Regularization in Deep Learning

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!

In this project I am going to use regularization in this deep learning model.

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

# import packages
import numpy as np
import matplotlib.pyplot as plt
# pleasr refer "reg_utils.py" and "testCases.py" in the same directory for the below "reg_utils" and "testCases"libraries
from reg_utils import sigmoid, relu, plot_decision_boundary, initialize_parameters, 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'

For this project, I am going 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. Later, I can give this recommendations to French Football Corporation as an AI expert :)

**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

Loading Data

def load_2D_dataset():
    data = scipy.io.loadmat('data.mat')
    train_X = data['X'].T
    train_Y = data['y'].T
    test_X = data['Xval'].T
    test_Y = data['yval'].T

    plt.scatter(train_X[0, :], train_X[1, :], c=train_Y[0], s=40, cmap=plt.cm.Spectral);
    
    return train_X, train_Y, test_X, test_Y


train_X, train_Y, test_X, test_Y = load_2D_dataset()

# Following 2D dataset from France's past 10 games.

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 is to 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'll try to regularize it and decide which model that I will choose to solve the French Football Corporation's problem.

1 - Non-regularized model

I am using 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()"

Before you proceed please refer "reg_utils.py" and "testCases.py" in the same directory. It will provide all the dropout functions that I am goinng to use tin this project.

In each part, I will run this model with the correct inputs so that it calls the functions that I've implemented. Take a look at the code below to familiarize yourself 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_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.

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: 0.1385164242326018

On the training set:
Accuracy: 0.9478672985781991
On the test set:
Accuracy: 0.915

The train accuracy is 94.7% while the test accuracy is 91.5%. This is the baseline model (I am going to observe the impact of regularization on this model). Run the following code to plot the decision boundary of the model.

plt.title("Model without 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)

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.

2 - L2 Regularization

The standard way to avoid overfitting is called L2 regularization. It consists of appropriately modifying the cost function, from: $$J = -\frac{1}{m} \sum\limits_{i = 1}^{m} \large{(}\small y^{(i)}\log\left(a^{L}\right) + (1-y^{(i)})\log\left(1- a^{L}\right) \large{)} \tag{1}$$ To: $$J_{regularized} = \small \underbrace{-\frac{1}{m} \sum\limits_{i = 1}^{m} \large{(}\small y^{(i)}\log\left(a^{L}\right) + (1-y^{(i)})\log\left(1- a^{L}\right) \large{)} }\text{cross-entropy cost} + \underbrace{\frac{1}{m} \frac{\lambda}{2} \sum\limits_l\sum\limits_k\sum\limits_j W{k,j}^{[l]2} }_\text{L2 regularization cost} \tag{2}$$

Let's modify the cost and observe the consequences.

Exercise: Implement compute_cost_with_regularization() 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 you have to do this for $W^{[1]}$, $W^{[2]}$ and $W^{[3]}$, then sum the three terms and multiply by $ \frac{1}{m} \frac{\lambda}{2} $.

# GRADED FUNCTION: compute_cost_with_regularization

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 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

A3, Y_assess, parameters = compute_cost_with_regularization_test_case()

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

cost = 1.7864859451590758

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

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

# GRADED FUNCTION: backward_propagation_with_regularization

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

X_assess, Y_assess, cache = backward_propagation_with_regularization_test_case()

grads = backward_propagation_with_regularization(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 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

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: 0.2684918873282238
Cost after iteration 20000: 0.26809163371273004

On the train set:
Accuracy: 0.9383886255924171
On the test set:
Accuracy: 0.93

WOW 😮, the test set accuracy increased to 93%. I have saved the French football team!

Now, I am not overfitting the training data anymore. Let's plot the decision boundary.

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)

Observations:

• The value of $\lambda$ is a hyperparameter that I can tune using a dev set.
• L2 regularization makes my 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 I 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.

3 - 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!

Figure 2 : Drop-out on the second hidden layer.
At each iteration, I 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: I have shut down on average 40% of the neurons. $3^{rd}$ layer: I have shut down on average 20% of the neurons.

When I shut some neurons down, I am actually modifying the model. The idea behind drop-out is that at each iteration, I have train a different model that uses only a subset of my neurons. With dropout, my 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

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

# GRADED FUNCTION: forward_propagation_with_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)
             # 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 = 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

Implementing the backward propagation with dropout. As before, I am training a 3 layer network and adding 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. You will have to carry out 2 Steps:

1. I have 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 have 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).

# GRADED FUNCTION: backward_propagation_with_dropout

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 I ahvve shut down each neurons of layer 1 and 2 with 24% 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


/Users/rajeshidumalla/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)
/Users/rajeshidumalla/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: 0.061016986574905605
Cost after iteration 20000: 0.060582435798513114

On the train set:
Accuracy: 0.9289099526066351
On the test set:
Accuracy: 0.95

Dropout works great! The test accuracy has increased again (to 95%)! My 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 me!

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)

Note:

• A common mistake when using dropout is to use it both in training and testing. You should use dropout (randomly eliminate nodes) only in training.
• Deep learning frameworks like tensorflow, PaddlePaddle, keras or caffe come with a dropout layer implementation.

**What I have learned about dropout:**

• Dropout is a regularization technique.
• I do only use dropout during training and not using dropout (randomly eliminate nodes) during test time.
• Applying 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:

3-layer NN without regularization:

• On the training set accuracy: 94.7%
• On the test set accracy: 91.5%

3-layer NN with L2-regularization:

• On the train set accuracy: 93.8%
• On the test set accuracy: 93%

3-layer NN with dropout:

• On the train set accuracy: 92.8%
• On the test set accuracy: 95%

I have learned 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 my system.

Key notes:

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