# Source code for objax.privacy.dpsgd.privacyaccountant

```# Copyright 2020 Google LLC
#
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#
# Unless required by applicable law or agreed to in writing, software
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
"""RDP analysis of the Sampled Gaussian Mechanism.

Mainly copied from the privacy analysis in TF Privacy.
"""
import math
from typing import Tuple, Union
import numpy as np
from scipy import special
import six

__all__ = ["analyze_dp", "convert_renyidp_to_dp", "analyze_renyi",
"analyze_dp_sample_without_replacement", "analyze_renyi_sample_without_replacement"]

def _log_add(logx: float, logy: float) -> float:
"""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: float, logy: float) -> float:
"""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_sub_sign(logx: float, logy: float) -> Tuple[bool, float]:
"""Returns log(exp(logx)-exp(logy)) and its sign."""
if logx > logy:
s = True
mag = logx + np.log(1 - np.exp(logy - logx))
elif logx < logy:
s = False
mag = logy + np.log(1 - np.exp(logx - logy))
else:
s = True
mag = -np.inf

return s, mag

def _log_comb(n, k) -> float:
return (special.gammaln(n + 1) - special.gammaln(k + 1)
- special.gammaln(n - k + 1))

def _compute_log_a_int(q: float, sigma: int, alpha: int) -> float:
"""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 = _log_comb(alpha, i) + i * math.log(q) + (alpha - i) * math.log(1 - q)

s = log_coef_i + (i * i - i) / (2 * (sigma ** 2))

return float(log_a)

def _compute_log_a_frac(q: float, sigma: float, alpha: float) -> float:
"""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:
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

def _compute_log_a(q: float, sigma: Union[float, int], alpha: float) -> float:
"""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: float) -> float:
"""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: Tuple[float, ...], rdp: Tuple[float, ...], eps: float) -> Tuple[float, float]:
"""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 eps < 0:
raise ValueError("Value of privacy loss bound epsilon must be >=0.")
if len(orders_vec) != len(rdp_vec):
raise ValueError("Input lists must have the same length.")

# Basic bound (see https://arxiv.org/abs/1702.07476 Proposition 3 in v3):
#   delta = min( np.exp((rdp_vec - eps) * (orders_vec - 1)) )

# Improved bound from https://arxiv.org/abs/2004.00010 Proposition 12 (in v4):
logdeltas = []  # work in log space to avoid overflows
for (a, r) in zip(orders_vec, rdp_vec):
if a < 1:
raise ValueError("Renyi divergence order must be >=1.")
if r < 0:
raise ValueError("Renyi divergence must be >=0.")
# For small alpha, we are better of with bound via KL divergence:
# delta <= sqrt(1-exp(-KL)).
# Take a min of the two bounds.
logdelta = 0.5 * math.log1p(-math.exp(-r))
if a > 1.01:
# This bound is not numerically stable as alpha->1.
# Thus we have a min value for alpha.
# The bound is also not useful for small alpha, so doesn't matter.
rdp_bound = (a - 1) * (r - eps + math.log1p(-1 / a)) - math.log(a)
logdelta = min(logdelta, rdp_bound)

logdeltas.append(logdelta)

idx_opt = np.argmin(logdeltas)
return min(math.exp(logdeltas[idx_opt]), 1.), orders_vec[idx_opt]

def _compute_eps(orders: Tuple[float, ...], rdp: Tuple[float, ...], delta: float) -> Tuple[float, float]:
"""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 delta <= 0:
raise ValueError("Privacy failure probability bound delta must be >0.")
if len(orders_vec) != len(rdp_vec):
raise ValueError("Input lists must have the same length.")

# Basic bound (see https://arxiv.org/abs/1702.07476 Proposition 3 in v3):
#   eps = min( rdp_vec - math.log(delta) / (orders_vec - 1) )

# Improved bound from https://arxiv.org/abs/2004.00010 Proposition 12 (in v4).
# Also appears in https://arxiv.org/abs/2001.05990 Equation 20 (in v1).
eps_vec = []
for (a, r) in zip(orders_vec, rdp_vec):
if a < 1:
raise ValueError("Renyi divergence order must be >=1.")
if r < 0:
raise ValueError("Renyi divergence must be >=0.")

if delta**2 + math.expm1(-r) >= 0:
# In this case, we can simply bound via KL divergence:
# delta <= sqrt(1-exp(-KL)).
eps = 0  # No need to try further computation if we have eps = 0.
elif a > 1.01:
# This bound is not numerically stable as alpha->1.
# Thus we have a min value of alpha.
# The bound is also not useful for small alpha, so doesn't matter.
eps = r + math.log1p(-1 / a) - math.log(delta * a) / (a - 1)
else:
# In this case we can't do anything. E.g., asking for delta = 0.
eps = np.inf
eps_vec.append(eps)

idx_opt = np.argmin(eps_vec)
return max(0, eps_vec[idx_opt]), orders_vec[idx_opt]

def _stable_inplace_diff_in_log(vec, signs, n=-1):
"""Replaces the first n-1 dims of vec with the log of abs difference operator.

Args:
vec: numpy array of floats with size larger than 'n'
signs: Optional numpy array of bools with the same size as vec in case one
needs to compute partial differences vec and signs jointly describe a
vector of real numbers' sign and abs in log scale.
n: Optonal upper bound on number of differences to compute. If negative, all
differences are computed.

Returns:
The first n-1 dimension of vec and signs will store the log-abs and sign of
the difference.

Raises:
ValueError: If input is malformed.
"""

assert vec.shape == signs.shape
if n < 0:
n = np.max(vec.shape) - 1
else:
assert np.max(vec.shape) >= n + 1
for j in range(0, n, 1):
if signs[j] == signs[j + 1]:  # When the signs are the same
# if the signs are both positive, then we can just use the standard one
signs[j], vec[j] = _log_sub_sign(vec[j + 1], vec[j])
# otherwise, we do that but toggle the sign
if not signs[j + 1]:
signs[j] = ~signs[j]
else:  # When the signs are different.
vec[j] = _log_add(vec[j], vec[j + 1])
signs[j] = signs[j + 1]

def _get_forward_diffs(fun, n):
"""Computes up to nth order forward difference evaluated at 0.

See Theorem 27 of https://arxiv.org/pdf/1808.00087.pdf

Args:
fun: Function to compute forward differences of.
n: Number of differences to compute.

Returns:
Pair (deltas, signs_deltas) of the log deltas and their signs.
"""
func_vec = np.zeros(n + 3)
signs_func_vec = np.ones(n + 3, dtype=bool)

# ith coordinate of deltas stores log(abs(ith order discrete derivative))
deltas = np.zeros(n + 2)
signs_deltas = np.zeros(n + 2, dtype=bool)
for i in range(1, n + 3, 1):
func_vec[i] = fun(1.0 * (i - 1))
for i in range(0, n + 2, 1):
# Diff in log scale
_stable_inplace_diff_in_log(func_vec, signs_func_vec, n=n + 2 - i)
deltas[i] = func_vec
signs_deltas[i] = signs_func_vec
return deltas, signs_deltas

def _analyze_renyi(q: float, sigma: float, alpha: float) -> float:
"""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)

[docs]def analyze_renyi(q: float, noise_multiplier: float, steps: int, orders: Union[float, Tuple[float, ...]]):
"""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 = _analyze_renyi(q, noise_multiplier, orders)
else:
rdp = np.array(
[_analyze_renyi(q, noise_multiplier, order) for order in orders])

return rdp * steps

[docs]def analyze_renyi_sample_without_replacement(q: float, noise_multiplier: float, steps: int,
orders: Union[float, Tuple[float, ...]]) -> float:
"""Compute RDP of Gaussian Mechanism using sampling without replacement.

This function applies to the following schemes:
1. Sampling w/o replacement: Sample a uniformly random subset of size m = q*n.
2. ``Replace one data point'' version of differential privacy, i.e., n is
considered public information.

Reference: Theorem 27 of https://arxiv.org/pdf/1808.00087.pdf (A strengthened
version applies subsampled-Gaussian mechanism)
- Wang, Balle, Kasiviswanathan. "Subsampled Renyi Differential Privacy and
Analytical Moments Accountant." AISTATS'2019.

Args:
q: The sampling proportion =  m / n.  Assume m is an integer <= n.
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 = _analyze_renyi_sample_without_replacement_scalar(
q, noise_multiplier, orders)
else:
rdp = np.array([
_analyze_renyi_sample_without_replacement_scalar(q, noise_multiplier,
order)
for order in orders
])

return rdp * steps

def _analyze_renyi_sample_without_replacement_scalar(q: float, sigma: float, alpha: Union[float, int]) -> float:
"""Compute RDP of the Sampled Gaussian mechanism at order alpha.

Args:
q: The sampling proportion =  m / n.  Assume m is an integer <= n.
sigma: The std of the additive Gaussian noise.
alpha: The order at which RDP is computed.

Returns:
RDP at alpha, can be np.inf.
"""

assert (q <= 1) and (q >= 0) and (alpha >= 1)

if q == 0:
return 0

if q == 1.:
return alpha / (2 * sigma**2)

if np.isinf(alpha):
return np.inf

if float(alpha).is_integer():
return _analyze_renyi_sample_without_replacement_int(q, sigma, int(alpha)) / (alpha - 1)
else:
# When alpha not an integer, we apply Corollary 10 of [WBK19] to interpolate
# the CGF and obtain an upper bound
alpha_f = math.floor(alpha)
alpha_c = math.ceil(alpha)

x = _analyze_renyi_sample_without_replacement_int(q, sigma, alpha_f)
y = _analyze_renyi_sample_without_replacement_int(q, sigma, alpha_c)
t = alpha - alpha_f
return ((1 - t) * x + t * y) / (alpha - 1)

def _analyze_renyi_sample_without_replacement_int(q: float, sigma: float, alpha: Union[float, int]) -> float:
"""Compute log(A_alpha) for integer alpha, subsampling without replacement.

When alpha is smaller than max_alpha, compute the bound Theorem 27 exactly,
otherwise compute the bound with Stirling approximation.

Args:
q: The sampling proportion = m / n.  Assume m is an integer <= n.
sigma: The std of the additive Gaussian noise.
alpha: The order at which RDP is computed.

Returns:
RDP at alpha, can be np.inf.
"""

max_alpha = 256
assert isinstance(alpha, six.integer_types)

if np.isinf(alpha):
return np.inf
elif alpha == 1:
return 0

def cgf(x):
# Return rdp(x+1)*x, the rdp of Gaussian mechanism is alpha/(2*sigma**2)
return x * 1.0 * (x + 1) / (2.0 * sigma**2)

def func(x):
# Return the rdp of Gaussian mechanism
return 1.0 * x / (2.0 * sigma**2)

# Initialize with 1 in the log space.
log_a = 0
# Calculates the log term when alpha = 2
log_f2m1 = func(2.0) + np.log(1 - np.exp(-func(2.0)))
if alpha <= max_alpha:
# We need forward differences of exp(cgf)
# The following line is the numerically stable way of implementing it.
# The output is in polar form with logarithmic magnitude
deltas, _ = _get_forward_diffs(cgf, alpha)
# Compute the bound exactly requires book keeping of O(alpha**2)

for i in range(2, alpha + 1):
if i == 2:
s = 2 * np.log(q) + _log_comb(alpha, 2) + np.minimum(
np.log(4) + log_f2m1,
func(2.0) + np.log(2))
elif i > 2:
delta_lo = deltas[int(2 * np.floor(i / 2.0)) - 1]
delta_hi = deltas[int(2 * np.ceil(i / 2.0)) - 1]
s = np.log(4) + 0.5 * (delta_lo + delta_hi)
s = np.minimum(s, np.log(2) + cgf(i - 1))
s += i * np.log(q) + _log_comb(alpha, i)
return float(log_a)
else:
# Compute the bound with stirling approximation. Everything is O(x) now.
for i in range(2, alpha + 1):
if i == 2:
s = 2 * np.log(q) + _log_comb(alpha, 2) + np.minimum(
np.log(4) + log_f2m1,
func(2.0) + np.log(2))
else:
s = np.log(2) + cgf(i - 1) + i * np.log(q) + _log_comb(alpha, i)

return log_a

def analyze_heterogeneous_renyi(sampling_probabilities: Tuple[float, ...], noise_multipliers: Tuple[float, ...],
steps_list: Tuple[int, ...], orders: Tuple[Union[int, float], ...]) -> float:
"""Computes RDP of Heteregoneous Applications of Sampled Gaussian Mechanisms.

Args:
sampling_probabilities: A list containing the sampling rates.
noise_multipliers: A list containing the noise multipliers: the ratio of the
standard deviation of the Gaussian noise to the l2-sensitivity of the
function to which it is added.
steps_list: A list containing the number of steps at each
`sampling_probability` and `noise_multiplier`.
orders: An array (or a scalar) of RDP orders.

Returns:
The RDPs at all orders. Can be `np.inf`.
"""
assert len(sampling_probabilities) == len(noise_multipliers)

rdp = 0
for q, noise_multiplier, steps in zip(sampling_probabilities,
noise_multipliers, steps_list):
rdp += analyze_renyi(q, noise_multiplier, steps, orders)

return rdp

[docs]def convert_renyidp_to_dp(orders: Tuple[float, ...], rdp: Tuple[float, ...], target_eps: float = None,
target_delta: float = None) -> Tuple[float, float, float]:
"""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
if target_delta is not None:
eps, opt_order = _compute_eps(orders, rdp, target_delta)
return eps, target_delta, opt_order

[docs]def analyze_dp(q: float,
noise_multiplier: float,
steps: int,
orders: Tuple[float, ...] = (1.25, 1.5, 1.75, 2., 2.25, 2.5, 3., 3.5, 4., 4.5, 5, 6,
7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20),
delta: float = 1e-05) -> float:
"""Compute and print results of DP-SGD analysis.

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.
delta: The target delta.

Returns:
eps

Raises:
ValueError: If target_delta are messed up.
"""
if noise_multiplier == 0:
return float('inf')

rdp = analyze_renyi(q, noise_multiplier, steps, orders)
eps, _, opt_order = convert_renyidp_to_dp(orders, rdp, target_delta=delta)

return eps

[docs]def analyze_dp_sample_without_replacement(q: float,
noise_multiplier: float,
steps: int,
orders: Tuple[float, ...] = (1.25, 1.5, 1.75, 2., 2.25, 2.5, 3., 3.5, 4., 4.5,
5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
18, 19, 20),
delta: float = 1e-05) -> float:
"""Compute and print results of DP-SGD analysis for sample without replacement.

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.
delta: The target delta.

Returns:
eps

Raises:
ValueError: If target_delta are messed up.
"""
if noise_multiplier == 0:
return float('inf')

rdp = analyze_renyi_sample_without_replacement(q, noise_multiplier, steps, orders)
eps, _, opt_order = convert_renyidp_to_dp(orders, rdp, target_delta=delta)

return eps
```