pytorch.py

#Training a 1-hidden layer neural network using the PyTorch library.


import  numpy as np 
import matplotlib.pyplot as plt 
from scipy.linalg import qr
import numdifftools as nd 
import torch
import torch.nn as nn
import torch.nn.functional as F
from torch.autograd import Variable
import torch.optim as optim


#PARAMETERS

n = 600 #number of datapoints
dimx = 13 #dimension of input
dimy = 4 #dimension of output
bs = 30 #batch size (default 30)
h = 4 #size of the hidden layer (default 4)
epochs = 30 #number of epochs (default 50)
lr = 1e-12 #learning rate (default 1e-5)
alpha = 0.9 #momentum parameter (default 0.9)
sigma = 0.5
mu = 0
ev = 1e6 #max eigenvalue (default 1e0)
criterion = nn.MSELoss() #loss


#FUNCTIONS

def random_normal_init(shape):
    return np.random.normal(size=shape, scale=0.05)

def xavier_init(shape):
    epsilon = np.sqrt(6/sum(shape))
    return np.random.uniform(low=-epsilon, high=epsilon, size=shape)

def newton_function(M):
    preds = np.matmul(x_enlarged,M)
    return np.mean((preds-y.reshape((n*dimy)))*(preds-y.reshape((n*dimy))))

class Net(nn.Module):

    def __init__(self):
        super(Net, self).__init__()
        self.fc1 = nn.Linear(dimx, h)
        self.fc2 = nn.Linear(h, dimy)

    def forward(self, x):
        x = x.view(-1, self.num_flat_features(x))
        x = self.fc1(x)
        x = self.fc2(x)
        return x

    def num_flat_features(self, x):
        size = x.size()[1:]  # all dimensions except the batch dimension
        num_features = 1
        for s in size:
            num_features *= s
        return num_features

#DATA

x = np.random.rand(n,dimx) #data
m = np.mean(x)
s = np.std(x)

Aux1 = np.random.randn(dimx,dimx)
U, R1 = qr(Aux1)
Aux2 = np.random.randn(dimy,dimy)
V, R2 = qr(Aux2)
diag = sigma*np.random.randn(min(dimx,dimy))+mu
D = np.zeros((dimx,dimy))
for i in range(min(dimx,dimy)):
    D[i,i] = diag[i]
D[0,0] += ev #introducing bad conditioning

A = np.matmul(np.matmul(U, D), V.T) #true relation that we try to learn
print('Condition number of the A matrix: {}'.format(np.linalg.cond(A)))
print(A.shape)

y = np.matmul(x,A) #truth (labels)



#1-STOCHASTIC GRADIENT DESCENT WITH MOMENTUM

print('SGD with momentum using the Pytorch package')

net = Net()
optimizer = optim.SGD(net.parameters(), lr=lr, momentum=alpha)

losses_gd = []
delta1 = 0
delta2 = 0

for i in range(epochs):

    print('Epoch: {}'.format(i))
    avg_loss = []
    np.random.shuffle(x)

    for j in range(int(n/bs)):

        batch = torch.from_numpy(x[j*bs:(j+1)*bs])
        batch = batch.type(torch.FloatTensor)
        labels = torch.from_numpy(y[j*bs:(j+1)*bs])
        labels = labels.type(torch.FloatTensor)

        batch, labels = Variable(batch), Variable(labels)

        outputs = net(batch)
        loss = criterion(outputs, labels)
        loss.backward()
        optimizer.step()

        avg_loss.append(loss.data[0])
        
    #print avg loss on the epoch
    print('loss on this epoch: {}'.format(np.mean(np.array(avg_loss))))
    losses_gd.append(np.mean(np.array(avg_loss)))

#PLOTTING THE LOSS

plt.plot(np.log(np.array(losses_gd)), label="Gradient descent")
plt.legend()
plt.title("Log-loss over epochs")
plt.show()


#2-NEWTON'S METHOD

print('Newton method')

losses_newton = []

W1 = random_normal_init((dimx,h))
W2 = random_normal_init((h,dimy))
x_enlarged = np.zeros((n*dimy,dimx*dimy))
for i in range(n*dimy):
    x_enlarged[i,(i % dimy)*dimx:((i % dimy) +1)*dimx] = x[int(i/dimy),:]
W = np.matmul(W1,W2)
W = W.reshape((dimx*dimy))

for i in range(epochs):

    print('Epoch number: {}'.format(i))

    gd = nd.Gradient(newton_function)
    
    Hess = nd.Hessian(newton_function)
    Hinv = np.linalg.solve(Hess(gd(W)),np.identity(dimx*dimy))

    W = W-np.matmul(Hinv, gd(W))

    loss = newton_function(W)
    print('loss on this epoch: {}'.format(loss))
    losses_newton.append(loss)

#PLOTTING THE LOSS

plt.plot(np.log(np.array(losses_gd)), label="Gradient descent")
plt.plot(np.log(np.array(losses_newton)), label="Newton-Raphson")
plt.title('Log-Loss over epochs')
plt.legend()
#plt.ylim(ymin=-50, ymax=50)
plt.show()