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# Copyright (C) IBM Corporation 2019
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"""
The classic Gaussian mechanism in differential privacy, and its derivatives.
"""
from math import erf
from numbers import Real, Integral
import numpy as np
from diffprivlib.mechanisms.base import DPMechanism, bernoulli_neg_exp
from diffprivlib.mechanisms.geometric import Geometric
from diffprivlib.mechanisms.laplace import Laplace
from diffprivlib.utils import copy_docstring
[docs]
class Gaussian(DPMechanism):
r"""The Gaussian mechanism in differential privacy.
First proposed by Dwork and Roth in "The algorithmic foundations of differential privacy" [DR14]_. Samples from the
Gaussian distribution are generated using two samples from `random.normalvariate` as detailed in [HB21b]_, to
prevent against reconstruction attacks due to limited floating point precision.
Parameters
----------
epsilon : float
Privacy parameter :math:`\epsilon` for the mechanism. Must be in (0, 1]. For ``epsilon > 1``, use
:class:`.GaussianAnalytic`.
delta : float
Privacy parameter :math:`\delta` for the mechanism. Must be in (0, 1].
sensitivity : float
The sensitivity of the mechanism. Must be in [0, ∞).
random_state : int or RandomState, optional
Controls the randomness of the mechanism. To obtain a deterministic behaviour during randomisation,
``random_state`` has to be fixed to an integer.
References
----------
.. [DR14] Dwork, Cynthia, and Aaron Roth. "The algorithmic foundations of differential privacy." Found. Trends
Theor. Comput. Sci. 9, no. 3-4 (2014): 211-407.
.. [HB21b] Holohan, Naoise, and Stefano Braghin. "Secure Random Sampling in Differential Privacy." arXiv preprint
arXiv:2107.10138 (2021).
"""
def __init__(self, *, epsilon, delta, sensitivity, random_state=None):
super().__init__(epsilon=epsilon, delta=delta, random_state=random_state)
self.sensitivity = self._check_sensitivity(sensitivity)
self._scale = np.sqrt(2 * np.log(1.25 / self.delta)) * self.sensitivity / self.epsilon
@classmethod
def _check_epsilon_delta(cls, epsilon, delta):
if epsilon == 0 or delta == 0:
raise ValueError("Neither Epsilon nor Delta can be zero")
if isinstance(epsilon, Real) and epsilon > 1.0:
raise ValueError("Epsilon cannot be greater than 1. If required, use GaussianAnalytic instead.")
return super()._check_epsilon_delta(epsilon, delta)
@classmethod
def _check_sensitivity(cls, sensitivity):
if not isinstance(sensitivity, Real):
raise TypeError("Sensitivity must be numeric")
if sensitivity < 0:
raise ValueError("Sensitivity must be non-negative")
return float(sensitivity)
def _check_all(self, value):
super()._check_all(value)
self._check_sensitivity(self.sensitivity)
if not isinstance(value, Real):
raise TypeError("Value to be randomised must be a number")
return True
[docs]
@copy_docstring(Laplace.bias)
def bias(self, value):
return 0.0
[docs]
@copy_docstring(Laplace.variance)
def variance(self, value):
self._check_all(0)
return self._scale ** 2
[docs]
@copy_docstring(Laplace.randomise)
def randomise(self, value):
self._check_all(value)
try:
standard_normal = (self._rng.normalvariate(0, 1) + self._rng.normalvariate(0, 1)) / np.sqrt(2)
except AttributeError: # random_state is a np.random.RandomState
standard_normal = (self._rng.standard_normal() + self._rng.standard_normal()) / np.sqrt(2)
return value + standard_normal * self._scale
[docs]
class GaussianAnalytic(Gaussian):
r"""The analytic Gaussian mechanism in differential privacy.
As first proposed by Balle and Wang in "Improving the Gaussian Mechanism for Differential Privacy: Analytical
Calibration and Optimal Denoising".
Paper link: https://arxiv.org/pdf/1805.06530.pdf
Parameters
----------
epsilon : float
Privacy parameter :math:`\epsilon` for the mechanism. Must be in (0, ∞].
delta : float
Privacy parameter :math:`\delta` for the mechanism. Must be in (0, 1].
sensitivity : float
The sensitivity of the mechanism. Must be in [0, ∞).
random_state : int or RandomState, optional
Controls the randomness of the mechanism. To obtain a deterministic behaviour during randomisation,
``random_state`` has to be fixed to an integer.
"""
def __init__(self, *, epsilon, delta, sensitivity, random_state=None):
super().__init__(epsilon=epsilon, delta=delta, sensitivity=sensitivity, random_state=random_state)
self._scale = self._find_scale()
@classmethod
def _check_epsilon_delta(cls, epsilon, delta):
if epsilon == 0 or delta == 0:
raise ValueError("Neither Epsilon nor Delta can be zero")
return DPMechanism._check_epsilon_delta(epsilon, delta) # pylint: disable=protected-access
def _check_all(self, value):
super()._check_all(value)
return True
def _find_scale(self):
if self.sensitivity / self.epsilon == 0:
return 0.0
epsilon = self.epsilon
delta = self.delta
def phi(val):
return (1 + erf(val / np.sqrt(2))) / 2
def b_plus(val):
return phi(np.sqrt(epsilon * val)) - np.exp(epsilon) * phi(- np.sqrt(epsilon * (val + 2))) - delta
def b_minus(val):
return phi(- np.sqrt(epsilon * val)) - np.exp(epsilon) * phi(- np.sqrt(epsilon * (val + 2))) - delta
delta_0 = b_plus(0)
if delta_0 < 0:
target_func = b_plus
else:
target_func = b_minus
# Find the starting interval by doubling the initial size until the target_func sign changes, as suggested
# in the paper
left = 0
right = 1
while target_func(left) * target_func(right) > 0:
left = right
right *= 2
# Binary search code copied from mechanisms.LaplaceBoundedDomain
old_interval_size = (right - left) * 2
while old_interval_size > right - left:
old_interval_size = right - left
middle = (right + left) / 2
if target_func(middle) * target_func(left) <= 0:
right = middle
if target_func(middle) * target_func(right) <= 0:
left = middle
alpha = np.sqrt(1 + (left + right) / 4) + (-1 if delta_0 < 0 else 1) * np.sqrt((left + right) / 4)
return alpha * self.sensitivity / np.sqrt(2 * self.epsilon)
[docs]
class GaussianDiscrete(DPMechanism):
r"""The Discrete Gaussian mechanism in differential privacy.
As proposed by Canonne, Kamath and Steinke, re-purposed for approximate :math:`(\epsilon,\delta)`-differential
privacy.
Paper link: https://arxiv.org/pdf/2004.00010.pdf
Parameters
----------
epsilon : float
Privacy parameter :math:`\epsilon` for the mechanism. Must be in (0, ∞].
delta : float
Privacy parameter :math:`\delta` for the mechanism. Must be in (0, 1].
sensitivity : int, default: 1
The sensitivity of the mechanism. Must be in [0, ∞).
random_state : int or RandomState, optional
Controls the randomness of the mechanism. To obtain a deterministic behaviour during randomisation,
``random_state`` has to be fixed to an integer.
"""
def __init__(self, *, epsilon, delta, sensitivity=1, random_state=None):
super().__init__(epsilon=epsilon, delta=delta, random_state=random_state)
self.sensitivity = self._check_sensitivity(sensitivity)
self._scale = self._find_scale()
@classmethod
def _check_epsilon_delta(cls, epsilon, delta):
if epsilon == 0 or delta == 0:
raise ValueError("Neither Epsilon nor Delta can be zero")
return super()._check_epsilon_delta(epsilon, delta)
@classmethod
def _check_sensitivity(cls, sensitivity):
if not isinstance(sensitivity, Integral):
raise TypeError("Sensitivity must be an integer")
if sensitivity < 0:
raise ValueError("Sensitivity must be non-negative")
return sensitivity
def _check_all(self, value):
super()._check_all(value)
self._check_sensitivity(self.sensitivity)
if not isinstance(value, Integral):
raise TypeError("Value to be randomised must be an integer")
return True
[docs]
@copy_docstring(Laplace.bias)
def bias(self, value):
return 0.0
@copy_docstring(Laplace.variance)
def variance(self, value):
raise NotImplementedError
[docs]
@copy_docstring(Geometric.randomise)
def randomise(self, value):
self._check_all(value)
if self._scale == 0:
return value
tau = 1 / (1 + np.floor(self._scale))
sigma2 = self._scale ** 2
while True:
geom_x = 0
while bernoulli_neg_exp(tau, self._rng):
geom_x += 1
bern_b = self._rng.random() < 0.5
if bern_b and not geom_x:
continue
lap_y = int((1 - 2 * bern_b) * geom_x)
bern_c = bernoulli_neg_exp((abs(lap_y) - tau * sigma2) ** 2 / 2 / sigma2, self._rng)
if bern_c:
return value + lap_y
def _find_scale(self):
"""Determine the scale of the mechanism's distribution given epsilon and delta.
"""
if self.sensitivity / self.epsilon == 0:
return 0
def objective(sigma, epsilon_, delta_, sensitivity_):
"""Function for which we are seeking its root. """
idx_0 = int(np.floor(epsilon_ * sigma ** 2 / sensitivity_ - sensitivity_ / 2))
idx_1 = int(np.floor(epsilon_ * sigma ** 2 / sensitivity_ + sensitivity_ / 2))
idx = 1
lhs, rhs, denom = float(idx_0 < 0), 0, 1
_term, diff = 1, 1
while _term > 0 and diff > 0:
_term = np.exp(-idx ** 2 / 2 / sigma ** 2)
if idx > idx_0:
lhs += _term
if idx_0 < -idx:
lhs += _term
if idx > idx_1:
diff = -rhs
rhs += _term
diff += rhs
denom += 2 * _term
idx += 1
if idx > 1e6:
raise ValueError("Infinite sum not converging, aborting. Try changing the epsilon and/or delta.")
return (lhs - np.exp(epsilon_) * rhs) / denom - delta_
epsilon = self.epsilon
delta = self.delta
sensitivity = self.sensitivity
# Begin by locating the root within an interval [2**i, 2**(i+1)]
guess_0 = 1
f_0 = objective(guess_0, epsilon, delta, sensitivity)
pwr = 1 if f_0 > 0 else -1
guess_1 = 2 ** pwr
f_1 = objective(guess_1, epsilon, delta, sensitivity)
while f_0 * f_1 > 0:
guess_0 *= 2 ** pwr
guess_1 *= 2 ** pwr
f_0 = f_1
f_1 = objective(guess_1, epsilon, delta, sensitivity)
# Find the root (sigma) using the bisection method
while not np.isclose(guess_0, guess_1, atol=1e-12, rtol=1e-6):
guess_mid = (guess_0 + guess_1) / 2
f_mid = objective(guess_mid, epsilon, delta, sensitivity)
if f_mid * f_0 <= 0:
f_1 = f_mid
guess_1 = guess_mid
if f_mid * f_1 <= 0:
f_0 = f_mid
guess_0 = guess_mid
return (guess_0 + guess_1) / 2