PyTorch Hello World

PyTorch (3 Part Series)

1. PyTorch Hello World

2. PyTorch Image Recognition with Dense Network

3. PyTorch Image Recognition with Convolutional Networks

I recently started working with PyTorch, a Python framework for neural networks and machine learning. Since machine learning involves processing large amounts of data, sometimes it can be hard to understand the results that one gets back from the network. Before getting into anything more complicated, let’s replicate a really basic backpropagation as a sanity check. To run the code in this article, you’ll need to install NumPy and PyTorch.

In neural networks primer, we saw how to manually calculate the forward and back propagation for a tiny network consisting of one input neuron, one hidden neuron, and one output neuron:

Sample calculation

We ran an input of 0.8 through the network, then backpropagated using 1 as the target value, with a learning rate of 0.1. We used sigmoid as the activation function and the quadratic cost function to compare the actual output from the network with the desired output.

The code below uses PyTorch to do the same thing:

import torch
import torch.nn as nn
import torch.optim as optim


class Net(nn.Module):
    def __init__(self):
        super(Net, self).__init__()
        self.hidden_layer = nn.Linear(1, 1)
        self.hidden_layer.weight = torch.nn.Parameter(torch.tensor([[1.58]]))
        self.hidden_layer.bias = torch.nn.Parameter(torch.tensor([-0.14]))

        self.output_layer = nn.Linear(1, 1)
        self.output_layer.weight = torch.nn.Parameter(torch.tensor([[2.45]]))
        self.output_layer.bias = torch.nn.Parameter(torch.tensor([-0.11]))

    def forward(self, x):
        x = torch.sigmoid(self.hidden_layer(x))
        x = torch.sigmoid(self.output_layer(x))
        return x


net = Net()
print(f"network topology: {net}")

print(f"w_l1 = {round(net.hidden_layer.weight.item(), 4)}")
print(f"b_l1 = {round(net.hidden_layer.bias.item(), 4)}")
print(f"w_l2 = {round(net.output_layer.weight.item(), 4)}")
print(f"b_l2 = {round(net.output_layer.bias.item(), 4)}")

# run input data forward through network
input_data = torch.tensor([0.8])
output = net(input_data)
print(f"a_l2 = {round(output.item(), 4)}")

# backpropagate gradient
target = torch.tensor([1.])
criterion = nn.MSELoss()
loss = criterion(output, target)
net.zero_grad()
loss.backward()

# update weights and biases
optimizer = optim.SGD(net.parameters(), lr=0.1)
optimizer.step()

print(f"updated_w_l1 = {round(net.hidden_layer.weight.item(), 4)}")
print(f"updated_b_l1 = {round(net.hidden_layer.bias.item(), 4)}")
print(f"updated_w_l2 = {round(net.output_layer.weight.item(), 4)}")
print(f"updated_b_l2 = {round(net.output_layer.bias.item(), 4)}")

output = net(input_data)
print(f"updated_a_l2 = {round(output.item(), 4)}")

Some notes on this code:

nn.Linear is used for fully connected, or dense, layers. For this simple case, we have a single input and a single output for each layer.
The forward method is called when we pass the input into the network with output = net(input_data).
By default, PyTorch sets up random weights and biases. However, here we initialize them directly since we want the results to match our manual calculation (shown later in the article).
In PyTorch, tensor is analogous to array in numpy.
criterion = nn.MSELoss() sets up the quadratic cost function - though it’s called the mean squared error loss function in PyTorch.
loss = criterion(output, target) calculates the cost, also known as the loss.
Next we use net.zero_grad() to reset the gradient to zero (otherwise the backpropagation is cumulative). It isn’t strictly necessary here, but it’s good to keep this in mind when running backpropagation in a loop.
loss.backward() computes the gradient, i.e. the derivative of the cost with respect to all of the weights and biases.
Finally we use this gradient to update the weights and biases in the network using the SGD (stochastic gradient descent) optimizer, with a learning rate of 0.1.

The results are below:

C:\Dev\python\pytorch>python backprop_pytorch.py
network topology: Net(
  (hidden_layer): Linear(in_features=1, out_features=1, bias=True)
  (output_layer): Linear(in_features=1, out_features=1, bias=True)
)
w_l1 = 1.58
b_l1 = -0.14
w_l2 = 2.45
b_l2 = -0.11
a_l2 = 0.8506
updated_w_l1 = 1.5814
updated_b_l1 = -0.1383
updated_w_l2 = 2.4529
updated_b_l2 = -0.1062
updated_a_l2 = 0.8515

We print out the network topology as well as the weights, biases, and output, both before and after the backpropagation step.

Below, let’s replicate this calculation with plain Python. This calculation is almost the same as the one we saw in the neural networks primer. The only difference is that PyTorch’s MSELoss function doesn’t have the extra division by 2, so in the code below, I’ve adjusted dc_da_l2 = 2 * (a_l2-1) to match what PyTorch does:

import numpy as np


def sigmoid(z_value):
    return 1.0/(1.0+np.exp(-z_value))


def z(w, a, b):
    return w * a + b


def sigmoid_prime(z_value):
    return sigmoid(z_value)*(1-sigmoid(z_value))


def dc_db(z_value, dc_da):
    return sigmoid_prime(z_value) * dc_da


def dc_dw(a_prev, dc_db_value):
    return a_prev * dc_db_value


def dc_da_prev(w, dc_db_value):
    return w * dc_db_value


a_l0 = 0.8
w_l1 = 1.58
b_l1 = -0.14
print(f"w_l1 = {round(w_l1, 4)}")
print(f"b_l1 = {round(b_l1, 4)}")

z_l1 = z(w_l1, a_l0, b_l1)
a_l1 = sigmoid(z_l1)

w_l2 = 2.45
b_l2 = -0.11
print(f"w_l2 = {round(w_l2, 4)}")
print(f"b_l2 = {round(b_l2, 4)}")

z_l2 = z(w_l2, a_l1, b_l2)
a_l2 = sigmoid(z_l2)
print(f"a_l2 = {round(a_l2, 4)}")

dc_da_l2 = 2 * (a_l2-1)
dc_db_l2 = dc_db(z_l2, dc_da_l2)
dc_dw_l2 = dc_dw(a_l1, dc_db_l2)
dc_da_l1 = dc_da_prev(w_l2, dc_db_l2)

step_size = 0.1
updated_b_l2 = b_l2 - dc_db_l2 * step_size
updated_w_l2 = w_l2 - dc_dw_l2 * step_size

dc_db_l1 = dc_db(z_l1, dc_da_l1)
dc_dw_l1 = dc_dw(a_l0, dc_db_l1)

updated_b_l1 = b_l1 - dc_db_l1 * step_size
updated_w_l1 = w_l1 - dc_dw_l1 * step_size

print(f"updated_w_l1 = {round(updated_w_l1, 4)}")
print(f"updated_b_l1 = {round(updated_b_l1, 4)}")

print(f"updated_w_l2 = {round(updated_w_l2, 4)}")
print(f"updated_b_l2 = {round(updated_b_l2, 4)}")

updated_z_l1 = z(updated_w_l1, a_l0, updated_b_l1)
updated_a_l1 = sigmoid(updated_z_l1)
updated_z_l2 = z(updated_w_l2, updated_a_l1, updated_b_l2)
updated_a_l2 = sigmoid(updated_z_l2)
print(f"updated_a_l2 = {round(updated_a_l2, 4)}")

Here are the results:

C:\Dev\python\pytorch>python backprop_manual_calculation.py
w_l1 = 1.58
b_l1 = -0.14
w_l2 = 2.45
b_l2 = -0.11
a_l2 = 0.8506
updated_w_l1 = 1.5814
updated_b_l1 = -0.1383
updated_w_l2 = 2.4529
updated_b_l2 = -0.1062
updated_a_l2 = 0.8515

We can see that the results match the ones from the PyTorch network! In the next article, we’ll use PyTorch to recognize digits from the MNIST database.

The code is available on github:

nestedsoftware / pytorch

Demonstrations of basic PyTorch usage. Includes MNIST recognition using dense as well as convolutional networks.

This project contains scripts to demonstrate basic PyTorch usage. The code requires python 3, numpy, and pytorch.

Manual vs. PyTorch Backprop Calculation

To compare a manual backprop calculation with the equivalent PyTorch version, run:

python backprop_manual_calculation.py
w_l1 = 1.58
b_l1 = -0.14
w_l2 = 2.45
b_l2 = -0.11
a_l2 = 0.8506
updated_w_l1 = 1.5814
updated_b_l1 = -0.1383
updated_w_l2 = 2.4529
updated_b_l2 = -0.1062
updated_a_l2 = 0.8515

and

python backprop_pytorch.py
network topology: Net(
  (hidden_layer): Linear(in_features=1, out_features=1, bias=True)
  (output_layer): Linear(in_features=1, out_features=1, bias=True)
)
w_l1 = 1.58
b_l1 = -0.14
w_l2 = 2.45
b_l2 = -0.11
a_l2 = 0.8506
updated_w_l1 = 1.5814
updated_b_l1 = -0.1383
updated_w_l2 = 2.4529
updated_b_l2 = -0.1062
updated_a_l2 = 0.8515

Blog post: PyTorch Hello World

MNIST Recognition

The next examples recognize MNIST digits using a dense network at first, and then several convolutional network designs (examples are adapted from Michael Nielsen's book, Neural Networks and Deep Learning).

I've added additional data normalization to the input since the original blog articles were published, using the code below (common.py):

normalization = transforms.Normalize((0.1305,), (0.3081,))
transformations = transforms.Compose([transforms.ToTensor(), normalization])

0.1305 is the average value of the input data and 0.3081 is the standard deviation relative to the values generated just by applying transforms.ToTensor() to the raw data. The data_normalization_calculations.md file shows an easy way to obtain these values.

To train a fully connected network on the MNIST dataset (as described in chapter 1 of Neural Networks and Deep Learning, run:

python pytorch_mnist.py
Test data results: 0.9758

Blog post: PyTorch Image Recognition with Dense Network

To train convolutional networks (as described in chapter 6), run the following.

Simple network:

python pytorch_mnist_convnet.py
Test data results: 0.9891

Two convolutional layers:

python pytorch_mnist_convnet.py --net 2conv
Test data results: 0.9913

Two convolutional layers with ReLU:

python pytorch_mnist_convnet.py --net relu --lr 0.03 --wd 0.00005
Test data results: 0.993

Two convolutional layers and extended training data:

python pytorch_mnist_convnet.py --net relu --lr 0.03 --wd 0.00005 --extend_data
Test data results: 0.9943

Final network:

python pytorch_mnist_convnet.py --net final --epochs 40 --lr 0.005 --extend_data
Test data results: 0.9964

Blog post: PyTorch Image Recognition with Convolutional Networks.

View on GitHub

Neural Networks Primer