Neural Network with one hidden layer


In [3]:
# Package imports
import numpy as np
import matplotlib.pyplot as plt
import sklearn
import sklearn.datasets
import sklearn.linear_model

%matplotlib inline

np.random.seed(1)

Logistic regression would'nt work for non-linear data ,it would still try to find a decision boundary which would still go on to try and find a linear decision boundary .

That is where Neural Networks come in they are able to find non-linear boundary , simply said they can learn complex functions.

Mathematically:

For one example $x^{(i)}$: $$z^{[1] (i)} = W^{[1]} x^{(i)} + b^{[1]}\tag{1}$$ $$a^{[1] (i)} = \tanh(z^{[1] (i)})\tag{2}$$ $$z^{[2] (i)} = W^{[2]} a^{[1] (i)} + b^{[2]}\tag{3}$$ $$\hat{y}^{(i)} = a^{[2] (i)} = \sigma(z^{ [2] (i)})\tag{4}$$ $$y^{(i)}_{prediction} = \begin{cases} 1 & \mbox{if } a^{[2](i)} > 0.5 \\ 0 & \mbox{otherwise } \end{cases}\tag{5}$$

Given the predictions on all the examples, you can also compute the cost $J$ as follows: $$J = - \frac{1}{m} \sum\limits_{i = 0}^{m} \large\left(\small y^{(i)}\log\left(a^{[2] (i)}\right) + (1-y^{(i)})\log\left(1- a^{[2] (i)}\right) \large \right) \small \tag{6}$$

Reminder: The general methodology to build a Neural Network is to:

1. Define the neural network structure ( # of input units,  # of hidden units, etc). 
2. Initialize the model's parameters
3. Loop:
    - Implement forward propagation
    - Compute loss
    - Implement backward propagation to get the gradients
    - Update parameters (gradient descent)

In [5]:
def layer_sizes(X, Y):
    """
    Arguments:
    X -- input dataset of shape (input size, number of examples)
    Y -- labels of shape (output size, number of examples)
    
    Returns:
    n_x -- the size of the input layer
    n_h -- the size of the hidden layer
    n_y -- the size of the output layer
    """
    
    n_x = X.shape[0]# size of input layer
    n_h = 4
    n_y = Y.shape[0] # size of output layer
    
    return (n_x, n_h, n_y)

In [7]:
def initialize_parameters(n_x, n_h, n_y):
    """
    Argument:
    n_x -- size of the input layer
    n_h -- size of the hidden layer
    n_y -- size of the output layer
    
    Returns:
    params -- python dictionary containing your parameters:
                    W1 -- weight matrix of shape (n_h, n_x)
                    b1 -- bias vector of shape (n_h, 1)
                    W2 -- weight matrix of shape (n_y, n_h)
                    b2 -- bias vector of shape (n_y, 1)
    """
    
    np.random.seed(2) # we set up a seed so that your output matches ours although the initialization is random.
    
    W1 = np.random.randn((n_h,n_x)) * 0.01
    #np.random.randn(x,y) = make a x*y matrix with random weights
    # i have no idea why but randn((x,y)) gives an integer datatype required error

    b1 = np.zeros((n_h,1))
    #np.zeros((x,1)) makes a vector with column x*1 zeros 
    #zeros((x,x)) definitely needs a tuple if you want to make a matrix
    W2 = np.random.randn((n_y,n_h)) * 0.01
    b2 = np.zeros((n_y,1))
    
    
    assert (W1.shape == (n_h, n_x))
    assert (b1.shape == (n_h, 1))
    assert (W2.shape == (n_y, n_h))
    assert (b2.shape == (n_y, 1))
    
    parameters = {"W1": W1,
                  "b1": b1,
                  "W2": W2,
                  "b2": b2}
    
    return parameters

In [1]:
#BT2217676  XOJKTBYZ

def forward_propagation(X, parameters):
    """
    Argument:
    X -- input data of size (n_x, m)
    parameters -- python dictionary containing your parameters (output of initialization function)
    
    Returns:
    A2 -- The sigmoid output of the second activation
    cache -- a dictionary containing "Z1", "A1", "Z2" and "A2"
    """
    # Retrieve each parameter from the dictionary "parameters"
    
    W1 = parameters["W1"]
    b1 = parameters["b1"]
    W2 = parameters["W2"]
    b2 = parameters["b2"]
    
    
    # Implement Forward Propagation to calculate A2 (probabilities)
    
    Z1 = np.dot(W1,X)+ b1
    A1 = np.tanh(Z1)
    Z2 = np.dot(W2,A1)+ b2
    A2 = sigmoid(Z2)
    
    
    assert(A2.shape == (1, X.shape[1]))
    
    cache = {"Z1": Z1,
             "A1": A1,
             "Z2": Z2,
             "A2": A2}
    
    return A2, cache

Now that you have computed $A^{[2]}$ (in the Python variable "A2"), which contains $a^{[2](i)}$ for every example, you can compute the cost function as follows:

$$ J = - \frac{1}{m} \sum\limits_{i = 0}^{m} \large{(} \small y^{(i)}\log\left(a^{[2] (i)}\right) + (1-y^{(i)})\log\left(1- a^{[2] (i)}\right) \large{)} \small\tag{13} $$

In [9]:
def compute_cost(A2, Y, parameters):
    """
    Computes the cross-entropy cost given in equation (13)
    
    Arguments:
    A2 -- The sigmoid output of the second activation, of shape (1, number of examples)
    Y -- "true" labels vector of shape (1, number of examples)
    parameters -- python dictionary containing your parameters W1, b1, W2 and b2
    
    Returns:
    cost -- cross-entropy cost given equation (13)
    """
    
    m = Y.shape[1] # number of example

    # Compute the cross-entropy cost
    # see how np.multiply works , i dont understand it now 
    logprobs = np.multiply(np.log(A2),Y) + np.multiply(np.log(1-A2),1-Y)
    cost = (-1./m)* np.sum(logprobs)
   
    
    cost = np.squeeze(cost)     # makes sure cost is the dimension we expect. 
                                # E.g., turns [[17]] into 17 
    assert(isinstance(cost, float))
    
    return cost

$\frac{\partial \mathcal{J} }{ \partial z_{2}^{(i)} } = \frac{1}{m} (a^{[2](i)} - y^{(i)})$

$\frac{\partial \mathcal{J} }{ \partial W_2 } = \frac{\partial \mathcal{J} }{ \partial z_{2}^{(i)} } a^{[1] (i) T} $

$\frac{\partial \mathcal{J} }{ \partial b_2 } = \sum_i{\frac{\partial \mathcal{J} }{ \partial z_{2}^{(i)}}}$

$\frac{\partial \mathcal{J} }{ \partial z_{1}^{(i)} } = W_2^T \frac{\partial \mathcal{J} }{ \partial z_{2}^{(i)} } * ( 1 - a^{[1] (i) 2}) $

$\frac{\partial \mathcal{J} }{ \partial W_1 } = \frac{\partial \mathcal{J} }{ \partial z_{1}^{(i)} } X^T $

$\frac{\partial \mathcal{J} _i }{ \partial b_1 } = \sum_i{\frac{\partial \mathcal{J} }{ \partial z_{1}^{(i)}}}$

  • Note that $*$ denotes elementwise multiplication.
  • The notation you will use is common in deep learning coding:
    • dW1 = $\frac{\partial \mathcal{J} }{ \partial W_1 }$
    • db1 = $\frac{\partial \mathcal{J} }{ \partial b_1 }$
    • dW2 = $\frac{\partial \mathcal{J} }{ \partial W_2 }$
    • db2 = $\frac{\partial \mathcal{J} }{ \partial b_2 }$
  • Tips:
    • To compute dZ1 you'll need to compute $g^{[1]'}(Z^{[1]})$. Since $g^{[1]}(.)$ is the tanh activation function, if $a = g^{[1]}(z)$ then $g^{[1]'}(z) = 1-a^2$. So you can compute $g^{[1]'}(Z^{[1]})$ using (1 - np.power(A1, 2)).

In [2]:
def backward_propagation(parameters, cache, X, Y):
    """
    Implement the backward propagation using the instructions above.
    
    Arguments:
    parameters -- python dictionary containing our parameters 
    cache -- a dictionary containing "Z1", "A1", "Z2" and "A2".
    X -- input data of shape (2, number of examples)
    Y -- "true" labels vector of shape (1, number of examples)
    
    Returns:
    grads -- python dictionary containing your gradients with respect to different parameters
    """
    m = X.shape[1]
    
    # First, retrieve W1 and W2 from the dictionary "parameters".
    
    W1 = parameters["W1"]
    W2 = parameters["W2"]
 
        
    # Retrieve also A1 and A2 from dictionary "cache".
    
    A1 = cache["A1"]
    A2 = cache["A2"]
   
    
    # Backward propagation: calculate dW1, db1, dW2, db2. 
    dZ2 = A2 - Y
    dW2 = (1./m)*np.dot(dZ2,A1.T)
    db2 = (1./m)*np.sum(dZ2,axis=1,keepdims= True) # keepdims=true so that you dont get (1,) or (,1) answers
    
    dZ1 = np.dot(W2.T,dZ2)* (1-np.power(A1,2)) #* denotes element wise product
    dW1 = (1./m)*np.dot(dZ1,X.T)
    db1 = (1./m)*np.sum(dZ1,axis=1,keepdims= True)
    
    
    grads = {"dW1": dW1,
             "db1": db1,
             "dW2": dW2,
             "db2": db2}
    
    return grads

In [3]:
def update_parameters(parameters, grads, learning_rate = 1.2):
    """
    Updates parameters using the gradient descent update rule given above
    
    Arguments:
    parameters -- python dictionary containing your parameters 
    grads -- python dictionary containing your gradients 
    
    Returns:
    parameters -- python dictionary containing your updated parameters 
    """
    # Retrieve each parameter from the dictionary "parameters"

    W1 = parameters["W1"]
    b1 = parameters["b1"]
    W2 = parameters["W2"]
    b2 = parameters["b2"]

    
    # Retrieve each gradient from the dictionary "grads"

    dW1 = grads["dW1"]
    db1 = grads["db1"]
    dW2 = grads["dW2"]
    db2 = grads["db2"]

    
    # Update rule for each parameter
   
    W1 = W1 - learning_rate * dW1
    b1 = b1 - learning_rate * db1
    W2 = W2 - learning_rate * dW2
    b2 = b2 - learning_rate * db2
    
    
    parameters = {"W1": W1,
                  "b1": b1,
                  "W2": W2,
                  "b2": b2}
    
    return parameters

In [4]:
def nn_model(X, Y, n_h, num_iterations = 10000, print_cost=False):
    """
    Arguments:
    X -- dataset of shape (2, number of examples)
    Y -- labels of shape (1, number of examples)
    n_h -- size of the hidden layer
    num_iterations -- Number of iterations in gradient descent loop
    print_cost -- if True, print the cost every 1000 iterations
    
    Returns:
    parameters -- parameters learnt by the model. They can then be used to predict.
    """
    
    np.random.seed(3)
    n_x = layer_sizes(X, Y)[0]
    n_y = layer_sizes(X, Y)[2]
    
    # Initialize parameters, then retrieve W1, b1, W2, b2. Inputs: "n_x, n_h, n_y". Outputs = "W1, b1, W2, b2, parameters".
    ### START CODE HERE ### (≈ 5 lines of code)
    parameters = initialize_parameters(X.shape[0],n_h,Y.shape[0])
    W1 = parameters["W1"]
    b1 = parameters["b1"]
    W2 = parameters["W2"]
    b2 = parameters["b2"]
    ### END CODE HERE ###
    
    # Loop (gradient descent)

    for i in range(0, num_iterations):
         
        
        # Forward propagation. Inputs: "X, parameters". Outputs: "A2, cache".
        A2, cache = forward_propagation(X,parameters)
        
        # Cost function. Inputs: "A2, Y, parameters". Outputs: "cost".
        cost = compute_cost(A2,Y,parameters)
 
        # Backpropagation. Inputs: "parameters, cache, X, Y". Outputs: "grads".
        grads = backward_propagation(parameters,cache,X,Y)
 
        # Gradient descent parameter update. Inputs: "parameters, grads". Outputs: "parameters".
        parameters = update_parameters(parameters,grads)
        
       
        
        # Print the cost every 1000 iterations
        if print_cost and i % 1000 == 0:
            print ("Cost after iteration %i: %f" %(i, cost))

    return parameters

In [7]:
def predict(parameters, X):
    """
    Using the learned parameters, predicts a class for each example in X
    
    Arguments:
    parameters -- python dictionary containing your parameters 
    X -- input data of size (n_x, m)
    
    Returns
    predictions -- vector of predictions of our model (red: 0 / blue: 1)
    """
    
    # Computes probabilities using forward propagation, and classifies to 0/1 using 0.5 as the threshold.
    
    A2, cache = forward_propagation(X,parameters)
    predictions = A2 > 0.5 #See what > does for a vector 
    
    
    return predictions

Some Interpretations

  • The larger models (with more hidden units) are able to fit the training set better, until eventually the largest models overfit the data.
  • The best hidden layer size seems to be around n_h = 5. Indeed, a value around here seems to fits the data well without also incurring noticable overfitting.
  • You will also learn later about regularization, which lets you use very large models (such as n_h = 50) without much overfitting.

Basically Deeper layers ie having more units in the same layer doesn't lead to higher accuracy after a certain number of units .

Check if deeper nets change Anything?