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optimization.py
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optimization.py
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import autograd.numpy as np
import autograd.numpy.random as npr
from autograd import grad, elementwise_grad, jacobian
from autograd.core import getval
from kalman_filter import Kalman
from copy import copy
from matplotlib import pyplot as plt
"""
This package provides KGD, KGD + Momentum, and Kalman RMSProp
from the paper https://arxiv.org/pdf/1810.12273.pdf.
"""
"""
------------------------------
Interface methods.
"""
def gd(g, x, callback=None, num_iters=200,
step_size=0.1, how='regular',
sigma_Q=0.01, sigma_R=2.0):
"""
Performs vanilla gradient descent with how='regular'
or Kalman Gradient Descent using how='filtered'.
"""
if type(step_size) is type(0.1):
alpha=lambda t:step_size
else:
alpha=step_size
if how=='regular':
return _gd_regular(g, x,
callback=callback,
num_iters=num_iters,
alpha=alpha,
)
elif how=='filtered':
return _gd_filtered(g, x,
callback=callback,
num_iters=num_iters,
alpha=alpha,
sigma_Q=sigma_Q,
sigma_R=sigma_R
)
else:
raise ValueError("Allowable hows are 'regular' and 'filtered'.")
def momgd(g, x, callback=None, num_iters=200,
step_size=0.1, mass=0.9, how='regular',
sigma_Q=0.01, sigma_R=2.0):
"""
Performs gradient descent with momentum using how='regular'
or Kalman gradient descent with momentum using how='filtered'.
"""
if type(step_size) is type(0.1):
alpha=lambda t:step_size
else:
alpha=step_size
if type(mass) is type(0.1):
mu=lambda t:mass
else:
mu=mass
if how=='regular':
return _momgd_regular(g, x,
callback=callback,
num_iters=num_iters,
alpha=alpha,
mu=mu
)
elif how=='filtered':
return _momgd_filtered(g, x,
callback=callback,
num_iters=num_iters,
alpha=alpha,
mu=mu,
sigma_Q=sigma_Q,
sigma_R=sigma_R
)
else:
raise ValueError("Allowable hows are 'regular' and 'filtered'.")
def rmsprop(g, x, callback=None, num_iters=100,
step_size=0.1, gamma=0.9, eps=10**-8, how='regular',
sigma_Q=0.01, sigma_R=2.0):
"""
Performs RMSProp using how='regular'
or Kalman RMSProp using how='filtered'.
"""
if type(step_size) is type(0.1):
alpha=lambda t:step_size
else:
alpha=step_size
if type(gamma) is type(0.1):
gamma_=lambda t:gamma
else:
gamma_=gamma
if how=='regular':
return _rmsprop_regular(g, x,
callback=callback,
num_iters=num_iters,
alpha=alpha,
gamma=gamma_,
eps=eps
)
elif how=='filtered':
return _rmsprop_filtered(g, x,
callback=callback,
num_iters=num_iters,
alpha=alpha,
gamma=gamma_,
eps=eps,
sigma_Q=sigma_Q,
sigma_R=sigma_R
)
else:
raise ValueError("Allowable hows are 'regular' and 'filtered'.")
"""
------------------------------------
Benchmark unfiltered optimization algorithms
"""
def _gd_regular(g, x, callback=None, num_iters=200,
alpha=lambda t:0.1):
for t in range(num_iters):
dx=g(x)
if callback: callback(x, t, dx)
x+=-alpha(t)*dx
return x
def _momgd_regular(g, x, callback=None, num_iters=200,
alpha=lambda t:0.1, mu=lambda t:0.9):
v=np.zeros(len(x))
for t in range(num_iters):
dx=g(x)
if callback: callback(x, t, dx)
v=mu(t) * v - (1-mu(t))* dx
x+= alpha(t) * v
return x
def _rmsprop_regular(g, x, callback=None, num_iters=200,
alpha=lambda t:0.1, gamma=lambda t:0.9, eps=1e-8):
r=np.ones(len(x))
for t in range(num_iters):
dx=g(x)
if callback: callback(x, t, dx)
r = gamma(t) * r + (1-gamma(t))* dx**2
x += - alpha(t)/(np.sqrt(r)+ eps) * dx
return x
"""
--------------------------------------
Filtered optimization algorithms.
"""
def _gd_filtered(g, x, callback=None, num_iters=200,
alpha=lambda t:0.1,
sigma_Q=0.01, sigma_R=2.0):
"""
Implements the Kalman Gradient Descent algorithm from
https://arxiv.org/pdf/1810.12273.pdf.
"""
# Initialize the Kalman filter
N=len(x)
dx=g(x)
z_0=np.hstack([x, dx]).reshape(2*N,1)
Q=sigma_Q*np.eye(2*N)#dynamics noise
R=sigma_R*np.eye(N)#observation noise
C=np.hstack([np.zeros((N,N)), np.eye(N)])
kf=Kalman(init_state=z_0,
init_cov=np.eye(2*N)*0.01,
C_t=C, Q_t=Q, R_t=R
)
for t in range(num_iters):
# evaluate the gradient
dx=g(x)
#construct transition matrix
A_t=np.block(
[[np.eye(N) , -alpha(t)*np.eye(N)],
[np.zeros((N,N)) , np.eye(N)]]
)
#increment the kalman filter
z_hat=kf.filter(dx.T.reshape(N,1), A_t=A_t)
x_hat, dx_hat=z_hat[:N], z_hat[N:]
if callback: callback(x, t, dx_hat.reshape(x.shape))
#perform the KGD update
x+=-alpha(t)*dx_hat.flatten()
return x
def _momgd_filtered(g, x, callback=None, num_iters=200,
alpha=lambda t:0.1, mu=lambda t:0.9,
sigma_Q=0.01, sigma_R=2.0):
"""
Implements the Kalman Gradient Descent + momentum algorithm from
https://arxiv.org/pdf/1810.12273.pdf.
"""
# Initialize the Kalman filter
N=len(x)
dx=g(x)
v=np.zeros(len(x))
z_0=np.hstack([v, x, dx]).reshape(3*N,1)
Q=sigma_Q * np.eye(3*N)#dynamics noise
R=sigma_R*np.eye(N)#observation noise
C=np.hstack([np.zeros((N,2*N)), np.eye(N)])
kf=Kalman(init_state=z_0,
init_cov=np.eye(3*N)*0.01,
C_t=C, Q_t=Q, R_t=R
)
for t in range(num_iters):
# evaluate the gradient
dx=g(x)
#construct transition matrix
A_t=np.block(
[[np.eye(N) * mu(t) , np.zeros((N,N)), (1-mu(t))*np.eye(N)],
[-alpha(t)* np.eye(N), np.eye(N), -alpha(t)*(1-mu(t)) * np.eye(N)],
[np.zeros((N,N)) , np.zeros((N,N)), np.eye(N)]]
)
#increment the kalman filter
z_hat=kf.filter(dx.T.reshape(N,1), A_t=A_t)
v_hat, x_hat, dx_hat=z_hat[:N], z_hat[N:2*N], z_hat[2*N:]
if callback: callback(x, t, dx_hat.reshape(x.shape))
#perform the KGD + momentum update
v=mu(t) * v - (1-mu(t))* dx_hat.flatten()#don't flatte, use .reshpape(x.shape)
x+= alpha(t) * v
return x
def _rmsprop_filtered(g, x, callback=None, num_iters=200,
alpha=lambda t:0.1, gamma=lambda t:0.9, eps=1e-8,
sigma_Q=0.01, sigma_R=2.0):
"""
Implements the Kalman RMSProp algorithm from
https://arxiv.org/pdf/1810.12273.pdf.
"""
# Initialize the Kalman filter
N=len(x)
dx=g(x)
r=np.ones(len(x))#should be ones.
z_0=np.hstack([r, x, dx]).reshape(3*N,1)
Q=sigma_Q * np.eye(3*N)#dynamics noise
R=sigma_R*np.eye(N)#observation noise
C=np.hstack([np.zeros((N,2*N)), np.eye(N)])
kf=Kalman(init_state=z_0,
init_cov=np.eye(3*N)*0.01,
C_t=C, Q_t=Q, R_t=R
)
for t in range(num_iters):
# evaluate the gradient
dx=g(x)
#construct transition matrix
beta_t= alpha(t) / np.sqrt((gamma(t)*r + (1-gamma(t))*(dx**2))+ eps)
A_t=np.block(
[[np.eye(N) * gamma(t) , np.zeros((N,N)), (1-gamma(t))*np.diag(dx)],
[np.zeros((N,N)), np.eye(N), -beta_t * np.eye(N)],
[np.zeros((N,N)) , np.zeros((N,N)), np.eye(N)]]
)
#increment the kalman filter
z_hat=kf.filter(dx.T.reshape(N,1), A_t=A_t)
r_hat, x_hat, dx_hat=z_hat[:N], z_hat[N:2*N], z_hat[2*N:]
#perform the Kalman RMSProp update
r = gamma(t) * r + (1-gamma(t))* dx_hat.flatten()**2
x += - alpha(t)/(np.sqrt(r)+ eps) * dx_hat.flatten()
if callback: callback(x, t, dx_hat.reshape(x.shape))
return x
if __name__=='__main__':
pass