def create_complex_mech(sigma1, sigma2, eps, coeffs):
gm1 = ExactGaussianMechanism(sigma1, name='GM1')
gm2 = ExactGaussianMechanism(sigma2, name='GM2')
SVT = PureDP_Mechanism(eps=eps, name='SVT')
# run gm1 for 3 rounds
# run gm2 for 5 times
# run SVT for once
# compose them with the transformation: compose.
compose = Composition()
composed_mech = compose([gm1, gm2, SVT], coeffs)
return composed_mech
def testcase_multi_para():
"""
create test cases when there are multi parameters (e.g., #coeff and sigma in Composed Gaussian mechanism)
"""
test_case = []
# each test_case = {'noise_para','delta','coeff','eps'} range sigma from 1 to 100, coeff is the number of composition
sigma_list = [
int(1.6**i) for i in range(int(math.floor(math.log(100, 1.6))) + 1)
]
for sigma in sigma_list:
for coeff in [1, 10, 100, 1000]:
for delta in [1e-3, 1e-4, 1e-5, 1e-6]:
gm = ExactGaussianMechanism(sigma, name='GM')
compose = Composition()
composed_mech = compose([gm], [coeff])
cur_test = {
'sigma': sigma,
'delta': delta,
'coeff': coeff,
'eps': composed_mech.get_approxDP(delta)
}
test_case.append(cur_test)
return test_case
import numpy as np
import time
## Example 1: RDP composition of a homogenous sequence of mechanisms
sigma1 = 5
gm1 = GaussianMechanism(sigma1, phi_off=False, name='phi_GM1')
compose = ComposeAFA()
composed_mech = compose([gm1], [10])
delta1 = 1e-6
eps1 = composed_mech.get_approxDP(delta1)
# RDP-based accountant.
gm2 = GaussianMechanism(sigma1, name='rdp_GM2')
compose_rdp = Composition()
composed_mech_rdp = compose_rdp([gm2], [10])
#Exact Gaussian mechanism.
gm3 = ExactGaussianMechanism(sigma1, name='exact_GM3')
compose_exact = ComposeGaussian()
composed_mech_exact = compose_exact([gm3], [10])
# Get name of the composed object, a structured description of the mechanism generated automatically
print('Mechanism name is \"', composed_mech.name, '\"')
print('Parameters are: ', composed_mech.params)
print('epsilon(delta) = ', eps1, ', at delta = ', delta1)
print('Results from rdp_based accountant, epsilon(delta) = ',
composed_mech_rdp.get_approxDP(delta1), ', at delta = ', delta1)
print('Results from AFA, epsilon(delta) = ', eps1, ', at delta = ', delta1)
print('Results from exact Gaussian accountant, epsilon(delta) = ',
from autodp.mechanism_zoo import ExactGaussianMechanism
from autodp.transformer_zoo import Composition, ComposeGaussian
import matplotlib.pyplot as plt
sigma1 = 5.0
sigma2 = 8.0
gm1 = ExactGaussianMechanism(sigma1, name='GM1')
gm2 = ExactGaussianMechanism(sigma2, name='GM2')
# run gm1 for 3 rounds
# run gm2 for 5 times
# compose them with the transformation: compose and
rdp_compose = Composition()
rdp_composed_mech = rdp_compose([gm1, gm2], [3, 5])
compose = ComposeGaussian()
composed_mech = compose([gm1, gm2], [3, 5])
# Query for eps given delta
delta1 = 1e-6
eps1 = composed_mech.get_approxDP(delta1)
eps1b = rdp_composed_mech.get_approxDP(delta1)
delta2 = 1e-4
eps2 = composed_mech.get_approxDP(delta2)
eps2b = rdp_composed_mech.get_approxDP(delta2)
# Get name of the composed object, a structured description of the mechanism generated automatically
print('Mechanism name is \"', composed_mech.name, '\"')
def exp2_gaussian():
"""
Compare privacy cost over composition, y-axis is delta, epsilon is fixed.
"""
#the number of compositions
klist = [80 * i for i in range(2, 15)][3:]
print('coeff list', klist)
doc = {}
eps = 2.0
for sigma in [30, 60]:
# RDP
eps_rdp = []
# Analytical Gaussian
eps_exact = []
# phi function
eps_phi = []
for coeff in klist:
rdp_gaussian = GaussianMechanism(sigma, name='Laplace')
exact_gaussian = GaussianMechanism(sigma,
coeff=coeff,
CDF_off=False)
phi_gaussian = GaussianMechanism(sigma,
coeff=coeff,
CDF_off=False,
cdf_approx=True)
compose = Composition()
composed_rdp_gaussian = compose([rdp_gaussian], [coeff])
eps_rdp.append(composed_rdp_gaussian.approx_delta(eps))
eps_exact.append(exact_gaussian.approx_delta(eps))
eps_phi.append(phi_gaussian.approx_delta(eps))
print('delta with analytical Gaussian mechanism', exact_gaussian)
print('delta with RDP analysis', eps_rdp)
print('delta with phi function method', eps_phi)
cur_result = {}
cur_result['rdp'] = eps_rdp
cur_result['gt'] = eps_exact
cur_result['phi'] = eps_phi
doc[str(sigma)] = cur_result
#with open(path, 'wb') as f:
# pickle.dump(doc, f)
import matplotlib.pyplot as plt
props = fm.FontProperties(family='Gill Sans',
fname='/Library/Fonts/GillSans.ttc')
f, ax = plt.subplots()
plt.figure(num=0, figsize=(12, 8), dpi=80, facecolor='w', edgecolor='k')
plt.plot(klist, doc['30']['rdp'], 'm', linewidth=2)
plt.plot(klist, doc['30']['gt'], 'D--', color='pink', linewidth=2)
plt.plot(klist, doc['30']['phi'], 'x-', color='darkred', linewidth=2)
plt.plot(klist, doc['60']['rdp'], color='darkorange', linewidth=2)
plt.plot(klist, doc['60']['gt'], 'D--', color='pink', linewidth=2)
plt.plot(klist, doc['60']['phi'], 'x-', color='darkblue', linewidth=2)
plt.yscale('log')
plt.legend([
r'RDP $\sigma=30$', 'Exact Accountant $\sigma=30$',
'Our AFA $\sigma=30$', 'RDP $\sigma=60$',
'Exact Accountant $\sigma=60$', 'Our AFA with $\sigma=60$'
],
loc='best',
fontsize=18)
plt.grid(True)
plt.xticks(fontsize=22)
plt.yticks(fontsize=22)
plt.xlabel(r'Number of Compositions $k$', fontsize=22)
plt.ylabel(r'$\delta$', fontsize=22)
ax.set_title('Title', fontproperties=props)
#plt.show()
plt.savefig('exp1_delta.pdf', bbox_inches='tight')
def exp5_subsample_fixed_eps():
"""
Evaluate poisson subsample Gaussian mechanism
sample probability = 0.02, fix delta to 1e-5 and compare epsilon over composition
x axis is # composition
y axis is epsilon(delta)
Evaluate four methods
BBGHS_RDP :eps_rdp
Our phi-function lower bound: eps_phi_left
Our phi-function upper bound: eps_phi_right
Double quadrature:eps_quadrature
"""
delta = 1e-5
import pickle
prob = 0.02
klist = [100 * i for i in range(2, 16)]
doc = {}
exp4_path = 'exp5.pkl'
if os.path.exists(exp4_path):
with open(exp4_path, 'rb') as f:
doc = pickle.load(f)
klist = klist[:]
eps_phi_left = doc['phi_left']
eps_phi_right = doc['phi_right']
eps_rdp = doc['rdp']
eps_quarture = doc['quarture']
else:
for sigma in [2]:
eps_rdp = []
eps_phi_left = []
eps_phi_right = []
eps_quarture = []
for coeff in klist:
gm1 = ExactGaussianMechanism(sigma, name='GM1')
compose = Composition()
poisson_sample = AmplificationBySampling(PoissonSampling=True)
composed_mech = compose(
[poisson_sample(gm1, prob, improved_bound_flag=True)],
[coeff])
#phi_subsample_left = SubSampleGaussian(sigma, prob,coeff, CDF_off=False, lower_bound = True)
#phi_subsample_right = SubSampleGaussian(sigma, prob,coeff, CDF_off=False, upper_bound = True)
phi_quarture = SubSampleGaussian(sigma,
prob,
coeff,
CDF_off=False)
eps_rdp.append(composed_mech.approxDP(delta))
eps_quarture.append(phi_quarture.get_approxDP(delta))
#eps_phi_left.append(phi_subsample_left.approxDP(delta))
#eps_phi_right.append(phi_subsample_right.approxDP(delta))
print('eps using double quarture', eps_quarture)
#print('eps using phi lower bound', eps_phi_left)
#print('eps using phi right bound', eps_phi_right)
print('eps using the optimal rdp conversion', eps_rdp)
cur_result = {}
cur_result['rdp'] = eps_rdp
cur_result['phi_left'] = eps_phi_left
cur_result['phi_right'] = eps_phi_right
cur_result['quarture'] = eps_quarture
with open(exp4_path, 'wb') as f:
pickle.dump(cur_result, f)
for sigma in [2]:
naive_rdp = []
eps_optimal_rdp = []
for coeff in klist:
gm1 = ExactGaussianMechanism(sigma, name='GM1')
compose = Composition()
poisson_sample = AmplificationBySampling(PoissonSampling=True)
composed_mech = compose(
[poisson_sample(gm1, prob, improved_bound_flag=True)], [coeff])
eps_optimal_rdp.append(composed_mech.approxDP(delta))
print('RDP optimal conversion', eps_optimal_rdp)
print('quarture result', eps_quarture)
# copy the results from fourier paper and our previous experimental results
eps_quarture = [
0.5732509555915974, 0.7058115513738935, 0.8194825434078763,
0.9209338664859984, 1.0136435019043732, 1.099702697374365,
1.1804867199047775, 1.2569025991868228, 1.3296691895163435,
1.3993259169003278, 1.4662601850161303, 1.5308113615711694,
1.5932321716160407, 1.6637669337344212
]
eps_phi_left = [
0.5553297955867375, 0.6832218356617155, 0.7928056642133938,
0.8906191704210357, 0.9799696230766813, 1.0628838022920961,
1.1406664339350132, 1.2142613396519286, 1.2843523223848659,
1.3513924639580124, 1.4158316082377795, 1.4779534607841263,
1.5380153993392631, 1.5962516845946515
]
eps_phi_right = [
0.5866220805156243, 0.7223973520962709, 0.8387942691051304,
0.9426878979377773, 1.0376345860832061, 1.1257547173351041,
1.2084450581608812, 1.2867150182217082, 1.3612194904959256,
1.4325246778969292, 1.5010566850456206, 1.567135674028485,
1.6310457295344567, 1.6929980332709857
]
fft_lower = [
0.5738987073827246, 0.7070330939276875, 0.8213662927173179,
0.92358900507979, 1.0171625920742302, 1.1041565229625558,
1.1859259173798518, 1.2634158671679698, 1.3373168083154394,
1.4081514706603082, 1.4763269642671626, 1.54216775460533,
1.605937390055278, 1.667853467197358
]
fft_higher = [
0.5738987073827246, 0.7070330939276875, 0.8213662927173179,
0.92358900507979, 1.0171625920742302, 1.1041565229625558,
1.1859259173798518, 1.2634158671679698, 1.3373168083154394,
1.4081514706603082, 1.4763269642671626, 1.54216775460533,
1.605937390055278, 1.667853467197358
]
import matplotlib.pyplot as plt
#epsusingphi[1.7183697391746319e-06, 1.735096619083328e-06, 2.090003729609751e-06]
#epsusingrdp[1.1702248059464182e-09, 4.4203574134371593e-10, 3.824438543631459e-10]
props = fm.FontProperties(family='Gill Sans',
fname='/Library/Fonts/GillSans.ttc')
f, ax = plt.subplots()
plt.figure(num=0, figsize=(12, 8), dpi=80, facecolor='w', edgecolor='k')
#plt.plot(klist,naive_rdp, 'g.-.', linewidth=2)
plt.plot(klist, eps_rdp, 'm.-.', linewidth=2)
#plt.plot(klist, eps_optimal_rdp, 'y.-', linewidth=2)
plt.plot(klist, eps_phi_left, 'cx-', linewidth=2)
plt.plot(klist, eps_phi_right, 'rx-', linewidth=2)
plt.plot(klist, eps_quarture, 'y.-', linewidth=2)
plt.plot(klist, fft_lower, 'gs--', linewidth=1)
plt.plot(klist, fft_higher, 'rs--', linewidth=1)
#plt.plot(klist, doc['20']['rdp'], '--k', linewidth=2)
#plt.plot(klist, doc['20']['eps'], '--r^', linewidth=2)
plt.legend([
r'BBGHS_RDP', 'our AFA lower bound', 'our AFA higher bound',
'Double quadrature', 'FA lower bound', 'FA higher bound'
],
loc='best',
fontsize=17)
plt.grid(True)
plt.xticks(fontsize=22)
plt.yticks(fontsize=22)
plt.xlabel(r'Number of Compositions $k$', fontsize=22)
plt.ylabel(r'$\epsilon$', fontsize=22)
ax.set_title('Title', fontproperties=props)
#plt.show()
plt.savefig('exp4_eps.pdf', bbox_inches='tight')
def exp4_subsample_fixed_eps():
"""
Evaluate poisson subsample Gaussian mechanism
sample probability = 0.02, fix epsilon to 1.0 and compare delta over composition
x axis is # composition
y axis is delta(epsilon)
Evaluate four methods
BBGHS_RDP :eps_rdp
Our phi-function lower bound: eps_phi_left
Our phi-function upper bound: eps_phi_right
Double quadrature:eps_quadrature
"""
eps = 1.0
import pickle
prob = 0.02
klist = [100 * i for i in range(2, 16)]
exp4_path = 'gamma_0.02_sigma_12.pkl'
if os.path.exists(exp4_path):
with open(exp4_path, 'rb') as f:
doc = pickle.load(f)
klist = klist[:]
eps_phi_left = doc['phi_left']
eps_phi_right = doc['phi_right']
eps_rdp = doc['rdp']
eps_quarture = doc['quarture']
else:
for sigma in [2]:
eps_rdp = []
eps_phi_left = []
eps_phi_right = []
eps_quarture = []
for coeff in klist:
gm1 = ExactGaussianMechanism(sigma, name='GM1')
compose = Composition()
poisson_sample = AmplificationBySampling(PoissonSampling=True)
composed_mech = compose(
[poisson_sample(gm1, prob, improved_bound_flag=True)],
[coeff])
#uncomment the code below to run upper and lower bounds, which is much slower
#phi_subsample_left = SubSampleGaussian(sigma, prob,coeff, CDF_off=False, lower_bound = True)
#phi_subsample_right = SubSampleGaussian(sigma, prob,coeff, CDF_off=False, upper_bound = True)
phi_quarture = SubSampleGaussian(sigma,
prob,
coeff,
CDF_off=False)
eps_rdp.append(composed_mech.approx_delta(eps))
eps_quarture.append(phi_quarture.get_approx_delta(eps))
#eps_phi_left.append(phi_subsample_left.approx_delta(eps))
#eps_phi_right.append(phi_subsample_right.approx_delta(eps))
print('eps using double quarture', eps_quarture)
print('eps using phi lower bound', eps_phi_left)
print('eps using phi right bound', eps_phi_right)
print('eps using rdp', eps_rdp)
cur_result = {}
cur_result['rdp'] = eps_rdp
#eps_phi = [8.126706321817492e-11, 1.680813696457811e-08, 3.7171456678093376e-07, 2.8469414374309687e-06, 1.2134517188197369e-05, 3.612046366420398e-05, 8.487142126792672e-05, 0.00016916885963894502]
# previous experimental records for the sigma = 2.0, eps =1.0
cur_result['phi_left'] = [
1.9705163375286992e-11, 6.54413434131078e-09,
1.8172415920878554e-07, 1.5912195465344071e-06,
7.408024600416465e-06, 2.3476597821679127e-05,
5.780615396550352e-05, 0.0001194871003954497,
0.00021761168363315566, 0.00036048564689211086,
0.0005551755399635068, 0.0008073269983187659,
0.0011211655514957834, 0.0014996039591825359
]
cur_result['phi_right'] = [
1.4208450705963226e-10, 2.7065873453364222e-08,
5.615768975610223e-07, 4.101636149422114e-06,
1.6856443667586166e-05, 4.875023579764651e-05,
0.00011190249522187185, 0.00021878046437442066,
0.00038078926980138466, 0.0006074760381444841,
0.0009062357815770332, 0.0012823422696145335,
0.0017391548126395213, 0.0022783994671059515
]
#cur_result['phi_left'] = eps_phi_left
#cur_result['phi_right'] = eps_phi_right
cur_result['quarture'] = eps_quarture
with open(exp4_path, 'wb') as f:
pickle.dump(cur_result, f)
# copy the results from Fourier accountant paper
fft_lower = [
7.887405682751439e-11, 1.621056159906838e-08, 3.547744440606434e-07,
2.6891898301208684e-06, 1.1344617219024322e-05, 3.342412117224878e-05,
7.773514229351955e-05, 0.00015336724020791973, 0.00026855202090630444,
0.00042999375546865035, 0.0006426099200177218, 0.0009095489321732455,
0.00123236372576394, 0.0016112544734264512
]
fft_higher = [
8.195861133559092e-11, 1.7175792871794577e-08, 3.8343699421628997e-07,
2.9650469214763064e-06, 1.2761325748530785e-05, 3.836009767606709e-05,
9.10255425945141e-05, 0.0001832372011979862, 0.00032737944462639086,
0.0005348503868262945, 0.0008155844687884689, 0.0011778834943758018,
0.001628443617837156, 0.002172490405255
]
for sigma in [2]:
eps_optimal_rdp = []
for coeff in klist:
gm1 = ExactGaussianMechanism(sigma, name='GM1')
compose = Composition()
poisson_sample = AmplificationBySampling(PoissonSampling=True)
composed_mech = compose(
[poisson_sample(gm1, prob, improved_bound_flag=True)], [coeff])
eps_optimal_rdp.append(composed_mech.approx_delta(eps))
print('optimal conversion', eps_optimal_rdp)
print('quarture result', eps_quarture)
print('RDP result', eps_rdp)
import matplotlib.pyplot as plt
props = fm.FontProperties(family='Gill Sans',
fname='/Library/Fonts/GillSans.ttc')
f, ax = plt.subplots()
plt.figure(num=0, figsize=(12, 8), dpi=80, facecolor='w', edgecolor='k')
#plt.plot(klist,naive_rdp, 'g.-.', linewidth=2)
plt.plot(klist, eps_rdp, 'm.-.', linewidth=2)
plt.plot(klist, eps_optimal_rdp, 'y.-', linewidth=2)
plt.plot(klist, eps_phi_left, 'cx--', linewidth=2)
plt.plot(klist, eps_phi_right, 'rx--', linewidth=2)
plt.plot(klist, eps_quarture, 'bx--', linewidth=2)
plt.plot(klist, fft_lower, 'gs-', linewidth=1)
plt.plot(klist, fft_higher, 'rs-', linewidth=1)
plt.ylim([1e-12, 1e-1])
plt.yscale('log')
#plt.plot(klist, doc['20']['rdp'], '--k', linewidth=2)
#plt.plot(klist, doc['20']['eps'], '--r^', linewidth=2)
plt.legend([
r'BBGHS_RDP_conversion', 'Optimal_RDP_Conversion',
'$\phi$-function_lower', '$\phi$-function_upper',
'Double quadrature method', 'FA lower bound', 'FA higher bound'
],
loc='best',
fontsize=17)
plt.grid(True)
plt.xticks(fontsize=20)
plt.yticks(fontsize=20)
plt.xlabel(
r'Number of Composition ($\epsilon=1.0, \sigma=2$, sample probability=$0.02$)',
fontsize=20)
plt.ylabel(r'$\delta$', fontsize=20)
ax.set_title('Title', fontproperties=props)
#plt.show()
plt.savefig('exp4_delta.pdf', bbox_inches='tight')
def exp_2a():
eps_a = [] #standard SVT
eps_e = [] #Laplace-SVT (via RDP)
eps_g = [] #Gaussian SVT c>1
eps_g_c = [] #c=1 for Gaussian SVT
eps_i = []
eps_kov = [] #generalized SVT
eps_noisy = []
k_list = [int(1.4**i) for i in range(int(math.floor(math.log(n,1.4)))+1)]
print(len(k_list))
query = np.zeros(n)
rho = np.random.normal(scale=sigma_1)
lap_rho = np.random.laplace(loc=0.0, scale=lambda_rho)
"""
compute eps for noisy screening
p = Prob[ nu > Margin]
q = Prob[ nu - 1 > margin]
count_gau counts #tops in Gaussian-SVT
count_lap counts #tops in Laplace-SVT
"""
count_gau = 0
count_lap = 0
# the following is for data-dependent screening in CVPR-20
p = scipy.stats.norm.logsf(margin, scale=sigma_2)
q = scipy.stats.norm.logsf(margin + 1, scale=sigma_2)
params = {}
params['logp'] = p
params['logq'] = q
per_screen_mech = NoisyScreenMechanism(params, name='NoisyScreen')
per_gaussian_mech = ExactGaussianMechanism(sigma_2,name='GM1')
index = []
compose = Composition()
for idx, qu in enumerate(query):
nu = np.random.normal(scale=sigma_2)
lap_nu = np.random.laplace(loc=0.0, scale=lambda_nu)
if nu >= rho + margin:
count_gau += 1
if lap_nu >= lap_rho + margin:
count_lap += 1
count_gau = max(count_gau, 1)
count_lap = max(count_lap, 1)
if idx in k_list:
index.append(idx)
print('number of queries passing threshold', count_gau)
#eps_a records the standard SVT
eps_a.append(eps_1 * count_lap + eps_1)
# compose data-dependent screening
screen_mech = compose([per_screen_mech], [idx])
gaussian_mech = compose([per_gaussian_mech], [idx])
# standard SVT with RDP calculation
param_lap_svt = {}
param_lap_svt['b'] = lambda_rho
param_lap_svt['k'] = idx
param_lap_svt['c'] = count_lap
lapsvtrdp_mech = LaplaceSVT_Mechanism(param_lap_svt)
eps_e.append(lapsvtrdp_mech.get_approxDP(delta))
# stage-wise generalized SVT, k is the maximum length of each chunk
k = int(idx / np.sqrt(count_gau))
generalized_mech = StageWiseMechanism({'sigma':sigma_1,'k':k, 'c':count_gau})
eps_kov.append(generalized_mech.get_approxDP(delta))
# Gaussian-SVT c>1 with RDP, k is the total length before algorithm stops
gaussianSVT_c = GaussianSVT_Mechanism({'sigma':sigma_1,'k':idx, 'c':count_gau}, rdp_c_1=False)
eps_g.append(gaussianSVT_c.get_approxDP(delta))
#Gaussian-SVT with c=1, we use average_k as the approximate maximum length of each chunk, margin is used in Proposition 10
average_k = int(idx / max(count_gau, 1))
params_SVT = {}
params_SVT['k'] = average_k
params_SVT['sigma'] = sigma_1
params_SVT['margin'] = margin
per_gaussianSVT_mech = GaussianSVT_Mechanism(params_SVT)
gaussianSVT_mech = compose([per_gaussianSVT_mech],[max(count_gau, 1)])
eps_g_c.append(gaussianSVT_mech.get_approxDP(delta))
eps_i.append(gaussian_mech.get_approxDP(delta)) # Gaussian Mechanism
eps_noisy.append(screen_mech.get_approxDP(delta))
import matplotlib
import matplotlib.pyplot as plt
font = {'family': 'times',
'weight': 'bold',
'size': 18}
props = fm.FontProperties(family='Gill Sans', fname='/Library/Fonts/GillSans.ttc')
f, ax = plt.subplots()
plt.figure(num=0, figsize=(12, 8), dpi=80, facecolor='w', edgecolor='k')
plt.loglog(index, eps_a, '-r', linewidth=2)
plt.loglog(index, eps_e, '--g^', linewidth=2)
plt.loglog(index, eps_g, '-c^', linewidth=2)
plt.loglog(index, eps_g_c, '-bs', linewidth=2)
plt.loglog(index, eps_i, '--k', linewidth=2)
plt.loglog(index, eps_noisy, color='brown', linewidth=2)
plt.loglog(index, eps_kov, color='hotpink', linewidth=2)
plt.legend(
['Laplace-SVT (Pure-DP from Lyu et al., 2017)', 'Laplace-SVT (via RDP)', 'Gaussian-SVT c>1 (RDP by Theorem 11)',
'Gaussian-SVT c=1 (RDP by Theorem 8)', 'Gaussian Mechanism', 'Noisy Screening (data-dependent RDP)',
'Stage-wise generalized SVT'], loc='best', fontsize=17)
plt.grid(True)
plt.xticks(fontsize=20)
plt.yticks(fontsize=20)
plt.xlabel(r'Iterations', fontsize=20)
plt.ylabel(r'$\epsilon$', fontsize=20)
ax.set_title('Title', fontproperties=props)
plt.savefig('exp2a.pdf', bbox_inches='tight')