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activations.py
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"""
activations.py
Contains bounds on various activation functions
Copyright (C) 2018, Akhilan Boopathy <[email protected]>
Lily Weng <[email protected]>
Pin-Yu Chen <[email protected]>
Sijia Liu <[email protected]>
Luca Daniel <[email protected]>
"""
from numba import njit
import numpy as np
#Functions for bounding various activation functions
@njit
def sigmoid(x):
return 1.0/(1.0+np.exp(-x))
@njit
def sigmoidd(x):
return np.exp(-x)/(1.0+np.exp(-x))**2
@njit
def sigmoidid(x):
return 2.0*np.arccosh(1.0/(2.0*np.sqrt(x)))
@njit
def sigmoidut(l, u):
act = sigmoid
actd = sigmoidd
actid = sigmoidid
upper = u
lower = 0
al = act(l)
for i in range(20):
guess = (upper + lower)/2
guesst = actd(guess)
guesss = (act(guess)-al)/(guess-l)
if guesss >= guesst:
upper = guess
else:
lower = guess
return upper
@njit
def sigmoidlt(l, u):
act = sigmoid
actd = sigmoidd
actid = sigmoidid
upper = 0
lower = l
au = act(u)
for i in range(20):
guess = (upper + lower)/2
guesst = actd(guess)
guesss = (au-act(guess))/(u-guess)
if guesss >= guesst:
lower = guess
else:
upper = guess
return lower
@njit
def tanh(x):
return np.tanh(x)
@njit
def tanhd(x):
return 1.0/np.cosh(x)**2
@njit
def tanhid(x):
return np.arccosh(1.0/np.sqrt(x))
@njit
def tanhut(l, u):
act = tanh
actd = tanhd
actid = tanhid
upper = u
lower = 0
al = act(l)
for i in range(20):
guess = (upper + lower)/2
guesst = actd(guess)
guesss = (act(guess)-al)/(guess-l)
if guesss >= guesst:
upper = guess
else:
lower = guess
return upper
@njit
def tanhlt(l, u):
act = tanh
actd = tanhd
actid = tanhid
upper = 0
lower = l
au = act(u)
for i in range(20):
guess = (upper + lower)/2
guesst = actd(guess)
guesss = (au-act(guess))/(u-guess)
if guesss >= guesst:
lower = guess
else:
upper = guess
return lower
@njit
def atan(x):
return np.arctan(x)
@njit
def atand(x):
return 1.0/(1.0+x**2)
@njit
def atanid(x):
return np.sqrt(1.0/x-1.0)
@njit
def atanut(l, u):
act = atan
actd = atand
actid = atanid
upper = u
lower = 0
al = act(l)
for i in range(20):
guess = (upper + lower)/2
guesst = actd(guess)
guesss = (act(guess)-al)/(guess-l)
if guesss >= guesst:
upper = guess
else:
lower = guess
return upper
@njit
def atanlt(l, u):
act = atan
actd = atand
actid = atanid
upper = 0
lower = l
au = act(u)
for i in range(20):
guess = (upper + lower)/2
guesst = actd(guess)
guesss = (au-act(guess))/(u-guess)
if guesss >= guesst:
lower = guess
else:
upper = guess
return lower
@njit
def relu_linear_bounds(LB, UB):
alpha_u = np.zeros(UB.shape, dtype=np.float32)
beta_u = np.zeros(UB.shape, dtype=np.float32)
alpha_l = np.zeros(LB.shape, dtype=np.float32)
beta_l = np.zeros(LB.shape, dtype=np.float32)
for i in range(LB.shape[0]):
for j in range(LB.shape[1]):
for k in range(LB.shape[2]):
## Original
if LB[i,j,k] > 0:
alpha_u[i,j,k] = 1
alpha_l[i,j,k] = 1
elif UB[i,j,k] <= 0:
pass #All zeros
else:
alpha_u[i,j,k] = UB[i,j,k]/(UB[i,j,k]-LB[i,j,k])
alpha_l[i,j,k] = UB[i,j,k]/(UB[i,j,k]-LB[i,j,k])
beta_u[i,j,k] = -alpha_u[i,j,k]*LB[i,j,k]
return alpha_u, alpha_l, beta_u, beta_l
@njit
def ada_linear_bounds(LB, UB):
alpha_u = np.zeros(UB.shape, dtype=np.float32)
beta_u = np.zeros(UB.shape, dtype=np.float32)
alpha_l = np.zeros(LB.shape, dtype=np.float32)
beta_l = np.zeros(LB.shape, dtype=np.float32)
for i in range(LB.shape[0]):
for j in range(LB.shape[1]):
for k in range(LB.shape[2]):
## Adaptive
if LB[i,j,k] >= 0:
alpha_u[i,j,k] = 1
alpha_l[i,j,k] = 1
elif UB[i,j,k] <= 0:
pass #All zeros
elif UB[i,j,k] >= -LB[i,j,k]:
alpha_u[i,j,k] = UB[i,j,k]/(UB[i,j,k]-LB[i,j,k])
alpha_l[i,j,k] = 1
beta_u[i,j,k] = -alpha_u[i,j,k]*LB[i,j,k]
else:
alpha_u[i,j,k] = UB[i,j,k]/(UB[i,j,k]-LB[i,j,k])
beta_u[i,j,k] = -alpha_u[i,j,k]*LB[i,j,k]
return alpha_u, alpha_l, beta_u, beta_l
@njit
def atan_linear_bounds(LB, UB):
alpha_u = np.zeros(UB.shape, dtype=np.float32)
beta_u = np.zeros(UB.shape, dtype=np.float32)
alpha_l = np.zeros(LB.shape, dtype=np.float32)
beta_l = np.zeros(LB.shape, dtype=np.float32)
for i in range(LB.shape[0]):
for j in range(LB.shape[1]):
for k in range(LB.shape[2]):
act = atan
actd = atand
actid = atanid
actut = atanut
actlt = atanlt
## General (Sigmoid-like functions)
if UB[i,j,k] == LB[i,j,k]:
alpha_u[i,j,k] = actd(UB[i,j,k])
alpha_l[i,j,k] = actd(LB[i,j,k])
beta_u[i,j,k] = act(UB[i,j,k])-actd(UB[i,j,k])*UB[i,j,k]
beta_l[i,j,k] = act(LB[i,j,k])-actd(LB[i,j,k])*LB[i,j,k]
elif LB[i,j,k] >= 0:
alpha = (act(UB[i,j,k])-act(LB[i,j,k]))/(UB[i,j,k]-LB[i,j,k])
d = (UB[i,j,k]+LB[i,j,k])/2#actid(alpha)
alpha_u[i,j,k] = actd(d)
alpha_l[i,j,k] = alpha
beta_u[i,j,k] = act(d)-actd(d)*d
beta_l[i,j,k] = act(LB[i,j,k])-alpha*LB[i,j,k]
elif UB[i,j,k] <= 0:
alpha = (act(UB[i,j,k])-act(LB[i,j,k]))/(UB[i,j,k]-LB[i,j,k])
d = (UB[i,j,k]+LB[i,j,k])/2#-actid(alpha)
alpha_u[i,j,k] = alpha
alpha_l[i,j,k] = actd(d)
beta_u[i,j,k] = act(LB[i,j,k])-alpha*LB[i,j,k]
beta_l[i,j,k] = act(d)-actd(d)*d
else:
du = actut(LB[i,j,k], UB[i,j,k])
dus = (act(du)-act(LB[i,j,k]))/(du-LB[i,j,k])
dut = actd(du)
if dut < dus:
alpha_u[i,j,k] = dut
beta_u[i,j,k] = act(du)-dut*du
else:
alpha_u[i,j,k] = dus
beta_u[i,j,k] = act(LB[i,j,k])-LB[i,j,k]*dus
dl = actlt(LB[i,j,k], UB[i,j,k])
dls = (act(dl)-act(UB[i,j,k]))/(dl-UB[i,j,k])
dlt = actd(dl)
if dlt < dls:
alpha_l[i,j,k] = dlt
beta_l[i,j,k] = act(dl)-dlt*dl
else:
alpha_l[i,j,k] = dls
beta_l[i,j,k] = act(UB[i,j,k])-UB[i,j,k]*dls
return alpha_u, alpha_l, beta_u, beta_l
@njit
def sigmoid_linear_bounds(LB, UB):
alpha_u = np.zeros(UB.shape, dtype=np.float32)
beta_u = np.zeros(UB.shape, dtype=np.float32)
alpha_l = np.zeros(LB.shape, dtype=np.float32)
beta_l = np.zeros(LB.shape, dtype=np.float32)
for i in range(LB.shape[0]):
for j in range(LB.shape[1]):
for k in range(LB.shape[2]):
act = sigmoid
actd = sigmoidd
actid = sigmoidid
actut = sigmoidut
actlt = sigmoidlt
## General (Sigmoid-like functions)
if UB[i,j,k] == LB[i,j,k]:
alpha_u[i,j,k] = actd(UB[i,j,k])
alpha_l[i,j,k] = actd(LB[i,j,k])
beta_u[i,j,k] = act(UB[i,j,k])-actd(UB[i,j,k])*UB[i,j,k]
beta_l[i,j,k] = act(LB[i,j,k])-actd(LB[i,j,k])*LB[i,j,k]
elif LB[i,j,k] >= 0:
alpha = (act(UB[i,j,k])-act(LB[i,j,k]))/(UB[i,j,k]-LB[i,j,k])
d = (UB[i,j,k]+LB[i,j,k])/2#actid(alpha)
alpha_u[i,j,k] = actd(d)
alpha_l[i,j,k] = alpha
beta_u[i,j,k] = act(d)-actd(d)*d
beta_l[i,j,k] = act(LB[i,j,k])-alpha*LB[i,j,k]
elif UB[i,j,k] <= 0:
alpha = (act(UB[i,j,k])-act(LB[i,j,k]))/(UB[i,j,k]-LB[i,j,k])
d = (UB[i,j,k]+LB[i,j,k])/2#-actid(alpha)
alpha_u[i,j,k] = alpha
alpha_l[i,j,k] = actd(d)
beta_u[i,j,k] = act(LB[i,j,k])-alpha*LB[i,j,k]
beta_l[i,j,k] = act(d)-actd(d)*d
else:
du = actut(LB[i,j,k], UB[i,j,k])
dus = (act(du)-act(LB[i,j,k]))/(du-LB[i,j,k])
dut = actd(du)
if dut < dus:
alpha_u[i,j,k] = dut
beta_u[i,j,k] = act(du)-dut*du
else:
alpha_u[i,j,k] = dus
beta_u[i,j,k] = act(LB[i,j,k])-LB[i,j,k]*dus
dl = actlt(LB[i,j,k], UB[i,j,k])
dls = (act(dl)-act(UB[i,j,k]))/(dl-UB[i,j,k])
dlt = actd(dl)
if dlt < dls:
alpha_l[i,j,k] = dlt
beta_l[i,j,k] = act(dl)-dlt*dl
else:
alpha_l[i,j,k] = dls
beta_l[i,j,k] = act(UB[i,j,k])-UB[i,j,k]*dls
return alpha_u, alpha_l, beta_u, beta_l
@njit
def tanh_linear_bounds(LB, UB):
alpha_u = np.zeros(UB.shape, dtype=np.float32)
beta_u = np.zeros(UB.shape, dtype=np.float32)
alpha_l = np.zeros(LB.shape, dtype=np.float32)
beta_l = np.zeros(LB.shape, dtype=np.float32)
for i in range(LB.shape[0]):
for j in range(LB.shape[1]):
for k in range(LB.shape[2]):
act = tanh
actd = tanhd
actid = tanhid
actut = tanhut
actlt = tanhlt
## General (Sigmoid-like functions)
if UB[i,j,k] == LB[i,j,k]:
alpha_u[i,j,k] = actd(UB[i,j,k])
alpha_l[i,j,k] = actd(LB[i,j,k])
beta_u[i,j,k] = act(UB[i,j,k])-actd(UB[i,j,k])*UB[i,j,k]
beta_l[i,j,k] = act(LB[i,j,k])-actd(LB[i,j,k])*LB[i,j,k]
elif LB[i,j,k] >= 0:
alpha = (act(UB[i,j,k])-act(LB[i,j,k]))/(UB[i,j,k]-LB[i,j,k])
d = (UB[i,j,k]+LB[i,j,k])/2#actid(alpha)
alpha_u[i,j,k] = actd(d)
alpha_l[i,j,k] = alpha
beta_u[i,j,k] = act(d)-actd(d)*d
beta_l[i,j,k] = act(LB[i,j,k])-alpha*LB[i,j,k]
elif UB[i,j,k] <= 0:
alpha = (act(UB[i,j,k])-act(LB[i,j,k]))/(UB[i,j,k]-LB[i,j,k])
d = (UB[i,j,k]+LB[i,j,k])/2#-actid(alpha)
alpha_u[i,j,k] = alpha
alpha_l[i,j,k] = actd(d)
beta_u[i,j,k] = act(LB[i,j,k])-alpha*LB[i,j,k]
beta_l[i,j,k] = act(d)-actd(d)*d
else:
du = actut(LB[i,j,k], UB[i,j,k])
dus = (act(du)-act(LB[i,j,k]))/(du-LB[i,j,k])
dut = actd(du)
if dut < dus:
alpha_u[i,j,k] = dut
beta_u[i,j,k] = act(du)-dut*du
else:
alpha_u[i,j,k] = dus
beta_u[i,j,k] = act(LB[i,j,k])-LB[i,j,k]*dus
dl = actlt(LB[i,j,k], UB[i,j,k])
dls = (act(dl)-act(UB[i,j,k]))/(dl-UB[i,j,k])
dlt = actd(dl)
if dlt < dls:
alpha_l[i,j,k] = dlt
beta_l[i,j,k] = act(dl)-dlt*dl
else:
alpha_l[i,j,k] = dls
beta_l[i,j,k] = act(UB[i,j,k])-UB[i,j,k]*dls
return alpha_u, alpha_l, beta_u, beta_l