This code is provided as supplementary material of the lecture Machine Learning and Optimization in Communications (MLOC).
This code illustrates
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import torch
import torch.nn as nn
import torch.optim as optim
import numpy as np
import matplotlib
import matplotlib.pyplot as plt
from ipywidgets import interactive
import ipywidgets as widgets
device = 'cuda' if torch.cuda.is_available() else 'cpu'
print("We are using the following device for learning:",device)
Specify the parameters of the transmission as the fiber length $L$ (in km), the fiber nonlinearity coefficienty $\gamma$ (given in 1/W/km) and the total noise power $P_n$ (given in dBM. The noise is due to amplified spontaneous emission in amplifiers along the link). We assume a model of a dispersion-less fiber affected by nonlinearity. The model, which is described for instance in [1] is given by an iterative application of the equation $$ x_{k+1} = x_k\exp\left(\jmath\frac{L}{K}\gamma|x_k|^2\right) + n_{k+1},\qquad 0 \leq k < K $$ where $x_0$ is the channel input (the modulated, complex symbols) and $x_K$ is the channel output. $K$ denotes the number of steps taken to simulate the channel Usually $K=50$ gives a good approximation.
[1] S. Li, C. Häger, N. Garcia, and H. Wymeersch, "Achievable Information Rates for Nonlinear Fiber Communication via End-to-end Autoencoder Learning," Proc. ECOC, Rome, Sep. 2018
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# Length of transmission (in km)
L = 5000
# fiber nonlinearity coefficient
gamma = 1.27
Pn = -21.3 # noise power (in dBm)
Kstep = 50 # number of steps used in the channel model
# noise variance per step
sigma_n = np.sqrt((10**((Pn-30)/10)) / Kstep / 2)
# def simulate_channel(x, Pin, constellation):
# # modulate bpsk
# input_power_linear = 10**((Pin-30)/10)
# norm_factor = 1 / np.sqrt(np.mean(np.abs(constellation)**2)/input_power_linear)
# modulated = constellation[x] * norm_factor
#
#
# temp = np.array(modulated, copy=True)
# for i in range(Kstep):
# power = np.absolute(temp)**2
# rotcoff = (L / Kstep) * gamma * power
#
# temp = temp * np.exp(1j*rotcoff) + sigma_n*(np.random.randn(len(x)) + 1j*np.random.randn(len(x)))
# return temp
Helper function to compute Symbol Error Rate (SER)
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# helper function to compute the symbol error rate
def SER(predictions, labels):
return (np.sum(np.argmax(predictions, 1) != labels) / predictions.shape[0])
Here, we define the parameters of the neural network and training, generate the validation set and a helping set to show the decision regions
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# set input power
Pin = 4
input_power_linear = 10**((Pin-30)/10)
# number of points in constellation
M = 16
# validation set. Training examples are generated on the fly
N_valid = 100000
# number of neurons in hidden layers at transmitter
hidden_neurons_TX_1 = 50
hidden_neurons_TX_2 = 50
hidden_neurons_TX_3 = 50
hidden_neurons_TX_4 = 50
hidden_neurons_TX = [hidden_neurons_TX_1, hidden_neurons_TX_2, hidden_neurons_TX_3, hidden_neurons_TX_4]
# number of neurons in hidden layers at receiver
hidden_neurons_RX_1 = 50
hidden_neurons_RX_2 = 50
hidden_neurons_RX_3 = 50
hidden_neurons_RX_4 = 50
hidden_neurons_RX = [hidden_neurons_RX_1, hidden_neurons_RX_2, hidden_neurons_RX_3, hidden_neurons_RX_4]
# Generate Validation Data
y_valid = np.random.randint(M,size=N_valid)
y_valid_onehot = np.eye(M)[y_valid]
# meshgrid for plotting
# assume that the worst case constellation is the one where all points lie on a straight line starting at the center and then are spreaded equidistantly. In this case, this is the scaling factor of the constellation points and we assume that there is an (M+1)th point which defines ext_max
ext_max = np.sqrt(M*input_power_linear)
mgx,mgy = np.meshgrid(np.linspace(-ext_max,ext_max,400), np.linspace(-ext_max,ext_max,400))
meshgrid = np.column_stack((np.reshape(mgx,(-1,1)),np.reshape(mgy,(-1,1))))
This is the main neural network/Autoencoder with transmitter, channel and receiver. Transmitter and receiver each with ELU activation function. Note that the final layer does not use a softmax function, as this function is already included in the CrossEntropyLoss.
Here the idea is to vary the batch size during training. In the first iterations, we start with a small batch size to rapidly get to a working solution. The closer we come towards the end of the training we increase the batch size. If keeping the abtch size small, it may happen that there are no misclassifications in a small batch and there is no incentive of the training to improve. A larger batch size will most likely contain errors in the batch and hence there will be incentive to keep on training and improving.
Here, the data is generated on the fly inside the graph, by using TensorFlows random number generation. As TensorFlow does not natively support complex numbers (at least in early versions), we decided to replace the complex number operations in the channel by a simple rotation matrix and treating real and imaginary parts separately.
We use the ELU activation function inside the neural network and employ the Adam optimization algorithm.
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class Autoencoder(nn.Module):
def __init__(self, hidden_neurons_TX, hidden_neurons_RX):
super(Autoencoder, self).__init__()
# Define Transmitter Layer: Linear function, M input neurons (symbols), 2 output neurons (real and imaginary part)
self.fcT1 = nn.Linear(M,hidden_neurons_TX[0])
self.fcT2 = nn.Linear(hidden_neurons_TX[0], hidden_neurons_TX[1])
self.fcT3 = nn.Linear(hidden_neurons_TX[1], hidden_neurons_TX[2])
self.fcT4 = nn.Linear(hidden_neurons_TX[2], hidden_neurons_TX[3])
self.fcT5 = nn.Linear(hidden_neurons_TX[3], 2)
# Define Receiver Layer: Linear function, 2 input neurons (real and imaginary part), M output neurons (symbols)
self.fcR1 = nn.Linear(2,hidden_neurons_RX[0])
self.fcR2 = nn.Linear(hidden_neurons_RX[0], hidden_neurons_RX[1])
self.fcR3 = nn.Linear(hidden_neurons_RX[1], hidden_neurons_RX[2])
self.fcR4 = nn.Linear(hidden_neurons_RX[2], hidden_neurons_RX[3])
self.fcR5 = nn.Linear(hidden_neurons_RX[3], M)
# Non-linearity (used in transmitter and receiver)
self.activation_function = nn.ELU()
def forward(self, x, input_power_linear):
# compute output
encoded = self.network_transmitter(x)
# compute normalization factor and normalize channel output
norm_factor = torch.sqrt(torch.mean(torch.mul(encoded,encoded))/input_power_linear * 2 )
modulated = encoded / norm_factor
received = self.channel_model(modulated)
logits = self.network_receiver(received)
return logits
def network_transmitter(self,batch_labels):
out = self.activation_function(self.fcT1(batch_labels))
out = self.activation_function(self.fcT2(out))
out = self.activation_function(self.fcT3(out))
out = self.activation_function(self.fcT4(out))
out = self.activation_function(self.fcT5(out))
return out
def network_receiver(self,inp):
out = self.activation_function(self.fcR1(inp))
out = self.activation_function(self.fcR2(out))
out = self.activation_function(self.fcR3(out))
out = self.activation_function(self.fcR4(out))
logits = self.activation_function(self.fcR5(out))
return logits
def channel_model(self,modulated):
# simulate the channel
for i in range(Kstep):
power = torch.norm(modulated, dim=1) ** 2
rotcoff = (L / Kstep) * gamma * power
# rotation matrix corresponding to exp(1j*rotcoff)
temp = torch.stack([modulated[:,0] * torch.cos(rotcoff) - modulated[:,1]*torch.sin(rotcoff), modulated[:,0]*torch.sin(rotcoff)+modulated[:,1]*torch.cos(rotcoff)], dim=1)
modulated = torch.add(temp, sigma_n*torch.randn(len(modulated),2).to(device))
return modulated
Now, carry out the training as such. First initialize the variables and then loop through the training. Here, the epochs are not defined in the classical way, as we do not have a training set per se. We generate new data on the fly and never reuse data.
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model = Autoencoder(hidden_neurons_TX, hidden_neurons_RX)
model.to(device)
softmax = nn.Softmax(dim=1)
# Cross Entropy loss
loss_fn = nn.CrossEntropyLoss()
# Adam Optimizer
optimizer = optim.Adam(model.parameters())
# Training parameters
num_epochs = 35
batches_per_epoch = 300
# Vary batch size during training
batch_size_per_epoch = np.linspace(100,10000,num=num_epochs)
validation_SERs = np.zeros(num_epochs)
validation_received = []
decision_region_evolution = []
constellations = []
print('Start Training')
for epoch in range(num_epochs):
batch_labels = torch.empty(int(batch_size_per_epoch[epoch]), device=device)
for step in range(batches_per_epoch):
# Generate training data: In most cases, you have a dataset and do not generate a training dataset during training loop
# sample new mini-batch directory on the GPU (if available)
batch_labels.random_(M)
batch_labels_onehot = torch.zeros(int(batch_size_per_epoch[epoch]), M, device=device)
batch_labels_onehot[range(batch_labels_onehot.shape[0]), batch_labels.long()]=1
# Propagate (training) data through the net
logits = model(batch_labels_onehot, input_power_linear)
# compute loss
loss = loss_fn(logits.squeeze(), batch_labels.long())
# compute gradients
loss.backward()
# Adapt weights
optimizer.step()
# reset gradients
optimizer.zero_grad()
# compute validation SER
out_valid = softmax(model(torch.Tensor(y_valid_onehot).to(device), input_power_linear))
validation_SERs[epoch] = SER(out_valid.detach().cpu().numpy().squeeze(), y_valid)
print('Validation SER after epoch %d: %f (loss %1.8f)' % (epoch, validation_SERs[epoch], loss.detach().cpu().numpy()))
# calculate and store received validation data
encoded = model.network_transmitter(torch.Tensor(y_valid_onehot).to(device))
norm_factor = torch.sqrt(torch.mean(torch.mul(encoded,encoded))/input_power_linear * 2 )
modulated = encoded / norm_factor
received = model.channel_model(modulated)
validation_received.append(received.detach().cpu().numpy())
# calculate and store constellation
encoded = model.network_transmitter(torch.eye(M).to(device))
norm_factor = torch.sqrt(torch.mean(torch.mul(encoded,encoded))/input_power_linear * 2 )
modulated = encoded / norm_factor
constellations.append(modulated.detach().cpu().numpy())
# store decision region for generating the animation
mesh_prediction = softmax(model.network_receiver(torch.Tensor(meshgrid).to(device)))
decision_region_evolution.append(0.195*mesh_prediction.detach().cpu().numpy() + 0.4)
print('Training finished')
Plot decision region and scatter plot of the validation set. Note that the validation set is only used for computing SERs and plotting, there is no feedback towards the training!
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cmap = matplotlib.cm.tab20
base = plt.cm.get_cmap(cmap)
color_list = base.colors
new_color_list = [[t/2 + 0.5 for t in color_list[k]] for k in range(len(color_list))]
# find minimum SER from validation set
min_SER_iter = np.argmin(validation_SERs)
print('Minimum SER obtained: %1.5f' % validation_SERs[min_SER_iter])
ext_max_plot = 1.05*max(max(abs(validation_received[min_SER_iter][:,0])), max(abs(validation_received[min_SER_iter][:,1])))
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%matplotlib inline
plt.figure(figsize=(19,6))
font = {'size' : 14}
plt.rc('font', **font)
plt.rc('text', usetex=True)
plt.subplot(131)
plt.scatter(constellations[min_SER_iter][:,0], constellations[min_SER_iter][:,1], c=range(M), cmap='tab20',s=50)
plt.axis('scaled')
plt.xlabel(r'$\Re\{r\}$',fontsize=14)
plt.ylabel(r'$\Im\{r\}$',fontsize=14)
plt.xlim((-ext_max_plot,ext_max_plot))
plt.ylim((-ext_max_plot,ext_max_plot))
plt.grid(which='both')
plt.title('Constellation',fontsize=16)
plt.subplot(132)
#plt.contourf(mgx,mgy,decision_region_evolution[-1].reshape(mgy.shape).T,cmap='coolwarm',vmin=0.3,vmax=0.7)
plt.scatter(validation_received[min_SER_iter][:,0], validation_received[min_SER_iter][:,1], c=y_valid, cmap='tab20',s=4)
plt.axis('scaled')
plt.xlabel(r'$\Re\{r\}$',fontsize=14)
plt.ylabel(r'$\Im\{r\}$',fontsize=14)
plt.xlim((-ext_max_plot,ext_max_plot))
plt.ylim((-ext_max_plot,ext_max_plot))
plt.title('Received',fontsize=16)
plt.subplot(133)
decision_scatter = np.argmax(decision_region_evolution[min_SER_iter], 1)
plt.scatter(meshgrid[:,0], meshgrid[:,1], c=decision_scatter, cmap=matplotlib.colors.ListedColormap(colors=new_color_list),s=4)
plt.scatter(validation_received[min_SER_iter][0:4000,0], validation_received[min_SER_iter][0:4000,1], c=y_valid[0:4000], cmap='tab20',s=4)
plt.axis('scaled')
plt.xlim((-ext_max_plot,ext_max_plot))
plt.ylim((-ext_max_plot,ext_max_plot))
plt.xlabel(r'$\Re\{r\}$',fontsize=14)
plt.ylabel(r'$\Im\{r\}$',fontsize=14)
plt.title('Decision regions',fontsize=16)
#plt.savefig('decision_region_AE_Pin%d.pdf' % Pin,bbox_inches='tight')
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Generate animation and save as a gif.
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%matplotlib notebook
%matplotlib notebook
# Generate animation
from matplotlib import animation, rc
#from matplotlib.animation import PillowWriter # Disable if you don't want to save any GIFs.
font = {'size' : 18}
plt.rc('font', **font)
fig = plt.figure(figsize=(14,6))
ax1 = fig.add_subplot(1,2,1)
ax2 = fig.add_subplot(1,2,2)
ax1.axis('scaled')
ax2.axis('scaled')
written = False
def animate(i):
ax1.clear()
ax1.scatter(constellations[i][:,0], constellations[i][:,1], c=range(M), cmap='tab20',s=50)
ax2.clear()
#ax2.scatter([0,0.02],[0.02,0], c=[1,2], cmap='tab20',s=100)
#decision_scatter = np.argmax(decision_region_evolution[i], 1)
decision_scatter = np.argmax(decision_region_evolution[i], 1)
ax2.scatter(meshgrid[:,0], meshgrid[:,1], c=decision_scatter, cmap=matplotlib.colors.ListedColormap(colors=new_color_list),s=4)
ax2.scatter(validation_received[i][0:4000,0], validation_received[i][0:4000,1], c=y_valid[0:4000], cmap='tab20',s=4)
#plt.scatter(meshgrid[:,0] * ext_max,meshgrid[:,1] * ext_max, c=decision_scatter, cmap=matplotlib.colors.ListedColormap(colors=new_color_list),s=4, marker='s')
#plt.scatter(X_valid[0:4000,0]*ext_max, X_valid[0:4000,1]*ext_max, c=y_valid[0:4000], cmap='tab20',s=4)
ax1.set_xlim(( -ext_max_plot, ext_max_plot))
ax1.set_ylim(( -ext_max_plot, ext_max_plot))
ax2.set_xlim(( -ext_max_plot, ext_max_plot))
ax2.set_ylim(( -ext_max_plot, ext_max_plot))
ax1.set_title('Constellation', fontsize=14)
ax2.set_title('Decision regions', fontsize=14)
ax1.set_xlabel(r'$\Re\{r\}$',fontsize=14)
ax1.set_ylabel(r'$\Im\{r\}$',fontsize=14)
ax2.set_xlabel(r'$\Re\{r\}$',fontsize=14)
ax2.set_ylabel(r'$\Im\{r\}$',fontsize=14)
anim = animation.FuncAnimation(fig, animate, frames=min_SER_iter+1, interval=200, blit=False)
fig.show()
#anim.save('learning_decision_AE_Pin%d_varbatch.gif' % Pin, writer=PillowWriter(fps=5))
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