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Lyapunov.py
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import numpy as np
import torch
import torch.nn as nn
import torch.optim as optim
import matplotlib.pyplot as plt
from torch.optim.optimizer import required
np.random.seed(11)
iter_num = 100
x_init = np.sqrt(2) / 2.0
y_init = np.sqrt(2) / 2.0
beta_lsd = 0.5
lr_lsd = 0.1
alpha = 0.51
lyapunov_coef = 7.7
def lyapunov(x):
return lyapunov_coef * (x[0] ** 2 + x[1] ** 2)
def f(x):
return x[0] * x[1]
class LSDOpt(optim.Optimizer):
def __init__(self, params, lr=required, momentum=0, alpha=0.1):
defaults = dict(lr=lr, momentum=momentum, alpha=alpha)
super(LSDOpt, self).__init__(params, defaults)
def step(self, vjps, effect=[1, 1], closure=None):
loss = None
if closure is not None:
loss = closure()
for group in self.param_groups:
momentum = group['momentum']
for index, p in enumerate(group['params']):
if p.grad is None:
continue
vjp = vjps[index]
grad = p.grad.data
param_state = self.state[p]
if 'previous_iterate' not in param_state:
prev_p = param_state['previous_iterate'] = p.data.detach()
else:
prev_p = param_state['previous_iterate']
param_state['previous_iterate'] = p.data
p.data = p.data + effect[index] * momentum * (
p.data - prev_p.data) + group['alpha'] * vjp - group['lr'] * grad
return loss
def LSD(net):
net.x.data = torch.FloatTensor([x_init])
net.y.data = torch.FloatTensor([y_init])
xys = [[net.x.data[0] + 0, net.y.data[0] + 0]]
vs = [[0.0, 0.0]]
lyapunov_values = []
opt = LSDOpt(net.parameters(), lr=lr_lsd, momentum=beta_lsd, alpha=alpha)
for i in range(iter_num):
xys += [[net.x.data[0], net.y.data[0]]]
# update x
loss = net()
opt.zero_grad()
loss.backward(create_graph=True)
_, y_t = xys[-1]
_, y_tm1 = xys[-2]
v_y = (y_t - y_tm1)
vjp_x = torch.autograd.grad(
-net.y.grad, net.x,
grad_outputs=torch.FloatTensor([v_y]),
create_graph=True)[0]
vjps = [vjp_x.data, 0]
net.y.grad.data *= 0
opt.step(vjps=vjps, effect=[1, 0])
# update y
loss = -net()
opt.zero_grad()
loss.backward(create_graph=True)
x_t, _ = xys[-1]
x_tm1, _ = xys[-2]
v_x = x_t - x_tm1
vjp_y = torch.autograd.grad(
-net.x.grad, net.y,
grad_outputs=torch.FloatTensor([v_x]),
create_graph=True)[0]
vjps = [0, vjp_y.data]
net.x.grad.data *= 0
opt.step(vjps=vjps, effect=[0, 1])
vs += [[v_x, v_y]]
lyapunov_values += [2 * (v_x**2 + v_y**2) + lyapunov_coef *
(net.x.data[0]**2 + net.y.data[0]**2)]
xys = np.array(xys)
vs = np.array(vs)
lyapunov_values = np.array(lyapunov_values)
return (xys, vs, lyapunov_values)
class Net(nn.Module):
def __init__(self):
super(Net, self).__init__()
self.x = torch.nn.Parameter(torch.FloatTensor([0]))
self.y = torch.nn.Parameter(torch.FloatTensor([0]))
def forward(self, xy=None):
if xy is not None:
x = xy[:, 0]
y = xy[:, 1]
return (x * y)
return (self.x * self.y)
net = Net()
xys_vjp, vs_vjp, lyapunov_values = LSD(net)
def plot_lyapunov_descent(lyapunov_values):
plt.figure(figsize=(5, 5))
plt.plot(lyapunov_values, color='#0A4D8C')
plt.show()
plt.close()
plot_lyapunov_descent(lyapunov_values)
def plot_2_D_lyapunov(xys_vjp):
x_lim = 1.0
y_lim = 1.0
x_0 = np.linspace(-x_lim, x_lim, 100)
x_1 = np.linspace(-y_lim, y_lim, 100)
zl = np.zeros((100, 100))
for i in range(100):
for j in range(100):
zl[j, i] = lyapunov(np.array(
[x_0[i], x_1[j]])) + vs_vjp[i][0]**2 + vs_vjp[i][1]**2
X_0, X_1 = np.meshgrid(x_0, x_1)
cs = plt.contourf(X_0, X_1, zl, 15, cmap='Blues')
plt.colorbar()
plt.contour(cs, color='k', linewidths=0.3, ls='-', alpha=0.7)
zs = []
for i in range(xys_vjp.shape[0]):
zs += [f(xys_vjp[i]) + np.linalg.norm(vs_vjp[i])]
zs = np.array(zs)
plt.plot(xys_vjp[:, 0], xys_vjp[:, 1], c='#DD1321', zorder=5)
plt.show()
plot_2_D_lyapunov(xys_vjp)