def scatter_plot(votes, threshold, sigma1, sigma2, order):
fig, ax = setup_plot()
x = []
y = []
for i, v in enumerate(votes):
if threshold is not None and sigma1 is not None:
q_step1 = math.exp(pate.compute_logpr_answered(threshold, sigma1, v))
else:
q_step1 = 1.
if random.random() < q_step1:
logq_step2 = pate.compute_logq_gaussian(v, sigma2)
x.append(max(v))
y.append(pate.rdp_gaussian(logq_step2, sigma2, order))
print('Selected {} queries.'.format(len(x)))
# Plot the data-independent curve:
# data_ind = pate.rdp_data_independent_gaussian(sigma, order)
# plt.plot([0, 5000], [data_ind, data_ind], color='tab:blue', linestyle='-', linewidth=2)
ax.set_yscale('log')
plt.xlim(xmin=0, xmax=5000)
plt.ylim(ymin=1e-300, ymax=1)
plt.yticks([1, 1e-100, 1e-200, 1e-300])
plt.scatter(x, y, s=1, alpha=0.5)
plt.ylabel(r'RDP at $\alpha={}$'.format(order), fontsize=16)
plt.xlabel(r'max count', fontsize=16)
ax.tick_params(labelsize=14)
plt.show()
def plot_rdp_curve_per_example(votes, sigmas):
orders = np.linspace(1., 100., endpoint=True, num=1000)
orders[0] = 1.001
fig, ax = setup_plot()
for i in range(votes.shape[0]):
for sigma in sigmas:
logq = pate.compute_logq_gaussian(votes[i,], sigma)
rdp = pate.rdp_gaussian(logq, sigma, orders)
ax.plot(
orders,
rdp,
alpha=1.,
label=r'Data-dependent bound, $\sigma$={}'.format(int(sigma)),
linewidth=5)
for sigma in sigmas:
ax.plot(
orders,
pate.rdp_data_independent_gaussian(sigma, orders),
alpha=.3,
label=r'Data-independent bound, $\sigma$={}'.format(int(sigma)),
linewidth=10)
plt.xlim(xmin=1, xmax=100)
plt.ylim(ymin=0)
plt.xticks([1, 20, 40, 60, 80, 100])
plt.yticks([0, .0025, .005, .0075, .01])
plt.xlabel(r'Order $\alpha$', fontsize=16)
plt.ylabel(r'RDP value $\varepsilon$ at $\alpha$', fontsize=16)
ax.tick_params(labelsize=14)
plt.legend(loc=0, fontsize=13)
plt.show()
def plot_rdp_curve_per_example(votes, sigmas):
orders = np.linspace(1., 100., endpoint=True, num=1000)
orders[0] = 1.001
fig, ax = setup_plot()
for i in range(votes.shape[0]):
for sigma in sigmas:
logq = pate.compute_logq_gaussian(votes[i,], sigma)
rdp = pate.rdp_gaussian(logq, sigma, orders)
ax.plot(
orders,
rdp,
alpha=1.,
label=r'Data-dependent bound, $\sigma$={}'.format(int(sigma)),
linewidth=5)
for sigma in sigmas:
ax.plot(
orders,
pate.rdp_data_independent_gaussian(sigma, orders),
alpha=.3,
label=r'Data-independent bound, $\sigma$={}'.format(int(sigma)),
linewidth=10)
plt.xlim(xmin=1, xmax=100)
plt.ylim(ymin=0)
plt.xticks([1, 20, 40, 60, 80, 100])
plt.yticks([0, .0025, .005, .0075, .01])
plt.xlabel(r'Order $\alpha$', fontsize=16)
plt.ylabel(r'RDP value $\varepsilon$ at $\alpha$', fontsize=16)
ax.tick_params(labelsize=14)
plt.legend(loc=0, fontsize=13)
plt.show()
def compute_privacy_cost_per_bins(bin_num, votes, sigma2, order):
"""Outputs average privacy cost per bin.
Args:
bin_num: Number of bins.
votes: A matrix of votes, where each row contains votes in one instance.
sigma2: The scale (std) of the Gaussian noise. (Same as sigma_2 in
Algorithms 1 and 2.)
order: The Renyi order for which privacy cost is computed.
Returns:
Expected eps of RDP (ignoring delta) per example in each bin.
"""
n = votes.shape[0]
bin_counts = np.zeros(bin_num)
bin_rdp = np.zeros(bin_num) # RDP at order=order
for i in xrange(n):
v = votes[i,]
logq = pate.compute_logq_gaussian(v, sigma2)
rdp_at_order = pate.rdp_gaussian(logq, sigma2, order)
bin_idx = int(math.floor(max(v) * bin_num / sum(v)))
assert 0 <= bin_idx < bin_num
bin_counts[bin_idx] += 1
bin_rdp[bin_idx] += rdp_at_order
if (i + 1) % 1000 == 0:
print('example {}'.format(i + 1))
sys.stdout.flush()
return bin_rdp / bin_counts
def scatter_plot(votes, threshold, sigma1, sigma2, order):
fig, ax = setup_plot()
x = []
y = []
for i, v in enumerate(votes):
if threshold is not None and sigma1 is not None:
q_step1 = math.exp(pate.compute_logpr_answered(threshold, sigma1, v))
else:
q_step1 = 1.
if random.random() < q_step1:
logq_step2 = pate.compute_logq_gaussian(v, sigma2)
x.append(max(v))
y.append(pate.rdp_gaussian(logq_step2, sigma2, order))
print('Selected {} queries.'.format(len(x)))
# Plot the data-independent curve:
# data_ind = pate.rdp_data_independent_gaussian(sigma, order)
# plt.plot([0, 5000], [data_ind, data_ind], color='tab:blue', linestyle='-', linewidth=2)
ax.set_yscale('log')
plt.xlim(xmin=0, xmax=5000)
plt.ylim(ymin=1e-300, ymax=1)
plt.yticks([1, 1e-100, 1e-200, 1e-300])
plt.scatter(x, y, s=1, alpha=0.5)
plt.ylabel(r'RDP at $\alpha={}$'.format(order), fontsize=16)
plt.xlabel(r'max count', fontsize=16)
ax.tick_params(labelsize=14)
plt.show()
def _compute_rdp(votes, baseline, threshold, sigma1, sigma2, delta, orders,
data_ind):
"""Computes the (data-dependent) RDP curve for Confident GNMax."""
rdp_cum = np.zeros(len(orders))
rdp_sqrd_cum = np.zeros(len(orders))
answered = 0
for i, v in enumerate(votes):
if threshold is None:
logq_step1 = 0 # No thresholding, always proceed to step 2.
rdp_step1 = np.zeros(len(orders))
else:
logq_step1 = pate.compute_logpr_answered(threshold, sigma1,
v - baseline[i, ])
if data_ind:
rdp_step1 = pate.compute_rdp_data_independent_threshold(
sigma1, orders)
else:
rdp_step1 = pate.compute_rdp_threshold(logq_step1, sigma1,
orders)
if data_ind:
rdp_step2 = pate.rdp_data_independent_gaussian(sigma2, orders)
else:
logq_step2 = pate.compute_logq_gaussian(v, sigma2)
rdp_step2 = pate.rdp_gaussian(logq_step2, sigma2, orders)
q_step1 = np.exp(logq_step1)
rdp = rdp_step1 + rdp_step2 * q_step1
# The expression below evaluates
# E[(cost_of_step_1 + Bernoulli(pr_of_step_2) * cost_of_step_2)^2]
rdp_sqrd = (rdp_step1**2 + 2 * rdp_step1 * q_step1 * rdp_step2 +
q_step1 * rdp_step2**2)
rdp_sqrd_cum += rdp_sqrd
rdp_cum += rdp
answered += q_step1
if ((i + 1) % 1000 == 0) or (i == votes.shape[0] - 1):
rdp_var = rdp_sqrd_cum / i - (rdp_cum /
i)**2 # Ignore Bessel's correction.
eps_total, order_opt = pate.compute_eps_from_delta(
orders, rdp_cum, delta)
order_opt_idx = np.searchsorted(orders, order_opt)
eps_std = ((i + 1) * rdp_var[order_opt_idx])**.5 # Std of the sum.
print(
'queries = {}, E[answered] = {:.2f}, E[eps] = {:.3f} (std = {:.5f}) '
'at order = {:.2f} (contribution from delta = {:.3f})'.format(
i + 1, answered, eps_total, eps_std, order_opt,
-math.log(delta) / (order_opt - 1)))
sys.stdout.flush()
_, order_opt = pate.compute_eps_from_delta(orders, rdp_cum, delta)
return order_opt
def _compute_rdp(votes, baseline, threshold, sigma1, sigma2, delta, orders,
data_ind):
"""Computes the (data-dependent) RDP curve for Confident GNMax."""
rdp_cum = np.zeros(len(orders))
rdp_sqrd_cum = np.zeros(len(orders))
answered = 0
for i, v in enumerate(votes):
if threshold is None:
logq_step1 = 0 # No thresholding, always proceed to step 2.
rdp_step1 = np.zeros(len(orders))
else:
logq_step1 = pate.compute_logpr_answered(threshold, sigma1,
v - baseline[i,])
if data_ind:
rdp_step1 = pate.compute_rdp_data_independent_threshold(sigma1, orders)
else:
rdp_step1 = pate.compute_rdp_threshold(logq_step1, sigma1, orders)
if data_ind:
rdp_step2 = pate.rdp_data_independent_gaussian(sigma2, orders)
else:
logq_step2 = pate.compute_logq_gaussian(v, sigma2)
rdp_step2 = pate.rdp_gaussian(logq_step2, sigma2, orders)
q_step1 = np.exp(logq_step1)
rdp = rdp_step1 + rdp_step2 * q_step1
# The expression below evaluates
# E[(cost_of_step_1 + Bernoulli(pr_of_step_2) * cost_of_step_2)^2]
rdp_sqrd = (
rdp_step1**2 + 2 * rdp_step1 * q_step1 * rdp_step2 +
q_step1 * rdp_step2**2)
rdp_sqrd_cum += rdp_sqrd
rdp_cum += rdp
answered += q_step1
if ((i + 1) % 1000 == 0) or (i == votes.shape[0] - 1):
rdp_var = rdp_sqrd_cum / i - (
rdp_cum / i)**2 # Ignore Bessel's correction.
eps_total, order_opt = pate.compute_eps_from_delta(orders, rdp_cum, delta)
order_opt_idx = np.searchsorted(orders, order_opt)
eps_std = ((i + 1) * rdp_var[order_opt_idx])**.5 # Std of the sum.
print(
'queries = {}, E[answered] = {:.2f}, E[eps] = {:.3f} (std = {:.5f}) '
'at order = {:.2f} (contribution from delta = {:.3f})'.format(
i + 1, answered, eps_total, eps_std, order_opt,
-math.log(delta) / (order_opt - 1)))
sys.stdout.flush()
_, order_opt = pate.compute_eps_from_delta(orders, rdp_cum, delta)
return order_opt
def _test_rdp_gaussian_as_function_of_q(self):
# Test for data-independent and data-dependent ranges over q.
# The following corresponds to orders 1.1, 2.5, 32, 250
# sigmas 1.5, 15, 1500, 15000.
# Hand calculated -log(q0)s arranged in a 'sigma major' ordering.
neglogq0s = [
2.8, 2.6, 427, None, 4.8, 4.0, 4.7, 275, 9.6, 8.8, 6.0, 4, 12,
11.2, 8.6, 6.4
]
idx_neglogq0s = 0 # To iterate through neglogq0s.
orders = [1.1, 2.5, 32, 250]
sigmas = [1.5, 15, 1500, 15000]
for sigma in sigmas:
for order in orders:
curr_neglogq0 = neglogq0s[idx_neglogq0s]
idx_neglogq0s += 1
if curr_neglogq0 is None: # sigma == 1.5 and order == 250:
continue
rdp_at_q0 = pate.rdp_gaussian(-curr_neglogq0, sigma, order)
# Data-dependent range. (Successively halve the value of q.)
logq_dds = (
-curr_neglogq0 -
np.array([0, np.log(2), np.log(4),
np.log(8)]))
# Check that in q_dds, rdp is decreasing.
for idx in range(len(logq_dds) - 1):
self.assertGreater(
pate.rdp_gaussian(logq_dds[idx], sigma, order),
pate.rdp_gaussian(logq_dds[idx + 1], sigma, order))
# Data-independent range.
q_dids = np.exp(-curr_neglogq0) + np.array(
[0.1, 0.2, 0.3, 0.4])
# Check that in q_dids, rdp is constant.
for q in q_dids:
self.assertEqual(
rdp_at_q0, pate.rdp_gaussian(np.log(q), sigma, order))
def _test_rdp_gaussian_value_errors(self):
# Test for ValueErrors.
with self.assertRaises(ValueError):
pate.rdp_gaussian(1.0, 1.0, np.array([2, 3, 4]))
with self.assertRaises(ValueError):
pate.rdp_gaussian(np.log(0.5), -1.0, np.array([2, 3, 4]))
with self.assertRaises(ValueError):
pate.rdp_gaussian(np.log(0.5), 1.0, np.array([1, 3, 4]))
def _test_compute_q0(self):
# Stub code to search a logq space and figure out logq0 by eyeballing
# results. This code does not run with the tests. Remove underscore to run.
sigma = 15
order = 250
logqs = np.arange(-290, -270, 1)
count = 0
for logq in logqs:
count += 1
sys.stdout.write("\t%0.5g: %0.10g" %
(logq, pate.rdp_gaussian(logq, sigma, order)))
sys.stdout.flush()
if count % 5 == 0:
print("")
def _test_rdp_gaussian_as_function_of_q(self):
# Test for data-independent and data-dependent ranges over q.
# The following corresponds to orders 1.1, 2.5, 32, 250
# sigmas 1.5, 15, 1500, 15000.
# Hand calculated -log(q0)s arranged in a 'sigma major' ordering.
neglogq0s = [
2.8, 2.6, 427, None, 4.8, 4.0, 4.7, 275, 9.6, 8.8, 6.0, 4, 12, 11.2,
8.6, 6.4
]
idx_neglogq0s = 0 # To iterate through neglogq0s.
orders = [1.1, 2.5, 32, 250]
sigmas = [1.5, 15, 1500, 15000]
for sigma in sigmas:
for order in orders:
curr_neglogq0 = neglogq0s[idx_neglogq0s]
idx_neglogq0s += 1
if curr_neglogq0 is None: # sigma == 1.5 and order == 250:
continue
rdp_at_q0 = pate.rdp_gaussian(-curr_neglogq0, sigma, order)
# Data-dependent range. (Successively halve the value of q.)
logq_dds = (-curr_neglogq0 - np.array(
[0, np.log(2), np.log(4), np.log(8)]))
# Check that in q_dds, rdp is decreasing.
for idx in range(len(logq_dds) - 1):
self.assertGreater(
pate.rdp_gaussian(logq_dds[idx], sigma, order),
pate.rdp_gaussian(logq_dds[idx + 1], sigma, order))
# Data-independent range.
q_dids = np.exp(-curr_neglogq0) + np.array([0.1, 0.2, 0.3, 0.4])
# Check that in q_dids, rdp is constant.
for q in q_dids:
self.assertEqual(rdp_at_q0, pate.rdp_gaussian(
np.log(q), sigma, order))
def compute_rdp_curve(votes, threshold, sigma1, sigma2, orders,
target_answered):
rdp_cum = np.zeros(len(orders))
answered = 0
for i, v in enumerate(votes):
v = sorted(v, reverse=True)
q_step1 = math.exp(pate.compute_logpr_answered(threshold, sigma1, v))
logq_step2 = pate.compute_logq_gaussian(v, sigma2)
rdp = pate.rdp_gaussian(logq_step2, sigma2, orders)
rdp_cum += q_step1 * rdp
answered += q_step1
if answered >= target_answered:
print('Processed {} queries to answer {}.'.format(i, target_answered))
return rdp_cum
assert False, 'Never reached {} answered queries.'.format(target_answered)
def compute_rdp_curve(votes, threshold, sigma1, sigma2, orders,
target_answered):
rdp_cum = np.zeros(len(orders))
answered = 0
for i, v in enumerate(votes):
v = sorted(v, reverse=True)
q_step1 = math.exp(pate.compute_logpr_answered(threshold, sigma1, v))
logq_step2 = pate.compute_logq_gaussian(v, sigma2)
rdp = pate.rdp_gaussian(logq_step2, sigma2, orders)
rdp_cum += q_step1 * rdp
answered += q_step1
if answered >= target_answered:
print('Processed {} queries to answer {}.'.format(i, target_answered))
return rdp_cum
assert False, 'Never reached {} answered queries.'.format(target_answered)
def plot_rdp_of_sigma(v, order):
sigmas = np.linspace(1., 1000., endpoint=True, num=1000)
fig, ax = setup_plot()
y = np.zeros(len(sigmas))
for i, sigma in enumerate(sigmas):
logq = pate.compute_logq_gaussian(v, sigma)
y[i] = pate.rdp_gaussian(logq, sigma, order)
ax.plot(sigmas, y, alpha=.8, linewidth=5)
plt.xlim(xmin=1, xmax=1000)
plt.ylim(ymin=0)
# plt.yticks([0, .0004, .0008, .0012])
ax.tick_params(labelleft='off')
plt.xlabel(r'Noise $\sigma$', fontsize=16)
plt.ylabel(r'RDP at order $\alpha={}$'.format(order), fontsize=16)
ax.tick_params(labelsize=14)
# plt.legend(loc=0, fontsize=13)
plt.show()
def plot_rdp_of_sigma(v, order):
sigmas = np.linspace(1., 1000., endpoint=True, num=1000)
fig, ax = setup_plot()
y = np.zeros(len(sigmas))
for i, sigma in enumerate(sigmas):
logq = pate.compute_logq_gaussian(v, sigma)
y[i] = pate.rdp_gaussian(logq, sigma, order)
ax.plot(sigmas, y, alpha=.8, linewidth=5)
plt.xlim(xmin=1, xmax=1000)
plt.ylim(ymin=0)
# plt.yticks([0, .0004, .0008, .0012])
ax.tick_params(labelleft='off')
plt.xlabel(r'Noise $\sigma$', fontsize=16)
plt.ylabel(r'RDP at order $\alpha={}$'.format(order), fontsize=16)
ax.tick_params(labelsize=14)
# plt.legend(loc=0, fontsize=13)
plt.show()
def plot_rdp_curve_per_example(votes, sigmas):
orders = np.linspace(1., 100., endpoint=True, num=1000)
orders[0] = 1.5
fig, ax = plt.subplots()
fig.set_figheight(4.5)
fig.set_figwidth(4.7)
styles = [':', '-']
labels = ['ex1', 'ex2']
for i in xrange(votes.shape[0]):
print(sorted(votes[i, ], reverse=True)[:10])
for sigma in sigmas:
logq = pate.compute_logq_gaussian(votes[i, ], sigma)
rdp = pate.rdp_gaussian(logq, sigma, orders)
ax.plot(orders,
rdp,
label=r'{} $\sigma$={}'.format(labels[i], int(sigma)),
linestyle=styles[i],
linewidth=5)
for sigma in sigmas:
ax.plot(orders,
pate.rdp_data_independent_gaussian(sigma, orders),
alpha=.3,
label=r'Data-ind bound $\sigma$={}'.format(int(sigma)),
linewidth=10)
plt.yticks([0, .01])
plt.xlabel(r'Order $\lambda$', fontsize=16)
plt.ylabel(r'RDP value $\varepsilon$ at $\lambda$', fontsize=16)
ax.tick_params(labelsize=14)
fout_name = os.path.join(FLAGS.figures_dir, 'rdp_flow1.pdf')
print('Saving the graph to ' + fout_name)
fig.savefig(fout_name, bbox_inches='tight')
plt.legend(loc=0, fontsize=13)
plt.show()
def analyze_gnmax_conf_data_dep(votes, threshold, sigma1, sigma2, delta):
# Short list of orders.
# orders = np.round(np.logspace(np.log10(20), np.log10(200), num=20))
# Long list of orders.
orders = np.concatenate((np.arange(20, 40, .2),
np.arange(40, 75, .5),
np.logspace(np.log10(75), np.log10(200), num=20)))
n = votes.shape[0]
num_classes = votes.shape[1]
num_teachers = int(sum(votes[0,]))
if threshold is not None and sigma1 is not None:
is_data_ind_step1 = pate.is_data_independent_always_opt_gaussian(
num_teachers, num_classes, sigma1, orders)
else:
is_data_ind_step1 = [True] * len(orders)
is_data_ind_step2 = pate.is_data_independent_always_opt_gaussian(
num_teachers, num_classes, sigma2, orders)
eps_partitioned = np.full(n, None, dtype=Partition)
order_opt = np.full(n, None, dtype=float)
ss_std_opt = np.full(n, None, dtype=float)
answered = np.zeros(n)
rdp_step1_total = np.zeros(len(orders))
rdp_step2_total = np.zeros(len(orders))
ls_total = np.zeros((len(orders), num_teachers))
answered_total = 0
for i in range(n):
v = votes[i,]
if threshold is not None and sigma1 is not None:
logq_step1 = pate.compute_logpr_answered(threshold, sigma1, v)
rdp_step1_total += pate.compute_rdp_threshold(logq_step1, sigma1, orders)
else:
logq_step1 = 0. # always answer
pr_answered = np.exp(logq_step1)
logq_step2 = pate.compute_logq_gaussian(v, sigma2)
rdp_step2_total += pr_answered * pate.rdp_gaussian(logq_step2, sigma2,
orders)
answered_total += pr_answered
rdp_ss = np.zeros(len(orders))
ss_std = np.zeros(len(orders))
for j, order in enumerate(orders):
if not is_data_ind_step1[j]:
ls_step1 = pate_ss.compute_local_sensitivity_bounds_threshold(v,
num_teachers, threshold, sigma1, order)
else:
ls_step1 = np.full(num_teachers, 0, dtype=float)
if not is_data_ind_step2[j]:
ls_step2 = pate_ss.compute_local_sensitivity_bounds_gnmax(
v, num_teachers, sigma2, order)
else:
ls_step2 = np.full(num_teachers, 0, dtype=float)
ls_total[j,] += ls_step1 + pr_answered * ls_step2
beta_ss = .49 / order
ss = pate_ss.compute_discounted_max(beta_ss, ls_total[j,])
sigma_ss = ((order * math.exp(2 * beta_ss)) / ss) ** (1 / 3)
rdp_ss[j] = pate_ss.compute_rdp_of_smooth_sensitivity_gaussian(
beta_ss, sigma_ss, order)
ss_std[j] = ss * sigma_ss
rdp_total = rdp_step1_total + rdp_step2_total + rdp_ss
answered[i] = answered_total
_, order_opt[i] = pate.compute_eps_from_delta(orders, rdp_total, delta)
order_idx = np.searchsorted(orders, order_opt[i])
# Since optimal orders are always non-increasing, shrink orders array
# and all cumulative arrays to speed up computation.
if order_idx < len(orders):
orders = orders[:order_idx + 1]
rdp_step1_total = rdp_step1_total[:order_idx + 1]
rdp_step2_total = rdp_step2_total[:order_idx + 1]
eps_partitioned[i] = Partition(step1=rdp_step1_total[order_idx],
step2=rdp_step2_total[order_idx],
ss=rdp_ss[order_idx],
delta=-math.log(delta) / (order_opt[i] - 1))
ss_std_opt[i] = ss_std[order_idx]
if i > 0 and (i + 1) % 1 == 0:
print('queries = {}, E[answered] = {:.2f}, E[eps] = {:.3f} +/- {:.3f} '
'at order = {:.2f}. Contributions: delta = {:.3f}, step1 = {:.3f}, '
'step2 = {:.3f}, ss = {:.3f}'.format(
i + 1,
answered[i],
sum(eps_partitioned[i]),
ss_std_opt[i],
order_opt[i],
eps_partitioned[i].delta,
eps_partitioned[i].step1,
eps_partitioned[i].step2,
eps_partitioned[i].ss))
sys.stdout.flush()
return eps_partitioned, answered, ss_std_opt, order_opt
def _find_optimal_smooth_sensitivity_parameters(votes, baseline, num_teachers,
threshold, sigma1, sigma2,
delta, ind_step1, ind_step2,
order):
"""Optimizes smooth sensitivity parameters by minimizing a cost function.
The cost function is
exact_eps + cost of GNSS + two stds of noise,
which captures that upper bound of the confidence interval of the sanitized
privacy budget.
Since optimization is done with full view of sensitive data, the results
cannot be released.
"""
rdp_cum = 0
answered_cum = 0
ls_cum = 0
# Define a plausible range for the beta values.
betas = np.arange(.3 / order, .495 / order, .01 / order)
cost_delta = math.log(1 / delta) / (order - 1)
for i, v in enumerate(votes):
if threshold is None:
log_pr_answered = 0
rdp1 = 0
ls_step1 = np.zeros(num_teachers)
else:
log_pr_answered = pate.compute_logpr_answered(
threshold, sigma1, v - baseline[i, ])
if ind_step1: # apply data-independent bound for step 1 (thresholding).
rdp1 = pate.compute_rdp_data_independent_threshold(
sigma1, order)
ls_step1 = np.zeros(num_teachers)
else:
rdp1 = pate.compute_rdp_threshold(log_pr_answered, sigma1,
order)
ls_step1 = pate_ss.compute_local_sensitivity_bounds_threshold(
v - baseline[i, ], num_teachers, threshold, sigma1, order)
pr_answered = math.exp(log_pr_answered)
answered_cum += pr_answered
if ind_step2: # apply data-independent bound for step 2 (GNMax).
rdp2 = pate.rdp_data_independent_gaussian(sigma2, order)
ls_step2 = np.zeros(num_teachers)
else:
logq_step2 = pate.compute_logq_gaussian(v, sigma2)
rdp2 = pate.rdp_gaussian(logq_step2, sigma2, order)
# Compute smooth sensitivity.
ls_step2 = pate_ss.compute_local_sensitivity_bounds_gnmax(
v, num_teachers, sigma2, order)
rdp_cum += rdp1 + pr_answered * rdp2
ls_cum += ls_step1 + pr_answered * ls_step2 # Expected local sensitivity.
if ind_step1 and ind_step2:
# Data-independent bounds.
cost_opt, beta_opt, ss_opt, sigma_ss_opt = None, 0., 0., np.inf
else:
# Data-dependent bounds.
cost_opt, beta_opt, ss_opt, sigma_ss_opt = np.inf, None, None, None
for beta in betas:
ss = pate_ss.compute_discounted_max(beta, ls_cum)
# Solution to the minimization problem:
# min_sigma {order * exp(2 * beta)/ sigma^2 + 2 * ss * sigma}
sigma_ss = ((order * math.exp(2 * beta)) / ss)**(1 / 3)
cost_ss = pate_ss.compute_rdp_of_smooth_sensitivity_gaussian(
beta, sigma_ss, order)
# Cost captures exact_eps + cost of releasing SS + two stds of noise.
cost = rdp_cum + cost_ss + 2 * ss * sigma_ss
if cost < cost_opt:
cost_opt, beta_opt, ss_opt, sigma_ss_opt = cost, beta, ss, sigma_ss
if ((i + 1) % 100 == 0) or (i == votes.shape[0] - 1):
eps_before_ss = rdp_cum + cost_delta
eps_with_ss = (eps_before_ss +
pate_ss.compute_rdp_of_smooth_sensitivity_gaussian(
beta_opt, sigma_ss_opt, order))
print(
'{}: E[answered queries] = {:.1f}, RDP at {} goes from {:.3f} to '
'{:.3f} +/- {:.3f} (ss = {:.4}, beta = {:.4f}, sigma_ss = {:.3f})'
.format(i + 1, answered_cum, order, eps_before_ss, eps_with_ss,
ss_opt * sigma_ss_opt, ss_opt, beta_opt, sigma_ss_opt))
sys.stdout.flush()
# Return optimal parameters for the last iteration.
return beta_opt, ss_opt, sigma_ss_opt
def analyze_gnmax_conf_data_dep(votes, threshold, sigma1, sigma2, delta):
# Short list of orders.
# orders = np.round(np.logspace(np.log10(20), np.log10(200), num=20))
# Long list of orders.
orders = np.concatenate((np.arange(20, 40, .2), np.arange(40, 75, .5),
np.logspace(np.log10(75), np.log10(200), num=20)))
n = votes.shape[0]
num_classes = votes.shape[1]
num_teachers = int(sum(votes[0, ]))
if threshold is not None and sigma1 is not None:
is_data_ind_step1 = pate.is_data_independent_always_opt_gaussian(
num_teachers, num_classes, sigma1, orders)
else:
is_data_ind_step1 = [True] * len(orders)
is_data_ind_step2 = pate.is_data_independent_always_opt_gaussian(
num_teachers, num_classes, sigma2, orders)
eps_partitioned = np.full(n, None, dtype=Partition)
order_opt = np.full(n, None, dtype=float)
ss_std_opt = np.full(n, None, dtype=float)
answered = np.zeros(n)
rdp_step1_total = np.zeros(len(orders))
rdp_step2_total = np.zeros(len(orders))
ls_total = np.zeros((len(orders), num_teachers))
answered_total = 0
for i in range(n):
v = votes[i, ]
if threshold is not None and sigma1 is not None:
logq_step1 = pate.compute_logpr_answered(threshold, sigma1, v)
rdp_step1_total += pate.compute_rdp_threshold(
logq_step1, sigma1, orders)
else:
logq_step1 = 0. # always answer
pr_answered = np.exp(logq_step1)
logq_step2 = pate.compute_logq_gaussian(v, sigma2)
rdp_step2_total += pr_answered * pate.rdp_gaussian(
logq_step2, sigma2, orders)
answered_total += pr_answered
rdp_ss = np.zeros(len(orders))
ss_std = np.zeros(len(orders))
for j, order in enumerate(orders):
if not is_data_ind_step1[j]:
ls_step1 = pate_ss.compute_local_sensitivity_bounds_threshold(
v, num_teachers, threshold, sigma1, order)
else:
ls_step1 = np.full(num_teachers, 0, dtype=float)
if not is_data_ind_step2[j]:
ls_step2 = pate_ss.compute_local_sensitivity_bounds_gnmax(
v, num_teachers, sigma2, order)
else:
ls_step2 = np.full(num_teachers, 0, dtype=float)
ls_total[j, ] += ls_step1 + pr_answered * ls_step2
beta_ss = .49 / order
ss = pate_ss.compute_discounted_max(beta_ss, ls_total[j, ])
sigma_ss = ((order * math.exp(2 * beta_ss)) / ss)**(1 / 3)
rdp_ss[j] = pate_ss.compute_rdp_of_smooth_sensitivity_gaussian(
beta_ss, sigma_ss, order)
ss_std[j] = ss * sigma_ss
rdp_total = rdp_step1_total + rdp_step2_total + rdp_ss
answered[i] = answered_total
_, order_opt[i] = pate.compute_eps_from_delta(orders, rdp_total, delta)
order_idx = np.searchsorted(orders, order_opt[i])
# Since optimal orders are always non-increasing, shrink orders array
# and all cumulative arrays to speed up computation.
if order_idx < len(orders):
orders = orders[:order_idx + 1]
rdp_step1_total = rdp_step1_total[:order_idx + 1]
rdp_step2_total = rdp_step2_total[:order_idx + 1]
eps_partitioned[i] = Partition(step1=rdp_step1_total[order_idx],
step2=rdp_step2_total[order_idx],
ss=rdp_ss[order_idx],
delta=-math.log(delta) /
(order_opt[i] - 1))
ss_std_opt[i] = ss_std[order_idx]
if i > 0 and (i + 1) % 1 == 0:
print(
'queries = {}, E[answered] = {:.2f}, E[eps] = {:.3f} +/- {:.3f} '
'at order = {:.2f}. Contributions: delta = {:.3f}, step1 = {:.3f}, '
'step2 = {:.3f}, ss = {:.3f}'.format(
i + 1, answered[i], sum(eps_partitioned[i]), ss_std_opt[i],
order_opt[i], eps_partitioned[i].delta,
eps_partitioned[i].step1, eps_partitioned[i].step2,
eps_partitioned[i].ss))
sys.stdout.flush()
return eps_partitioned, answered, ss_std_opt, order_opt
def run_analysis(votes, mechanism, noise_scale, params):
"""Computes data-dependent privacy.
Args:
votes: A matrix of votes, where each row contains votes in one instance.
mechanism: A name of the mechanism ('lnmax', 'gnmax', or 'gnmax_conf')
noise_scale: A mechanism privacy parameter.
params: Other privacy parameters.
Returns:
Four lists: cumulative privacy cost epsilon, how privacy budget is split,
how many queries were answered, optimal order.
"""
def compute_partition(order_opt, eps):
order_opt_idx = np.searchsorted(orders, order_opt)
if mechanism == 'gnmax_conf':
p = (rdp_select_cum[order_opt_idx],
rdp_cum[order_opt_idx] - rdp_select_cum[order_opt_idx],
-math.log(delta) / (order_opt - 1))
else:
p = (rdp_cum[order_opt_idx], -math.log(delta) / (order_opt - 1))
return [x / eps for x in p] # Ensures that sum(x) == 1
# Short list of orders.
# orders = np.round(np.concatenate((np.arange(2, 50 + 1, 1),
# np.logspace(np.log10(50), np.log10(1000), num=20))))
# Long list of orders.
orders = np.concatenate((np.arange(2, 100 + 1, .5),
np.logspace(np.log10(100), np.log10(500), num=100)))
delta = 1e-8
n = votes.shape[0]
eps_total = np.zeros(n)
partition = [None] * n
order_opt = np.full(n, np.nan, dtype=float)
answered = np.zeros(n, dtype=float)
rdp_cum = np.zeros(len(orders))
rdp_sqrd_cum = np.zeros(len(orders))
rdp_select_cum = np.zeros(len(orders))
answered_sum = 0
for i in range(n):
v = votes[i,]
if mechanism == 'lnmax':
logq_lnmax = pate.compute_logq_laplace(v, noise_scale)
rdp_query = pate.rdp_pure_eps(logq_lnmax, 2. / noise_scale, orders)
rdp_sqrd = rdp_query ** 2
pr_answered = 1
elif mechanism == 'gnmax':
logq_gmax = pate.compute_logq_gaussian(v, noise_scale)
rdp_query = pate.rdp_gaussian(logq_gmax, noise_scale, orders)
rdp_sqrd = rdp_query ** 2
pr_answered = 1
elif mechanism == 'gnmax_conf':
logq_step1 = pate.compute_logpr_answered(params['t'], params['sigma1'], v)
logq_step2 = pate.compute_logq_gaussian(v, noise_scale)
q_step1 = np.exp(logq_step1)
logq_step1_min = min(logq_step1, math.log1p(-q_step1))
rdp_gnmax_step1 = pate.rdp_gaussian(logq_step1_min,
2 ** .5 * params['sigma1'], orders)
rdp_gnmax_step2 = pate.rdp_gaussian(logq_step2, noise_scale, orders)
rdp_query = rdp_gnmax_step1 + q_step1 * rdp_gnmax_step2
# The expression below evaluates
# E[(cost_of_step_1 + Bernoulli(pr_of_step_2) * cost_of_step_2)^2]
rdp_sqrd = (
rdp_gnmax_step1 ** 2 + 2 * rdp_gnmax_step1 * q_step1 * rdp_gnmax_step2
+ q_step1 * rdp_gnmax_step2 ** 2)
rdp_select_cum += rdp_gnmax_step1
pr_answered = q_step1
else:
raise ValueError(
'Mechanism must be one of ["lnmax", "gnmax", "gnmax_conf"]')
rdp_cum += rdp_query
rdp_sqrd_cum += rdp_sqrd
answered_sum += pr_answered
answered[i] = answered_sum
eps_total[i], order_opt[i] = pate.compute_eps_from_delta(
orders, rdp_cum, delta)
partition[i] = compute_partition(order_opt[i], eps_total[i])
if i > 0 and (i + 1) % 1000 == 0:
rdp_var = rdp_sqrd_cum / i - (
rdp_cum / i) ** 2 # Ignore Bessel's correction.
order_opt_idx = np.searchsorted(orders, order_opt[i])
eps_std = ((i + 1) * rdp_var[order_opt_idx]) ** .5 # Std of the sum.
print(
'queries = {}, E[answered] = {:.2f}, E[eps] = {:.3f} (std = {:.5f}) '
'at order = {:.2f} (contribution from delta = {:.3f})'.format(
i + 1, answered_sum, eps_total[i], eps_std, order_opt[i],
-math.log(delta) / (order_opt[i] - 1)))
sys.stdout.flush()
return eps_total, partition, answered, order_opt
def _find_optimal_smooth_sensitivity_parameters(
votes, baseline, num_teachers, threshold, sigma1, sigma2, delta, ind_step1,
ind_step2, order):
"""Optimizes smooth sensitivity parameters by minimizing a cost function.
The cost function is
exact_eps + cost of GNSS + two stds of noise,
which captures that upper bound of the confidence interval of the sanitized
privacy budget.
Since optimization is done with full view of sensitive data, the results
cannot be released.
"""
rdp_cum = 0
answered_cum = 0
ls_cum = 0
# Define a plausible range for the beta values.
betas = np.arange(.3 / order, .495 / order, .01 / order)
cost_delta = math.log(1 / delta) / (order - 1)
for i, v in enumerate(votes):
if threshold is None:
log_pr_answered = 0
rdp1 = 0
ls_step1 = np.zeros(num_teachers)
else:
log_pr_answered = pate.compute_logpr_answered(threshold, sigma1,
v - baseline[i,])
if ind_step1: # apply data-independent bound for step 1 (thresholding).
rdp1 = pate.compute_rdp_data_independent_threshold(sigma1, order)
ls_step1 = np.zeros(num_teachers)
else:
rdp1 = pate.compute_rdp_threshold(log_pr_answered, sigma1, order)
ls_step1 = pate_ss.compute_local_sensitivity_bounds_threshold(
v - baseline[i,], num_teachers, threshold, sigma1, order)
pr_answered = math.exp(log_pr_answered)
answered_cum += pr_answered
if ind_step2: # apply data-independent bound for step 2 (GNMax).
rdp2 = pate.rdp_data_independent_gaussian(sigma2, order)
ls_step2 = np.zeros(num_teachers)
else:
logq_step2 = pate.compute_logq_gaussian(v, sigma2)
rdp2 = pate.rdp_gaussian(logq_step2, sigma2, order)
# Compute smooth sensitivity.
ls_step2 = pate_ss.compute_local_sensitivity_bounds_gnmax(
v, num_teachers, sigma2, order)
rdp_cum += rdp1 + pr_answered * rdp2
ls_cum += ls_step1 + pr_answered * ls_step2 # Expected local sensitivity.
if ind_step1 and ind_step2:
# Data-independent bounds.
cost_opt, beta_opt, ss_opt, sigma_ss_opt = None, 0., 0., np.inf
else:
# Data-dependent bounds.
cost_opt, beta_opt, ss_opt, sigma_ss_opt = np.inf, None, None, None
for beta in betas:
ss = pate_ss.compute_discounted_max(beta, ls_cum)
# Solution to the minimization problem:
# min_sigma {order * exp(2 * beta)/ sigma^2 + 2 * ss * sigma}
sigma_ss = ((order * math.exp(2 * beta)) / ss)**(1 / 3)
cost_ss = pate_ss.compute_rdp_of_smooth_sensitivity_gaussian(
beta, sigma_ss, order)
# Cost captures exact_eps + cost of releasing SS + two stds of noise.
cost = rdp_cum + cost_ss + 2 * ss * sigma_ss
if cost < cost_opt:
cost_opt, beta_opt, ss_opt, sigma_ss_opt = cost, beta, ss, sigma_ss
if ((i + 1) % 100 == 0) or (i == votes.shape[0] - 1):
eps_before_ss = rdp_cum + cost_delta
eps_with_ss = (
eps_before_ss + pate_ss.compute_rdp_of_smooth_sensitivity_gaussian(
beta_opt, sigma_ss_opt, order))
print('{}: E[answered queries] = {:.1f}, RDP at {} goes from {:.3f} to '
'{:.3f} +/- {:.3f} (ss = {:.4}, beta = {:.4f}, sigma_ss = {:.3f})'.
format(i + 1, answered_cum, order, eps_before_ss, eps_with_ss,
ss_opt * sigma_ss_opt, ss_opt, beta_opt, sigma_ss_opt))
sys.stdout.flush()
# Return optimal parameters for the last iteration.
return beta_opt, ss_opt, sigma_ss_opt
