Regularization

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Abstract

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 generalize to new examples that it has never seen! This article will use regularization in our deep learning models.

Implement

Problem Statement: You have just been hired as an AI expert by the French Football Corporation. They would like you 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.

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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 you the following 2D dataset from France’s past 10 games.

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train_X, train_Y, test_X, test_Y = load_2D_dataset()

output_3_0

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

Our 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-regularized model. Then I will show you how to regularize it and decide which model we will choose to solve the French Football Corporation’s problem.

Non-regularized model

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

  • in regularization 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 regularization. Then, I will implement:

  • L2 regularization — functions: “compute_cost_with_regularization()“ and “backward_propagation_with_regularization()“
  • 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:

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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_regularization(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_regularization(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.

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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)
plt.title("Model without regularization")
axes = plt.gca()
axes.set_xlim([-0.75,0.40])
axes.set_ylim([-0.75,0.65])
#to plot the decision boundary of our model.
plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y)

Output:

Cost after iteration 0: 0.6557412523481002
Cost after iteration 10000: 0.16329987525724216
Cost after iteration 20000: 0.13851642423255986
On the training set:
Accuracy: 0.947867298578
On the test set:
Accuracy: 0.915

png

png

The train accuracy is 94.8% while the test accuracy is 91.5%. This is the baseline model (I will observe the impact of regularization on this model).

The non-regularized model is obviously overfitting the training set. It is fitting the noisy points! Lets now look at two techniques to reduce overfitting.

L2 Regularization

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

To:

Let’s modify out cost and observe the consequences.

compute_cost_with_regularization

I will implement compute_cost_with_regularization() which computes the cost given by formula (2). To calculate $\sum\limitsk\sum\limits_j W{k,j}^{[l]2}$ , I will use np.sum(np.square(Wl))

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def compute_cost_with_regularization(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 = (1 / m) * (lambd / 2) * (np.sum(np.square(W1))+np.sum(np.square(W2))+np.sum(np.square(W3))) 
    
    cost = cross_entropy_cost + L2_regularization_cost
    
    return cost

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

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

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

    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 -- regularization 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 / m) * W3
    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 / m) * W2
    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 / m) * W1
    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

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

  • compute_cost_with_regularization instead of compute_cost
  • backward_propagation_with_regularization instead of backward_propagation
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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)
plt.title("Model with L2-regularization")
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)
Cost after iteration 0: 0.6974484493131264
Cost after iteration 10000: 0.2684918873282239
Cost after iteration 20000: 0.2680916337127301

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Congrats, the test set accuracy increased to 93%. We have saved the French football team!

Observations

  • The value of $\lambda$ is a hyperparameter that we can tune using a dev set.
  • L2 regularization 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-regularization actually doing?

L2-regularization relies on the assumption that a model with small weights is simpler than a model with large weights. Thus, by penalizing the square values of the weights in the cost function you drive all the weights 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.

What you should remember——the implications of L2-regularization on

  • The cost computation:
    • A regularization 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.

Dropout

Finally, dropout is a widely used regularization 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!

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.

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.

Forward propagation with dropout

I will 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.

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

  1. In lecture, we dicussed 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 vectorized 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. Hint: 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) I would do: X = (X < 0.5). Note that 0 and 1 are respectively equivalent to False and True.
  3. Set $A^{[1]}$ to $A^{[1]} * D^{[1]}$. (I am 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 I am 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.)
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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)
    # Steps 1-4 below correspond to the Steps 1-4 described above.
    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 = np.multiply(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)
    ### (approx. 4 lines)
    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

Backward propagation with dropout

I will implement the backward propagation with dropout. As before, I am 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.

Instruction:
Backpropagation with dropout is actually quite easy. I 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).
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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)
    ### (≈ 2 lines of code)
    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)
    ### (≈ 2 lines of code)
    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

Let’s now run the model with dropout (keep_prob = 0.86). It means at every iteration I 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.
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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)
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)

Output:

Cost after iteration 0: 0.6543912405149825
Cost after iteration 10000: 0.06101698657490559
Cost after iteration 20000: 0.060582435798513114
On the train set:
Accuracy: 0.928909952607
On the test set:
Accuracy: 0.95

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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 we!

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——we will soon learn some of these frameworks.
    What you should remember about dropout:

  • Dropout is a regularization 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.

Conclusions

Here are the results of our three models:

image-20181214160805789

Note that regularization 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 your system.
What we want you to remember from this article:

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

github

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