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Copy pathlossdependent_SE.py
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136 lines (115 loc) · 4.64 KB
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from numba import njit
from math import sqrt, erf, exp, atan, pi
import numpy as np
@njit
def safe_erf(x):
if x > 6.0:
return 1.0
elif x < -6.0:
return -1.0
else:
return erf(x)
@njit
def safe_exp(x):
if x < -700.0: # underflow threshold
return 0.0
else:
return exp(x)
# --- Square loss ---
@njit
def compute_square_conjugates(alpha, epsilon, m, q, V, nodes, weights):
den = 1 + V
m_hat = alpha * (1 - epsilon) * sqrt(2 / pi) / den
q_hat = alpha * (1 + q - 2 * sqrt(2 / pi) * (1 - epsilon) * m) / den**2
V_hat = alpha / den
return m_hat, q_hat, V_hat
@njit
def compute_square_conjugates_kernel_zerolambda(alpha, epsilon, m, q, V, nodes, weights):
m_hat = alpha * (1 - epsilon) * sqrt(2 / pi) / V
q_hat = alpha * (1 + q - 2 * sqrt(2 / pi) * (1 - epsilon) * m) / V**2
V_hat = alpha / V
return m_hat, q_hat, V_hat
# --- Hinge loss ---
@njit
def compute_hinge_conjugates(alpha, epsilon, m, q, V, nodes, weights):
# Utils
eta = m**2 / q
one_minus_eta = 1.0 - eta
sqrt2q = sqrt(2 * q)
sqrt_2pi = sqrt(2.0 * pi)
sqrt_2q_one_minus_eta = sqrt(2.0 * q * one_minus_eta)
# m_hat
m_hat_term1 = 1.0 + (safe_erf(1.0 / sqrt_2q_one_minus_eta) + safe_erf((1.0 - V) / sqrt_2q_one_minus_eta) * (V - 1.0)) / V
m_hat_term2 = sqrt_2q_one_minus_eta / (V * sqrt(pi)) * (safe_exp(-1.0 / (2.0 * q * one_minus_eta)) - safe_exp(-(1.0 - V)**2 / (2.0 * q * one_minus_eta)))
m_hat = alpha * (1.0 - epsilon) / sqrt_2pi * (m_hat_term1 + m_hat_term2)
# q_hat
q_hat_term1 = alpha * (0.5 + (0.5 * safe_erf(1 / sqrt2q) * (1 + q) + 0.5 * safe_erf((1 - V) / sqrt2q) * (V**2 - 1 - q) + sqrt(q / (2 * pi)) * (safe_exp(-1 / (2 * q)) - safe_exp(-(1 - V)**2 / (2 * q)) * (1 + V))) / V**2)
a1 = -(1 - V) / sqrt(q)
b1 = 0.0
integral1 = 0.0
for i in range(len(nodes)):
t = 0.5 * (b1 - a1) * nodes[i] + 0.5 * (b1 + a1)
f = safe_exp(-t**2 / 2) / sqrt(2 * pi) * safe_erf(sqrt(eta) * t / sqrt(2 * one_minus_eta))
integral1 += weights[i] * f
integral1 *= 0.5 * (b1 - a1)
a2 = (1 - V) / sqrt(q)
b2 = 1 / sqrt(q)
integral2 = 0.0
for i in range(len(nodes)):
t = 0.5 * (b2 - a2) * nodes[i] + 0.5 * (b2 + a2)
f = safe_exp(-t**2 / 2) / sqrt(2 * pi) * safe_erf(sqrt(eta) * t / sqrt(2 * one_minus_eta)) * (1 - sqrt(q) * t)**2
integral2 += weights[i] * f
integral2 *= 0.5 * (b2 - a2)
arcterm = -atan(sqrt(eta) / sqrt(one_minus_eta)) / pi
q_hat_term2 = alpha * (1 - epsilon) * (arcterm - integral1 + integral2 / V**2)
q_hat = q_hat_term1 + q_hat_term2
# V_hat
V_hat_term1 = alpha / (2 * V) * (safe_erf(1 / sqrt2q) - safe_erf((1 - V) / sqrt2q))
a = (1 - V) / sqrt(q)
b = 1 / sqrt(q)
integral = 0.0
for i in range(len(nodes)):
t = 0.5 * (b - a) * nodes[i] + 0.5 * (b + a)
f = safe_exp(-t**2 / 2) / sqrt(2 * pi) * safe_erf(sqrt(eta) * t / sqrt(2 * one_minus_eta))
integral += weights[i] * f
integral *= 0.5 * (b - a)
V_hat_term2 = alpha * (1 - epsilon) / V * integral
V_hat = V_hat_term1 + V_hat_term2
return m_hat, q_hat, V_hat
@njit
def compute_hinge_conjugates_kernel_zerolambda(alpha, epsilon, m, q, V, nodes, weights):
# Utils
eta = m**2 / q
one_minus_eta = 1.0 - eta
sqrt2q = sqrt(2 * q)
sqrt_2pi = sqrt(2.0 * pi)
sqrt_2q_one_minus_eta = sqrt(2.0 * q * one_minus_eta)
# m_hat
m_hat_term1 = sqrt_2pi * (erf(1.0 / sqrt_2q_one_minus_eta) + 1)
m_hat_term2 = 2 * sqrt(q) * sqrt(one_minus_eta) * exp(-1 / (2.0 * q * one_minus_eta))
m_hat = alpha * (1 - epsilon) / (2 * pi * V) * (m_hat_term1 + m_hat_term2)
# q_hat
q_hat_term1 = (1 + q) * (1 + erf(1 / sqrt2q)) / 2 + sqrt(q / (2 * pi)) * exp(-1 / (2 * q))
a1 = -10
b1 = 1 / sqrt(q)
integral1 = 0.0
for i in range(len(nodes)):
t = 0.5 * (b1 - a1) * nodes[i] + 0.5 * (b1 + a1)
f = exp(-t**2 / 2) / sqrt_2pi * erf(sqrt(eta) * t / sqrt(2 * one_minus_eta)) * (1 - sqrt(q) * t)**2
integral1 += weights[i] * f
integral1 *= 0.5 * (b1 - a1)
q_hat_term2 = (1 - epsilon) * integral1
q_hat = alpha / V**2 * (q_hat_term1 + q_hat_term2)
# V_hat
V_hat_term1 = (1 + erf(1 / sqrt2q)) / 2
a2 = -10
b2 = 1 / sqrt(q)
integral2 = 0.0
for i in range(len(nodes)):
t = 0.5 * (b2 - a2) * nodes[i] + 0.5 * (b2 + a2)
f = exp(-t**2 / 2) / sqrt_2pi * erf(sqrt(eta) * t / sqrt(2 * one_minus_eta))
integral2 += weights[i] * f
integral2 *= 0.5 * (b2 - a2)
V_hat_term2 = (1 - epsilon) * integral2
V_hat = alpha / V * (V_hat_term1 + V_hat_term2)
return m_hat, q_hat, V_hat