The following problems are taken from a few assignments from the coursera courses Introduction to Deep Learning (by Higher School of Economics) and Neural Networks and Deep Learning (by Prof Andrew Ng, deeplearning.ai). The problem descriptions are taken straightaway from the assignments.

1. Linear models, Optimization

In this assignment a linear classifier will be implemented and it will be trained using stochastic gradient descent with numpy. Two-dimensional classification To make things more intuitive, let’s solve a 2D classification problem with synthetic data.

Features As we can notice the data above isn’t linearly separable. Hence we should add features (or use non-linear model). Note that decision line between two classes have form of circle, since that we can add quadratic features to make the problem linearly separable. The idea under this displayed on image below: Here are some test results for the implemented expand function, that is used for adding quadratic features: # simple test on random numbers dummy_X = np . array ([ [ 0 , 0 ], [ 1 , 0 ], [ 2.61 , - 1.28 ], [ - 0.59 , 2.1 ] ]) # call expand function dummy_expanded = expand ( dummy_X ) # what it should have returned: x0 x1 x0^2 x1^2 x0*x1 1 dummy_expanded_ans = np . array ([[ 0. , 0. , 0. , 0. , 0. , 1. ], [ 1. , 0. , 1. , 0. , 0. , 1. ], [ 2.61 , - 1.28 , 6.8121 , 1.6384 , - 3.3408 , 1. ], [ - 0.59 , 2.1 , 0.3481 , 4.41 , - 1.239 , 1. ]])

Logistic regression

To classify objects we will obtain probability of object belongs to class ‘1’. To predict probability we will use output of linear model and logistic function:

def probability ( X , w ): """ Given input features and weights return predicted probabilities of y==1 given x, P(y=1|x), see description above :param X: feature matrix X of shape [n_samples,6] (expanded) :param w: weight vector w of shape [6] for each of the expanded features :returns: an array of predicted probabilities in [0,1] interval. """ return 1. / ( 1 + np . exp ( - np . dot ( X , w )))

In logistic regression the optimal parameters w are found by cross-entropy minimization:

def compute_loss ( X , y , w ): """ Given feature matrix X [n_samples,6], target vector [n_samples] of 1/0, and weight vector w [6], compute scalar loss function using formula above. """ return - np . mean ( y * np . log ( probability ( X , w )) + ( 1 - y ) * np . log ( 1 - probability ( X , w )))

Since we train our model with gradient descent, we should compute gradients. To be specific, we need the following derivative of loss function over each weight:

Here is the derivation (can be found here too):

def compute_grad ( X , y , w ): """ Given feature matrix X [n_samples,6], target vector [n_samples] of 1/0, and weight vector w [6], compute vector [6] of derivatives of L over each weights. """ return np.dot((probability(X, w) - y), X) / X.shape[0]

Training In this section we’ll use the functions we wrote to train our classifier using stochastic gradient descent. We shall try to change hyper-parameters like batch size, learning rate and so on to find the best one.

Mini-batch SGD Stochastic gradient descent just takes a random example on each iteration, calculates a gradient of the loss on it and makes a step: w = np . array ([ 0 , 0 , 0 , 0 , 0 , 1 ]) # initialize eta = 0.05 # learning rate n_iter = 100 batch_size = 4 loss = np . zeros ( n_iter ) for i in range ( n_iter ): ind = np . random . choice ( X_expanded . shape [ 0 ], batch_size ) loss [ i ] = compute_loss ( X_expanded , y , w ) dw = compute_grad(X_expanded[ind, :], y[ind], w) w = w - eta*dw The following animation shows how the decision surface and the cross-entropy loss function changes with different batches with SGD where batch-size=4.

SGD with momentum

Momentum is a method that helps accelerate SGD in the relevant direction and dampens oscillations as can be seen in image below. It does this by adding a fraction α of the update vector of the past time step to the current update vector.

eta = 0.05 # learning rate alpha = 0.9 # momentum nu = np.zeros_like(w) n_iter = 100 batch_size = 4 loss = np . zeros ( n_iter ) for i in range ( n_iter ): ind = np . random . choice ( X_expanded . shape [ 0 ], batch_size ) loss [ i ] = compute_loss ( X_expanded , y , w ) dw = compute_grad(X_expanded[ind, :], y[ind], w) nu = alpha*nu + eta*dw w = w - nu

The following animation shows how the decision surface and the cross-entropy loss function changes with different batches with SGD + momentum where batch-size=4. As can be seen, the loss function drops much faster, leading to a faster convergence.

RMSprop

We also need to implement RMSPROP algorithm, which use squared gradients to adjust learning rate as follows:

eta = 0.05 # learning rate alpha = 0.9 # momentum G = np.zeros_like(w) eps = 1e-8 n_iter = 100 batch_size = 4 loss = np . zeros ( n_iter ) for i in range ( n_iter ): ind = np . random . choice ( X_expanded . shape [ 0 ], batch_size ) loss [ i ] = compute_loss ( X_expanded , y , w ) dw = compute_grad(X_expanded[ind, :], y[ind], w) G = alpha*G + (1-alpha)*dw**2 w = w - eta*dw / np.sqrt(G + eps)

The following animation shows how the decision surface and the cross-entropy loss function changes with different batches with SGD + RMSProp where batch-size=4. As can be seen again, the loss function drops much faster, leading to a faster convergence.

2. Planar data classification with a neural network with one hidden layer, an implementation from scratch In this assignment a neural net with a single hidden layer will be trained from scratch. We shall see a big difference between this model and the one implemented using logistic regression. We shall learn how to: Implement a 2-class classification neural network with a single hidden layer

Use units with a non-linear activation function, such as tanh

Compute the cross entropy loss

Implement forward and backward propagation Dataset The following figure visualizes a “flower” 2-class dataset that we shall work on, the colors indicates the class labels. We have m = 400 training examples. Simple Logistic Regression Before building a full neural network, lets first see how logistic regression performs on this problem. We can use sklearn’s built-in functions to do that, by running the code below to train a logistic regression classifier on the dataset. # Train the logistic regression classifier clf = sklearn . linear_model . LogisticRegressionCV (); clf . fit ( X . T , Y . T ); We can now plot the decision boundary of the model and accuracy with the following code. # Plot the decision boundary for logistic regression plot_decision_boundary ( lambda x : clf . predict ( x ), X , Y ) plt . title ( "Logistic Regression" ) # Print accuracy LR_predictions = clf . predict ( X . T ) print ( 'Accuracy of logistic regression: %d ' % float (( np . dot ( Y , LR_predictions ) + np . dot ( 1 - Y , 1 - LR_predictions )) / float ( Y . size ) * 100 ) + '% ' + "(percentage of correctly labelled datapoints)" ) Accuracy of logistic regression: 47 % (percentage of correctly labelled datapoints) Accuracy 47% Interpretation: The dataset is not linearly separable, so logistic regression doesn’t perform well. Hopefully a neural network will do better. Let’s try this now! Neural Network model Logistic regression did not work well on the “flower dataset”. We are going to train a Neural Network with a single hidden layer, by implementing the network with python numpy from scratch. Here is our model:

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)

Defining the neural network structure Define three variables and the function layer_sizes: - n_x: the size of the input layer - n_h: the size of the hidden layer (set this to 4) - n_y: the size of the output layer

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 """

Initialize the model’s parameters Implement the function initialize_parameters() . Instructions: Make sure the parameters’ sizes are right. Refer to the neural network figure above if needed.

We will initialize the weights matrices with random values. Use: np.random.randn(a,b) * 0.01 to randomly initialize a matrix of shape (a,b).

We will initialize the bias vectors as zeros. Use: np.zeros((a,b)) to initialize a matrix of shape (a,b) with zeros.



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 the 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) """

The Loop Implement forward_propagation() . Instructions: Look above at the mathematical representation of the classifier.

We can use the function sigmoid() .

. We can use the function np.tanh() . It is part of the numpy library.

. It is part of the numpy library. The steps we have to implement are: Retrieve each parameter from the dictionary “parameters” (which is the output of initialize_parameters() ) by using parameters[".."] . Implement Forward Propagation. Compute Z [ 1 ] , A [ 1 ] , Z [ 2 ] Z[1],A[1],Z[2] and A [ 2 ] A[2] (the vector of all the predictions on all the examples in the training set).

Values needed in the backpropagation are stored in “ cache “. The cache will be given as an input to the backpropagation function.

def forward_propagation ( X , parameters ): """ Argument: X -- input data of size (n_x, m) parameters -- python dictionary containing the parameters (output of initialization function) Returns: A2 -- The sigmoid output of the second activation cache -- a dictionary containing "Z1", "A1", "Z2" and "A2" """

Implement compute_cost() to compute the value of the cost J. There are many ways to implement the cross-entropy loss.

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 the parameters W1, b1, W2 and b2 Returns: cost -- cross-entropy cost given equation (13) """

Using the cache computed during forward propagation, we can now implement backward propagation. Implement the function backward_propagation() . Instructions: Backpropagation is usually the hardest (most mathematical) part in deep learning. The following figure is taken from is the slide from the lecture on backpropagation. We’ll want to use the six equations on the right of this slide, since we are building a vectorized implementation.

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 the gradients with respect to different parameters """

Implement the update rule. Use gradient descent. We have to use (dW1, db1, dW2, db2) in order to update (W1, b1, W2, b2). General gradient descent rule: θ=θ−α(∂J/∂θ) where α is the learning rate and θ

represents a parameter. Illustration: The gradient descent algorithm with a good learning rate (converging) and a bad learning rate (diverging). Images courtesy of Adam Harley.

def update_parameters ( parameters , grads , learning_rate = 1.2 ): """ Updates parameters using the gradient descent update rule given above Arguments: parameters -- python dictionary containing our parameters grads -- python dictionary containing our gradients Returns: parameters -- python dictionary containing our updated parameters """

Integrate previous parts in nn_model() Build the neural network model in nn_model() . Instructions: The neural network model has to use the previous functions in the right order. 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. """

Predictions Use the model to predict by building predict(). Use forward propagation to predict results. def predict ( parameters , X ): """ Using the learned parameters, predicts a class for each example in X Arguments: parameters -- python dictionary containing our parameters X -- input data of size (n_x, m) Returns predictions -- vector of predictions of our model (red: 0 / blue: 1) """ It is time to run the model and see how it performs on a planar dataset. Run the following code to test our model with a single hidden layer of nh hidden units. # Build a model with a n_h-dimensional hidden layer parameters = nn_model ( X , Y , n_h = 4 , num_iterations = 10000 , print_cost = True ) # Plot the decision boundary plot_decision_boundary ( lambda x : predict ( parameters , x . T ), X , Y ) plt . title ( "Decision Boundary for hidden layer size " + str ( 4 )) Cost after iteration 0: 0.693048 Cost after iteration 1000: 0.288083 Cost after iteration 2000: 0.254385 Cost after iteration 3000: 0.233864 Cost after iteration 4000: 0.226792 Cost after iteration 5000: 0.222644 Cost after iteration 6000: 0.219731 Cost after iteration 7000: 0.217504 Cost after iteration 8000: 0.219471 Cost after iteration 9000: 0.218612

Cost after iteration 9000 0.218607 # Print accuracy predictions = predict ( parameters , X ) print ( 'Accuracy: %d ' % float (( np . dot ( Y , predictions . T ) + np . dot ( 1 - Y , 1 - predictions . T )) / float ( Y . size ) * 100 ) + '%' ) Accuracy: 90% Accuracy is really high compared to Logistic Regression. The model has learnt the leaf patterns of the flower! Neural networks are able to learn even highly non-linear decision boundaries, unlike logistic regression. Now, let’s try out several hidden layer sizes. We can observe different behaviors of the model for various hidden layer sizes. The results are shown below.

Tuning hidden layer size Accuracy for 1 hidden units: 67.5 % Accuracy for 2 hidden units: 67.25 % Accuracy for 3 hidden units: 90.75 % Accuracy for 4 hidden units: 90.5 % Accuracy for 5 hidden units: 91.25 % Accuracy for 20 hidden units: 90.0 % Accuracy for 50 hidden units: 90.25 %

Interpretation: 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.

We shall also learn later about regularization, which lets us use very large models (such as n_h = 50) without much overfitting.

3. Getting deeper with Keras

Tensorflow is a powerful and flexible tool, but coding large neural architectures with it is tedious.

There are plenty of deep learning toolkits that work on top of it like Slim, TFLearn, Sonnet, Keras.

Choice is matter of taste and particular task

We’ll be using Keras to predict handwritten digits with the mnist dataset.

The following figure shows 225 sample images from the dataset.

The pretty keras Using only the following few lines of code we can learn a simple deep neural net with 3 dense hidden layers and with Relu activation, with dropout 0.5 after each dense layer.

import keras from keras.models import Sequential import keras.layers as ll model = Sequential ( name = "mlp" ) model . add ( ll . InputLayer ([ 28 , 28 ])) model . add ( ll . Flatten ()) # network body model . add ( ll . Dense ( 128 )) model . add ( ll . Activation ( 'relu' )) model . add ( ll . Dropout ( 0.5 )) model . add ( ll . Dense ( 128 )) model . add ( ll . Activation ( 'relu' )) model . add ( ll . Dropout ( 0.5 )) model . add ( ll . Dense ( 128 )) model . add ( ll . Activation ( 'relu' )) model . add ( ll . Dropout ( 0.5 )) # output layer: 10 neurons for each class with softmax model . add ( ll . Dense ( 10 , activation = 'softmax' )) # categorical_crossentropy is our good old crossentropy # but applied for one-hot-encoded vectors model . compile ( "adam" , "categorical_crossentropy" , metrics = [ "accuracy" ]) The following shows the summary of the model:

_________________________________________________________________ Layer (type) Output Shape Param # ================================================================= input_12 (InputLayer) (None, 28, 28) 0 _________________________________________________________________ flatten_12 (Flatten) (None, 784) 0 _________________________________________________________________ dense_35 (Dense) (None, 128) 100480 _________________________________________________________________ activation_25 (Activation) (None, 128) 0 _________________________________________________________________ dropout_22 (Dropout) (None, 128) 0 _________________________________________________________________ dense_36 (Dense) (None, 128) 16512 _________________________________________________________________ activation_26 (Activation) (None, 128) 0 _________________________________________________________________ dropout_23 (Dropout) (None, 128) 0 _________________________________________________________________ dense_37 (Dense) (None, 128) 16512 _________________________________________________________________ activation_27 (Activation) (None, 128) 0 _________________________________________________________________ dropout_24 (Dropout) (None, 128) 0 _________________________________________________________________ dense_38 (Dense) (None, 10) 1290 ================================================================= Total params: 134,794 Trainable params: 134,794 Non-trainable params: 0 _________________________________________________________________

Model interface Keras models follow Scikit-learn‘s interface of fit/predict with some notable extensions. Let’s take a tour. # fit(X,y) ships with a neat automatic logging. # Highly customizable under the hood. model . fit ( X_train , y_train , validation_data = ( X_val , y_val ), epochs = 13 ); Train on 50000 samples, validate on 10000 samples Epoch 1/13 50000/50000 [==============================] - 14s - loss: 0.1489 - acc: 0.9587 - val_loss: 0.0950 - val_acc: 0.9758 Epoch 2/13 50000/50000 [==============================] - 12s - loss: 0.1543 - acc: 0.9566 - val_loss: 0.0957 - val_acc: 0.9735 Epoch 3/13 50000/50000 [==============================] - 11s - loss: 0.1509 - acc: 0.9586 - val_loss: 0.0985 - val_acc: 0.9752 Epoch 4/13 50000/50000 [==============================] - 11s - loss: 0.1515 - acc: 0.9577 - val_loss: 0.0967 - val_acc: 0.9752 Epoch 5/13 50000/50000 [==============================] - 11s - loss: 0.1471 - acc: 0.9596 - val_loss: 0.1008 - val_acc: 0.9737 Epoch 6/13 50000/50000 [==============================] - 11s - loss: 0.1488 - acc: 0.9598 - val_loss: 0.0989 - val_acc: 0.9749 Epoch 7/13 50000/50000 [==============================] - 11s - loss: 0.1495 - acc: 0.9592 - val_loss: 0.1011 - val_acc: 0.9748 Epoch 8/13 50000/50000 [==============================] - 11s - loss: 0.1434 - acc: 0.9604 - val_loss: 0.1005 - val_acc: 0.9761 Epoch 9/13 50000/50000 [==============================] - 11s - loss: 0.1514 - acc: 0.9590 - val_loss: 0.0951 - val_acc: 0.9759 Epoch 10/13 50000/50000 [==============================] - 11s - loss: 0.1424 - acc: 0.9613 - val_loss: 0.0995 - val_acc: 0.9739 Epoch 11/13 50000/50000 [==============================] - 11s - loss: 0.1408 - acc: 0.9625 - val_loss: 0.0977 - val_acc: 0.9751 Epoch 12/13 50000/50000 [==============================] - 11s - loss: 0.1413 - acc: 0.9601 - val_loss: 0.0938 - val_acc: 0.9753 Epoch 13/13 50000/50000 [==============================] - 11s - loss: 0.1430 - acc: 0.9619 - val_loss: 0.0981 - val_acc: 0.9761 As we could see, with a simple model without any convolution layers we could obtain more than 97.5% accuracy on the validation dataset. The following figures show the weights learnt at different layers. Some Tips & tricks to improve accuracy Here are some tips on what we can do to improve accuracy: Network size More neurons, More layers, (docs) Nonlinearities in the hidden layers tanh, relu, leaky relu, etc Larger networks may take more epochs to train, so don’t discard the net just because it could didn’t beat the baseline in 5 epochs.

Early Stopping Training for 100 epochs regardless of anything is probably a bad idea. Some networks converge over 5 epochs, others – over 500. Way to go: stop when validation score is 10 iterations past maximum

Faster optimization rmsprop, nesterov_momentum, adam, adagrad and so on. Converge faster and sometimes reach better optima It might make sense to tweak learning rate/momentum, other learning parameters, batch size and number of epochs

Regularize to prevent overfitting Add some L2 weight norm to the loss function, theano will do the rest Can be done manually or via – https://keras.io/regularizers/

to prevent overfitting Data augmemntation – getting 5x as large dataset for free is a great deal https://keras.io/preprocessing/image/ Zoom-in+slice = move Rotate+zoom(to remove black stripes) any other perturbations Simple way to do that (if we have PIL/Image): from scipy.misc import imrotate,imresize and a few slicing Stay realistic. There’s usually no point in flipping dogs upside down as that is not the way we usually see them.

– getting 5x as large dataset for free is a great deal