def RDP_pureDP(params, alpha):
"""
This function generically converts pure DP to Renyi DP.
It implements Lemma 1.4 of Bun et al.'s CDP paper.
With an additional cap at eps.
:param params: pure DP parameter
:param alpha: The order of the Renyi Divergence
:return:Evaluation of the RDP's epsilon
"""
eps = params['eps']
# assert(eps>=0)
# if alpha < 1:
# # Pure DP needs to have identical support, thus - log(q(p>0)) = 0.
# return 0
# else:
# return np.minimum(eps,alpha*eps*eps/2)
assert (alpha >= 0)
if alpha == 1:
# Calculate this by l'Hospital rule
return eps * (math.cosh(eps) - 1) / math.sinh(eps)
elif np.isinf(alpha):
return eps
elif alpha > 1:
# in the proof of Lemma 4 of Bun et al. (2016)
s, mag = utils.stable_log_diff_exp(
utils.stable_log_sinh(alpha * eps),
utils.stable_log_sinh((alpha - 1) * eps))
return (mag - utils.stable_log_sinh(eps)) / (alpha - 1)
else:
return min(alpha * eps * eps / 2,
eps * (math.cosh(eps) - 1) / math.sinh(eps))
def rdp(alpha):
assert (alpha >= 0)
if alpha == 1:
# Calculate this by l'Hospital rule
return eps * (math.cosh(eps) - 1) / math.sinh(eps)
elif np.isinf(alpha):
return eps
elif alpha > 1:
# in the proof of Lemma 4 of Bun et al. (2016)
s, mag = utils.stable_log_diff_exp(
utils.stable_log_sinh(alpha * eps),
utils.stable_log_sinh((alpha - 1) * eps))
return (mag - utils.stable_log_sinh(eps)) / (alpha - 1)
else:
return min(alpha * eps * eps / 2,
eps * (math.cosh(eps) - 1) / math.sinh(eps))