# Copyright 2018 The TensorFlow Authors. All Rights Reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
# ==============================================================================
"""RDP analysis of the Sampled Gaussian Mechanism.
Functionality for computing Renyi differential privacy (RDP) of an additive
Sampled Gaussian Mechanism (SGM). Its public interface consists of two methods:
compute_rdp(q, noise_multiplier, T, orders) computes RDP for SGM iterated
T times.
get_privacy_spent(orders, rdp, target_eps, target_delta) computes delta
(or eps) given RDP at multiple orders and
a target value for eps (or delta).
Example use:
Suppose that we have run an SGM applied to a function with l2-sensitivity 1.
Its parameters are given as a list of tuples (q1, sigma1, T1), ...,
(qk, sigma_k, Tk), and we wish to compute eps for a given delta.
The example code would be:
max_order = 32
orders = range(2, max_order + 1)
rdp = np.zeros_like(orders, dtype=float)
for q, sigma, T in parameters:
rdp += rdp_accountant.compute_rdp(q, sigma, T, orders)
eps, _, opt_order = rdp_accountant.get_privacy_spent(rdp, target_delta=delta)
"""
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
import math
import sys
import numpy as np
from scipy import special
import six
########################
# LOG-SPACE ARITHMETIC #
########################
def _log_add(logx, logy):
"""Add two numbers in the log space."""
a, b = min(logx, logy), max(logx, logy)
if a == -np.inf: # adding 0
return b
# Use exp(a) + exp(b) = (exp(a - b) + 1) * exp(b)
return math.log1p(math.exp(a - b)) + b # log1p(x) = log(x + 1)
def _log_sub(logx, logy):
"""Subtract two numbers in the log space. Answer must be non-negative."""
if logx < logy:
raise ValueError("The result of subtraction must be non-negative.")
if logy == -np.inf: # subtracting 0
return logx
if logx == logy:
return -np.inf # 0 is represented as -np.inf in the log space.
try:
# Use exp(x) - exp(y) = (exp(x - y) - 1) * exp(y).
return math.log(math.expm1(logx - logy)) + logy # expm1(x) = exp(x) - 1
except OverflowError:
return logx
def _log_print(logx):
"""Pretty print."""
if logx < math.log(sys.float_info.max):
return "{}".format(math.exp(logx))
else:
return "exp({})".format(logx)
def _compute_log_a_int(q, sigma, alpha):
"""Compute log(A_alpha) for integer alpha. 0 < q < 1."""
assert isinstance(alpha, six.integer_types)
# Initialize with 0 in the log space.
log_a = -np.inf
for i in range(alpha + 1):
log_coef_i = (
math.log(special.binom(alpha, i)) + i * math.log(q) +
(alpha - i) * math.log(1 - q))
s = log_coef_i + (i * i - i) / (2 * (sigma**2))
log_a = _log_add(log_a, s)
return float(log_a)
def _compute_log_a_frac(q, sigma, alpha):
"""Compute log(A_alpha) for fractional alpha. 0 < q < 1."""
# The two parts of A_alpha, integrals over (-inf,z0] and [z0, +inf), are
# initialized to 0 in the log space:
log_a0, log_a1 = -np.inf, -np.inf
i = 0
z0 = sigma**2 * math.log(1 / q - 1) + .5
while True: # do ... until loop
coef = special.binom(alpha, i)
log_coef = math.log(abs(coef))
j = alpha - i
log_t0 = log_coef + i * math.log(q) + j * math.log(1 - q)
log_t1 = log_coef + j * math.log(q) + i * math.log(1 - q)
log_e0 = math.log(.5) + _log_erfc((i - z0) / (math.sqrt(2) * sigma))
log_e1 = math.log(.5) + _log_erfc((z0 - j) / (math.sqrt(2) * sigma))
log_s0 = log_t0 + (i * i - i) / (2 * (sigma**2)) + log_e0
log_s1 = log_t1 + (j * j - j) / (2 * (sigma**2)) + log_e1
if coef > 0:
log_a0 = _log_add(log_a0, log_s0)
log_a1 = _log_add(log_a1, log_s1)
else:
log_a0 = _log_sub(log_a0, log_s0)
log_a1 = _log_sub(log_a1, log_s1)
i += 1
if max(log_s0, log_s1) < -30:
break
return _log_add(log_a0, log_a1)
def _compute_log_a(q, sigma, alpha):
"""Compute log(A_alpha) for any positive finite alpha."""
if float(alpha).is_integer():
return _compute_log_a_int(q, sigma, int(alpha))
else:
return _compute_log_a_frac(q, sigma, alpha)
def _log_erfc(x):
"""Compute log(erfc(x)) with high accuracy for large x."""
try:
return math.log(2) + special.log_ndtr(-x * 2**.5)
except NameError:
# If log_ndtr is not available, approximate as follows:
r = special.erfc(x)
if r == 0.0:
# Using the Laurent series at infinity for the tail of the erfc function:
# erfc(x) ~ exp(-x^2-.5/x^2+.625/x^4)/(x*pi^.5)
# To verify in Mathematica:
# Series[Log[Erfc[x]] + Log[x] + Log[Pi]/2 + x^2, {x, Infinity, 6}]
return (-math.log(math.pi) / 2 - math.log(x) - x**2 - .5 * x**-2 +
.625 * x**-4 - 37. / 24. * x**-6 + 353. / 64. * x**-8)
else:
return math.log(r)
def _compute_delta(orders, rdp, eps):
"""Compute delta given a list of RDP values and target epsilon.
Args:
orders: An array (or a scalar) of orders.
rdp: A list (or a scalar) of RDP guarantees.
eps: The target epsilon.
Returns:
Pair of (delta, optimal_order).
Raises:
ValueError: If input is malformed.
"""
orders_vec = np.atleast_1d(orders)
rdp_vec = np.atleast_1d(rdp)
if len(orders_vec) != len(rdp_vec):
raise ValueError("Input lists must have the same length.")
deltas = np.exp((rdp_vec - eps) * (orders_vec - 1))
idx_opt = np.argmin(deltas)
return min(deltas[idx_opt], 1.), orders_vec[idx_opt]
def _compute_eps(orders, rdp, delta):
"""Compute epsilon given a list of RDP values and target delta.
Args:
orders: An array (or a scalar) of orders.
rdp: A list (or a scalar) of RDP guarantees.
delta: The target delta.
Returns:
Pair of (eps, optimal_order).
Raises:
ValueError: If input is malformed.
"""
orders_vec = np.atleast_1d(orders)
rdp_vec = np.atleast_1d(rdp)
if len(orders_vec) != len(rdp_vec):
raise ValueError("Input lists must have the same length.")
eps = rdp_vec - math.log(delta) / (orders_vec - 1)
idx_opt = np.nanargmin(eps) # Ignore NaNs
return eps[idx_opt], orders_vec[idx_opt]
def _compute_rdp(q, sigma, alpha):
"""Compute RDP of the Sampled Gaussian mechanism at order alpha.
Args:
q: The sampling rate.
sigma: The std of the additive Gaussian noise.
alpha: The order at which RDP is computed.
Returns:
RDP at alpha, can be np.inf.
"""
if q == 0:
return 0
if q == 1.:
return alpha / (2 * sigma**2)
if np.isinf(alpha):
return np.inf
return _compute_log_a(q, sigma, alpha) / (alpha - 1)
def compute_rdp(q, noise_multiplier, steps, orders):
"""Compute RDP of the Sampled Gaussian Mechanism.
Args:
q: The sampling rate.
noise_multiplier: The ratio of the standard deviation of the Gaussian noise
to the l2-sensitivity of the function to which it is added.
steps: The number of steps.
orders: An array (or a scalar) of RDP orders.
Returns:
The RDPs at all orders, can be np.inf.
"""
if np.isscalar(orders):
rdp = _compute_rdp(q, noise_multiplier, orders)
else:
rdp = np.array([_compute_rdp(q, noise_multiplier, order)
for order in orders])
return rdp * steps
def get_privacy_spent(orders, rdp, target_eps=None, target_delta=None):
"""Compute delta (or eps) for given eps (or delta) from RDP values.
Args:
orders: An array (or a scalar) of RDP orders.
rdp: An array of RDP values. Must be of the same length as the orders list.
target_eps: If not None, the epsilon for which we compute the corresponding
delta.
target_delta: If not None, the delta for which we compute the corresponding
epsilon. Exactly one of target_eps and target_delta must be None.
Returns:
eps, delta, opt_order.
Raises:
ValueError: If target_eps and target_delta are messed up.
"""
if target_eps is None and target_delta is None:
raise ValueError(
"Exactly one out of eps and delta must be None. (Both are).")
if target_eps is not None and target_delta is not None:
raise ValueError(
"Exactly one out of eps and delta must be None. (None is).")
if target_eps is not None:
delta, opt_order = _compute_delta(orders, rdp, target_eps)
return target_eps, delta, opt_order
else:
eps, opt_order = _compute_eps(orders, rdp, target_delta)
return eps, target_delta, opt_order
def compute_rdp_from_ledger(ledger, orders):
"""Compute RDP of Sampled Gaussian Mechanism from ledger.
Args:
ledger: A formatted privacy ledger.
orders: An array (or a scalar) of RDP orders.
Returns:
RDP at all orders, can be np.inf.
"""
total_rdp = np.zeros_like(orders, dtype=float)
for sample in ledger:
# Compute equivalent z from l2_clip_bounds and noise stddevs in sample.
# See https://arxiv.org/pdf/1812.06210.pdf for derivation of this formula.
effective_z = sum([
(q.noise_stddev / q.l2_norm_bound)**-2 for q in sample.queries])**-0.5
total_rdp += compute_rdp(
sample.selection_probability, effective_z, 1, orders)
return total_rdp
