GaussianMechanism: Gaussian Mechanism

In DPpack: Differentially Private Statistical Analysis and Machine Learning

Gaussian Mechanism

Description

This function implements the Gaussian mechanism for differential privacy by adding noise to the true value(s) of a function according to specified values of epsilon, delta, and l2-global sensitivity(-ies). Global sensitivity calculated based either on bounded or unbounded differential privacy can be used \insertCiteKifer2011DPpack. If true.values is a vector, the provided epsilon and delta are divided such that (epsilon, delta)-level differential privacy is satisfied across all function values. In the case that each element of true.values comes from its own function with different corresponding sensitivities, a vector of sensitivities may be provided. In this case, if desired, the user can specify how to divide epsilon and delta among the function values using alloc.proportions.

Usage

GaussianMechanism(
  true.values,
  eps,
  delta,
  sensitivities,
  type.DP = "aDP",
  alloc.proportions = NULL,
  analytic = FALSE,
  tol = 1e-12
)

Arguments

true.values	Real number or numeric vector corresponding to the true value(s) of the desired function.
eps	Positive real number defining the epsilon privacy parameter.
delta	Positive real number defining the delta privacy parameter.
sensitivities	Real number or numeric vector corresponding to the l2-global sensitivity(-ies) of the function(s) generating true.values. This value must be of length 1 or of the same length as true.values. If it is of length 1 and true.values is a vector, this indicates that the given sensitivity applies simultaneously to all elements of true.values and that the privacy budget need not be allocated (alloc.proportions is unused in this case). If it is of the same length as true.values, this indicates that each element of true.values comes from its own function with different corresponding sensitivities. In this case, the l2-norm of the provided sensitivities is used to generate the Gaussian noise.
type.DP	String indicating the type of differential privacy desired for the Gaussian mechanism. Can be either 'pDP' for probabilistic DP \insertCiteLiu2019aDPpack or 'aDP' for approximate DP \insertCiteDwork2006bDPpack. Note that if 'aDP' is chosen, epsilon must be strictly less than 1.
alloc.proportions	Optional numeric vector giving the allocation proportions of epsilon and delta to the function values in the case of vector-valued sensitivities. For example, if sensitivities is of length two and alloc.proportions = c(.75, .25), then 75% of the privacy budget eps (and 75% of delta) is allocated to the noise computation for the first element of true.values, and the remaining 25% is allocated to the noise computation for the second element of true.values. This ensures (eps, delta)-level privacy across all computations. Input does not need to be normalized, meaning alloc.proportions = c(3,1) produces the same result as the example above.
analytic	Indicates whether to use the analytic Gaussian mechanism to compute the noise scale \insertCiteBalle2018DPpack. Defaults to FALSE.
tol	Error tolerance for binary search used in determining the noise parameter for the analytic Gaussian mechanism. Unused if analytic is FALSE. Defaults to 1e-12.

Value

Sanitized function values based on the bounded and/or unbounded definitions of differential privacy, sanitized via the Gaussian mechanism.

References

\insertRef

Dwork2006aDPpack

\insertRef

Kifer2011DPpack

\insertRef

Balle2018DPpack

\insertRef

Liu2019aDPpack

\insertRef

Dwork2006bDPpack

Examples

# Simulate dataset
n <- 100
c0 <- 5 # Lower bound
c1 <- 10 # Upper bound
D1 <- stats::runif(n, c0, c1)

# Privacy budget
epsilon <- 0.9 # eps must be in (0, 1) for approximate differential privacy
delta <- 0.01
sensitivity <- (c1-c0)/n

# Approximate differential privacy
private.mean.approx <- GaussianMechanism(mean(D1), epsilon, delta,
                                         sensitivity)
private.mean.approx

# Probabilistic differential privacy
private.mean.prob <- GaussianMechanism(mean(D1), epsilon, delta, sensitivity,
                                       type.DP = 'pDP')
private.mean.prob

# Analytic Gaussian mechanism
epsilon <- 1.1 # epsilon can be > 1 for analytic Gaussian mechanism
private.mean.analytic <- GaussianMechanism(mean(D1), epsilon, delta,
                                           sensitivity, analytic=TRUE)
private.mean.analytic

# Simulate second dataset
d0 <- 3 # Lower bound
d1 <- 6 # Upper bound
D2 <- stats::runif(n, d0, d1)
D <- matrix(c(D1,D2),ncol=2)
sensitivities <- c((c1-c0)/n, (d1-d0)/n)
epsilon <- 0.9 # Total privacy budget for all means
delta <- 0.01

# Here, sensitivities are summed and the result is used to generate Laplace
# noise. This is essentially the same as allocating epsilon proportional to
# the corresponding sensitivity. The results satisfy (0.9,0.01)-approximate
# differential privacy.
private.means <- GaussianMechanism(apply(D, 2, mean), epsilon, delta,
                                   sensitivities)
private.means

# Here, privacy budget is explicitly split so that 75% is given to the first
# vector element and 25% is given to the second.
private.means <- GaussianMechanism(apply(D, 2, mean), epsilon, delta,
                                   sensitivities,
                                   alloc.proportions = c(0.75, 0.25))
private.means

DPpack documentation built on Oct. 20, 2024, 9:07 a.m.