he 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.
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. ]])
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 weightsreturn 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:
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]
In this section we’ll use the functions you 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.
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]) # initializeeta = 0.05 # learning raten_iter = 100batch_size = 4loss = 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.
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*dww = 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.
We also need to implement RMSPROP algorithm, which use squared gradients to adjust learning rate as follows:
eta = 0.05 # learning ratealpha = 0.9 # momentumG = np.zeros_like(w)
eps = 1e-8
n_iter = 100batch_size = 4loss = 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**2w = 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.
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:
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.
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 | 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!
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)
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
"""
Implement the function initialize_parameters()
.
Instructions:
np.random.randn(a,b) * 0.01
to randomly initialize a matrix of shape (a,b).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 your parameters:
W1 -- weight matrix of shape (n_h, n_x)
b1 -- bias vector of shape (n_h, 1)
W2 -- weight matrix of shape (n_y, n_h)
b2 -- bias vector of shape (n_y, 1)
"""
Implement forward_propagation()
.
Instructions:
sigmoid()
.np.tanh()
. It is part of the numpy library.initialize_parameters()
) by using parameters[".."]
.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 your parameters (output of initialization function) Returns: A2 -- The sigmoid output of the second activation cache -- a dictionary containing "Z1", "A1", "Z2" and "A2" """
def compute_cost(A2, Y, parameters):
"""
Computes the cross-entropy cost given in equation (13)
Arguments: A2 -- The sigmoid output of the second activation, of shape (1, number of examples)
Y -- "true" labels vector of shape (1, number of examples)
parameters -- python dictionary containing your parameters W1, b1, W2 and b2
Returns: cost -- cross-entropy cost given equation (13)
"""
Using the cache computed during forward propagation, we can now implement backward propagation.
Implement the function backward_propagation()
.
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 activationcache -- a dictionary containing "Z1", "A1", "Z2" and "A2""""
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 your 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 your parameters
grads -- python dictionary containing your gradients
Returns: parameters -- python dictionary containing your updated parameters
"""
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.
"""
def predict(parameters, X): """ Using the learned parameters, predicts a class for each example in X Arguments: parameters -- python dictionary containing your parameters X -- input data of size (n_x, m) Returns predictions -- vector of predictions of our model (red: 0 / blue: 1) """
It is time to run the model and see how it performs on a planar dataset. Run the following code to test your 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.
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:
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 your 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
_________________________________________________________________
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: 0.9758Epoch 2/13 50000/50000 [==============================] - 12s - loss: 0.1543 - acc: 0.9566 - val_loss: 0.0957 - val_acc: 0.9735Epoch 3/13 50000/50000 [==============================] - 11s - loss: 0.1509 - acc: 0.9586 - val_loss: 0.0985 - val_acc: 0.9752Epoch 4/13 50000/50000 [==============================] - 11s - loss: 0.1515 - acc: 0.9577 - val_loss: 0.0967 - val_acc: 0.9752Epoch 5/13 50000/50000 [==============================] - 11s - loss: 0.1471 - acc: 0.9596 - val_loss: 0.1008 - val_acc: 0.9737Epoch 6/13 50000/50000 [==============================] - 11s - loss: 0.1488 - acc: 0.9598 - val_loss: 0.0989 - val_acc: 0.9749Epoch 7/13 50000/50000 [==============================] - 11s - loss: 0.1495 - acc: 0.9592 - val_loss: 0.1011 - val_acc: 0.9748Epoch 8/13 50000/50000 [==============================] - 11s - loss: 0.1434 - acc: 0.9604 - val_loss: 0.1005 - val_acc: 0.9761Epoch 9/13 50000/50000 [==============================] - 11s - loss: 0.1514 - acc: 0.9590 - val_loss: 0.0951 - val_acc: 0.9759Epoch 10/13 50000/50000 [==============================] - 11s - loss: 0.1424 - acc: 0.9613 - val_loss: 0.0995 - val_acc: 0.9739Epoch 11/13 50000/50000 [==============================] - 11s - loss: 0.1408 - acc: 0.9625 - val_loss: 0.0977 - val_acc: 0.9751Epoch 12/13 50000/50000 [==============================] - 11s - loss: 0.1413 - acc: 0.9601 - val_loss: 0.0938 - val_acc: 0.9753Epoch 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.
Here are some tips on what we can do to improve accuracy:
from scipy.misc import imrotate,imresize
Comment
@John Smethurst Yes SGD is online approach, we need not wait for the entire data as we need in case of full Batch GD. Although in SGD the cost function does not decrease continuously at each step (using momentum we can make the convergence much sharper). Gradient Descent in general can converge at local minimum if the cost function is not convex, AI / simulated annealing is a hill-climbing type approach that applies some heuristic to get rid of the local minimum.
Just looked up stochastic gradient descent, never seen it before. Did this give a large efficiency saving over doing just standard gradient descent?
I remember coding a simulated annealing algorithm years ago:
https://en.wikipedia.org/wiki/Simulated_annealing
I never really bench marked it though.
I might do a bit of research later trying investigating choice of loss function/optimisation method etc. Perhaps you know of something already?
Regards
John S
Actually I was planning to include some more problems, but did not get time. Thanks, will serve its purpose if it helps someone.
You have really done a massive amount in one article!!
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