An Introduction to Neural Networks

Here’s something that might surprise you: neural networks aren’t that complicated! The term “neural network” gets used as a buzzword a lot, but in reality they’re often much simpler than people imagine.

This post is intended for complete beginners and assumes ZERO prior knowledge of machine learning. We’ll understand how neural networks work while implementing one from scratch in Python.

Let’s get started!

1. Building Blocks: Neurons

First, we have to talk about neurons, the basic unit of a neural network. A neuron takes inputs, does some math with them, and produces one output. Here’s what a 2-input neuron looks like:

Screenshot 2019-06-02 at 11.12.54 AM

3 things are happening here. First, each input is multiplied by a weight:

x1→x1∗w1

x2→x2∗w2

Next, all the weighted inputs are added together with a bias b:

(x1∗w1)+(x2∗w2)+b

Finally, the sum is passed through an activation function:

y=f(x1∗w1+x2∗w2+b)

The activation function is used to turn an unbounded input into an output that has a nice, predictable form. A commonly used activation function is the sigmoid function:

The sigmoid function only outputs numbers in the range (0,1). You can think of it as compressing (−∞,+∞)to (0,1) – big negative numbers become ~, and big positive numbers become ~1.

A Simple Example

Assume we have a 2-input neuron that uses the sigmoid activation function and has the following parameters:

b=4

w=[0,1] is just a way of writing w1=0, w2=1 in vector form. Now, let’s give the neuron an input of x=[2,3]. We’ll use the dot product to write things more concisely:

(w⋅x)+b=((w1∗x1)+(w2∗x2))+b

                                                                     =0∗2+1∗3+4

                                                                      =7

The neuron outputs 0.999 given the inputs x=[2,3]. That’s it! This process of passing inputs forward to get an output is known as feedforward.

Coding a Neuron

Time to implement a neuron! We’ll use NumPy, a popular and powerful computing library for Python, to help us do math:


import numpy as np

def sigmoid(x):
# Our activation function: f(x) = 1 / (1 + e^(-x))
return 1 / (1 + np.exp(-x))

class Neuron:
def __init__(self, weights, bias):
self.weights = weights
self.bias = bias

def feedforward(self, inputs):
# Weight inputs, add bias, then use the activation function
total = np.dot(self.weights, inputs) + self.bias
return sigmoid(total)

weights = np.array([0, 1]) # w1 = 0, w2 = 1
bias = 4 # b = 4
n = Neuron(weights, bias)

x = np.array([2, 3]) # x1 = 2, x2 = 3
print(n.feedforward(x)) # 0.9990889488055994

Recognize those numbers? That’s the example we just did! We get the same answer of 0.999.

2. Combining Neurons into a Neural Network

A neural network is nothing more than a bunch of neurons connected together. Here’s what a simple neural network might look like:

Screenshot 2019-06-02 at 11.29.39 AM

This network has 2 inputs, a hidden layer with 2 neurons (h1 and h2), and an output layer with 1 neuron (o1). Notice that the inputs for o1 are the outputs from and h2 – that’s what makes this a network.

A hidden layer is any layer between the input (first) layer and output (last) layer. There can be multiple hidden layers!

An Example: Feedforward

Let’s use the network pictured above and assume all neurons have the same weights w=[0,1], the same bias b=0, and the same sigmoid activation function. Let h1,h2,o1 denote the outputs of the neurons they represent.

What happens if we pass in the input ?

                                                            h1=h2=f(w⋅x+b)

          =f((0∗2)+(1∗3)+0)

                                                                        =f(3)

                                                                        =0.9526

The output of the neural network for input x=[2,3] is 0.7216. Pretty simple, right?

A neural network can have any number of layers with any number of neurons in those layers. The basic idea stays the same: feed the input(s) forward through the neurons in the network to get the output(s) at the end. For simplicity, we’ll keep using the network pictured above for the rest of this post.

Coding a Neural Network: Feedforward

Let’s implement feedforward for our neural network. Here’s the image of the network again for reference:

Screenshot 2019-06-02 at 11.29.39 AM


import numpy as np

# ... code from previous section here

class OurNeuralNetwork:
'''
A neural network with:
- 2 inputs
- a hidden layer with 2 neurons (h1, h2)
- an output layer with 1 neuron (o1)
Each neuron has the same weights and bias:
- w = [0, 1]
- b = 0
'''
def __init__(self):
weights = np.array([0, 1])
bias = 0

# The Neuron class here is from the previous section
self.h1 = Neuron(weights, bias)
self.h2 = Neuron(weights, bias)
self.o1 = Neuron(weights, bias)

def feedforward(self, x):
out_h1 = self.h1.feedforward(x)
out_h2 = self.h2.feedforward(x)

# The inputs for o1 are the outputs from h1 and h2
out_o1 = self.o1.feedforward(np.array([out_h1, out_h2]))

return out_o1

network = OurNeuralNetwork()
x = np.array([2, 3])
print(network.feedforward(x)) # 0.7216325609518421

We got 0.7216 again! Looks like it works.

3. Training a Neural Network, Part 1

Say we have the following measurements:

Name Weight (lb) Height (in) Gender
Alice 133 65 F
Bob 160 72 M
Charlie 152 70 M
Diana 120 60 F

Let’s train our network to predict someone’s gender given their weight and height:

Screenshot 2019-06-02 at 11.40.05 AM

We’ll represent Male with a 0 and Female with a 1, and we’ll also shift the data to make it easier to use:

Name Weight (minus 135) Height (minus 66) Gender
Alice -2 -1 1
Bob 25 6 0
Charlie 17 4 0
Diana -15 -6 1

I arbitrarily chose the shift amounts (135 and 66) to make the numbers look nice. Normally, you’d shift by the mean.

Loss

Before we train our network, we first need a way to quantify how “good” it’s doing so that it can try to do “better”. That’s what the loss is.

We’ll use the mean squared error (MSE) loss:

Screenshot 2019-06-02 at 5.43.24 PM

Let’s break this down:

  • is the number of samples, which is 4 (Alice, Bob, Charlie, Diana).
  • represents the variable being predicted, which is Gender.
  • is the true value of the variable (the “correct answer”). For example, for Alice would be 1 (Female).
  • ypred is the predicted value of the variable. It’s whatever our network outputs.

 

is known as the squared error. Our loss function is simply taking the average over all squared errors (hence the name mean squared error). The better our predictions are, the lower our loss will be!

Better predictions = Lower loss.

Training a network = trying to minimize its loss.

An Example Loss Calculation

Let’s say our network always outputs – in other words, it’s confident all humans are Male 🤔. What would our loss be?

Name ytrue ypred (ytrue−ypred)
Alice 1 0 1
Bob 0 0 0
Charlie 0 0 0
Diana 1 0 1

MSE=1/4(1+0+0+1)=0.5

Code: MSE Loss

Here’s some code to calculate loss for us:

import numpy as np

def mse_loss(y_true, y_pred):
  # y_true and y_pred are numpy arrays of the same length.
  return ((y_true - y_pred) ** 2).mean()

y_true = np.array([1, 0, 0, 1])
y_pred = np.array([0, 0, 0, 0])

print(mse_loss(y_true, y_pred)) # 0.5

Nice. Onwards!

4. Training a Neural Network, Part 2

We now have a clear goal: minimize the loss of the neural network. We know we can change the network’s weights and biases to influence its predictions, but how do we do so in a way that decreases loss?

This section uses a bit of multivariable calculus. If you’re not comfortable with calculus, feel free to skip over the math parts.

For simplicity, let’s pretend we only have Alice in our dataset:

Name Weight (minus 135) Height (minus 66) Gender
Alice -2 -1 1

Then the mean squared error loss is just Alice’s squared error:

Screenshot 2019-06-02 at 5.47.50 PM

Another way to think about loss is as a function of weights and biases. Let’s label each weight and bias in our network:

Screenshot 2019-06-02 at 5.48.20 PM

Then, we can write loss as a multivariable function:

L(w1,w2,w3,w4,w5,w6,b1,b2,b3)

Imagine we wanted to tweak w1. How would loss L change if we changed w1? That’s a question the partial derivative can answer. How do we calculate it?

Here’s where the math starts to get more complex. Don’t be discouraged! I recommend getting a pen and paper to follow along – it’ll help you understand.

To start, let’s rewrite the partial derivative in terms of ∂ypred/∂w1 instead:

Screenshot 2019-06-02 at 5.58.30 PM

This works because of the Chain Rule.

Screenshot 2019-06-02 at 5.59.09 PM.png

here is weight, and x2 is height. This is the second time we’ve seen (the derivate of the sigmoid function) now! Let’s derive it:

Screenshot 2019-06-02 at 6.00.13 PM

We’ll use this nice form for f′(x) later.

We’re done! We’ve managed to break down ∂L/∂w1 into several parts we can calculate:

Screenshot 2019-06-02 at 6.00.59 PM

This system of calculating partial derivatives by working backwards is known as backpropagation, or “backprop”.

Phew. That was a lot of symbols – it’s alright if you’re still a bit confused. Let’s do an example to see this in action!

Example: Calculating the Partial Derivative

We’re going to continue pretending only Alice is in our dataset:

Name Weight (minus 135) Height (minus 66) Gender
Alice -2 -1 1

Let’s initialize all the weights to 1 and all the biases to 0. If we do a feedforward pass through the network, we get:

Screenshot 2019-06-02 at 6.01.39 PM

The network outputs , which doesn’t strongly favor Male (0) or Female (1). Let’s calculate ∂L/∂w1:

Screenshot 2019-06-02 at 6.02.40 PM

Reminder: we derived for our sigmoid activation function earlier.

We did it! This tells us that if we were to increase w1, L would increase a tiiiny bit as a result.

Training: Stochastic Gradient Descent

We have all the tools we need to train a neural network now! We’ll use an optimization algorithm called stochastic gradient descent (SGD) that tells us how to change our weights and biases to minimize loss. It’s basically just this update equation:

Screenshot 2019-06-02 at 6.08.03 PM

η is a constant called the learning rate that controls how fast we train. All we’re doing is subtracting η∂L/∂w1 from w1:

  • If ∂L/∂w1 is positive, will decrease, which makes decrease.
  • If ∂L/∂w1 is negative, will increase, which makes decrease.

If we do this for every weight and bias in the network, the loss will slowly decrease and our network will improve.

Our training process will look like this:

 

  • Choose one sample from our dataset. This is what makes it stochasticgradient descent – we only operate on one sample at a time.
  • Calculate all the partial derivatives of loss with respect to weights or biases (e.g. ∂L∂w1, ∂L∂w2, etc).
  • Use the update equation to update each weight and bias.
  • Go back to step 1.

 

Let’s see it in action!

Code: A Complete Neural Network

It’s finally time to implement a complete neural network:

Name Weight (minus 135) Height (minus 66) Gender
Alice -2 -1 1
Bob 25 6 0
Charlie 17 4 0
Diana -15 -6 1

Screenshot 2019-06-02 at 6.12.13 PM

import numpy as np

def sigmoid(x):
  # Sigmoid activation function: f(x) = 1 / (1 + e^(-x))
  return 1 / (1 + np.exp(-x))

def deriv_sigmoid(x):
  # Derivative of sigmoid: f'(x) = f(x) * (1 - f(x))
  fx = sigmoid(x)
  return fx * (1 - fx)

def mse_loss(y_true, y_pred):
  # y_true and y_pred are numpy arrays of the same length.
  return ((y_true - y_pred) ** 2).mean()

class OurNeuralNetwork:
  '''
  A neural network with:
    - 2 inputs
    - a hidden layer with 2 neurons (h1, h2)
    - an output layer with 1 neuron (o1)

  *** DISCLAIMER ***:
  The code below is intended to be simple and educational, NOT optimal.
  Real neural net code looks nothing like this. DO NOT use this code.
  Instead, read/run it to understand how this specific network works.
  '''
  def __init__(self):
    # Weights
    self.w1 = np.random.normal()
    self.w2 = np.random.normal()
    self.w3 = np.random.normal()
    self.w4 = np.random.normal()
    self.w5 = np.random.normal()
    self.w6 = np.random.normal()

    # Biases
    self.b1 = np.random.normal()
    self.b2 = np.random.normal()
    self.b3 = np.random.normal()

  def feedforward(self, x):
    # x is a numpy array with 2 elements.
    h1 = sigmoid(self.w1 * x[0] + self.w2 * x[1] + self.b1)
    h2 = sigmoid(self.w3 * x[0] + self.w4 * x[1] + self.b2)
    o1 = sigmoid(self.w5 * h1 + self.w6 * h2 + self.b3)
    return o1

  def train(self, data, all_y_trues):
    '''
    - data is a (n x 2) numpy array, n = # of samples in the dataset.
    - all_y_trues is a numpy array with n elements.
      Elements in all_y_trues correspond to those in data.
    '''
    learn_rate = 0.1
    epochs = 1000 # number of times to loop through the entire dataset

    for epoch in range(epochs):
      for x, y_true in zip(data, all_y_trues):
        # --- Do a feedforward (we'll need these values later)
        sum_h1 = self.w1 * x[0] + self.w2 * x[1] + self.b1
        h1 = sigmoid(sum_h1)

        sum_h2 = self.w3 * x[0] + self.w4 * x[1] + self.b2
        h2 = sigmoid(sum_h2)

        sum_o1 = self.w5 * h1 + self.w6 * h2 + self.b3
        o1 = sigmoid(sum_o1)
        y_pred = o1

        # --- Calculate partial derivatives.
        # --- Naming: d_L_d_w1 represents "partial L / partial w1"
        d_L_d_ypred = -2 * (y_true - y_pred)

        # Neuron o1
        d_ypred_d_w5 = h1 * deriv_sigmoid(sum_o1)
        d_ypred_d_w6 = h2 * deriv_sigmoid(sum_o1)
        d_ypred_d_b3 = deriv_sigmoid(sum_o1)

        d_ypred_d_h1 = self.w5 * deriv_sigmoid(sum_o1)
        d_ypred_d_h2 = self.w6 * deriv_sigmoid(sum_o1)

        # Neuron h1
        d_h1_d_w1 = x[0] * deriv_sigmoid(sum_h1)
        d_h1_d_w2 = x[1] * deriv_sigmoid(sum_h1)
        d_h1_d_b1 = deriv_sigmoid(sum_h1)

        # Neuron h2
        d_h2_d_w3 = x[0] * deriv_sigmoid(sum_h2)
        d_h2_d_w4 = x[1] * deriv_sigmoid(sum_h2)
        d_h2_d_b2 = deriv_sigmoid(sum_h2)

        # --- Update weights and biases
        # Neuron h1
        self.w1 -= learn_rate * d_L_d_ypred * d_ypred_d_h1 * d_h1_d_w1
        self.w2 -= learn_rate * d_L_d_ypred * d_ypred_d_h1 * d_h1_d_w2
        self.b1 -= learn_rate * d_L_d_ypred * d_ypred_d_h1 * d_h1_d_b1

        # Neuron h2
        self.w3 -= learn_rate * d_L_d_ypred * d_ypred_d_h2 * d_h2_d_w3
        self.w4 -= learn_rate * d_L_d_ypred * d_ypred_d_h2 * d_h2_d_w4
        self.b2 -= learn_rate * d_L_d_ypred * d_ypred_d_h2 * d_h2_d_b2

        # Neuron o1
        self.w5 -= learn_rate * d_L_d_ypred * d_ypred_d_w5
        self.w6 -= learn_rate * d_L_d_ypred * d_ypred_d_w6
        self.b3 -= learn_rate * d_L_d_ypred * d_ypred_d_b3

      # --- Calculate total loss at the end of each epoch
      if epoch % 10 == 0:
        y_preds = np.apply_along_axis(self.feedforward, 1, data)
        loss = mse_loss(all_y_trues, y_preds)
        print("Epoch %d loss: %.3f" % (epoch, loss))

# Define dataset
data = np.array([
  [-2, -1],  # Alice
  [25, 6],   # Bob
  [17, 4],   # Charlie
  [-15, -6], # Diana
])
all_y_trues = np.array([
  1, # Alice
  0, # Bob
  0, # Charlie
  1, # Diana
])

# Train our neural network!
network = OurNeuralNetwork()
network.train(data, all_y_trues)

Our loss steadily decreases as the network learns:

Screenshot 2019-06-02 at 6.13.44 PM

We can now use the network to predict genders:

# Make some predictions
emily = np.array([-7, -3]) # 128 pounds, 63 inches
frank = np.array([20, 2])  # 155 pounds, 68 inches
print("Emily: %.3f" % network.feedforward(emily)) # 0.951 - F
print("Frank: %.3f" % network.feedforward(frank)) # 0.039 - M

Now What?

You made it! A quick recap of what we did:

  • Introduced neurons, the building blocks of neural networks.
  • Used the sigmoid activation function in our neurons.
  • Saw that neural networks are just neurons connected together.
  • Created a dataset with Weight and Height as inputs (or features) and Gender as the output (or label).
  • Learned about loss functions and the mean squared error (MSE) loss.
  • Realized that training a network is just minimizing its loss.
  • Used backpropagation to calculate partial derivatives.
  • Used stochastic gradient descent (SGD) to train our network.

There’s still much more to do:

Thanks for reading!

 

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