lim_{n\to\infty}C_{S,B,n}.$ Since the sequence of sets is decreasing and the measure in question is a probability (hence bounded above by 1), we have

Therefore, for each pair $D\sim D^{\prime}$ and for every $\epsilon>0$, there exists an $n_{0}$ so that for all $n\geq n_{0}$

The number $n_{0}$ depends on whichever is the slowest sequence to converge. Finally we obtain

Since this holds for all $\epsilon>0$ we conclude that $P_{D}(C_{S,B})\leq e^{\alpha}P_{D^{\prime}}(C_{S,B})+\beta.$ ∎

In principle, if it were possible for a computer to release a complete description of the function $\widetilde{f}_{D}$ then this result would demonstrate the privacy guarantee achieved by our algorithm. In practise a computer algorithm which runs in a finite amount of time may only output a finite set of points, hence this result is mainly of theoretical interest. However, in the case in which the functions to be output are continuous, and the restriction is made that $P_{D}$ are measures over $C$ (the continuous functions on the unit interval), another description of the $\sigma$-field becomes available. Namely, the restriction of $\mathcal{F}$ to the elements of $C$ corresponds to the borel $\sigma$-field over $C$ with the topology induced by the uniform norm ($\|f\|_{\infty}=\sup_{t}|f(t)|$). Therefore in the case of continuous functions, differential privacy over $\mathcal{F}_{0}$ hence leads to differential privacy throughout the borel $\sigma$-field.

In summary, we find that if every finite dimensional projection of the released function satisfies differential privacy, then so does every countable-dimensional projection. We now explore techniques which achieve the differential privacy over these $\sigma$-fields.

2 Differential Privacy via the Exponential Mechanism

demonstrate that such a technique achieves the $\alpha$-differential privacy, which is strictly stronger than the $(\alpha,\beta)$-differential privacy we consider here. Although this technique is conceptually appealing for its simplicity, it remains challenging to use in practise since the set of functions $G$ may need to be very large in order to ensure the utility of the released function (in the sense of expected error). Since the algorithm which outputs from $P_{D}$ must obtain the normalization constant to the distribution above, it must evidently compute the probabilities for each $g_{i}$, which may be extremely time consuming. Note that techniques such as importance sampling are also difficult to bring to bear against this problem when it is important to maintain utility.

The technique given above can be interpreted as outputting a discrete random variable, and fulfilling privacy definition with respect to the $\sigma$-field consisting of the powerset of $G$. This implies the privacy with respect to the cylinder sets, since the restriction of each cylinder set to the elements of $G$ corresponds some subset of $G$.

3 Differential Privacy via Gaussian Process Noise

We propose to use measures $P_{D}$ over functions, which are Gaussian processes. The reason is that there is a strong connection between these measures over the infinite dimensional function space, and the Gaussian measures over finite dimensional vector spaces such as those used in Proposition 2.4. Therefore, with some additional technical machinery which we will illustrate next, it is possible to move from differentially private measures over vectors to those over functions.

For any finite subset $S\subset T$, the random vector $\{X_{t}:t\in S\}$ has a normal distribution with the means, variances, and covariances given by the above functions. Such a “finite dimensional distribution” may be regarded as a projection of the Gaussian process. Below we propose particular mean and covariance functions for which Proposition 2.4 will hold for all finite dimensional distributions. These will require some smoothness properties of the family of functions $\{f_{D}\}$. We first demonstrate the technical machinery which allows us to move from finite dimensional distributions to distributions on the function space, and then we give differentially private measures on function spaces of one dimension. Finally, we extend our results to multiple dimensions.

Let $G$ be a sample path of a Gaussian process having mean zero and covariance function $K$. Let $\{f_{D}:D\in\mathcal{D}\}$ be a family of functions indexed by databases. Then the release of

is $(\alpha,\beta)$-differentially private (with respect to the cylinder $\sigma$-field $\mathcal{F}$) whenever

Finally note that for any $A\in\mathcal{F}_{0}$, we may write $A=C_{X_{n},B}$ for some finite $n$, some vector $X_{n}=(x_{1},\ldots,x_{n})\in T^{n}$ and some borel set $B$. Then

Combining this with the above gives the requisite privacy statement for all $A\in\mathcal{F}_{0}$. Proposition 3.2 carries this to $\mathcal{F}$. ∎

4 Functions in a Reproducing Kernel Hilbert Space

When the family of functions lies in the reproducing kernel Hilbert space (RKHS) which corresponds to the covariance kernel of the Gaussian process, then establishing upper bounds of the form (9) is simple. Below, we give some basic definitions for RKHSs, and refer the reader to for a more detailed account. We first recall that the RKHS is generated from the closure of those functions which can be represented as finite linear combinations of the kernel, i.e.,

and the corresponding norm of $f$ is $\|f\|_{\mathcal{H}}=\sqrt{\left\langle f,f\right\rangle_{\mathcal{H}}}.$ This gives rise to the “reproducing” nature of the Hilbert space, namely, $\left\langle K_{x},K_{y}\right\rangle_{\mathcal{H}}=K(x,y)$. Furthermore, the functionals $\left\langle K_{x},\cdot\right\rangle_{\mathcal{H}}$ correspond to point evaluation, i.e.

The RKHS $\mathcal{H}$ is then the closure of $\mathcal{H}_{0}$ with respect to the RKHS norm. We now present the main theorem which suggests an upper bound of the form required in Proposition 3.3.

For $f\in\mathcal{H}$, where $\mathcal{H}$ is the RKHS corresponding to the kernel $K$, and for any finite sequence $x_{1},\ldots,x_{n}$ of distinct points in $T$, we have:

The proof is in the appendix. Together with Proposition 3.3, this result implies the following.

For $\{f_{D}:D\in\mathcal{D}\}\subseteq\mathcal{H}$, the release of

is $(\alpha,\beta)$-differentially private (with respect to the cylinder $\sigma$-field) whenever

and when $G$ is the sample path of a Gaussian process having mean zero and covariance function $K$, given by the reproducing kernel of $\mathcal{H}$.

Examples

We now give some examples in which the above technique may be used to construct private versions of functions in an RKHS.

Let $f_{D}$ be the kernel density estimator, where $D$ is regarded as a sequence of points $x_{i}\in T$ as $i=1,\ldots,n$ drawn from a distribution with density $f$. Let $h$ denote the bandwidth. Assuming a Gaussian kernel, the estimator is

Let $D\sim D^{\prime}$ so that $D^{\prime}=x_{1},\ldots,x_{n-1},x_{n}^{\prime}$ (no loss of generality is incurred by demanding that the data sequences differ in their last element). Then,

If we use the Gaussian kernel as the covariance function for the Gaussian process then upper bounding the RKHS norm of this function is trivial. Thus, let $K(x,y)=\text{exp}\left\{-\frac{\|x-y\|_{2}^{2}}{2h^{2}}\right\}$. Then $f_{D}-f_{D^{\prime}}=\frac{1}{n(2\pi h^{2})^{d/2}}\left(K_{x_{n}}-K_{x_{n}^{\prime}}\right)$ and

where $G$ is a sample path of a Gaussian process having mean zero and covariance $K$, then differential privacy is demonstrated by corollary 3.5. We may compare the utility of the released estimator to that of the non-private version. Under standard smoothness assumptions on $f$, it is well-known (see ) that the risk is

for some constants $c_{1}$ and $c_{2}$. The optimal bandwidth is $h\asymp(1/n)^{1/(4+d)}$ in which case $R=O(n^{-\frac{4}{4+d}})$.

For the differentially private function it is easy to see that

Therefore, at least in terms of rates, no accuracy has been lost.

The above demonstration of privacy also holds when the kernel is replaced by a non-isotropic Gaussian kernel. In this case the kernel density estimate may take the form

where $H$ is a positive definite matrix and $|H|$ is the determinant. For example it may be required to employ a different choice of bandwidth for each coordinate of the space, in which case $H$ would be a diagonal matrix having non-equal entries on the diagonal. So long as $H$ is fixed a-priori, privacy may be established by adding a Gaussian process having mean zero and covariance given by

As above, the sensitivity is upper bounded, as

Therefore it satisfies the $(\alpha,\beta)$-DP to release

where $G$ is a sample path of a Gaussian process having mean zero and covariance $K$.

1.2 Private Choice of Bandwidth

Note that the above assumed that $h$ (or $H$) was fixed a-priori by the user. In usual statistical settings $h$ is a parameter that is tuned depending on the data (not simply set to the correct order of growth as a function of $n$). Thus rather than fixed $h$ the user would use $\widehat{h}$ which depends on the data itself. In order to do this it is necessary to find a differentially private version of $\widehat{h}$ and then to employ the composition property of differential privacy (citation).

The typical way that the bandwidth is selected is by employing the leave-one-out cross validation. This consists of choosing a grid of candidate values for $h$, evaluating the leave one out log likelihood for each value, and then choosing whichever is the maximizer. This technique may be amenable to private analysis via the “exponential mechanism” of (citation), however it would evidently require that $T$ be a compact set which is known a-priori. An alternative is to use a “rule of thumb” (see ) for determining the bandwidth which is given by

In which $IQR_{j}$ is the observed interquartile range of the data along the $j^{th}$ coordinate. Thus this method gives a diagonal matrix $H$ as in the above section. To make a private version $\widetilde{h}_{j}$ we may use the technique of in which a differentially private algorithm for the interquartile range was developed.

2 Functions in a Sobolev Space

The above technique worked easily since we chose a particular RKHS in which we knew the kernel density estimator to live. What’s more, since the functions themselves lay in the generating set of functions for that space, the determination of the norm of the difference $f_{D}-f_{D^{\prime}}$ was extremely simple. In general we may not be so lucky that the family of functions is amenable to such analysis. In this section we demonstrate a more broadly applicable technique which may be used whenever the functions are sufficiently smooth. Consider the Sobolev space

This is a RKHS with the kernel $K(x,y)=\text{exp}\left\{-\gamma\ |x-y|\right\}$ for positive constant $\gamma$. The norm in this space is given by

See e.g., (p. 316) and for details. Thus for a family of functions in one dimension which lay in the Sobolev space $H^{1}$, we may determine a noise level necessary to achieve the differential privacy by bounding the above quantity for the difference of two functions. For functions over higher dimensional domains (as $^{d}$ for some $d>1$) we may construct an RKHS by taking the $d$-fold tensor product of the above RKHS (see, in particular for details on the construction). The resulting space has the reproducing kernel

and is the completion of the set of functions

The norm over this set of functions is given by:

The norm over the completed space agrees with the above on $\mathcal{G}_{0}$. The explicit form is obtained by substituting (10) into the right hand side of (11) and replacing all instances of $\prod_{j=1}^{d}f_{j}(x_{j})$ with $f(x_{1},\ldots,x_{j})$. Thus the norm in the completed space is defined for all $f$ possessing all first partial derivatives which are all in $\mathcal{L}_{2}$.

We revisit the example of a kernel density estimator (with an isotropic Gaussian kernel). We note that this isotropic kernel function is in the set $G_{0}$ defined above, as

Therefore, choosing $\gamma=1/h$ leads to

Therefore, we observe a technique which attains higher generality than the ad-hoc analysis of the preceding section. However this is at the expense of the noise level, which grows at a higher rate as $d$ increases. An example of the technique applied to the same kernel density estimation problem as above is given in Figure 2.

3 Minimizers of Regularized Functionals in an RKHS

The construction of the following section is due to , who were interested in determining the sensitivity of certain kernel machines (among other algorithms) with the aim of bounding the generalization error of the output classifiers. noted that these bounds are useful for establishing the noise level required for differential privacy of support vector machines. They are also useful for our approach to privacy in a function space.

We will now demonstrate that for (12), whenever the loss function is admissible, the minimizers on adjacent datasets may be bounded close together in RKHS norm. Denote the part of the optimization due to the loss function:

where $\eta\in$ and we use $\delta_{D^{\prime},D}=f_{D^{\prime}}-f_{D}$. This also holds when $f_{D}$ and $f_{D^{\prime}}$ swap places. Summing the resulting inequality with the above and rearranging yields

Due to the definition of $f_{D},f_{D^{\prime}}$ as the minimizers of their respective functionals we have

Moving the loss function term to the other side and using the Lipschitz property we finally obtain that

What’s more, the reproducing property together with Cauchy-Schwarz inequality yields

Algorithms

There are two main modes in which functions $f_{D}$ could be released by the holder of the data $D$ to the outside parties. The first is a “batch” setting in which the parties designate some finite collection of points $x_{1}\ldots,x_{n}\in T$. The database owner computes $\widetilde{f}_{D}(x_{i})$ for each $i$ and return the vector of results. At this point the entire transaction would end with only the collection of pairs $(x_{i},\widetilde{f}_{D}(x_{i}))$ being known to the outsiders. An alternative is the “online” setting in which outside users repeatedly specify points in $x_{i}\in T$, the database owner replies with $\widetilde{f}_{D}(x_{i})$, but unlike the former setting he remains available to respond to more requests for function evaluations. We name these settings “batch” and “online” for their resemblance of the batch and online settings typically considered in machine learning algorithms.

The batch method is nothing more than sampling a multivariate Gaussian, since the set $x_{1},\ldots,x_{n}\in T$ specifies the finite dimensional distribution of the Gaussian process from which to sample. The released vector is simply

In the online setting, the data owner upon receiving a request for evaluation at $x_{i}$ would sample the gaussian process conditioned on the samples already produced at $x_{1},\ldots,x_{i-1}$. Let

The database owner may track the inverse matrix $C_{i}^{-1}$ and after each request update it into $C_{i+1}^{-1}$ by making use of Schurs Complements combined with the matrix inversion lemma. Nevertheless we note that as $i$ increases the computational complexity of answering the request will in general grow. In the very least, the construction of $V_{i}$ takes time proportional to $i$. This may make this approach problematic to implement in practise. However we note that when using the covariance kernel

that a more efficient algorithm presents itself. This is the kernel considered in section 4.2. Due to the above form of $K$, we find that for $x<y<z$ we have: $K(x,z)=K(x,y)K(y,z)$. Therefore in using the above algorithm we would find that $V_{i}$ is always contained in the span of at most two rows of $C_{i}$. This is most evident when, for instance, $x_{i}<\min_{j<i}x_{j}$. In this case let $m=\arg\min_{j<i}x_{j}$ $V_{i}=K(x_{i},x_{m})C_{i}(m)$, in which $C_{i}(m)$ means the $m^{th}$ row of $C_{i}$. Therefore $C_{i}^{-1}V_{i}$ will be a sparse vector with exactly one non-zero entry (taking value $K(x,x_{m})$) in the $m^{th}$ position. Similar algebra applies whenever $x_{i}$ falls between two previous points, in which case $V_{i}$ lays in the span of the two rows corresponding to the closest point on the left and the closest on the right. Using the above kernel with some choice of $\gamma$ let

Let $\xi(x_{i})=\widetilde{f}_{D}(x_{i})-f_{D}(x_{i})$ represent the noise process. We find that the conditional distribution of $\xi(x_{i})$ to be Normal with mean and variance given by:

where $x_{(1)}<x_{(2)}<\cdots<x_{(i-1)}$ are the points $x_{1},\ldots,x_{i-1}$ after being sorted into increasing order. In using the above algorithm it is only necessary for the data owner to store the values $x_{i}$ and $\widetilde{f}_{D}(x_{i})$. When using the proper data structures e.g., a sorted doubly linked list for the $x_{i}$ it is possible to determine the mean and variance using the above technique in time proportional to $\log(i)$ which is a significant improvement over the general linear time scheme above (note that the linked list is suggested since then it is possible to update the list in constant time).

Conclusion

We have shown how to add random noise to a function in such a way that differential privacy is preserved. It would be interesting to study this method in the many applications of functional data analysis .

On a more theoretical note, we have not addressed the issue of lower bounds. Specifically, we can ask: Given that we want to release a differentially private function, what is the least amount of noise that must necessarily be added in order to preserve differential privacy? This question has been addressed in detail for real-valued, count-valued and vector-valued data. However, those techniques apply to the case of $\beta=0$ whereupon the family $\{P_{D}\}$ are all mutually absolutely continuous. In the case of $\beta>0$ which we consider this no longer applies and so the determination of lower bounds is complicated (for example, since quantities such as the KL divergence are no longer bounded).

Appendix

Proof of Proposition 3.4. Note that invertibility of the matrix is safely assumed due to Mercer’s theorem. Denote the matrix by $M^{-1}$. Denote by $P$ the operator $\mathcal{H}\to\mathcal{H}$ defined by

We find this operator to be idempotent in the sense that $P=P^{2}$:

$P$ is also self-adjoint due to the symmetry of $M$, i.e.

The latter term is nothing more than the left hand side in the statement. In summary the quantity in the statement of the theorem is just the square RKHS norm in the restriction of $\mathcal{H}$ to the subspace spanned by the functions $K_{x_{i}}$. $\Box$

References