# MIT License
#
# Copyright (C) IBM Corporation 2019
#
# Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated
# documentation files (the "Software"), to deal in the Software without restriction, including without limitation the
# rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit
# persons to whom the Software is furnished to do so, subject to the following conditions:
#
# The above copyright notice and this permission notice shall be included in all copies or substantial portions of the
# Software.
#
# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE
# WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
# TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
# SOFTWARE.
"""
The Bingham mechanism in differential privacy, for estimating the first eigenvector of a covariance matrix.
"""
import secrets
from numbers import Real
import numpy as np
from diffprivlib.mechanisms.base import DPMechanism
from diffprivlib.utils import copy_docstring
[docs]
class Bingham(DPMechanism):
r"""
The Bingham mechanism in differential privacy.
Used to estimate the first eigenvector (associated with the largest eigenvalue) of a covariance matrix.
Paper link: http://eprints.whiterose.ac.uk/123206/7/simbingham8.pdf
Parameters
----------
epsilon : float
Privacy parameter :math:`\epsilon` for the mechanism. Must be in (0, ∞].
sensitivity : float, 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, sensitivity=1.0, random_state=None):
super().__init__(epsilon=epsilon, delta=0, random_state=random_state)
self.sensitivity = self._check_sensitivity(sensitivity)
if isinstance(self._rng, secrets.SystemRandom):
self._rng = np.random.default_rng()
@classmethod
def _check_epsilon_delta(cls, epsilon, delta):
if not delta == 0:
raise ValueError("Delta must be zero")
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, np.ndarray):
raise TypeError(f"Value to be randomised must be a numpy array, got {type(value)}")
if value.ndim != 2:
raise ValueError(f"Array must be 2-dimensional, got {value.ndim} dimensions")
if value.shape[0] != value.shape[1]:
raise ValueError(f"Array must be square, got {value.shape[0]} x {value.shape[1]}")
if not np.allclose(value, value.T):
raise ValueError("Array must be symmetric, supplied array is not.")
return True
@copy_docstring(DPMechanism.bias)
def bias(self, value):
raise NotImplementedError
@copy_docstring(DPMechanism.variance)
def variance(self, value):
raise NotImplementedError
[docs]
def randomise(self, value):
"""Randomise `value` with the mechanism.
Parameters
----------
value : numpy array
The data to be randomised.
Returns
-------
numpy array
The randomised eigenvector.
"""
self._check_all(value)
eigvals, eigvecs = np.linalg.eigh(value)
dims = value.shape[0]
if dims == 1:
return np.ones((1, 1))
if self.sensitivity / self.epsilon == 0:
return eigvecs[:, eigvals.argmax()]
value_translated = self.epsilon * (eigvals.max() * np.eye(dims) - value) / 4 / self.sensitivity
translated_eigvals = np.linalg.eigvalsh(value_translated)
left, right, mid = 1, dims, (1 + dims) / 2
old_interval_size = (right - left) * 2
while right - left < old_interval_size:
old_interval_size = right - left
mid = (right + left) / 2
f_mid = np.array([1 / (mid + 2 * eig) for eig in translated_eigvals]).sum()
if f_mid <= 1:
right = mid
if f_mid >= 1:
left = mid
b_const = mid
omega = np.eye(dims) + 2 * value_translated / b_const
omega_inv = np.linalg.inv(omega)
norm_const = np.exp(-(dims - b_const) / 2) * ((dims / b_const) ** (dims / 2))
while True:
rnd_vec = self._rng.multivariate_normal(np.zeros(dims), omega_inv / 4, size=4).sum(axis=0)
unit_vec = rnd_vec / np.linalg.norm(rnd_vec)
prob = np.exp(-unit_vec.dot(value_translated).dot(unit_vec)) / norm_const\
/ ((unit_vec.dot(omega).dot(unit_vec)) ** (dims / 2))
if self._rng.random() <= prob:
return unit_vec