# MIT License
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# Copyright (C) IBM Corporation 2019
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"""
Implementation of the standard exponential mechanism, and its derivative, the hierarchical mechanism.
"""
from numbers import Real
import numpy as np
from numpy.random import random
from diffprivlib.mechanisms.base import DPMechanism
from diffprivlib.mechanisms.binary import Binary
from diffprivlib.utils import copy_docstring
class Exponential(DPMechanism):
"""
The exponential mechanism for achieving differential privacy on categorical inputs, as first proposed by McSherry
and Talwar.
The exponential mechanism achieves differential privacy by randomly choosing an output value for a given input
value, with greater probability given to values 'closer' to the input, as measured by a given utility function.
Paper link: https://www.cs.drexel.edu/~greenie/privacy/mdviadp.pdf
"""
def __init__(self):
super().__init__()
self._domain_values = None
self._utility_values = None
self._normalising_constant = None
self._sensitivity = None
self._balanced_tree = False
def __repr__(self):
output = super().__repr__()
output += ".set_utility(" + str(self.get_utility_list()) + ")" if self._utility_values is not None else ""
return output
def set_utility(self, utility_list):
"""Sets the utility function of the mechanism. The utility function is used to determine the probability of
selecting an output for a given input.
The utility function is set by `utility_list`, which is a list of pairwise 'distances' between values in the
mechanism's domain. As the mechanisms's domain is set by the values in `utility_list`, all possible pairs in
`utility_list` must be accounted for. The utility function is symmetric, meaning the distance from `a` to
`b` is the same as the distance from `b` to `a`. Setting the second distance will overwrite the first.
Parameters
----------
utility_list : list of tuples
The utility list of the mechanism. Must be specified as a list of tuples, of the form ("value1", "value2",
utility), where each `value` is a string and `utility` is a strictly positive float. A `utility` must be
specified for every pair of values given in the `utility_list`.
Returns
-------
self : class
Raises
------
TypeError
If the `value` components of each tuple are not strings of if the `utility` component is not a float.
ValueError
If the `utility` component is zero or negative.
"""
if not isinstance(utility_list, list):
raise TypeError("Utility must be given in a list")
self._normalising_constant = None
utility_values = {}
domain_values = []
sensitivity = 0
for _utility_sub_list in utility_list:
value1, value2, utility_value = _utility_sub_list
if not isinstance(value1, str) or not isinstance(value2, str):
raise TypeError("Utility keys must be strings")
if not isinstance(utility_value, Real):
raise TypeError("Utility value must be a number")
if utility_value < 0.0:
raise ValueError("Utility values must be non-negative")
sensitivity = max(sensitivity, utility_value)
if value1 not in domain_values:
domain_values.append(value1)
if value2 not in domain_values:
domain_values.append(value2)
if value1 == value2:
continue
if value1 < value2:
utility_values[(value1, value2)] = utility_value
else:
utility_values[(value2, value1)] = utility_value
self._utility_values = utility_values
self._sensitivity = sensitivity
self._domain_values = domain_values
self._check_utility_full(domain_values)
return self
def _check_utility_full(self, domain_values):
for val1 in domain_values:
for val2 in domain_values:
if val1 >= val2:
continue
if (val1, val2) not in self._utility_values:
raise ValueError("Utility value for (%s) missing" % (val1 + ", " + val2))
return True
def get_utility_list(self):
"""Gets the utility list of the mechanism, in the same form as accepted by `.set_utility_list`.
Returns
-------
utility_list : list of tuples (str, str, float), or None
Returns a list of tuples of the form ("value1", "value2", utility), or `None` if the utility has not yet
been set.
"""
if self._utility_values is None:
return None
utility_list = []
for _key, _utility in self._utility_values.items():
value1, value2 = _key
utility_list.append((value1, value2, _utility))
return utility_list
def _build_normalising_constant(self, re_eval=False):
balanced_tree = True
first_constant_value = None
normalising_constant = {}
for _base_leaf in self._domain_values:
constant_value = 0.0
for _target_leaf in self._domain_values:
constant_value += self._get_prob(_base_leaf, _target_leaf)
normalising_constant[_base_leaf] = constant_value
if first_constant_value is None:
first_constant_value = constant_value
elif not np.isclose(constant_value, first_constant_value):
balanced_tree = False
# If the tree is balanced, we can eliminate the doubling factor
if balanced_tree and not re_eval:
self._balanced_tree = True
return self._build_normalising_constant(True)
return normalising_constant
def _get_utility(self, value1, value2):
if value1 == value2:
return 0
if value1 > value2:
return self._get_utility(value1=value2, value2=value1)
return self._utility_values[(value1, value2)]
def _get_prob(self, value1, value2):
if value1 == value2:
return 1.0
balancing_factor = 1 if self._balanced_tree else 2
return np.exp(- self._epsilon * self._get_utility(value1, value2) / balancing_factor / self._sensitivity)
@copy_docstring(Binary.check_inputs)
def check_inputs(self, value):
super().check_inputs(value)
if self._utility_values is None:
raise ValueError("Utility function must be set")
if self._normalising_constant is None:
self._normalising_constant = self._build_normalising_constant()
if not isinstance(value, str):
raise TypeError("Value to be randomised must be a string")
if value not in self._domain_values:
raise ValueError("Value \"%s\" not in domain" % value)
return True
def set_epsilon_delta(self, epsilon, delta):
r"""Sets the value of :math:`\epsilon` and :math:`\delta` to be used by the mechanism.
For the exponential mechanism, `delta` must be zero and `epsilon` must be strictly positive.
Parameters
----------
epsilon : float
The value of epsilon for achieving :math:`(\epsilon,\delta)`-differential privacy with the mechanism. Must
have `epsilon > 0`.
delta : float
For the exponential mechanism, `delta` must be zero.
Returns
-------
self : class
Raises
------
ValueError
If `epsilon` is zero or negative, or if `delta` is non-zero.
"""
if not delta == 0:
raise ValueError("Delta must be zero")
self._normalising_constant = None
return super().set_epsilon_delta(epsilon, delta)
@copy_docstring(DPMechanism.get_bias)
def get_bias(self, value):
raise NotImplementedError
@copy_docstring(DPMechanism.get_variance)
def get_variance(self, value):
raise NotImplementedError
@copy_docstring(Binary.randomise)
def randomise(self, value):
self.check_inputs(value)
unif_rv = random() * self._normalising_constant[value]
cum_prob = 0
_target_value = None
for _target_value in self._normalising_constant.keys():
cum_prob += self._get_prob(value, _target_value)
if unif_rv <= cum_prob:
return _target_value
return _target_value
class ExponentialHierarchical(Exponential):
"""
Adaptation of the exponential mechanism to hierarchical data. Simplifies the process of specifying utility values,
as the values can be inferred from the hierarchy.
"""
def __init__(self):
super().__init__()
self._list_hierarchy = None
def __repr__(self):
output = super().__repr__()
output += ".set_hierarchy(" + str(self._list_hierarchy) + ")" if self._list_hierarchy is not None else ""
return output
def _build_hierarchy(self, nested_list, parent_node=None):
if parent_node is None:
parent_node = []
hierarchy = {}
for _i, _value in enumerate(nested_list):
if isinstance(_value, str):
hierarchy[_value] = parent_node + [_i]
elif not isinstance(_value, list):
raise TypeError("All leaves of the hierarchy must be a string " +
"(see node " + (parent_node + [_i]).__str__() + ")")
else:
hierarchy.update(self._build_hierarchy(_value, parent_node + [_i]))
self._check_hierarchy_height(hierarchy)
return hierarchy
@staticmethod
def _check_hierarchy_height(hierarchy):
hierarchy_height = None
for _value, _hierarchy_locator in hierarchy.items():
if hierarchy_height is None:
hierarchy_height = len(_hierarchy_locator)
elif len(_hierarchy_locator) != hierarchy_height:
raise ValueError("Leaves of the hierarchy must all be at the same level " +
"(node %s is at level %d instead of hierarchy height %d)" %
(_hierarchy_locator.__str__(), len(_hierarchy_locator), hierarchy_height))
@staticmethod
def _build_utility_list(hierarchy):
if not isinstance(hierarchy, dict):
raise TypeError("Hierarchy for _build_utility_list must be a dict")
utility_list = []
hierarchy_height = None
for _root_value, _root_hierarchy_locator in hierarchy.items():
if hierarchy_height is None:
hierarchy_height = len(_root_hierarchy_locator)
for _target_value, _target_hierarchy_locator in hierarchy.items():
if _root_value >= _target_value:
continue
i = 0
while (i < len(_root_hierarchy_locator) and
_root_hierarchy_locator[i] == _target_hierarchy_locator[i]):
i += 1
utility_list.append([_root_value, _target_value, hierarchy_height - i])
return utility_list
def set_hierarchy(self, list_hierarchy):
"""Sets the hierarchy of the hierarchical exponential mechanism.
The hierarchy is specified as a list of lists, where each leaf node is a string, and lies at the same depth as
each other leaf node. The utility between each leaf node is then calculated as
Parameters
----------
list_hierarchy : nested list of str
The hierarchy as specified as a nested list of string. Each string must be a leaf node, and each leaf node
must lie at the same depth in the hierarchy.
Returns
-------
self : class
"""
if not isinstance(list_hierarchy, list):
raise TypeError("Hierarchy must be a list")
self._list_hierarchy = list_hierarchy
hierarchy = self._build_hierarchy(list_hierarchy)
self.set_utility(self._build_utility_list(hierarchy))
return self
@copy_docstring(DPMechanism.get_bias)
def get_bias(self, value):
raise NotImplementedError
@copy_docstring(DPMechanism.get_variance)
def get_variance(self, value):
raise NotImplementedError
