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# Copyright (C) IBM Corporation 2019
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"""
The classic Laplace mechanism in differential privacy, and its derivatives.
"""
from numbers import Real
import numpy as np
from numpy.random import random
from diffprivlib.mechanisms.base import DPMechanism, TruncationAndFoldingMixin
from diffprivlib.utils import copy_docstring
class Laplace(DPMechanism):
r"""
The classic Laplace mechanism in differential privacy, as first proposed by Dwork, McSherry, Nissim and Smith.
Paper link: https://link.springer.com/content/pdf/10.1007/11681878_14.pdf
Includes extension to (relaxed) :math:`(\epsilon,\delta)`-differential privacy, as proposed by Holohan et al.
Paper link: https://arxiv.org/pdf/1402.6124.pdf
"""
def __init__(self):
super().__init__()
self._sensitivity = None
def __repr__(self):
output = super().__repr__()
output += ".set_sensitivity(" + str(self._sensitivity) + ")" if self._sensitivity is not None else ""
return output
def set_sensitivity(self, sensitivity):
"""Sets the sensitivity of the mechanism.
Parameters
----------
sensitivity : float
The sensitivity of the mechanism. Must satisfy `sensitivity` > 0.
Returns
-------
self : class
"""
if not isinstance(sensitivity, Real):
raise TypeError("Sensitivity must be numeric")
if sensitivity < 0:
raise ValueError("Sensitivity must be non-negative")
self._sensitivity = float(sensitivity)
return self
def check_inputs(self, value):
"""Checks that all parameters of the mechanism have been initialised correctly, and that the mechanism is ready
to be used.
Parameters
----------
value : float
The value to be checked
Returns
-------
True if the mechanism is ready to be used.
Raises
------
Exception
If parameters have not been set correctly, or if `value` falls outside the domain of the mechanism.
"""
super().check_inputs(value)
if not isinstance(value, Real):
raise TypeError("Value to be randomised must be a number")
if self._sensitivity is None:
raise ValueError("Sensitivity must be set")
return True
def get_bias(self, value):
"""Returns the bias of the mechanism at a given `value`.
Parameters
----------
value : int or float
The value at which the bias of the mechanism is sought.
Returns
-------
bias : float or None
The bias of the mechanism at `value`.
"""
return 0.0
def get_variance(self, value):
"""Returns the variance of the mechanism at a given `value`.
Parameters
----------
value : float
The value at which the variance of the mechanism is sought.
Returns
-------
bias : float
The variance of the mechanism at `value`.
"""
self.check_inputs(0)
return 2 * (self._sensitivity / (self._epsilon - np.log(1 - self._delta))) ** 2
def randomise(self, value):
"""Randomise `value` with the mechanism.
Parameters
----------
value : float
The value to be randomised.
Returns
-------
float
The randomised value.
"""
self.check_inputs(value)
scale = self._sensitivity / (self._epsilon - np.log(1 - self._delta))
unif_rv = random() - 0.5
return value - scale * np.sign(unif_rv) * np.log(1 - 2 * np.abs(unif_rv))
class LaplaceTruncated(Laplace, TruncationAndFoldingMixin):
"""
The truncated Laplace mechanism, where values outside a pre-described domain are mapped to the closest point
within the domain.
"""
def __init__(self):
super().__init__()
TruncationAndFoldingMixin.__init__(self)
def __repr__(self):
output = super().__repr__()
output += TruncationAndFoldingMixin.__repr__(self)
return output
@copy_docstring(Laplace.get_bias)
def get_bias(self, value):
self.check_inputs(value)
shape = self._sensitivity / self._epsilon
return shape / 2 * (np.exp((self._lower_bound - value) / shape) - np.exp((value - self._upper_bound) / shape))
@copy_docstring(Laplace.get_variance)
def get_variance(self, value):
self.check_inputs(value)
shape = self._sensitivity / self._epsilon
variance = value ** 2 + shape * (self._lower_bound * np.exp((self._lower_bound - value) / shape)
- self._upper_bound * np.exp((value - self._upper_bound) / shape))
variance += (shape ** 2) * (2 - np.exp((self._lower_bound - value) / shape)
- np.exp((value - self._upper_bound) / shape))
variance -= (self.get_bias(value) + value) ** 2
return variance
@copy_docstring(Laplace.check_inputs)
def check_inputs(self, value):
super().check_inputs(value)
TruncationAndFoldingMixin.check_inputs(self, value)
return True
@copy_docstring(Laplace.randomise)
def randomise(self, value):
TruncationAndFoldingMixin.check_inputs(self, value)
noisy_value = super().randomise(value)
return self._truncate(noisy_value)
class LaplaceFolded(Laplace, TruncationAndFoldingMixin):
"""
The folded Laplace mechanism, where values outside a pre-described domain are folded around the domain until they
fall within.
"""
def __init__(self):
super().__init__()
TruncationAndFoldingMixin.__init__(self)
def __repr__(self):
output = super().__repr__()
output += TruncationAndFoldingMixin.__repr__(self)
return output
@copy_docstring(Laplace.get_bias)
def get_bias(self, value):
self.check_inputs(value)
shape = self._sensitivity / self._epsilon
bias = shape * (np.exp((self._lower_bound + self._upper_bound - 2 * value) / shape) - 1)
bias /= np.exp((self._lower_bound - value) / shape) + np.exp((self._upper_bound - value) / shape)
return bias
@copy_docstring(DPMechanism.get_variance)
def get_variance(self, value):
raise NotImplementedError
@copy_docstring(Laplace.check_inputs)
def check_inputs(self, value):
super().check_inputs(value)
TruncationAndFoldingMixin.check_inputs(self, value)
return True
@copy_docstring(Laplace.randomise)
def randomise(self, value):
TruncationAndFoldingMixin.check_inputs(self, value)
noisy_value = super().randomise(value)
return self._fold(noisy_value)
class LaplaceBoundedDomain(LaplaceTruncated):
"""
The bounded Laplace mechanism on a bounded domain. The mechanism draws values directly from the domain, without any
post-processing.
"""
def __init__(self):
super().__init__()
self._scale = None
def _find_scale(self):
if self._epsilon is None or self._delta is None:
raise ValueError("Epsilon and Delta must be set before calling _find_scale().")
eps = self._epsilon
delta = self._delta
diam = self._upper_bound - self._lower_bound
delta_q = self._sensitivity
def _delta_c(shape):
if shape == 0:
return 2.0
return (2 - np.exp(- delta_q / shape) - np.exp(- (diam - delta_q) / shape)) / (1 - np.exp(- diam / shape))
def _f(shape):
return delta_q / (eps - np.log(_delta_c(shape)) - np.log(1 - delta))
left = delta_q / (eps - np.log(1 - delta))
right = _f(left)
old_interval_size = (right - left) * 2
while old_interval_size > right - left:
old_interval_size = right - left
middle = (right + left) / 2
if _f(middle) >= middle:
left = middle
if _f(middle) <= middle:
right = middle
return (right + left) / 2
def _cdf(self, value):
# Allow for infinite epsilon
if self._scale == 0:
return 0 if value < 0 else 1
if value < 0:
return 0.5 * np.exp(value / self._scale)
return 1 - 0.5 * np.exp(-value / self._scale)
def get_effective_epsilon(self):
r"""Gets the effective epsilon of the mechanism, only for strict :math:`\epsilon`-differential privacy. Returns
``None`` if :math:`\delta` is non-zero.
Returns
-------
float
The effective :math:`\epsilon` parameter of the mechanism. Returns ``None`` if `delta` is non-zero.
"""
if self._scale is None:
self._scale = self._find_scale()
if self._delta > 0.0:
return None
return self._sensitivity / self._scale
@copy_docstring(Laplace.get_bias)
def get_bias(self, value):
self.check_inputs(value)
if self._scale is None:
self._scale = self._find_scale()
bias = (self._scale - self._lower_bound + value) / 2 * np.exp((self._lower_bound - value) / self._scale) \
- (self._scale + self._upper_bound - value) / 2 * np.exp((value - self._upper_bound) / self._scale)
bias /= 1 - np.exp((self._lower_bound - value) / self._scale) / 2 \
- np.exp((value - self._upper_bound) / self._scale) / 2
return bias
@copy_docstring(Laplace.get_variance)
def get_variance(self, value):
self.check_inputs(value)
if self._scale is None:
self._scale = self._find_scale()
variance = value**2
variance -= (np.exp((self._lower_bound - value) / self._scale) * (self._lower_bound ** 2)
+ np.exp((value - self._upper_bound) / self._scale) * (self._upper_bound ** 2)) / 2
variance += self._scale * (self._lower_bound * np.exp((self._lower_bound - value) / self._scale)
- self._upper_bound * np.exp((value - self._upper_bound) / self._scale))
variance += (self._scale ** 2) * (2 - np.exp((self._lower_bound - value) / self._scale)
- np.exp((value - self._upper_bound) / self._scale))
variance /= 1 - (np.exp(-(value - self._lower_bound) / self._scale)
+ np.exp(-(self._upper_bound - value) / self._scale)) / 2
variance -= (self.get_bias(value) + value) ** 2
return variance
@copy_docstring(Laplace.randomise)
def randomise(self, value):
self.check_inputs(value)
if self._scale is None:
self._scale = self._find_scale()
value = min(value, self._upper_bound)
value = max(value, self._lower_bound)
unif_rv = random()
unif_rv *= self._cdf(self._upper_bound - value) - self._cdf(self._lower_bound - value)
unif_rv += self._cdf(self._lower_bound - value)
unif_rv -= 0.5
unif_rv = min(unif_rv, 0.5 - 1e-10)
return value - self._scale * np.sign(unif_rv) * np.log(1 - 2 * np.abs(unif_rv))
class LaplaceBoundedNoise(Laplace):
"""
The Laplace mechanism with bounded noise, only applicable for approximate differential privacy (delta > 0).
"""
def __init__(self):
super().__init__()
self._scale = None
self._noise_bound = None
def set_epsilon_delta(self, epsilon, delta):
r"""Set the privacy parameters :math:`\epsilon` and :math:`\delta` for the mechanism.
Epsilon must be strictly positive, `epsilon` > 0. `delta` must be strictly in the interval (0, 0.5).
- For zero `epsilon`, use :class:`.Uniform`.
- For zero `delta`, use :class:`.Laplace`.
Parameters
----------
epsilon : float
The value of epsilon for achieving :math:`(\epsilon,\delta)`-differential privacy with the mechanism. Must
have `epsilon > 0`.
delta : float
The value of delta for achieving :math:`(\epsilon,\delta)`-differential privacy with the mechanism. Must
have `0 < delta < 0.5`.
Returns
-------
self : class
"""
if epsilon == 0:
raise ValueError("Epsilon must be strictly positive. For zero epsilon, use :class:`.Uniform`.")
if isinstance(delta, Real) and not 0 < delta < 0.5:
raise ValueError("Delta must be strictly in (0,0.5). For zero delta, use :class:`.Laplace`.")
return DPMechanism.set_epsilon_delta(self, epsilon, delta)
def _cdf(self, value):
if self._scale == 0:
return 0 if value < 0 else 1
if value < 0:
return 0.5 * np.exp(value / self._scale)
return 1 - 0.5 * np.exp(-value / self._scale)
@copy_docstring(Laplace.get_bias)
def get_bias(self, value):
return 0.0
@copy_docstring(DPMechanism.get_variance)
def get_variance(self, value):
raise NotImplementedError
@copy_docstring(Laplace.randomise)
def randomise(self, value):
self.check_inputs(value)
if self._scale is None or self._noise_bound is None:
self._scale = self._sensitivity / self._epsilon
self._noise_bound = -1 if self._scale == 0 else \
self._scale * np.log(1 + (np.exp(self._epsilon) - 1) / 2 / self._delta)
unif_rv = random()
unif_rv *= self._cdf(self._noise_bound) - self._cdf(- self._noise_bound)
unif_rv += self._cdf(- self._noise_bound)
unif_rv -= 0.5
return value - self._scale * (np.sign(unif_rv) * np.log(1 - 2 * np.abs(unif_rv)))
