Regularisation

Deep Learning models have so much flexibility and capacity that overfitting can be a serious problem, if the training dataset is not big enough. Sure it does well on the training set, but the learned network doesn’t generalise to new examples that it has never seen!

Goals of this Notebook is to:

  • Use regularisation in our deep learning models.

Let’s first import the packages I am going to use.

# import packages
import numpy as np
import matplotlib.pyplot as plt
from reg_utils import sigmoid, relu, plot_decision_boundary, initialize_parameters, load_2D_dataset, predict_dec
from reg_utils import compute_cost, predict, forward_propagation, backward_propagation, update_parameters
import sklearn
import sklearn.datasets
import scipy.io
from testCases import *

%matplotlib inline
plt.rcParams['figure.figsize'] = (7.0, 4.0) # set default size of plots
plt.rcParams['image.interpolation'] = 'nearest'
plt.rcParams['image.cmap'] = 'gray'

Problem Statement and Scenario: I have just been hired as an AI expert by the French Football Corporation. They would like me to recommend positions where France’s goal keeper should kick the ball so that the French team’s players can then hit it with their head.

**Figure 1** : **Football field**
The goal keeper kicks the ball in the air, the players of each team are fighting to hit the ball with their head

They give me the following 2D dataset from France’s past 10 games.

train_X, train_Y, test_X, test_Y = load_2D_dataset()

png

Each dot corresponds to a position on the football field where a football player has hit the ball with his/her head after the French goal keeper has shot the ball from the left side of the football field.

  • If the dot is blue, it means the French player managed to hit the ball with his/her head
  • If the dot is red, it means the other team’s player hit the ball with their head

My goal: Use a deep learning model to find the positions on the field where the goalkeeper should kick the ball.

Analysis of the dataset: This dataset is a little noisy, but it looks like a diagonal line separating the upper left half (blue) from the lower right half (red) would work well.

I will first try a non-regularised model. Then I’ll learn how to regularise it and decide which model you will choose to solve the French Football Corporation’s problem.

1 - Non-regularised model

I will use the following neural network below. This model can be used:

  • in regularisation mode – by setting the lambd input to a non-zero value. We use “lambd” instead of “lambda” because “lambda” is a reserved keyword in Python.
  • in dropout mode – by setting the keep_prob to a value less than one

I will first try the model without any regularisation. Then, I will implement:

  • L2 regularisation – functions: “compute_cost_with_regularisation()” and “backward_propagation_with_regularisation()
  • Dropout – functions: “forward_propagation_with_dropout()” and “backward_propagation_with_dropout()

In each part, I will run this model with the correct inputs so that it calls the functions I’ve implemented. Let’s look at the code below to familiarise ourselves with the model.

def model(X, Y, learning_rate = 0.3, num_iterations = 30000, print_cost = True, lambd = 0, keep_prob = 1):
    """
    Implements a three-layer neural network: LINEAR->RELU->LINEAR->RELU->LINEAR->SIGMOID.

    Arguments:
    X -- input data, of shape (input size, number of examples)
    Y -- true "label" vector (1 for blue dot / 0 for red dot), of shape (output size, number of examples)
    learning_rate -- learning rate of the optimization
    num_iterations -- number of iterations of the optimization loop
    print_cost -- If True, print the cost every 10000 iterations
    lambd -- regularization hyperparameter, scalar
    keep_prob - probability of keeping a neuron active during drop-out, scalar.

    Returns:
    parameters -- parameters learned by the model. They can then be used to predict.
    """

    grads = {}
    costs = []                            # to keep track of the cost
    m = X.shape[1]                        # number of examples
    layers_dims = [X.shape[0], 20, 3, 1]

    # Initialize parameters dictionary.
    parameters = initialize_parameters(layers_dims)

    # Loop (gradient descent)

    for i in range(0, num_iterations):

        # Forward propagation: LINEAR -> RELU -> LINEAR -> RELU -> LINEAR -> SIGMOID.
        if keep_prob == 1:
            a3, cache = forward_propagation(X, parameters)
        elif keep_prob < 1:
            a3, cache = forward_propagation_with_dropout(X, parameters, keep_prob)

        # Cost function
        if lambd == 0:
            cost = compute_cost(a3, Y)
        else:
            cost = compute_cost_with_regularisation(a3, Y, parameters, lambd)

        # Backward propagation.
        assert(lambd==0 or keep_prob==1)    # it is possible to use both L2 regularization and dropout,
                                            # but this assignment will only explore one at a time
        if lambd == 0 and keep_prob == 1:
            grads = backward_propagation(X, Y, cache)
        elif lambd != 0:
            grads = backward_propagation_with_regularisation(X, Y, cache, lambd)
        elif keep_prob < 1:
            grads = backward_propagation_with_dropout(X, Y, cache, keep_prob)

        # Update parameters.
        parameters = update_parameters(parameters, grads, learning_rate)

        # Print the loss every 10000 iterations
        if print_cost and i % 10000 == 0:
            print("Cost after iteration {}: {}".format(i, cost))
        if print_cost and i % 1000 == 0:
            costs.append(cost)

    # plot the cost
    plt.plot(costs)
    plt.ylabel('cost')
    plt.xlabel('iterations (x1,000)')
    plt.title("Learning rate =" + str(learning_rate))
    plt.show()

    return parameters

Let’s train the model without any regularization, and observe the accuracy on the train/test sets.

parameters = model(train_X, train_Y)
print ("On the training set:")
predictions_train = predict(train_X, train_Y, parameters)
print ("On the test set:")
predictions_test = predict(test_X, test_Y, parameters)
Cost after iteration 0: 0.6557412523481002
Cost after iteration 10000: 0.16329987525724213
Cost after iteration 20000: 0.13851642423253263

png

On the training set:
Accuracy: 0.947867298578
On the test set:
Accuracy: 0.915

The train accuracy is 94.8% while the test accuracy is 91.5%. This is the baseline model. The following code plot the decision boundary of our model.

plt.title("Model without regularisation")
axes = plt.gca()
axes.set_xlim([-0.75,0.40])
axes.set_ylim([-0.75,0.65])
plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y)

png

The non-regularised model is obviously overfitting the training set. It is fitting the noisy points! Let’s now look at two techniques to reduce overfitting.

2 - L2 Regularisation

The standard way to avoid overfitting is called L2 regularisation. It consists of appropriately modifying our cost function, from: To:

Let’s modify our cost and observe the consequences.

Let’s implement compute_cost_with_regularisation() which computes the cost given by formula (2). To calculate $\sum\limits_k\sum\limits_j W_{k,j}^{[l]2}$ , use :

np.sum(np.square(Wl))

Note that I will do this for $W^{[1]}$, $W^{[2]}$ and $W^{[3]}$, then sum the three terms and multiply by $ \frac{1}{m} \frac{\lambda}{2} $.

def compute_cost_with_regularisation(A3, Y, parameters, lambd):
    """
    Implement the cost function with L2 regularization. See formula (2) above.

    Arguments:
    A3 -- post-activation, output of forward propagation, of shape (output size, number of examples)
    Y -- "true" labels vector, of shape (output size, number of examples)
    parameters -- python dictionary containing parameters of the model

    Returns:
    cost - value of the regularized loss function (formula (2))
    """
    m = Y.shape[1]
    W1 = parameters["W1"]
    W2 = parameters["W2"]
    W3 = parameters["W3"]

    cross_entropy_cost = compute_cost(A3, Y) # This gives you the cross-entropy part of the cost
    L2_regularization_cost = (lambd/(2*m))*(np.sum(np.square(W1))+np.sum(np.square(W2))+np.sum(np.square(W3)))

    cost = cross_entropy_cost + L2_regularization_cost

    return cost
A3, Y_assess, parameters = compute_cost_with_regularization_test_case()

print("cost = " + str(compute_cost_with_regularisation(A3, Y_assess, parameters, lambd = 0.1)))
cost = 1.78648594516

Of course, because I changed the cost, I have to change backward propagation as well! All the gradients have to be computed with respect to this new cost.

Let’s implement the changes needed in backward propagation to take into account regularisation. The changes only concern dW1, dW2 and dW3. For each, I have to add the regularisation term’s gradient ($\frac{d}{dW} ( \frac{1}{2}\frac{\lambda}{m} W^2) = \frac{\lambda}{m} W$).

def backward_propagation_with_regularisation(X, Y, cache, lambd):
    """
    Implements the backward propagation of our baseline model to which we added an L2 regularisation.

    Arguments:
    X -- input dataset, of shape (input size, number of examples)
    Y -- "true" labels vector, of shape (output size, number of examples)
    cache -- cache output from forward_propagation()
    lambd -- regularisation hyperparameter, scalar

    Returns:
    gradients -- A dictionary with the gradients with respect to each parameter, activation and pre-activation variables
    """

    m = X.shape[1]
    (Z1, A1, W1, b1, Z2, A2, W2, b2, Z3, A3, W3, b3) = cache

    dZ3 = A3 - Y

    dW3 = 1./m * np.dot(dZ3, A2.T) + lambd*W3/m
    db3 = 1./m * np.sum(dZ3, axis=1, keepdims = True)

    dA2 = np.dot(W3.T, dZ3)
    dZ2 = np.multiply(dA2, np.int64(A2 > 0))
    dW2 = 1./m * np.dot(dZ2, A1.T) + lambd*W2/m
    db2 = 1./m * np.sum(dZ2, axis=1, keepdims = True)

    dA1 = np.dot(W2.T, dZ2)
    dZ1 = np.multiply(dA1, np.int64(A1 > 0))
    dW1 = 1./m * np.dot(dZ1, X.T) + lambd*W1/m
    db1 = 1./m * np.sum(dZ1, axis=1, keepdims = True)

    gradients = {"dZ3": dZ3, "dW3": dW3, "db3": db3,"dA2": dA2,
                 "dZ2": dZ2, "dW2": dW2, "db2": db2, "dA1": dA1,
                 "dZ1": dZ1, "dW1": dW1, "db1": db1}

    return gradients
X_assess, Y_assess, cache = backward_propagation_with_regularization_test_case()

grads = backward_propagation_with_regularisation(X_assess, Y_assess, cache, lambd = 0.7)
print ("dW1 = "+ str(grads["dW1"]))
print ("dW2 = "+ str(grads["dW2"]))
print ("dW3 = "+ str(grads["dW3"]))
dW1 = [[-0.25604646  0.12298827 -0.28297129]
 [-0.17706303  0.34536094 -0.4410571 ]]
dW2 = [[ 0.79276486  0.85133918]
 [-0.0957219  -0.01720463]
 [-0.13100772 -0.03750433]]
dW3 = [[-1.77691347 -0.11832879 -0.09397446]]

Let’s now run the model with L2 regularisation $(\lambda = 0.7)$. The model() function will call:

  • compute_cost_with_regularisation instead of compute_cost
  • backward_propagation_with_regularisation instead of backward_propagation
parameters = model(train_X, train_Y, lambd = 0.7)
print ("On the train set:")
predictions_train = predict(train_X, train_Y, parameters)
print ("On the test set:")
predictions_test = predict(test_X, test_Y, parameters)
Cost after iteration 0: 0.6974484493131264
Cost after iteration 10000: 0.26849188732822393
Cost after iteration 20000: 0.2680916337127301

png

On the train set:
Accuracy: 0.938388625592
On the test set:
Accuracy: 0.93

Yay! The test set accuracy increased to 93%. We have saved the French football team! We are not overfitting the training data anymore. Let’s plot the decision boundary.

plt.title("Model with L2-regularisation")
axes = plt.gca()
axes.set_xlim([-0.75,0.40])
axes.set_ylim([-0.75,0.65])
plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y)

png

Observations:

  • The value of $\lambda$ is a hyperparameter that we can tune using a dev set.
  • L2 regularisation makes our decision boundary smoother. If $\lambda$ is too large, it is also possible to “oversmooth”, resulting in a model with high bias.

What is L2-regularisation actually doing?:

L2-regularisation relies on the assumption that a model with small weights is simpler than a model with large weights. Thus, by penalising the square values of the weights in the cost function all the weights are driven to smaller values. It becomes too costly for the cost to have large weights! This leads to a smoother model in which the output changes more slowly as the input changes.

Key points for this section are: – the implications of L2-regularisation on:

  • The cost computation:
    • A regularisation term is added to the cost
  • The backpropagation function:
    • There are extra terms in the gradients with respect to weight matrices
  • Weights end up smaller (“weight decay”):
    • Weights are pushed to smaller values.

3 - Dropout

Finally, dropout is a widely used regularisation technique that is specific to deep learning. It randomly shuts down some neurons in each iteration. Watch these two videos to see what this means!


Figure 2 : Drop-out on the second hidden layer.
At each iteration, we shut down (= set to zero) each neuron of a layer with probability $1 - keep\_prob$ or keep it with probability $keep\_prob$ (50% here). The dropped neurons don't contribute to the training in both the forward and backward propagations of the iteration.
Figure 3 : Drop-out on the first and third hidden layers.
$1^{st}$ layer: we shut down on average 40% of the neurons. $3^{rd}$ layer: we shut down on average 20% of the neurons.

When we shut some neurons down, we actually modify our model. The idea behind drop-out is that at each iteration, we train a different model that uses only a subset of our neurons. With dropout, our neurons thus become less sensitive to the activation of one other specific neuron, because that other neuron might be shut down at any time.

3.1 - Forward propagation with dropout

Let’s implement the forward propagation with dropout. I am using a 3 layer neural network, and will add dropout to the first and second hidden layers. We will not apply dropout to the input layer or output layer.

The below cell will implement the following:

I want to shut down some neurons in the first and second layers. To do that, I am going to carry out 4 Steps:

  1. Creating a variable $d^{[1]}$ with the same shape as $a^{[1]}$ using np.random.rand() to randomly get numbers between 0 and 1. Here, I will use a vectorised implementation, so create a random matrix $D^{[1]} = [d^{1} d^{1} … d^{1}] $ of the same dimension as $A^{[1]}$.
  2. Set each entry of $D^{[1]}$ to be 0 with probability (1-keep_prob) or 1 with probability (keep_prob), by thresholding values in $D^{[1]}$ appropriately. Trick: to set all the entries of a matrix X to 0 (if entry is less than 0.5) or 1 (if entry is more than 0.5) you would do: X = (X < 0.5). Note that 0 and 1 are respectively equivalent to False and True.
  3. Set $A^{[1]}$ to $A^{[1]} * D^{[1]}$. (We are shutting down some neurons). We can think of $D^{[1]}$ as a mask, so that when it is multiplied with another matrix, it shuts down some of the values.
  4. Divide $A^{[1]}$ by keep_prob. By doing this you are assuring that the result of the cost will still have the same expected value as without drop-out. (This technique is also called inverted dropout.)
def forward_propagation_with_dropout(X, parameters, keep_prob = 0.5):
    """
    Implements the forward propagation: LINEAR -> RELU + DROPOUT -> LINEAR -> RELU + DROPOUT -> LINEAR -> SIGMOID.

    Arguments:
    X -- input dataset, of shape (2, number of examples)
    parameters -- python dictionary containing your parameters "W1", "b1", "W2", "b2", "W3", "b3":
                    W1 -- weight matrix of shape (20, 2)
                    b1 -- bias vector of shape (20, 1)
                    W2 -- weight matrix of shape (3, 20)
                    b2 -- bias vector of shape (3, 1)
                    W3 -- weight matrix of shape (1, 3)
                    b3 -- bias vector of shape (1, 1)
    keep_prob - probability of keeping a neuron active during drop-out, scalar

    Returns:
    A3 -- last activation value, output of the forward propagation, of shape (1,1)
    cache -- tuple, information stored for computing the backward propagation
    """

    np.random.seed(1)

    # retrieve parameters
    W1 = parameters["W1"]
    b1 = parameters["b1"]
    W2 = parameters["W2"]
    b2 = parameters["b2"]
    W3 = parameters["W3"]
    b3 = parameters["b3"]

    # LINEAR -> RELU -> LINEAR -> RELU -> LINEAR -> SIGMOID
    Z1 = np.dot(W1, X) + b1
    A1 = relu(Z1)

    D1 = np.random.rand(A1.shape[0],A1.shape[1])      # Step 1: initialize matrix D1 = np.random.rand(..., ...)
    D1 = (D1<keep_prob)                               # Step 2: convert entries of D1 to 0 or 1 (using keep_prob as the threshold)
    A1 = A1*D1                                        # Step 3: shut down some neurons of A1
    A1 = A1/keep_prob                                 # Step 4: scale the value of neurons that haven't been shut down

    Z2 = np.dot(W2, A1) + b2
    A2 = relu(Z2)

    D2 = np.random.rand(A2.shape[0],A2.shape[1])      # Step 1: initialize matrix D2 = np.random.rand(..., ...)
    D2 = (D2<keep_prob)                               # Step 2: convert entries of D2 to 0 or 1 (using keep_prob as the threshold)
    A2 = A2*D2                                        # Step 3: shut down some neurons of A2
    A2 = A2/keep_prob                                 # Step 4: scale the value of neurons that haven't been shut down

    Z3 = np.dot(W3, A2) + b3
    A3 = sigmoid(Z3)

    cache = (Z1, D1, A1, W1, b1, Z2, D2, A2, W2, b2, Z3, A3, W3, b3)

    return A3, cache
X_assess, parameters = forward_propagation_with_dropout_test_case()

A3, cache = forward_propagation_with_dropout(X_assess, parameters, keep_prob = 0.7)
print ("A3 = " + str(A3))
A3 = [[ 0.36974721  0.00305176  0.04565099  0.49683389  0.36974721]]

3.2 - Backward propagation with dropout

Let’s implement the backward propagation with dropout. As before, we are training a 3 layer network. Add dropout to the first and second hidden layers, using the masks $D^{[1]}$ and $D^{[2]}$ stored in the cache.

The below cell will implement the following:

Backpropagation with dropout is actually quite easy. We will have to carry out 2 Steps:

  1. I had previously shut down some neurons during forward propagation, by applying a mask $D^{[1]}$ to A1. In backpropagation, I will have to shut down the same neurons, by reapplying the same mask $D^{[1]}$ to dA1.
  2. During forward propagation, I had divided A1 by keep_prob. In backpropagation, I’ll therefore have to divide dA1 by keep_prob again (the calculus interpretation is that if $A^{[1]}$ is scaled by keep_prob, then its derivative $dA^{[1]}$ is also scaled by the same keep_prob).
def backward_propagation_with_dropout(X, Y, cache, keep_prob):
    """
    Implements the backward propagation of our baseline model to which we added dropout.

    Arguments:
    X -- input dataset, of shape (2, number of examples)
    Y -- "true" labels vector, of shape (output size, number of examples)
    cache -- cache output from forward_propagation_with_dropout()
    keep_prob - probability of keeping a neuron active during drop-out, scalar

    Returns:
    gradients -- A dictionary with the gradients with respect to each parameter, activation and pre-activation variables
    """

    m = X.shape[1]
    (Z1, D1, A1, W1, b1, Z2, D2, A2, W2, b2, Z3, A3, W3, b3) = cache

    dZ3 = A3 - Y
    dW3 = 1./m * np.dot(dZ3, A2.T)
    db3 = 1./m * np.sum(dZ3, axis=1, keepdims = True)
    dA2 = np.dot(W3.T, dZ3)

    dA2 = dA2*D2            # Step 1: Apply mask D2 to shut down the same neurons as during the forward propagation
    dA2 = dA2/keep_prob     # Step 2: Scale the value of neurons that haven't been shut down

    dZ2 = np.multiply(dA2, np.int64(A2 > 0))
    dW2 = 1./m * np.dot(dZ2, A1.T)
    db2 = 1./m * np.sum(dZ2, axis=1, keepdims = True)

    dA1 = np.dot(W2.T, dZ2)

    dA1 = dA1*D1            # Step 1: Apply mask D1 to shut down the same neurons as during the forward propagation
    dA1 = dA1/keep_prob     # Step 2: Scale the value of neurons that haven't been shut down

    dZ1 = np.multiply(dA1, np.int64(A1 > 0))
    dW1 = 1./m * np.dot(dZ1, X.T)
    db1 = 1./m * np.sum(dZ1, axis=1, keepdims = True)

    gradients = {"dZ3": dZ3, "dW3": dW3, "db3": db3,"dA2": dA2,
                 "dZ2": dZ2, "dW2": dW2, "db2": db2, "dA1": dA1,
                 "dZ1": dZ1, "dW1": dW1, "db1": db1}

    return gradients
X_assess, Y_assess, cache = backward_propagation_with_dropout_test_case()

gradients = backward_propagation_with_dropout(X_assess, Y_assess, cache, keep_prob = 0.8)

print ("dA1 = " + str(gradients["dA1"]))
print ("dA2 = " + str(gradients["dA2"]))
dA1 = [[ 0.36544439  0.         -0.00188233  0.         -0.17408748]
 [ 0.65515713  0.         -0.00337459  0.         -0.        ]]
dA2 = [[ 0.58180856  0.         -0.00299679  0.         -0.27715731]
 [ 0.          0.53159854 -0.          0.53159854 -0.34089673]
 [ 0.          0.         -0.00292733  0.         -0.        ]]

Let’s now run the model with dropout (keep_prob = 0.86). It means at every iteration we shut down each neurons of layer 1 and 2 with 14% probability. The function model() will now call:

  • forward_propagation_with_dropout instead of forward_propagation.
  • backward_propagation_with_dropout instead of backward_propagation.
parameters = model(train_X, train_Y, keep_prob = 0.86, learning_rate = 0.3)

print ("On the train set:")
predictions_train = predict(train_X, train_Y, parameters)
print ("On the test set:")
predictions_test = predict(test_X, test_Y, parameters)
Cost after iteration 0: 0.6543912405149825


/home/jovyan/work/week5/Regularization/reg_utils.py:236: RuntimeWarning: divide by zero encountered in log
  logprobs = np.multiply(-np.log(a3),Y) + np.multiply(-np.log(1 - a3), 1 - Y)
/home/jovyan/work/week5/Regularization/reg_utils.py:236: RuntimeWarning: invalid value encountered in multiply
  logprobs = np.multiply(-np.log(a3),Y) + np.multiply(-np.log(1 - a3), 1 - Y)


Cost after iteration 10000: 0.06101698657490562
Cost after iteration 20000: 0.060582435798513114

png

On the train set:
Accuracy: 0.928909952607
On the test set:
Accuracy: 0.95

Dropout works great! The test accuracy has increased again (to 95%)! Our model is not overfitting the training set and does a great job on the test set. The French football team will be forever grateful to us!

Run the code below to plot the decision boundary.

plt.title("Model with dropout")
axes = plt.gca()
axes.set_xlim([-0.75,0.40])
axes.set_ylim([-0.75,0.65])
plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y)

png

Note:

  • A common mistake when using dropout is to use it both in training and testing. We should use dropout (randomly eliminate nodes) only in training.
  • Deep learning frameworks like tensorflow, PaddlePaddle, keras or caffe come with a dropout layer implementation. Don’t stress - you will soon learn some of these frameworks.

Key points for this section are:

  • Dropout is a regularisation technique.
  • We only use dropout during training. Don’t use dropout (randomly eliminate nodes) during test time.
  • Apply dropout both during forward and backward propagation.
  • During training time, divide each dropout layer by keep_prob to keep the same expected value for the activations. For example, if keep_prob is 0.5, then we will on average shut down half the nodes, so the output will be scaled by 0.5 since only the remaining half are contributing to the solution. Dividing by 0.5 is equivalent to multiplying by 2. Hence, the output now has the same expected value. We can check that this works even when keep_prob is other values than 0.5.

4 - Conclusions

Here are the results of our three models:

**model** **train accuracy** **test accuracy**
3-layer NN without regularisation 95% 91.5%
3-layer NN with L2-regularisation 94% 93%
3-layer NN with dropout 93% 95%

Note that regularisation hurts training set performance! This is because it limits the ability of the network to overfit to the training set. But since it ultimately gives better test accuracy, it is helping our system.

Key points for this section are:

  • Regularization will help you reduce overfitting.
  • Regularization will drive your weights to lower values.
  • L2 regularization and Dropout are two very effective regularization techniques.

last edited: 31/05/19

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