def subsample_func_int(x):
# output the cgf of the subsampled mechanism
mm = int(x)
eps_inf = func(np.inf)
moments_two = 2 * np.log(prob) + utils.logcomb(mm,2) \
+ np.minimum(np.log(4) + func(2.0) + np.log(1-np.exp(-func(2.0))),
func(2.0) + np.minimum(np.log(2),
2 * (eps_inf+np.log(1-np.exp(-eps_inf)))))
moment_bound = lambda j: np.minimum(j * (eps_inf + np.log(1-np.exp(-eps_inf))),
np.log(2)) + cgf(j - 1) \
+ j * np.log(prob) + utils.logcomb(mm, j)
moments = [moment_bound(j) for j in range(3, mm + 1, 1)]
return np.minimum(
(x - 1) * func(x),
utils.stable_logsumexp([0, moments_two] + moments))
def subsample_func_int(x):
# This function evaluates the CGF at alpha = x, i.e., lamb = x- 1
if np.isinf(func(x)):
return np.inf
if prob == 1.0:
return func(x)
mm = int(x)
fastbound = fast_poission_subsampled_cgf_upperbound(
func, mm, prob)
if x <= self.alphas[-1]: # compute the bound exactly.
moments = [
cgf(1) + 2 * np.log(prob) +
(mm - 2) * np.log(1 - prob) + self.logBinomC[mm, 2]
]
moments = moments + [
cgf(j - 1 + 1) + j * np.log(prob) +
(mm - j) * np.log(1 - prob) + self.logBinomC[mm, j]
for j in range(3, mm + 1, 1)
]
return utils.stable_logsumexp(
[(mm - 1) * np.log(1 - prob) +
np.log(1 + (mm - 1) * prob)] + moments)
elif mm <= self.m_lin_max:
moments = [
cgf(1) + 2 * np.log(prob) +
(mm - 2) * np.log(1 - prob) + utils.logcomb(mm, 2)
]
moments = moments + [
cgf(j - 1 + 1) + j * np.log(prob) +
(mm - j) * np.log(1 - prob) + utils.logcomb(mm, j)
for j in range(3, mm + 1, 1)
]
return utils.stable_logsumexp(
[(mm - 1) * np.log(1 - prob) +
np.log(1 + (mm - 1) * prob)] + moments)
else:
return fastbound
def subsample_func_int(x):
# This function evaluates teh CGF at alpha = x, i.e., lamb = x- 1
deltas_local, signs_deltas_local = self.deltas_cache[(func,
prob)]
if np.isinf(func(x)):
return np.inf
mm = int(x)
fastupperbound = fast_subsampled_cgf_upperbound(
func, mm, prob, deltas_local)
fastupperbound2 = general_upperbound(func, mm, prob)
if self.approx == True:
if fastupperbound2 < 0:
print('general rdp is negative', x)
return fastupperbound2
if mm <= self.alphas[
-1]: # compute the bound exactly. Requires book keeping of O(x^2)
moments = [
np.minimum(
np.minimum((j) * np.log(np.exp(func(np.inf)) - 1) +
np.minimum(cgf(j - 1), np.log(4)),
np.log(2) + cgf(j - 1)),
np.log(4) + 0.5 *
deltas_local[int(2 * np.floor(j / 2.0)) - 1] +
0.5 * deltas_local[int(2 * np.ceil(j / 2.0)) - 1])
+ j * np.log(prob) + self.logBinomC[int(mm), j]
for j in range(2, int(mm + 1), 1)
]
return np.minimum(fastupperbound,
utils.stable_logsumexp([0] + moments))
elif mm <= self.m_lin_max: # compute the bound with stirling approximation. Everything is O(x) now.
moment_bound = lambda j: np.minimum(
j * np.log(np.exp(func(np.inf)) - 1) + np.minimum(
cgf(j - 1), np.log(4)),
np.log(2) + cgf(j - 1)) + j * np.log(
prob) + utils.logcomb(mm, j)
moments = [moment_bound(j) for j in range(2, mm + 1, 1)]
return np.minimum(fastupperbound,
utils.stable_logsumexp([0] + moments))
else: # Compute the O(1) upper bound
return fastupperbound
def subsample_func_int(x):
# This function evaluates teh CGF at alpha = x, i.e., lamb = x- 1
if np.isinf(func(x)):
return np.inf
mm = int(x)
#
fastbound = fast_poission_subsampled_cgf_upperbound(
func, mm, prob)
k = self.alphas[-1]
fastbound_k = fast_k_subsample_upperbound(func, mm, prob, k)
if self.approx == True:
return fastbound_k
#fastbound = min(fastbound, fastbound_k)
if x <= self.alphas[-1]: # compute the bound exactly.
moments = [
cgf(j - 1) + j * np.log(prob) +
(mm - j) * np.log(1 - prob) + self.logBinomC[mm, j]
for j in range(2, mm + 1, 1)
]
return utils.stable_logsumexp(
[(mm - 1) * np.log(1 - prob) +
np.log(1 + (mm - 1) * prob)] + moments)
elif mm <= self.m_lin_max:
moments = [
cgf(j - 1) + j * np.log(prob) +
(mm - j) * np.log(1 - prob) + utils.logcomb(mm, j)
for j in range(2, mm + 1, 1)
]
return utils.stable_logsumexp(
[(mm - 1) * np.log(1 - prob) +
np.log(1 + (mm - 1) * prob)] + moments)
else:
return fastbound
def moment_bound(j): return np.minimum(j * (eps_inf + np.log(1-np.exp(-eps_inf))),
np.log(2)) + cgf(j - 1) \
+ j * np.log(prob) + utils.logcomb(mm, j)
moments = [moment_bound(j) for j in range(3, mm + 1, 1)]
def subsample_func_int(x):
# This function evaluates teh CGF at alpha = x, i.e., lamb = x- 1
deltas_local, signs_deltas_local = self.deltas_cache[(
func, prob)]
if np.isinf(func(x)):
return np.inf
mm = int(x)
eps_inf = func(np.inf)
moments_two = 2 * np.log(prob) + utils.logcomb(mm, 2) \
+ np.minimum(
np.log(4) + func(2.0) + np.log(1 - np.exp(-func(2.0))),
func(2.0) + np.minimum(np.log(2),
2 * (eps_inf + np.log(1 - np.exp(-eps_inf)))))
moment_bound = lambda j: np.minimum(np.log(4) + 0.5*deltas_local[int(2*np.floor(j/2.0))-1]
+ 0.5*deltas_local[int(2*np.ceil(j/2.0))-1],
np.minimum(j * (eps_inf + np.log(1 - np.exp(-eps_inf))),
np.log(2))
+ cgf(j - 1)) \
+ j * np.log(prob) + utils.logcomb(mm, j)
moment_bound_linear = lambda j: np.minimum(j * (eps_inf + np.log(1-np.exp(-eps_inf))),
np.log(2)) + cgf(j - 1) \
+ j * np.log(prob) + utils.logcomb(mm, j)
fastupperbound = fast_subsampled_cgf_upperbound(
func, mm, prob, deltas_local)
if mm <= self.alphas[
-1]: # compute the bound exactly. Requires book keeping of O(x^2)
#
# moments = [ np.minimum(np.minimum((j)*np.log(np.exp(func(np.inf))-1) + np.minimum(cgf(j-1),np.log(4)),
# np.log(2) + cgf(j-1)),
# np.log(4) + 0.5*deltas_local[int(2*np.floor(j/2.0))-1]
# + 0.5*deltas_local[int(2*np.ceil(j/2.0))-1]) + j*np.log(prob)
# +self.logBinomC[int(mm), j] for j in range(2,int(mm+1),1)]
moments = [
moment_bound(j) for j in range(3, mm + 1, 1)
]
return np.minimum(
fastupperbound,
utils.stable_logsumexp([0, moments_two] + moments))
elif mm <= self.m_lin_max: # compute the bound with stirling approximation. Everything is O(x) now.
# moment_bound = lambda j: np.minimum(j * np.log(np.exp(func(np.inf)) - 1)
# + np.minimum(cgf(j - 1), np.log(4)), np.log(2)
# + cgf(j - 1)) + j * np.log(prob) + utils.logcomb(mm, j)
# moments = [moment_bound(j) for j in range(2,mm+1,1)]
moments = [
moment_bound_linear(j)
for j in range(3, mm + 1, 1)
]
return np.minimum(
fastupperbound,
utils.stable_logsumexp([0, moments_two] + moments))
else: # Compute the O(1) upper bound
return fastupperbound
