- + -
-One can identify a set of key steps when using neural networks to solve supervised learning problems: - -
- + -
-Here we will be using the MNIST dataset, which is readily available through the scikit-learn -package. You may also find it for example here. -The MNIST (Modified National Institute of Standards and Technology) database is a large database -of handwritten digits that is commonly used for training various image processing systems. -The MNIST dataset consists of 70 000 images of size 28x28 pixels, each labeled from 0 to 9. -The scikit-learn dataset we will use consists of a selection of 1797 images of size \( 8\times 8 \) collected and processed from this database. +One can identify a set of key steps when using neural networks to solve supervised learning problems: -
-To feed data into a feed-forward neural network we need to represent -the inputs as a feature matrix \( X = (n_{inputs}, n_{features}) \). Each -row represents an input, in this case a handwritten digit, and -each column represents a feature, in this case a pixel. The -correct answers, also known as labels or targets are -represented as a 1D array of integers -\( Y = (n_{inputs}) = (5, 3, 1, 8,...) \). +
-As an example, say we want to build a neural network using supervised learning to predict Body-Mass Index (BMI) from -measurements of height (in m) -and weight (in kg). If we have measurements of 5 people the feature matrix could be for example: - -$$ X = \begin{bmatrix} -1.85 & 81\\ -1.71 & 65\\ -1.95 & 103\\ -1.55 & 42\\ -1.63 & 56 -\end{bmatrix} ,$$ - -
-and the targets would be: - -$$ Y = (23.7, 22.2, 27.1, 17.5, 21.1) $$ - -
-Since each input image is a 2D matrix, we need to flatten the image -(i.e. "unravel" the 2D matrix into a 1D array) to turn the data into a -feature matrix. This means we lose all spatial information in the -image, such as locality and translational invariance. More complicated -architectures such as Convolutional Neural Networks can take advantage -of such information, and are most commonly applied when analyzing -images. - -
- - -
# import necessary packages
-import numpy as np
-import matplotlib.pyplot as plt
-from sklearn import datasets
-
-
-# ensure the same random numbers appear every time
-np.random.seed(0)
-
-# display images in notebook
-%matplotlib inline
-plt.rcParams['figure.figsize'] = (12,12)
-
-
-# download MNIST dataset
-digits = datasets.load_digits()
-
-# define inputs and labels
-inputs = digits.images
-labels = digits.target
-
-print("inputs = (n_inputs, pixel_width, pixel_height) = " + str(inputs.shape))
-print("labels = (n_inputs) = " + str(labels.shape))
-
-
-# flatten the image
-# the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64
-n_inputs = len(inputs)
-inputs = inputs.reshape(n_inputs, -1)
-print("X = (n_inputs, n_features) = " + str(inputs.shape))
-
-
-# choose some random images to display
-indices = np.arange(n_inputs)
-random_indices = np.random.choice(indices, size=5)
-
-for i, image in enumerate(digits.images[random_indices]):
- plt.subplot(1, 5, i+1)
- plt.axis('off')
- plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
- plt.title("Label: %d" % digits.target[random_indices[i]])
-plt.show()
-
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs036.html b/doc/pub/NeuralNet/html/._NeuralNet-bs036.html index b82e20c62..f1c2fc395 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs036.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs036.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
-Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions. +Here we will be using the MNIST dataset, which is readily available through the scikit-learn +package. You may also find it for example here. +The MNIST (Modified National Institute of Standards and Technology) database is a large database +of handwritten digits that is commonly used for training various image processing systems. +The MNIST dataset consists of 70 000 images of size 28x28 pixels, each labeled from 0 to 9. +The scikit-learn dataset we will use consists of a selection of 1797 images of size \( 8\times 8 \) collected and processed from this database.
-We will reserve \( 80 \% \) of our dataset for training and \( 20 \% \) for testing. +To feed data into a feed-forward neural network we need to represent +the inputs as a feature matrix \( X = (n_{inputs}, n_{features}) \). Each +row represents an input, in this case a handwritten digit, and +each column represents a feature, in this case a pixel. The +correct answers, also known as labels or targets are +represented as a 1D array of integers +\( Y = (n_{inputs}) = (5, 3, 1, 8,...) \).
-It is important that the train and test datasets are drawn randomly from our dataset, to ensure -no bias in the sampling. -Say you are taking measurements of weather data to predict the weather in the coming 5 days. -You don't want to train your model on measurements taken from the hours 00.00 to 12.00, and then test it on data -collected from 12.00 to 24.00. +As an example, say we want to build a neural network using supervised learning to predict Body-Mass Index (BMI) from +measurements of height (in m) +and weight (in kg). If we have measurements of 5 people the feature matrix could be for example: + +$$ X = \begin{bmatrix} +1.85 & 81\\ +1.71 & 65\\ +1.95 & 103\\ +1.55 & 42\\ +1.63 & 56 +\end{bmatrix} ,$$ + +
+and the targets would be: + +$$ Y = (23.7, 22.2, 27.1, 17.5, 21.1) $$ + +
+Since each input image is a 2D matrix, we need to flatten the image +(i.e. "unravel" the 2D matrix into a 1D array) to turn the data into a +feature matrix. This means we lose all spatial information in the +image, such as locality and translational invariance. More complicated +architectures such as Convolutional Neural Networks can take advantage +of such information, and are most commonly applied when analyzing +images.
-
from sklearn.model_selection import train_test_split
+# import necessary packages
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn import datasets
-# one-liner from scikit-learn library
-train_size = 0.8
-test_size = 1 - train_size
-X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
- test_size=test_size)
-# equivalently in numpy
-def train_test_split_numpy(inputs, labels, train_size, test_size):
- n_inputs = len(inputs)
- inputs_shuffled = inputs.copy()
- labels_shuffled = labels.copy()
-
- np.random.shuffle(inputs_shuffled)
- np.random.shuffle(labels_shuffled)
-
- train_end = int(n_inputs*train_size)
- X_train, X_test = inputs_shuffled[:train_end], inputs_shuffled[train_end:]
- Y_train, Y_test = labels_shuffled[:train_end], labels_shuffled[train_end:]
-
- return X_train, X_test, Y_train, Y_test
+# ensure the same random numbers appear every time
+np.random.seed(0)
-#X_train, X_test, Y_train, Y_test = train_test_split_numpy(inputs, labels, train_size, test_size)
+# display images in notebook
+%matplotlib inline
+plt.rcParams['figure.figsize'] = (12,12)
-print("Number of training images: " + str(len(X_train)))
-print("Number of test images: " + str(len(X_test)))
+
+# download MNIST dataset
+digits = datasets.load_digits()
+
+# define inputs and labels
+inputs = digits.images
+labels = digits.target
+
+print("inputs = (n_inputs, pixel_width, pixel_height) = " + str(inputs.shape))
+print("labels = (n_inputs) = " + str(labels.shape))
+
+
+# flatten the image
+# the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64
+n_inputs = len(inputs)
+inputs = inputs.reshape(n_inputs, -1)
+print("X = (n_inputs, n_features) = " + str(inputs.shape))
+
+
+# choose some random images to display
+indices = np.arange(n_inputs)
+random_indices = np.random.choice(indices, size=5)
+
+for i, image in enumerate(digits.images[random_indices]):
+ plt.subplot(1, 5, i+1)
+ plt.axis('off')
+ plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
+ plt.title("Label: %d" % digits.target[random_indices[i]])
+plt.show()
@@ -329,7 +377,7 @@ X_train, X_test, Y_train, Y_test = train_tes
-Our simple feed-forward neural network will consist of an input layer, a single hidden layer and an output layer. The activation \( y \) of each neuron is a weighted sum of inputs, passed through an activation function. In case of the simple perceptron model we have - -$$ z = \sum_{i=1}^n w_i a_i ,$$ - -$$ y = f(z) ,$$ +Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions.
-where \( f \) is the activation function, \( a_i \) represents input from neuron \( i \) in the preceding layer -and \( w_i \) is the weight to input \( i \). -The activation of the neurons in the input layer is just the features (e.g. a pixel value). +We will reserve \( 80 \% \) of our dataset for training and \( 20 \% \) for testing.
-The simplest activation function for a neuron is the Heaviside function: - -$$ f(z) = -\begin{cases} -1, & z > 0\\ -0, & \text{otherwise} -\end{cases} -$$ +It is important that the train and test datasets are drawn randomly from our dataset, to ensure +no bias in the sampling. +Say you are taking measurements of weather data to predict the weather in the coming 5 days. +You don't want to train your model on measurements taken from the hours 00.00 to 12.00, and then test it on data +collected from 12.00 to 24.00.
-A feed-forward neural network with this activation is known as a perceptron. -For a binary classifier (i.e. two classes, 0 or 1, dog or not-dog) we can also use this in our output layer. -This activation can be generalized to \( k \) classes (using e.g. the one-against-all strategy), -and we call these architectures multiclass perceptrons. -
-However, it is now common to use the terms Single Layer Perceptron (SLP) (1 hidden layer) and -Multilayer Perceptron (MLP) (2 or more hidden layers) to refer to feed-forward neural networks with any activation function. + +
from sklearn.model_selection import train_test_split
-
-Typical choices for activation functions include the sigmoid function, hyperbolic tangent, and Rectified Linear Unit (ReLU).
-We will be using the sigmoid function \( \sigma(x) \):
+# one-liner from scikit-learn library
+train_size = 0.8
+test_size = 1 - train_size
+X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
+ test_size=test_size)
-$$ f(x) = \sigma(x) = \frac{1}{1 + e^{-x}} ,$$
+# equivalently in numpy
+def train_test_split_numpy(inputs, labels, train_size, test_size):
+ n_inputs = len(inputs)
+ inputs_shuffled = inputs.copy()
+ labels_shuffled = labels.copy()
+
+ np.random.shuffle(inputs_shuffled)
+ np.random.shuffle(labels_shuffled)
+
+ train_end = int(n_inputs*train_size)
+ X_train, X_test = inputs_shuffled[:train_end], inputs_shuffled[train_end:]
+ Y_train, Y_test = labels_shuffled[:train_end], labels_shuffled[train_end:]
+
+ return X_train, X_test, Y_train, Y_test
-
-which is inspired by probability theory (see logistic regression) and was most commonly used until about 2011. See the discussion below concerning other activation functions.
+#X_train, X_test, Y_train, Y_test = train_test_split_numpy(inputs, labels, train_size, test_size)
+print("Number of training images: " + str(len(X_train)))
+print("Number of test images: " + str(len(X_test)))
+
@@ -326,7 +331,7 @@ which is inspired by probability theory (see logistic regression) and was most c
- + -
-For a soft binary classifier, we could use a single neuron and interpret the output as either being the probability of being in class 0 or the probability of being in class 1. Alternatively we could use 2 neurons, and interpret each neuron as the probability of being in each class. +Our simple feed-forward neural network will consist of an input layer, a single hidden layer and an output layer. The activation \( y \) of each neuron is a weighted sum of inputs, passed through an activation function. In case of the simple perceptron model we have + +$$ z = \sum_{i=1}^n w_i a_i ,$$ + +$$ y = f(z) ,$$
-Since we are doing multiclass classification, with 10 categories, it is natural to use 10 neurons in the output layer. We number the neurons \( j = 0,1,...,9 \). The activation of each output neuron \( j \) will be according to the softmax function: - -$$ P(\text{class \( j \)} \mid \text{input \( \hat{a} \)}) = \frac{\exp{(\hat{a}^T \hat{w}_j)}} -{\sum_{c=0}^{9} \exp{(\hat{a}^T \hat{w}_c)}} ,$$ +where \( f \) is the activation function, \( a_i \) represents input from neuron \( i \) in the preceding layer +and \( w_i \) is the weight to input \( i \). +The activation of the neurons in the input layer is just the features (e.g. a pixel value).
-i.e. each neuron \( j \) outputs the probability of being in class \( j \) given an input from the hidden layer \( \hat{a} \), with \( \hat{w}_j \) the weights of neuron \( j \) to the inputs. -The denominator is a normalization factor to ensure the outputs (probabilities) sum up to 1. -The exponent is just the weighted sum of inputs as before: +The simplest activation function for a neuron is the Heaviside function: -$$ z_j = \sum_{i=1}^n w_ {ij} a_i+b_j.$$ +$$ f(z) = +\begin{cases} +1, & z > 0\\ +0, & \text{otherwise} +\end{cases} +$$
-Since each neuron in the output layer is connected to the 50 inputs from the hidden layer we have 50x10 = 500 -weights to the output layer. +A feed-forward neural network with this activation is known as a perceptron. +For a binary classifier (i.e. two classes, 0 or 1, dog or not-dog) we can also use this in our output layer. +This activation can be generalized to \( k \) classes (using e.g. the one-against-all strategy), +and we call these architectures multiclass perceptrons. + +
+However, it is now common to use the terms Single Layer Perceptron (SLP) (1 hidden layer) and +Multilayer Perceptron (MLP) (2 or more hidden layers) to refer to feed-forward neural networks with any activation function. + +
+Typical choices for activation functions include the sigmoid function, hyperbolic tangent, and Rectified Linear Unit (ReLU). +We will be using the sigmoid function \( \sigma(x) \): + +$$ f(x) = \sigma(x) = \frac{1}{1 + e^{-x}} ,$$ + +
+which is inspired by probability theory (see logistic regression) and was most commonly used until about 2011. See the discussion below concerning other activation functions.
@@ -325,7 +328,7 @@ weights to the output layer.
-Typically weights are initialized with small values distributed around zero, drawn from a uniform -or normal distribution. Setting all weights to zero means all neurons give the same output, making the network useless. +For a soft binary classifier, we could use a single neuron and interpret the output as either being the probability of being in class 0 or the probability of being in class 1. Alternatively we could use 2 neurons, and interpret each neuron as the probability of being in each class.
-Adding a bias value to the weighted sum of inputs allows the neural network to represent a greater range -of values. Without it, any input with the value 0 will be mapped to zero (before being passed through the activation). The bias unit has an output of 1, and a weight to each neuron \( j \), \( b_j \): +Since we are doing multiclass classification, with 10 categories, it is natural to use 10 neurons in the output layer. We number the neurons \( j = 0,1,...,9 \). The activation of each output neuron \( j \) will be according to the softmax function: -$$ z_j = \sum_{i=1}^n w_ {ij} a_i + b_j.$$ +$$ P(\text{class \( j \)} \mid \text{input \( \hat{a} \)}) = \frac{\exp{(\hat{a}^T \hat{w}_j)}} +{\sum_{c=0}^{9} \exp{(\hat{a}^T \hat{w}_c)}} ,$$
-The bias weights \( \hat{b} \) are often initialized to zero, but a small value like \( 0.01 \) ensures all neurons have some output which can be backpropagated in the first training cycle. +i.e. each neuron \( j \) outputs the probability of being in class \( j \) given an input from the hidden layer \( \hat{a} \), with \( \hat{w}_j \) the weights of neuron \( j \) to the inputs. +The denominator is a normalization factor to ensure the outputs (probabilities) sum up to 1. +The exponent is just the weighted sum of inputs as before: + +$$ z_j = \sum_{i=1}^n w_ {ij} a_i+b_j.$$ +
+Since each neuron in the output layer is connected to the 50 inputs from the hidden layer we have 50x10 = 500 +weights to the output layer. - -
# building our neural network
-
-n_inputs, n_features = X_train.shape
-n_hidden_neurons = 50
-n_categories = 10
-
-# we make the weights normally distributed using numpy.random.randn
-
-# weights and bias in the hidden layer
-hidden_weights = np.random.randn(n_features, n_hidden_neurons)
-hidden_bias = np.zeros(n_hidden_neurons) + 0.01
-
-# weights and bias in the output layer
-output_weights = np.random.randn(n_hidden_neurons, n_categories)
-output_bias = np.zeros(n_categories) + 0.01
-
@@ -316,7 +327,7 @@ output_bias = np48
-Denote \( F \) the number of features, \( H \) the number of hidden neurons and \( C \) the number of categories.
-For each input image we calculate a weighted sum of input features (pixel values) to each neuron \( j \) in the hidden layer \( l \):
-
-$$ z_{j}^{l} = \sum_{i=1}^{F} w_{ij}^{l} x_i + b_{j}^{l},$$
+Typically weights are initialized with small values distributed around zero, drawn from a uniform
+or normal distribution. Setting all weights to zero means all neurons give the same output, making the network useless.
-this is then passed through our activation function
+Adding a bias value to the weighted sum of inputs allows the neural network to represent a greater range
+of values. Without it, any input with the value 0 will be mapped to zero (before being passed through the activation). The bias unit has an output of 1, and a weight to each neuron \( j \), \( b_j \):
-$$ a_{j}^{l} = f(z_{j}^{l}) .$$
+$$ z_j = \sum_{i=1}^n w_ {ij} a_i + b_j.$$
-We calculate a weighted sum of inputs (activations in the hidden layer) to each neuron \( j \) in the output layer:
-
-$$ z_{j}^{L} = \sum_{i=1}^{H} w_{ij}^{L} a_{i}^{l} + b_{j}^{L}.$$
-
+The bias weights \( \hat{b} \) are often initialized to zero, but a small value like \( 0.01 \) ensures all neurons have some output which can be backpropagated in the first training cycle.
-Finally we calculate the output of neuron \( j \) in the output layer using the softmax function:
-$$ a_{j}^{L} = \frac{\exp{(z_j^{L})}}
-{\sum_{c=0}^{C-1} \exp{(z_c^{L})}} .$$
+
+
@@ -307,7 +318,7 @@ $$ a_{j}^{L} = \frac{\exp{(z_j^{L})}}
-Since our data has the dimensions \( X = (n_{inputs}, n_{features}) \) and our weights to the hidden
-layer have the dimensions
-\( W_{hidden} = (n_{features}, n_{hidden}) \),
-we can easily feed the network all our training data in one go by taking the matrix product
+Denote \( F \) the number of features, \( H \) the number of hidden neurons and \( C \) the number of categories.
+For each input image we calculate a weighted sum of input features (pixel values) to each neuron \( j \) in the hidden layer \( l \):
-$$ X W^{h} = (n_{inputs}, n_{hidden}),$$
+$$ z_{j}^{l} = \sum_{i=1}^{F} w_{ij}^{l} x_i + b_{j}^{l},$$
-and obtain a matrix that holds the weighted sum of inputs to the hidden layer
-for each input image and each hidden neuron.
-We also add the bias to obtain a matrix of weighted sums to the hidden layer \( Z^{h} \):
+this is then passed through our activation function
-$$ \hat{z}^{l} = \hat{X} \hat{W}^{l} + \hat{b}^{l} ,$$
+$$ a_{j}^{l} = f(z_{j}^{l}) .$$
-meaning the same bias (1D array with size equal number of hidden neurons) is added to each input image.
-This is then passed through the activation:
+We calculate a weighted sum of inputs (activations in the hidden layer) to each neuron \( j \) in the output layer:
-$$ \hat{a}^{l} = f(\hat{z}^l) .$$
+$$ z_{j}^{L} = \sum_{i=1}^{H} w_{ij}^{L} a_{i}^{l} + b_{j}^{L}.$$
-This is fed to the output layer:
+Finally we calculate the output of neuron \( j \) in the output layer using the softmax function:
-$$ \hat{z}^{L} = \hat{a}^{L} \hat{W}^{L} + \hat{b}^{L} .$$
+$$ a_{j}^{L} = \frac{\exp{(z_j^{L})}}
+{\sum_{c=0}^{C-1} \exp{(z_c^{L})}} .$$
-
-Finally we receive our output values for each image and each category by passing it through the softmax function:
-
-$$ output = softmax (\hat{z}^{L}) = (n_{inputs}, n_{categories}) .$$
-
-
-
-
-
@@ -355,7 +309,7 @@ predictions = predict(X_train)
-To measure how well our neural network is doing we need to introduce a cost function.
-We will call the function that gives the error of a single sample output the loss function, and the function
-that gives the total error of our network across all samples the cost function.
-A typical choice for multiclass classification is the cross-entropy loss, also known as the negative log likelihood.
+Since our data has the dimensions \( X = (n_{inputs}, n_{features}) \) and our weights to the hidden
+layer have the dimensions
+\( W_{hidden} = (n_{features}, n_{hidden}) \),
+we can easily feed the network all our training data in one go by taking the matrix product
+
+$$ X W^{h} = (n_{inputs}, n_{hidden}),$$
-In multiclass classification it is common to treat each integer label as a so called one-hot vector:
+and obtain a matrix that holds the weighted sum of inputs to the hidden layer
+for each input image and each hidden neuron.
+We also add the bias to obtain a matrix of weighted sums to the hidden layer \( Z^{h} \):
-$$ y = 5 \quad \rightarrow \quad \hat{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$$
-
-$$ y = 1 \quad \rightarrow \quad \hat{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,$$
+$$ \hat{z}^{l} = \hat{X} \hat{W}^{l} + \hat{b}^{l} ,$$
-i.e. a binary bit string of length \( C \), where \( C = 10 \) is the number of classes in the MNIST dataset.
+meaning the same bias (1D array with size equal number of hidden neurons) is added to each input image.
+This is then passed through the activation:
+
+$$ \hat{a}^{l} = f(\hat{z}^l) .$$
-Let \( y_{ic} \) denote the \( c \)-th component of the \( i \)-th one-hot vector.
-We define the cost function \( \mathcal{C} \) as a sum over the cross-entropy loss for each point \( \hat{x}_i \) in the dataset.
+This is fed to the output layer:
+
+$$ \hat{z}^{L} = \hat{a}^{L} \hat{W}^{L} + \hat{b}^{L} .$$
-In the one-hot representation only one of the terms in the loss function is non-zero, namely the
-probability of the correct category \( c' \)
-(i.e. the category \( c' \) such that \( y_{ic'} = 1 \)). This means that the cross entropy loss only punishes you for how wrong
-you got the correct label. The probability of category \( c \) is given by the softmax function. The vector \( \hat{\theta} \) represents the parameters of our network, i.e. all the weights and biases.
+Finally we receive our output values for each image and each category by passing it through the softmax function:
+$$ output = softmax (\hat{z}^{L}) = (n_{inputs}, n_{categories}) .$$
+
+
+
+
+
@@ -311,7 +357,7 @@ you got the correct label. The probability of category \( c \) is given by the s
-The network is trained by finding the weights and biases that minimize the cost function. One of the most widely used classes of methods is gradient descent and its generalizations. The idea behind gradient descent
-is simply to adjust the weights in the direction where the gradient of the cost function is large and negative. This ensures we flow toward a local minimum of the cost function.
-Each parameter \( \theta \) is iteratively adjusted according to the rule
-
-$$ \theta_{i+1} = \theta_i - \eta \nabla \mathcal{C}(\theta_i) ,$$
+To measure how well our neural network is doing we need to introduce a cost function.
+We will call the function that gives the error of a single sample output the loss function, and the function
+that gives the total error of our network across all samples the cost function.
+A typical choice for multiclass classification is the cross-entropy loss, also known as the negative log likelihood.
-where \( \eta \) is known as the learning rate, which controls how big a step we take towards the minimum.
-This update can be repeated for any number of iterations, or until we are satisfied with the result.
+In multiclass classification it is common to treat each integer label as a so called one-hot vector:
+
+$$ y = 5 \quad \rightarrow \quad \hat{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$$
+
+$$ y = 1 \quad \rightarrow \quad \hat{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,$$
-A simple and effective improvement is a variant called Batch Gradient Descent.
-Instead of calculating the gradient on the whole dataset, we calculate an approximation of the gradient
-on a subset of the data called a minibatch.
-If there are \( N \) data points and we have a minibatch size of \( M \), the total number of batches
-is \( N/M \).
-We denote each minibatch \( B_k \), with \( k = 1, 2,...,N/M \). The gradient then becomes:
-
-$$ \nabla \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) \quad \rightarrow \quad
-\frac{1}{M} \sum_{i \in B_k} \nabla \mathcal{L}_i(\theta) ,$$
+i.e. a binary bit string of length \( C \), where \( C = 10 \) is the number of classes in the MNIST dataset.
-i.e. instead of averaging the loss over the entire dataset, we average over a minibatch.
+Let \( y_{ic} \) denote the \( c \)-th component of the \( i \)-th one-hot vector.
+We define the cost function \( \mathcal{C} \) as a sum over the cross-entropy loss for each point \( \hat{x}_i \) in the dataset.
-This has two important benefits:
-
-
@@ -320,7 +313,7 @@ The various optmization methods, with codes and algorithms, are discussed in o
-It is common to add an extra term to the cost function, proportional
-to the size of the weights. This is equivalent to constraining the
-size of the weights, so that they do not grow out of control.
-Constraining the size of the weights means that the weights cannot
-grow arbitrarily large to fit the training data, and in this way
-reduces overfitting.
+The network is trained by finding the weights and biases that minimize the cost function. One of the most widely used classes of methods is gradient descent and its generalizations. The idea behind gradient descent
+is simply to adjust the weights in the direction where the gradient of the cost function is large and negative. This ensures we flow toward a local minimum of the cost function.
+Each parameter \( \theta \) is iteratively adjusted according to the rule
+
+$$ \theta_{i+1} = \theta_i - \eta \nabla \mathcal{C}(\theta_i) ,$$
-We will measure the size of the weights using the so called L2-norm, meaning our cost function becomes:
+where \( \eta \) is known as the learning rate, which controls how big a step we take towards the minimum.
+This update can be repeated for any number of iterations, or until we are satisfied with the result.
+
+
+A simple and effective improvement is a variant called Batch Gradient Descent.
+Instead of calculating the gradient on the whole dataset, we calculate an approximation of the gradient
+on a subset of the data called a minibatch.
+If there are \( N \) data points and we have a minibatch size of \( M \), the total number of batches
+is \( N/M \).
+We denote each minibatch \( B_k \), with \( k = 1, 2,...,N/M \). The gradient then becomes:
$$ \nabla \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) \quad \rightarrow \quad
-\frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) + \lambda \lvert \lvert \hat{w} \rvert \rvert_2^2
-= \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}(\theta) + \lambda \sum_{ij} w_{ij}^2,$$
+\frac{1}{M} \sum_{i \in B_k} \nabla \mathcal{L}_i(\theta) ,$$
-i.e. we sum up all the weights squared. The factor \( \lambda \) is known as a regularization parameter.
+i.e. instead of averaging the loss over the entire dataset, we average over a minibatch.
-In order to train the model, we need to calculate the derivative of
-the cost function with respect to every bias and weight in the
-network. In total our network has \( (64 + 1)\times 50=3250 \) weights in
-the hidden layer and \( (50 + 1)\times 10=510 \) weights to the output
-layer (\( +1 \) for the bias), and the gradient must be calculated for
-every parameter. We use the backpropagation algorithm discussed
-above. This is a clever use of the chain rule that allows us to
-calculate the gradient efficently.
+This has two important benefits:
+
+
@@ -313,7 +322,7 @@ calculate the gradient efficently.
-To more efficently train our network these equations are implemented using matrix operations.
-The error in the output layer is calculated simply as, with \( \hat{t} \) being our targets,
-
-$$ \delta_L = \hat{t} - \hat{y} = (n_{inputs}, n_{categories}) .$$
+It is common to add an extra term to the cost function, proportional
+to the size of the weights. This is equivalent to constraining the
+size of the weights, so that they do not grow out of control.
+Constraining the size of the weights means that the weights cannot
+grow arbitrarily large to fit the training data, and in this way
+reduces overfitting.
-The gradient for the output weights is calculated as
+We will measure the size of the weights using the so called L2-norm, meaning our cost function becomes:
-$$ \nabla W_{L} = \hat{a}^T \delta_L = (n_{hidden}, n_{categories}) ,$$
+$$ \nabla \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) \quad \rightarrow \quad
+\frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) + \lambda \lvert \lvert \hat{w} \rvert \rvert_2^2
+= \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}(\theta) + \lambda \sum_{ij} w_{ij}^2,$$
-where \( \hat{a} = (n_{inputs}, n_{hidden}) \). This simply means that we are summing up the gradients for each input.
-Since we are going backwards we have to transpose the activation matrix.
+i.e. we sum up all the weights squared. The factor \( \lambda \) is known as a regularization parameter.
-The gradient with respect to the output bias is then
+In order to train the model, we need to calculate the derivative of
+the cost function with respect to every bias and weight in the
+network. In total our network has \( (64 + 1)\times 50=3250 \) weights in
+the hidden layer and \( (50 + 1)\times 10=510 \) weights to the output
+layer (\( +1 \) for the bias), and the gradient must be calculated for
+every parameter. We use the backpropagation algorithm discussed
+above. This is a clever use of the chain rule that allows us to
+calculate the gradient efficently.
-$$ \nabla \hat{b}_{L} = \sum_{i=1}^{n_{inputs}} \delta_L = (n_{categories}) .$$
-
-
-The error in the hidden layer is
-
-$$ \Delta_h = \delta_L W_{L}^T \circ f'(z_{h}) = \delta_L W_{L}^T \circ a_{h} \circ (1 - a_{h}) = (n_{inputs}, n_{hidden}) ,$$
-
-
-where \( f'(a_{h}) \) is the derivative of the activation in the hidden layer. The matrix products mean
-that we are summing up the products for each neuron in the output layer. The symbol \( \circ \) denotes
-the Hadamard product, meaning element-wise multiplication.
-
-
-This again gives us the gradients in the hidden layer:
-
-$$ \nabla W_{h} = X^T \delta_h = (n_{features}, n_{hidden}) ,$$
-
-$$ \nabla b_{h} = \sum_{i=1}^{n_{inputs}} \delta_h = (n_{hidden}) .$$
-
-
-
-
-
@@ -394,7 +315,7 @@ lmbd = 0.0154
-As we can see the network does not seem to be learning at all. It seems to be just guessing the label for each image.
-In order to obtain a network that does something useful, we will have to do a bit more work.
+To more efficently train our network these equations are implemented using matrix operations.
+The error in the output layer is calculated simply as, with \( \hat{t} \) being our targets,
+
+$$ \delta_L = \hat{t} - \hat{y} = (n_{inputs}, n_{categories}) .$$
-The choice of hyperparameters such as learning rate and regularization parameter is hugely influential for the performance of the network. Typically a grid-search is performed, wherein we test different hyperparameters separated by orders of magnitude. For example we could test the learning rates \( \eta = 10^{-6}, 10^{-5},...,10^{-1} \) with different regularization parameters \( \lambda = 10^{-6},...,10^{-0} \).
+The gradient for the output weights is calculated as
+
+$$ \nabla W_{L} = \hat{a}^T \delta_L = (n_{hidden}, n_{categories}) ,$$
-Next, we haven't implemented minibatching yet, which introduces stochasticity and is though to act as an important regularizer on the weights. We call a feed-forward + backward pass with a minibatch an iteration, and a full training period
-going through the entire dataset (\( n/M \) batches) an epoch.
+where \( \hat{a} = (n_{inputs}, n_{hidden}) \). This simply means that we are summing up the gradients for each input.
+Since we are going backwards we have to transpose the activation matrix.
-If this does not improve network performance, you may want to consider altering the network architecture, adding more neurons or hidden layers.
-Andrew Ng goes through some of these considerations in this video. You can find a summary of the video here.
+The gradient with respect to the output bias is then
+$$ \nabla \hat{b}_{L} = \sum_{i=1}^{n_{inputs}} \delta_L = (n_{categories}) .$$
+
+
+The error in the hidden layer is
+
+$$ \Delta_h = \delta_L W_{L}^T \circ f'(z_{h}) = \delta_L W_{L}^T \circ a_{h} \circ (1 - a_{h}) = (n_{inputs}, n_{hidden}) ,$$
+
+
+where \( f'(a_{h}) \) is the derivative of the activation in the hidden layer. The matrix products mean
+that we are summing up the products for each neuron in the output layer. The symbol \( \circ \) denotes
+the Hadamard product, meaning element-wise multiplication.
+
+
+This again gives us the gradients in the hidden layer:
+
+$$ \nabla W_{h} = X^T \delta_h = (n_{features}, n_{hidden}) ,$$
+
+$$ \nabla b_{h} = \sum_{i=1}^{n_{inputs}} \delta_h = (n_{hidden}) .$$
+
+
+
+
+
@@ -300,7 +396,7 @@ Andrew Ng goes through some of these considerations in this 55
-It is very natural to think of the network as an object, with specific instances of the network
-being realizations of this object with different hyperparameters. An implementation using Python classes provides a clean structure and interface, and the full implementation of our neural network is given below.
+As we can see the network does not seem to be learning at all. It seems to be just guessing the label for each image.
+In order to obtain a network that does something useful, we will have to do a bit more work.
+The choice of hyperparameters such as learning rate and regularization parameter is hugely influential for the performance of the network. Typically a grid-search is performed, wherein we test different hyperparameters separated by orders of magnitude. For example we could test the learning rates \( \eta = 10^{-6}, 10^{-5},...,10^{-1} \) with different regularization parameters \( \lambda = 10^{-6},...,10^{-0} \).
-
-
+Next, we haven't implemented minibatching yet, which introduces stochasticity and is though to act as an important regularizer on the weights. We call a feed-forward + backward pass with a minibatch an iteration, and a full training period
+going through the entire dataset (\( n/M \) batches) an epoch.
- ):
- self.X_data_full = X_data
- self.Y_data_full = Y_data
+
+If this does not improve network performance, you may want to consider altering the network architecture, adding more neurons or hidden layers.
+Andrew Ng goes through some of these considerations in this video. You can find a summary of the video here.
- self.n_inputs = X_data.shape[0]
- self.n_features = X_data.shape[1]
- self.n_hidden_neurons = n_hidden_neurons
- self.n_categories = n_categories
-
- self.epochs = epochs
- self.batch_size = batch_size
- self.iterations = self.n_inputs // self.batch_size
- self.eta = eta
- self.lmbd = lmbd
-
- self.create_biases_and_weights()
-
- def create_biases_and_weights(self):
- self.hidden_weights = np.random.randn(self.n_features, self.n_hidden_neurons)
- self.hidden_bias = np.zeros(self.n_hidden_neurons) + 0.01
-
- self.output_weights = np.random.randn(self.n_hidden_neurons, self.n_categories)
- self.output_bias = np.zeros(self.n_categories) + 0.01
-
- def feed_forward(self):
- # feed-forward for training
- self.z_h = np.matmul(self.X_data, self.hidden_weights) + self.hidden_bias
- self.a_h = sigmoid(self.z_h)
-
- self.z_o = np.matmul(self.a_h, self.output_weights) + self.output_bias
-
- exp_term = np.exp(self.z_o)
- self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
-
- def feed_forward_out(self, X):
- # feed-forward for output
- z_h = np.matmul(X, self.hidden_weights) + self.hidden_bias
- a_h = sigmoid(z_h)
-
- z_o = np.matmul(a_h, self.output_weights) + self.output_bias
-
- exp_term = np.exp(z_o)
- probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
- return probabilities
-
- def backpropagation(self):
- error_output = self.probabilities - self.Y_data
- error_hidden = np.matmul(error_output, self.output_weights.T) * self.a_h * (1 - self.a_h)
-
- self.output_weights_gradient = np.matmul(self.a_h.T, error_output)
- self.output_bias_gradient = np.sum(error_output, axis=0)
-
- self.hidden_weights_gradient = np.matmul(self.X_data.T, error_hidden)
- self.hidden_bias_gradient = np.sum(error_hidden, axis=0)
-
- if self.lmbd > 0.0:
- self.output_weights_gradient += self.lmbd * self.output_weights
- self.hidden_weights_gradient += self.lmbd * self.hidden_weights
-
- self.output_weights -= self.eta * self.output_weights_gradient
- self.output_bias -= self.eta * self.output_bias_gradient
- self.hidden_weights -= self.eta * self.hidden_weights_gradient
- self.hidden_bias -= self.eta * self.hidden_bias_gradient
-
- def predict(self, X):
- probabilities = self.feed_forward_out(X)
- return np.argmax(probabilities, axis=1)
-
- def predict_probabilities(self, X):
- probabilities = self.feed_forward_out(X)
- return probabilities
-
- def train(self):
- data_indices = np.arange(self.n_inputs)
-
- for i in range(self.epochs):
- for j in range(self.iterations):
- # pick datapoints with replacement
- chosen_datapoints = np.random.choice(
- data_indices, size=self.batch_size, replace=False
- )
-
- # minibatch training data
- self.X_data = self.X_data_full[chosen_datapoints]
- self.Y_data = self.Y_data_full[chosen_datapoints]
-
- self.feed_forward()
- self.backpropagation()
-
@@ -393,7 +302,7 @@ being realizations of this object with different hyperparameters. An implementat
-To measure the performance of our network we evaluate how well it does it data it has never seen before, i.e. the test data.
-We measure the performance of the network using the accuracy score.
-The accuracy is as you would expect just the number of images correctly labeled divided by the total number of images. A perfect classifier will have an accuracy score of \( 1 \).
-
-$$ \text{Accuracy} = \frac{\sum_{i=1}^n I(\hat{y}_i = y_i)}{n} ,$$
-
-
-where \( I \) is the indicator function, \( 1 \) if \( \hat{y}_i = y_i \) and \( 0 \) otherwise.
+It is very natural to think of the network as an object, with specific instances of the network
+being realizations of this object with different hyperparameters. An implementation using Python classes provides a clean structure and interface, and the full implementation of our neural network is given below.
-
@@ -315,7 +395,7 @@ test_predict = dnn57
-We now perform a grid search to find the optimal hyperparameters for the network.
-Note that we are only using 1 layer with 50 neurons, and human performance is estimated to be around \( 98\% \) (\( 2\% \) error rate).
+To measure the performance of our network we evaluate how well it does it data it has never seen before, i.e. the test data.
+We measure the performance of the network using the accuracy score.
+The accuracy is as you would expect just the number of images correctly labeled divided by the total number of images. A perfect classifier will have an accuracy score of \( 1 \).
+
+$$ \text{Accuracy} = \frac{\sum_{i=1}^n I(\hat{y}_i = y_i)}{n} ,$$
+
+
+where \( I \) is the indicator function, \( 1 \) if \( \hat{y}_i = y_i \) and \( 0 \) otherwise.
-
@@ -313,7 +317,7 @@ DNN_numpy = np.
+We now perform a grid search to find the optimal hyperparameters for the network.
+Note that we are only using 1 layer with 50 neurons, and human performance is estimated to be around \( 98\% \) (\( 2\% \) error rate).
-
@@ -322,7 +315,7 @@ plt.show()
-scikit-learn focuses more
-on traditional machine learning methods, such as regression,
-clustering, decision trees, etc. As such, it has only two types of
-neural networks: Multi Layer Perceptron outputting continuous values,
-MPLRegressor, and Multi Layer Perceptron outputting labels,
-MLPClassifier. We will see how simple it is to use these classes.
-
-
-scikit-learn implements a few improvements from our neural network,
-such as early stopping, a varying learning rate, different
-optimization methods, etc. We would therefore expect a better
-performance overall.
+
-
@@ -319,7 +324,7 @@ DNN_scikit = np
+scikit-learn focuses more
+on traditional machine learning methods, such as regression,
+clustering, decision trees, etc. As such, it has only two types of
+neural networks: Multi Layer Perceptron outputting continuous values,
+MPLRegressor, and Multi Layer Perceptron outputting labels,
+MLPClassifier. We will see how simple it is to use these classes.
+
+
+scikit-learn implements a few improvements from our neural network,
+such as early stopping, a varying learning rate, different
+optimization methods, etc. We would therefore expect a better
+performance overall.
+
-
@@ -322,7 +321,7 @@ plt.show()
-Now we want to build on the experience gained from our neural network implementation in NumPy and scikit-learn
-and use it to construct a neural network in Tensorflow. Once we have constructed a neural network in NumPy
-and Tensorflow, building one in Keras is really quite trivial, though the performance may suffer.
-
-In our previous example we used only one hidden layer, and in this we will use two. From this it should be quite
-clear how to build one using an arbitrary number of hidden layers, using data structures such as Python lists or
-NumPy arrays.
+
+
@@ -295,7 +324,7 @@ NumPy arrays.
-Tensorflow is an open source library machine learning library
-developed by the Google Brain team for internal use. It was released
-under the Apache 2.0 open source license in November 9, 2015.
+Now we want to build on the experience gained from our neural network implementation in NumPy and scikit-learn
+and use it to construct a neural network in Tensorflow. Once we have constructed a neural network in NumPy
+and Tensorflow, building one in Keras is really quite trivial, though the performance may suffer.
-Tensorflow is a computational framework that allows you to construct
-machine learning models at different levels of abstraction, from
-high-level, object-oriented APIs like Keras, down to the C++ kernels
-that Tensorflow is built upon. The higher levels of abstraction are
-simpler to use, but less flexible, and our choice of implementation
-should reflect the problems we are trying to solve.
+In our previous example we used only one hidden layer, and in this we will use two. From this it should be quite
+clear how to build one using an arbitrary number of hidden layers, using data structures such as Python lists or
+NumPy arrays.
-
-Tensorflow uses so-called graphs to represent your computation
-in terms of the dependencies between individual operations, such that you first build a Tensorflow graph
-to represent your model, and then create a Tensorflow session to run the graph.
-
-
-In this guide we will analyze the same data as we did in our NumPy and
-scikit-learn tutorial, gathered from the MNIST database of images. We
-will give an introduction to the lower level Python Application
-Program Interfaces (APIs), and see how we use them to build our graph.
-Then we will build (effectively) the same graph in Keras, to see just
-how simple solving a machine learning problem can be.
-
-
-To install tensorflow on Unix/Linux systems, use pip as
-
-
-
-
-and/or if you use anaconda, just write (or install from the graphical user interface)
-
-
-
-
@@ -325,7 +297,7 @@ and/or if you use anaconda, just write (or install from the graphical use
+Tensorflow is an open source library machine learning library
+developed by the Google Brain team for internal use. It was released
+under the Apache 2.0 open source license in November 9, 2015.
+
+
+Tensorflow is a computational framework that allows you to construct
+machine learning models at different levels of abstraction, from
+high-level, object-oriented APIs like Keras, down to the C++ kernels
+that Tensorflow is built upon. The higher levels of abstraction are
+simpler to use, but less flexible, and our choice of implementation
+should reflect the problems we are trying to solve.
+
+
+Tensorflow uses so-called graphs to represent your computation
+in terms of the dependencies between individual operations, such that you first build a Tensorflow graph
+to represent your model, and then create a Tensorflow session to run the graph.
+
+
+In this guide we will analyze the same data as we did in our NumPy and
+scikit-learn tutorial, gathered from the MNIST database of images. We
+will give an introduction to the lower level Python Application
+Program Interfaces (APIs), and see how we use them to build our graph.
+Then we will build (effectively) the same graph in Keras, to see just
+how simple solving a machine learning problem can be.
+
+
+To install tensorflow on Unix/Linux systems, use pip as
-
+and/or if you use anaconda, just write (or install from the graphical user interface)
+
-
@@ -346,7 +327,7 @@ X_train, X_test, Y_train, Y_test = train_tes
-
+
+
+
@@ -424,6 +347,8 @@ MathJax.Hub.Config({
-
+
-
-
-
+ def create_accuracy(self):
+ with tf.name_scope('accuracy'):
+ probabilities = tf.nn.softmax(self.z_out)
+ predictions = tf.argmax(probabilities, axis=1)
+ labels = tf.argmax(self.Y, axis=1)
+
+ correct_predictions = tf.equal(predictions, labels)
+ correct_predictions = tf.cast(correct_predictions, tf.float32)
+ self.accuracy = tf.reduce_mean(correct_predictions)
+
+ def create_optimiser(self):
+ with tf.name_scope('optimizer'):
+ self.optimizer = tf.train.GradientDescentOptimizer(learning_rate=self.eta).minimize(self.loss, global_step=self.global_step)
+
+ def weight_variable(self, shape, name='', dtype=tf.float32):
+ initial = tf.truncated_normal(shape, stddev=0.1)
+ return tf.Variable(initial, name=name, dtype=dtype)
+
+ def bias_variable(self, shape, name='', dtype=tf.float32):
+ initial = tf.constant(0.1, shape=shape)
+ return tf.Variable(initial, name=name, dtype=dtype)
+
+ def fit(self):
+ data_indices = np.arange(self.n_inputs)
-
-
@@ -356,6 +425,7 @@ writer.add_graph(tf64
-Keras is a high level neural network
-that supports Tensorflow, CTNK and Theano as backends.
-If you have Tensorflow installed Keras is available through the tf.keras module.
-If you have Anaconda installed you may run the following command
-
-
-
-
-Alternatively, if you have Tensorflow or one of the other supported backends install you may use the pip package manager:
+
-
-or look up the instructions here.
-
-
-
-
-
-
@@ -333,10 +306,10 @@ test_accuracy = npfor i in range(len(eta_vals)):
for j in range(len(lmbd_vals)):
- DNN = DNN_keras[i][j]
+ DNN = DNN_tf[i][j]
- train_accuracy[i][j] = DNN.evaluate(X_train, Y_train)[1]
- test_accuracy[i][j] = DNN.evaluate(X_test, Y_test)[1]
+ train_accuracy[i][j] = DNN.train_accuracy
+ test_accuracy[i][j] = DNN.test_accuracy
fig, ax = plt.subplots(figsize = (10, 10))
@@ -354,6 +327,14 @@ ax.set_xlabel(&
plt.show()
+
+
+
-The Back propagation algorithm we derived above works by going from
-the output layer to the input layer, propagating the error gradient on
-the way. Once the algorithm has computed the gradient of the cost
-function with regards to each parameter in the network, it uses these
-gradients to update each parameter with a Gradient Descent (GD) step.
+Keras is a high level neural network
+that supports Tensorflow, CTNK and Theano as backends.
+If you have Tensorflow installed Keras is available through the tf.keras module.
+If you have Anaconda installed you may run the following command
+
+
+
+
+Alternatively, if you have Tensorflow or one of the other supported backends install you may use the pip package manager:
-Unfortunately for us, the gradients often get smaller and smaller as the
-algorithm progresses down to the first hidden layers. As a result, the
-GD update leaves the lower layer connection weights
-virtually unchanged, and training never converges to a good
-solution. This is known in the literature as
-the vanishing gradients problem.
+
+
+
+or look up the instructions here.
-In other cases, the opposite can happen, namely the the gradients can grow bigger and
-bigger. The result is that many of the layers get large updates of the
-weights the
-algorithm diverges. This is the exploding gradients problem, which is
-mostly encountered in recurrent neural networks. More generally, deep
-neural networks suffer from unstable gradients, different layers may
-learn at widely different speeds
+
+
+
+
+
+
+
+
@@ -305,6 +377,7 @@ learn at widely different speeds
-Although this unfortunate behavior has been empirically observed for
-quite a while (it was one of the reasons why deep neural networks were
-mostly abandoned for a long time), it is only around 2010 that
-significant progress was made in understanding it.
+The Back propagation algorithm we derived above works by going from
+the output layer to the input layer, propagating the error gradient on
+the way. Once the algorithm has computed the gradient of the cost
+function with regards to each parameter in the network, it uses these
+gradients to update each parameter with a Gradient Descent (GD) step.
-A paper titled Understanding the Difficulty of Training Deep
-Feedforward Neural Networks by Xavier Glorot and Yoshua Bengio found that
-the problems with the popular logistic
-sigmoid activation function and the weight initialization technique
-that was most popular at the time, namely random initialization using
-a normal distribution with a mean of 0 and a standard deviation of
-1.
+Unfortunately for us, the gradients often get smaller and smaller as the
+algorithm progresses down to the first hidden layers. As a result, the
+GD update leaves the lower layer connection weights
+virtually unchanged, and training never converges to a good
+solution. This is known in the literature as
+the vanishing gradients problem.
-They showed that with this activation function and this
-initialization scheme, the variance of the outputs of each layer is
-much greater than the variance of its inputs. Going forward in the
-network, the variance keeps increasing after each layer until the
-activation function saturates at the top layers. This is actually made
-worse by the fact that the logistic function has a mean of 0.5, not 0
-(the hyperbolic tangent function has a mean of 0 and behaves slightly
-better than the logistic function in deep networks).
+In other cases, the opposite can happen, namely the the gradients can grow bigger and
+bigger. The result is that many of the layers get large updates of the
+weights the
+algorithm diverges. This is the exploding gradients problem, which is
+mostly encountered in recurrent neural networks. More generally, deep
+neural networks suffer from unstable gradients, different layers may
+learn at widely different speeds
@@ -305,6 +306,7 @@ better than the logistic function in deep networks).
-Looking at the logistic activation function, when inputs become large
-(negative or positive), the function saturates at 0 or 1, with a
-derivative extremely close to 0. Thus when backpropagation kicks in,
-it has virtually no gradient to propagate back through the network,
-and what little gradient exists keeps getting diluted as
-backpropagation progresses down through the top layers, so there is
-really nothing left for the lower layers.
+Although this unfortunate behavior has been empirically observed for
+quite a while (it was one of the reasons why deep neural networks were
+mostly abandoned for a long time), it is only around 2010 that
+significant progress was made in understanding it.
-In their paper, Glorot and Bengio propose a way to significantly
-alleviate this problem. We need the signal to flow properly in both
-directions: in the forward direction when making predictions, and in
-the reverse direction when backpropagating gradients. We don’t want
-the signal to die out, nor do we want it to explode and saturate. For
-the signal to flow properly, the authors argue that we need the
-variance of the outputs of each layer to be equal to the variance of
-its inputs, and we also need the gradients to have equal variance
-before and after flowing through a layer in the reverse direction.
+A paper titled Understanding the Difficulty of Training Deep
+Feedforward Neural Networks by Xavier Glorot and Yoshua Bengio found that
+the problems with the popular logistic
+sigmoid activation function and the weight initialization technique
+that was most popular at the time, namely random initialization using
+a normal distribution with a mean of 0 and a standard deviation of
+1.
-One of the insights in the 2010 paper by Glorot and Bengio was that
-the vanishing/exploding gradients problems were in part due to a poor
-choice of activation function. Until then most people had assumed that
-if Nature had chosen to use roughly sigmoid activation functions in
-biological neurons, they must be an excellent choice. But it turns out
-that other activation functions behave much better in deep neural
-networks, in particular the ReLU activation function, mostly because
-it does not saturate for positive values (and also because it is quite
-fast to compute).
+They showed that with this activation function and this
+initialization scheme, the variance of the outputs of each layer is
+much greater than the variance of its inputs. Going forward in the
+network, the variance keeps increasing after each layer until the
+activation function saturates at the top layers. This is actually made
+worse by the fact that the logistic function has a mean of 0.5, not 0
+(the hyperbolic tangent function has a mean of 0 and behaves slightly
+better than the logistic function in deep networks).
@@ -310,6 +306,7 @@ fast to compute).
-The ReLU activation function suffers from a problem known as the dying
-ReLUs: during training, some neurons effectively die, meaning they
-stop outputting anything other than 0.
+Looking at the logistic activation function, when inputs become large
+(negative or positive), the function saturates at 0 or 1, with a
+derivative extremely close to 0. Thus when backpropagation kicks in,
+it has virtually no gradient to propagate back through the network,
+and what little gradient exists keeps getting diluted as
+backpropagation progresses down through the top layers, so there is
+really nothing left for the lower layers.
-In some cases, you may find that half of your network’s neurons are
-dead, especially if you used a large learning rate. During training,
-if a neuron’s weights get updated such that the weighted sum of the
-neuron’s inputs is negative, it will start outputting 0. When this
-happen, the neuron is unlikely to come back to life since the gradient
-of the ReLU function is 0 when its input is negative.
+In their paper, Glorot and Bengio propose a way to significantly
+alleviate this problem. We need the signal to flow properly in both
+directions: in the forward direction when making predictions, and in
+the reverse direction when backpropagating gradients. We don’t want
+the signal to die out, nor do we want it to explode and saturate. For
+the signal to flow properly, the authors argue that we need the
+variance of the outputs of each layer to be equal to the variance of
+its inputs, and we also need the gradients to have equal variance
+before and after flowing through a layer in the reverse direction.
-To solve this problem, nowadays practitioners use a variant of the ReLU
-function, such as the leaky ReLU discussed above or the so-called
-exponential linear unit (ELU) function
-
-$$
-ELU(z) = \left\{\begin{array}{cc} \alpha\left( \exp{(z)}-1\right) & z < 0,\\ z & z \ge 0.\end{array}\right.
-$$
+One of the insights in the 2010 paper by Glorot and Bengio was that
+the vanishing/exploding gradients problems were in part due to a poor
+choice of activation function. Until then most people had assumed that
+if Nature had chosen to use roughly sigmoid activation functions in
+biological neurons, they must be an excellent choice. But it turns out
+that other activation functions behave much better in deep neural
+networks, in particular the ReLU activation function, mostly because
+it does not saturate for positive values (and also because it is quite
+fast to compute).
@@ -300,6 +311,7 @@ $$
-In general it seems that the ELU activation function is better than
-the leaky ReLU function (and its variants), which is better than
-ReLU. ReLU performs better than \( \tanh \) which in turn performs better
-than the logistic function.
+The ReLU activation function suffers from a problem known as the dying
+ReLUs: during training, some neurons effectively die, meaning they
+stop outputting anything other than 0.
-If runtime
-performance is an issue, then you may opt for the leaky ReLU function over the
-ELU function If you don’t
-want to tweak yet another hyperparameter, you may just use the default
-\( \alpha \) of \( 0.01 \) for the leaky ReLU, and \( 1 \) for ELU. If you have
-spare time and computing power, you can use cross-validation or
-bootstrap to evaluate other activation functions.
+In some cases, you may find that half of your network’s neurons are
+dead, especially if you used a large learning rate. During training,
+if a neuron’s weights get updated such that the weighted sum of the
+neuron’s inputs is negative, it will start outputting 0. When this
+happen, the neuron is unlikely to come back to life since the gradient
+of the ReLU function is 0 when its input is negative.
+
+
+To solve this problem, nowadays practitioners use a variant of the ReLU
+function, such as the leaky ReLU discussed above or the so-called
+exponential linear unit (ELU) function
+
+$$
+ELU(z) = \left\{\begin{array}{cc} \alpha\left( \exp{(z)}-1\right) & z < 0,\\ z & z \ge 0.\end{array}\right.
+$$
@@ -292,6 +301,7 @@ bootstrap to evaluate other activation functions.
-The first thing we would like to do is divide the data into two or three
-parts. A training set, a validation or dev (development) set, and a
-test set. The test set is the data on which we want to make
-predictions. The dev set is a subset of the training data we use to
-check how well we are doing out-of-sample, after training the model on
-the training dataset. We use the validation error as a proxy for the
-test error in order to make tweaks to our model. It is crucial that we
-do not use any of the test data to train the algorithm. This is a
-cardinal sin in ML. Then:
-
-
-However, sometimes
-the training data and test data differ in subtle ways because, for
-example, they are collected using slightly different methods, or
-because it is cheaper to collect data in one way versus another. In
-this case, there can be a mismatch between the training and test
-data. This can lead to the neural network overfitting these small
-differences between the test and training sets, and a poor performance
-on the test set despite having a good performance on the validation
-set. To rectify this, Andrew Ng suggests making two validation or dev
-sets, one constructed from the training data and one constructed from
-the test data. The difference between the performance of the algorithm
-on these two validation sets quantifies the train-test mismatch. This
-can serve as another important diagnostic when using DNNs for
-supervised learning.
+If runtime
+performance is an issue, then you may opt for the leaky ReLU function over the
+ELU function If you don’t
+want to tweak yet another hyperparameter, you may just use the default
+\( \alpha \) of \( 0.01 \) for the leaky ReLU, and \( 1 \) for ELU. If you have
+spare time and computing power, you can use cross-validation or
+bootstrap to evaluate other activation functions.
@@ -313,6 +293,7 @@ supervised learning.
-Like all statistical methods, supervised learning using neural
-networks has important limitations. This is especially important when
-one seeks to apply these methods, especially to physics problems. Like
-all tools, DNNs are not a universal solution. Often, the same or
-better performance on a task can be achieved by using a few
-hand-engineered features (or even a collection of random
-features).
-
-
-Here we list some of the important limitations of supervised neural network based models.
+The first thing we would like to do is divide the data into two or three
+parts. A training set, a validation or dev (development) set, and a
+test set. The test set is the data on which we want to make
+predictions. The dev set is a subset of the training data we use to
+check how well we are doing out-of-sample, after training the model on
+the training dataset. We use the validation error as a proxy for the
+test error in order to make tweaks to our model. It is crucial that we
+do not use any of the test data to train the algorithm. This is a
+cardinal sin in ML. Then:
+However, sometimes
+the training data and test data differ in subtle ways because, for
+example, they are collected using slightly different methods, or
+because it is cheaper to collect data in one way versus another. In
+this case, there can be a mismatch between the training and test
+data. This can lead to the neural network overfitting these small
+differences between the test and training sets, and a poor performance
+on the test set despite having a good performance on the validation
+set. To rectify this, Andrew Ng suggests making two validation or dev
+sets, one constructed from the training data and one constructed from
+the test data. The difference between the performance of the algorithm
+on these two validation sets quantifies the train-test mismatch. This
+can serve as another important diagnostic when using DNNs for
+supervised learning.
+
+
+Like all statistical methods, supervised learning using neural
+networks has important limitations. This is especially important when
+one seeks to apply these methods, especially to physics problems. Like
+all tools, DNNs are not a universal solution. Often, the same or
+better performance on a task can be achieved by using a few
+hand-engineered features (or even a collection of random
+features).
+
+
+Here we list some of the important limitations of supervised neural network based models.
+
+
+
+
One can identify a set of key steps when using neural networks to solve supervised learning problems:
@@ -1405,7 +1433,7 @@ One can identify a set of key steps when using neural networks to solve supervis
Here we will be using the MNIST dataset, which is readily available through the scikit-learn
@@ -1505,7 +1533,7 @@ plt.show()
Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions.
@@ -1555,7 +1583,7 @@ X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=t
Our simple feed-forward neural network will consist of an input layer, a single hidden layer and an output layer. The activation \( y \) of each neuron is a weighted sum of inputs, passed through an activation function. In case of the simple perceptron model we have
@@ -1609,7 +1637,7 @@ which is inspired by probability theory (see logistic regression) and was most c
Typically weights are initialized with small values distributed around zero, drawn from a uniform
@@ -1700,7 +1728,7 @@ output_bias = np.zeros(n_categories) + 0.01
Denote \( F \) the number of features, \( H \) the number of hidden neurons and \( C \) the number of categories.
@@ -1735,7 +1763,7 @@ $$ a_{j}^{L} = \frac{\exp{(z_j^{L})}}
Since our data has the dimensions \( X = (n_{inputs}, n_{features}) \) and our weights to the hidden
@@ -1821,7 +1849,7 @@ predictions = predict(X_train)
To measure how well our neural network is doing we need to introduce a cost function.
@@ -1856,7 +1884,7 @@ you got the correct label. The probability of category \( c \) is given by the s
The network is trained by finding the weights and biases that minimize the cost function. One of the most widely used classes of methods is gradient descent and its generalizations. The idea behind gradient descent
@@ -1902,7 +1930,7 @@ The various optmization methods, with codes and algorithms, are discussed in o
It is common to add an extra term to the cost function, proportional
@@ -1937,7 +1965,7 @@ calculate the gradient efficently.
To more efficently train our network these equations are implemented using matrix operations.
@@ -2064,7 +2092,7 @@ lmbd = 0.01
As we can see the network does not seem to be learning at all. It seems to be just guessing the label for each image.
@@ -2084,7 +2112,7 @@ Andrew Ng goes through some of these considerations in this Full object-oriented implementation
+
It is very natural to think of the network as an object, with specific instances of the network
@@ -2198,7 +2226,7 @@ being realizations of this object with different hyperparameters. An implementat
To measure the performance of our network we evaluate how well it does it data it has never seen before, i.e. the test data.
@@ -2236,7 +2264,7 @@ test_predict = dnn.predict(X_test)
We now perform a grid search to find the optimal hyperparameters for the network.
@@ -2270,7 +2298,7 @@ DNN_numpy = np.zeros((len(eta_vals),
-
@@ -2313,7 +2341,7 @@ plt.show()
scikit-learn focuses more
@@ -2353,7 +2381,7 @@ DNN_scikit = np.zeros((len(eta_vals),
-
@@ -2396,7 +2424,7 @@ plt.show()
Now we want to build on the experience gained from our neural network implementation in NumPy and scikit-learn
@@ -2411,7 +2439,7 @@ NumPy arrays.
Tensorflow is an open source library machine learning library
@@ -2457,7 +2485,7 @@ and/or if you use anaconda, just write (or install from the graphical use
@@ -2524,7 +2552,7 @@ X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=t
@@ -2748,7 +2776,7 @@ writer.add_graph(tf.get_default_graph())
Keras is a high level neural network
@@ -2848,7 +2876,7 @@ plt.show()
The Back propagation algorithm we derived above works by going from
@@ -2877,7 +2905,7 @@ learn at widely different speeds
Although this unfortunate behavior has been empirically observed for
@@ -2907,7 +2935,7 @@ better than the logistic function in deep networks).
Looking at the logistic activation function, when inputs become large
@@ -2943,7 +2971,7 @@ fast to compute).
The ReLU activation function suffers from a problem known as the dying
@@ -2972,7 +3000,7 @@ $$
In general it seems that the ELU activation function is better than
@@ -2992,7 +3020,7 @@ bootstrap to evaluate other activation functions.
The first thing we would like to do is divide the data into two or three
@@ -3035,7 +3063,7 @@ supervised learning.
Like all statistical methods, supervised learning using neural
diff --git a/doc/pub/NeuralNet/html/NeuralNet-solarized.html b/doc/pub/NeuralNet/html/NeuralNet-solarized.html
index fa2aa10d3..e4faff3f1 100644
--- a/doc/pub/NeuralNet/html/NeuralNet-solarized.html
+++ b/doc/pub/NeuralNet/html/NeuralNet-solarized.html
@@ -112,54 +112,55 @@ div { text-align: justify; text-justify: inter-word; }
'___sec30'),
('Defining the cost function', 2, None, '___sec31'),
('Example: binary classification problem', 2, None, '___sec32'),
+ ('The Softmax function', 2, None, '___sec33'),
('Developing a code for doing neural networks with back '
'propagation',
2,
None,
- '___sec33'),
- ('Collect and pre-process data', 2, None, '___sec34'),
- ('Train and test datasets', 2, None, '___sec35'),
- ('Define model and architecture', 2, None, '___sec36'),
- ('Layers', 2, None, '___sec37'),
- ('Weights and biases', 2, None, '___sec38'),
- ('Feed-forward pass', 2, None, '___sec39'),
- ('Matrix multiplications', 2, None, '___sec40'),
- ('Choose cost function and optimizer', 2, None, '___sec41'),
- ('Optimizing the cost function', 2, None, '___sec42'),
- ('Regularization', 2, None, '___sec43'),
- ('Matrix multiplication', 2, None, '___sec44'),
- ('Improving performance', 2, None, '___sec45'),
- ('Full object-oriented implementation', 2, None, '___sec46'),
- ('Evaluate model performance on test data', 2, None, '___sec47'),
- ('Adjust hyperparameters', 2, None, '___sec48'),
- ('Visualization', 2, None, '___sec49'),
- ('scikit-learn implementation', 2, None, '___sec50'),
- ('Visualization', 2, None, '___sec51'),
+ '___sec34'),
+ ('Collect and pre-process data', 2, None, '___sec35'),
+ ('Train and test datasets', 2, None, '___sec36'),
+ ('Define model and architecture', 2, None, '___sec37'),
+ ('Layers', 2, None, '___sec38'),
+ ('Weights and biases', 2, None, '___sec39'),
+ ('Feed-forward pass', 2, None, '___sec40'),
+ ('Matrix multiplications', 2, None, '___sec41'),
+ ('Choose cost function and optimizer', 2, None, '___sec42'),
+ ('Optimizing the cost function', 2, None, '___sec43'),
+ ('Regularization', 2, None, '___sec44'),
+ ('Matrix multiplication', 2, None, '___sec45'),
+ ('Improving performance', 2, None, '___sec46'),
+ ('Full object-oriented implementation', 2, None, '___sec47'),
+ ('Evaluate model performance on test data', 2, None, '___sec48'),
+ ('Adjust hyperparameters', 2, None, '___sec49'),
+ ('Visualization', 2, None, '___sec50'),
+ ('scikit-learn implementation', 2, None, '___sec51'),
+ ('Visualization', 2, None, '___sec52'),
('Building neural networks in Tensorflow and Keras',
2,
None,
- '___sec52'),
- ('Tensorflow', 2, None, '___sec53'),
- ('Collect and pre-process data', 2, None, '___sec54'),
- ('Using TensorFlow backend', 2, None, '___sec55'),
- ('Optimizing and using gradient descent', 2, None, '___sec56'),
- ('Using Keras', 2, None, '___sec57'),
- ('Which activation function should I use?', 2, None, '___sec58'),
+ '___sec53'),
+ ('Tensorflow', 2, None, '___sec54'),
+ ('Collect and pre-process data', 2, None, '___sec55'),
+ ('Using TensorFlow backend', 2, None, '___sec56'),
+ ('Optimizing and using gradient descent', 2, None, '___sec57'),
+ ('Using Keras', 2, None, '___sec58'),
+ ('Which activation function should I use?', 2, None, '___sec59'),
('Is the Logistic activation function (Sigmoid) our choice?',
2,
None,
- '___sec59'),
- ('The derivative of the Logistic funtion', 2, None, '___sec60'),
- ('The RELU function family', 2, None, '___sec61'),
- ('Which activation function should we use?', 2, None, '___sec62'),
+ '___sec60'),
+ ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+ ('The RELU function family', 2, None, '___sec62'),
+ ('Which activation function should we use?', 2, None, '___sec63'),
('A top-down perspective on Neural networks',
2,
None,
- '___sec63'),
+ '___sec64'),
('Limitations of supervised learning with deep networks',
2,
None,
- '___sec64')]}
+ '___sec65')]}
end of tocinfo -->
+
-
One can identify a set of key steps when using neural networks to solve supervised learning problems:
@@ -1332,7 +1355,7 @@ One can identify a set of key steps when using neural networks to solve supervis
Here we will be using the MNIST dataset, which is readily available through the scikit-learn
@@ -1427,7 +1450,7 @@ plt.show()
Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions.
@@ -1476,7 +1499,7 @@ X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=t
Our simple feed-forward neural network will consist of an input layer, a single hidden layer and an output layer. The activation \( y \) of each neuron is a weighted sum of inputs, passed through an activation function. In case of the simple perceptron model we have
@@ -1522,7 +1545,7 @@ which is inspired by probability theory (see logistic regression) and was most c
-
-
Typically weights are initialized with small values distributed around zero, drawn from a uniform
@@ -1603,7 +1626,7 @@ output_bias = np.zeros(n_categories) + 0.01
Denote \( F \) the number of features, \( H \) the number of hidden neurons and \( C \) the number of categories.
@@ -1630,7 +1653,7 @@ $$ a_{j}^{L} = \frac{\exp{(z_j^{L})}}
-
Since our data has the dimensions \( X = (n_{inputs}, n_{features}) \) and our weights to the hidden
@@ -1705,7 +1728,7 @@ predictions = predict(X_train)
To measure how well our neural network is doing we need to introduce a cost function.
@@ -1736,7 +1759,7 @@ you got the correct label. The probability of category \( c \) is given by the s
The network is trained by finding the weights and biases that minimize the cost function. One of the most widely used classes of methods is gradient descent and its generalizations. The idea behind gradient descent
@@ -1776,7 +1799,7 @@ The various optmization methods, with codes and algorithms, are discussed in o
-
It is common to add an extra term to the cost function, proportional
@@ -1809,7 +1832,7 @@ calculate the gradient efficently.
To more efficently train our network these equations are implemented using matrix operations.
@@ -1923,7 +1946,7 @@ lmbd = 0.01
As we can see the network does not seem to be learning at all. It seems to be just guessing the label for each image.
@@ -1943,7 +1966,7 @@ Andrew Ng goes through some of these considerations in this Full object-oriented implementation
+
It is very natural to think of the network as an object, with specific instances of the network
@@ -2056,7 +2079,7 @@ being realizations of this object with different hyperparameters. An implementat
To measure the performance of our network we evaluate how well it does it data it has never seen before, i.e. the test data.
@@ -2091,7 +2114,7 @@ test_predict = dnn.predict(X_test)
We now perform a grid search to find the optimal hyperparameters for the network.
@@ -2124,7 +2147,7 @@ DNN_numpy = np.zeros((len(eta_vals),
@@ -2166,7 +2189,7 @@ plt.show()
scikit-learn focuses more
@@ -2205,7 +2228,7 @@ DNN_scikit = np.zeros((len(eta_vals),
@@ -2247,7 +2270,7 @@ plt.show()
Now we want to build on the experience gained from our neural network implementation in NumPy and scikit-learn
@@ -2262,7 +2285,7 @@ NumPy arrays.
Tensorflow is an open source library machine learning library
@@ -2307,7 +2330,7 @@ and/or if you use anaconda, just write (or install from the graphical use
@@ -2373,7 +2396,7 @@ X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=t
@@ -2596,7 +2619,7 @@ writer.add_graph(tf.get_default_graph())
Keras is a high level neural network
@@ -2695,7 +2718,7 @@ plt.show()
-
The Back propagation algorithm we derived above works by going from
@@ -2724,7 +2747,7 @@ learn at widely different speeds
-
Although this unfortunate behavior has been empirically observed for
@@ -2754,7 +2777,7 @@ better than the logistic function in deep networks).
Looking at the logistic activation function, when inputs become large
@@ -2790,7 +2813,7 @@ fast to compute).
The ReLU activation function suffers from a problem known as the dying
@@ -2817,7 +2840,7 @@ $$
In general it seems that the ELU activation function is better than
@@ -2837,7 +2860,7 @@ bootstrap to evaluate other activation functions.
-
The first thing we would like to do is divide the data into two or three
@@ -2879,7 +2902,7 @@ supervised learning.
Like all statistical methods, supervised learning using neural
diff --git a/doc/pub/NeuralNet/html/NeuralNet.html b/doc/pub/NeuralNet/html/NeuralNet.html
index b0a0523e3..eda365b42 100644
--- a/doc/pub/NeuralNet/html/NeuralNet.html
+++ b/doc/pub/NeuralNet/html/NeuralNet.html
@@ -117,54 +117,55 @@ div { text-align: justify; text-justify: inter-word; }
'___sec30'),
('Defining the cost function', 2, None, '___sec31'),
('Example: binary classification problem', 2, None, '___sec32'),
+ ('The Softmax function', 2, None, '___sec33'),
('Developing a code for doing neural networks with back '
'propagation',
2,
None,
- '___sec33'),
- ('Collect and pre-process data', 2, None, '___sec34'),
- ('Train and test datasets', 2, None, '___sec35'),
- ('Define model and architecture', 2, None, '___sec36'),
- ('Layers', 2, None, '___sec37'),
- ('Weights and biases', 2, None, '___sec38'),
- ('Feed-forward pass', 2, None, '___sec39'),
- ('Matrix multiplications', 2, None, '___sec40'),
- ('Choose cost function and optimizer', 2, None, '___sec41'),
- ('Optimizing the cost function', 2, None, '___sec42'),
- ('Regularization', 2, None, '___sec43'),
- ('Matrix multiplication', 2, None, '___sec44'),
- ('Improving performance', 2, None, '___sec45'),
- ('Full object-oriented implementation', 2, None, '___sec46'),
- ('Evaluate model performance on test data', 2, None, '___sec47'),
- ('Adjust hyperparameters', 2, None, '___sec48'),
- ('Visualization', 2, None, '___sec49'),
- ('scikit-learn implementation', 2, None, '___sec50'),
- ('Visualization', 2, None, '___sec51'),
+ '___sec34'),
+ ('Collect and pre-process data', 2, None, '___sec35'),
+ ('Train and test datasets', 2, None, '___sec36'),
+ ('Define model and architecture', 2, None, '___sec37'),
+ ('Layers', 2, None, '___sec38'),
+ ('Weights and biases', 2, None, '___sec39'),
+ ('Feed-forward pass', 2, None, '___sec40'),
+ ('Matrix multiplications', 2, None, '___sec41'),
+ ('Choose cost function and optimizer', 2, None, '___sec42'),
+ ('Optimizing the cost function', 2, None, '___sec43'),
+ ('Regularization', 2, None, '___sec44'),
+ ('Matrix multiplication', 2, None, '___sec45'),
+ ('Improving performance', 2, None, '___sec46'),
+ ('Full object-oriented implementation', 2, None, '___sec47'),
+ ('Evaluate model performance on test data', 2, None, '___sec48'),
+ ('Adjust hyperparameters', 2, None, '___sec49'),
+ ('Visualization', 2, None, '___sec50'),
+ ('scikit-learn implementation', 2, None, '___sec51'),
+ ('Visualization', 2, None, '___sec52'),
('Building neural networks in Tensorflow and Keras',
2,
None,
- '___sec52'),
- ('Tensorflow', 2, None, '___sec53'),
- ('Collect and pre-process data', 2, None, '___sec54'),
- ('Using TensorFlow backend', 2, None, '___sec55'),
- ('Optimizing and using gradient descent', 2, None, '___sec56'),
- ('Using Keras', 2, None, '___sec57'),
- ('Which activation function should I use?', 2, None, '___sec58'),
+ '___sec53'),
+ ('Tensorflow', 2, None, '___sec54'),
+ ('Collect and pre-process data', 2, None, '___sec55'),
+ ('Using TensorFlow backend', 2, None, '___sec56'),
+ ('Optimizing and using gradient descent', 2, None, '___sec57'),
+ ('Using Keras', 2, None, '___sec58'),
+ ('Which activation function should I use?', 2, None, '___sec59'),
('Is the Logistic activation function (Sigmoid) our choice?',
2,
None,
- '___sec59'),
- ('The derivative of the Logistic funtion', 2, None, '___sec60'),
- ('The RELU function family', 2, None, '___sec61'),
- ('Which activation function should we use?', 2, None, '___sec62'),
+ '___sec60'),
+ ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+ ('The RELU function family', 2, None, '___sec62'),
+ ('Which activation function should we use?', 2, None, '___sec63'),
('A top-down perspective on Neural networks',
2,
None,
- '___sec63'),
+ '___sec64'),
('Limitations of supervised learning with deep networks',
2,
None,
- '___sec64')]}
+ '___sec65')]}
end of tocinfo -->
+
-
One can identify a set of key steps when using neural networks to solve supervised learning problems:
@@ -1337,7 +1360,7 @@ One can identify a set of key steps when using neural networks to solve supervis
Here we will be using the MNIST dataset, which is readily available through the scikit-learn
@@ -1432,7 +1455,7 @@ plt.show()
Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions.
@@ -1481,7 +1504,7 @@ X_train, X_test, Y_train, Y_test = train_tes
Our simple feed-forward neural network will consist of an input layer, a single hidden layer and an output layer. The activation \( y \) of each neuron is a weighted sum of inputs, passed through an activation function. In case of the simple perceptron model we have
@@ -1527,7 +1550,7 @@ which is inspired by probability theory (see logistic regression) and was most c
-
-
Typically weights are initialized with small values distributed around zero, drawn from a uniform
@@ -1608,7 +1631,7 @@ output_bias = np
Denote \( F \) the number of features, \( H \) the number of hidden neurons and \( C \) the number of categories.
@@ -1635,7 +1658,7 @@ $$ a_{j}^{L} = \frac{\exp{(z_j^{L})}}
-
Since our data has the dimensions \( X = (n_{inputs}, n_{features}) \) and our weights to the hidden
@@ -1710,7 +1733,7 @@ predictions = predict(X_train)
To measure how well our neural network is doing we need to introduce a cost function.
@@ -1741,7 +1764,7 @@ you got the correct label. The probability of category \( c \) is given by the s
The network is trained by finding the weights and biases that minimize the cost function. One of the most widely used classes of methods is gradient descent and its generalizations. The idea behind gradient descent
@@ -1781,7 +1804,7 @@ The various optmization methods, with codes and algorithms, are discussed in o
-
It is common to add an extra term to the cost function, proportional
@@ -1814,7 +1837,7 @@ calculate the gradient efficently.
To more efficently train our network these equations are implemented using matrix operations.
@@ -1928,7 +1951,7 @@ lmbd = 0.01
As we can see the network does not seem to be learning at all. It seems to be just guessing the label for each image.
@@ -1948,7 +1971,7 @@ Andrew Ng goes through some of these considerations in this Full object-oriented implementation
+
It is very natural to think of the network as an object, with specific instances of the network
@@ -2061,7 +2084,7 @@ being realizations of this object with different hyperparameters. An implementat
To measure the performance of our network we evaluate how well it does it data it has never seen before, i.e. the test data.
@@ -2096,7 +2119,7 @@ test_predict = dnnAdjust hyperparameters
+
We now perform a grid search to find the optimal hyperparameters for the network.
@@ -2129,7 +2152,7 @@ DNN_numpy = np.
@@ -2171,7 +2194,7 @@ plt.show()
scikit-learn focuses more
@@ -2210,7 +2233,7 @@ DNN_scikit = np
@@ -2252,7 +2275,7 @@ plt.show()
Now we want to build on the experience gained from our neural network implementation in NumPy and scikit-learn
@@ -2267,7 +2290,7 @@ NumPy arrays.
Tensorflow is an open source library machine learning library
@@ -2312,7 +2335,7 @@ and/or if you use anaconda, just write (or install from the graphical use
@@ -2378,7 +2401,7 @@ X_train, X_test, Y_train, Y_test = train_tes
@@ -2601,7 +2624,7 @@ writer.add_graph(tfUsing Keras
+
Keras is a high level neural network
@@ -2700,7 +2723,7 @@ plt.show()
-
The Back propagation algorithm we derived above works by going from
@@ -2729,7 +2752,7 @@ learn at widely different speeds
-
Although this unfortunate behavior has been empirically observed for
@@ -2759,7 +2782,7 @@ better than the logistic function in deep networks).
Looking at the logistic activation function, when inputs become large
@@ -2795,7 +2818,7 @@ fast to compute).
The ReLU activation function suffers from a problem known as the dying
@@ -2822,7 +2845,7 @@ $$
In general it seems that the ELU activation function is better than
@@ -2842,7 +2865,7 @@ bootstrap to evaluate other activation functions.
-
The first thing we would like to do is divide the data into two or three
@@ -2884,7 +2907,7 @@ supervised learning.
Like all statistical methods, supervised learning using neural
diff --git a/doc/pub/NeuralNet/ipynb/NeuralNet.ipynb b/doc/pub/NeuralNet/ipynb/NeuralNet.ipynb
index c0c8483f6..3d3b3a872 100644
--- a/doc/pub/NeuralNet/ipynb/NeuralNet.ipynb
+++ b/doc/pub/NeuralNet/ipynb/NeuralNet.ipynb
@@ -1576,6 +1576,58 @@
"In case we use another activation function than the logistic one, we need to evaluate other derivatives. \n",
"\n",
"\n",
+ "## The Softmax function\n",
+ "In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation $z_i^l$, that is we need"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\frac{\\partial f(z_i^l)}{\\partial w_{jk}^l} =\n",
+ "\\frac{\\partial f(z_i^l)}{\\partial z_j^l} \\frac{\\partial z_j^l}{\\partial w_{jk}^l}= \\frac{\\partial f(z_i^l)}{\\partial z_j^l}a_k^{-1}.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "For the Softmax function we have"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "f(z_i^l) = \\frac{\\exp{(z_i^l)}}{\\sum_{k=1}^K\\exp{(z_k^l}}.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Its derivative with respect to $z_j^l$ gives"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\frac{\\partial f(z_i^l)}{\\partial z_j^l}= f(z_i^l)\\left(\\delta_{ij}-f(z_i^l)\\right),\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "which in case of the simply binary model reduces to having $i=j$. \n",
+ "\n",
"\n",
"## Developing a code for doing neural networks with back propagation\n",
"\n",
diff --git a/doc/pub/NeuralNet/ipynb/ipynb-NeuralNet-src.tar.gz b/doc/pub/NeuralNet/ipynb/ipynb-NeuralNet-src.tar.gz
index 8fa98fbb3..461b35ece 100644
Binary files a/doc/pub/NeuralNet/ipynb/ipynb-NeuralNet-src.tar.gz and b/doc/pub/NeuralNet/ipynb/ipynb-NeuralNet-src.tar.gz differ
diff --git a/doc/pub/NeuralNet/pdf/NeuralNet-minted.pdf b/doc/pub/NeuralNet/pdf/NeuralNet-minted.pdf
index 8ba639a9f..5ee22181f 100644
Binary files a/doc/pub/NeuralNet/pdf/NeuralNet-minted.pdf and b/doc/pub/NeuralNet/pdf/NeuralNet-minted.pdf differ
diff --git a/doc/src/NeuralNet/NeuralNet.do.txt b/doc/src/NeuralNet/NeuralNet.do.txt
index 36df52b55..f75c8ff27 100644
--- a/doc/src/NeuralNet/NeuralNet.do.txt
+++ b/doc/src/NeuralNet/NeuralNet.do.txt
@@ -1007,6 +1007,29 @@ where we have defined the targets $t_i$. The derivatives of the cost function wi
In case we use another activation function than the logistic one, we need to evaluate other derivatives.
+!split
+===== The Softmax function =====
+In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation $z_i^l$, that is we need
+!bt
+\[
+\frac{\partial f(z_i^l)}{\partial w_{jk}^l} =
+\frac{\partial f(z_i^l)}{\partial z_j^l} \frac{\partial z_j^l}{\partial w_{jk}^l}= \frac{\partial f(z_i^l)}{\partial z_j^l}a_k^{-1}.
+\]
+!et
+For the Softmax function we have
+!bt
+\[
+f(z_i^l) = \frac{\exp{(z_i^l)}}{\sum_{k=1}^K\exp{(z_k^l}}.
+\]
+!et
+Its derivative with respect to $z_j^l$ gives
+!bt
+\[
+\frac{\partial f(z_i^l)}{\partial z_j^l}= f(z_i^l)\left(\delta_{ij}-f(z_i^l)\right),
+\]
+!et
+which in case of the simply binary model reduces to having $i=j$.
+
!split
===== Developing a code for doing neural networks with back propagation =====
Feed-forward pass
+Weights and biases
# building our neural network
+n_inputs, n_features = X_train.shape
+n_hidden_neurons = 50
+n_categories = 10
+
+# we make the weights normally distributed using numpy.random.randn
+
+# weights and bias in the hidden layer
+hidden_weights = np.random.randn(n_features, n_hidden_neurons)
+hidden_bias = np.zeros(n_hidden_neurons) + 0.01
+
+# weights and bias in the output layer
+output_weights = np.random.randn(n_hidden_neurons, n_categories)
+output_bias = np.zeros(n_categories) + 0.01
+
Matrix multiplications
+Feed-forward pass
# setup the feed-forward pass, subscript h = hidden layer
-
-def sigmoid(x):
- return 1/(1 + np.exp(-x))
-
-def feed_forward(X):
- # weighted sum of inputs to the hidden layer
- z_h = np.matmul(X, hidden_weights) + hidden_bias
- # activation in the hidden layer
- a_h = sigmoid(z_h)
-
- # weighted sum of inputs to the output layer
- z_o = np.matmul(a_h, output_weights) + output_bias
- # softmax output
- # axis 0 holds each input and axis 1 the probabilities of each category
- exp_term = np.exp(z_o)
- probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
-
- return probabilities
-
-probabilities = feed_forward(X_train)
-print("probabilities = (n_inputs, n_categories) = " + str(probabilities.shape))
-print("probability that image 0 is in category 0,1,2,...,9 = \n" + str(probabilities[0]))
-print("probabilities sum up to: " + str(probabilities[0].sum()))
-print()
-
-# we obtain a prediction by taking the class with the highest likelihood
-def predict(X):
- probabilities = feed_forward(X)
- return np.argmax(probabilities, axis=1)
-
-predictions = predict(X_train)
-print("predictions = (n_inputs) = " + str(predictions.shape))
-print("prediction for image 0: " + str(predictions[0]))
-print("correct label for image 0: " + str(Y_train[0]))
-
Choose cost function and optimizer
+Matrix multiplications
# setup the feed-forward pass, subscript h = hidden layer
+
+def sigmoid(x):
+ return 1/(1 + np.exp(-x))
+
+def feed_forward(X):
+ # weighted sum of inputs to the hidden layer
+ z_h = np.matmul(X, hidden_weights) + hidden_bias
+ # activation in the hidden layer
+ a_h = sigmoid(z_h)
+
+ # weighted sum of inputs to the output layer
+ z_o = np.matmul(a_h, output_weights) + output_bias
+ # softmax output
+ # axis 0 holds each input and axis 1 the probabilities of each category
+ exp_term = np.exp(z_o)
+ probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+
+ return probabilities
+
+probabilities = feed_forward(X_train)
+print("probabilities = (n_inputs, n_categories) = " + str(probabilities.shape))
+print("probability that image 0 is in category 0,1,2,...,9 = \n" + str(probabilities[0]))
+print("probabilities sum up to: " + str(probabilities[0].sum()))
+print()
+
+# we obtain a prediction by taking the class with the highest likelihood
+def predict(X):
+ probabilities = feed_forward(X)
+ return np.argmax(probabilities, axis=1)
+
+predictions = predict(X_train)
+print("predictions = (n_inputs) = " + str(predictions.shape))
+print("prediction for image 0: " + str(predictions[0]))
+print("correct label for image 0: " + str(Y_train[0]))
+
Optimizing the cost function
+Choose cost function and optimizer
-
-
-The various optmization methods, with codes and algorithms, are discussed in our lectures on Gradient descent approaches.
+In the one-hot representation only one of the terms in the loss function is non-zero, namely the
+probability of the correct category \( c' \)
+(i.e. the category \( c' \) such that \( y_{ic'} = 1 \)). This means that the cross entropy loss only punishes you for how wrong
+you got the correct label. The probability of category \( c \) is given by the softmax function. The vector \( \hat{\theta} \) represents the parameters of our network, i.e. all the weights and biases.
Regularization
+Optimizing the cost function
+
+
+The various optmization methods, with codes and algorithms, are discussed in our lectures on Gradient descent approaches.
Matrix multiplication
+Regularization
# to categorical turns our integer vector into a onehot representation
-from sklearn.metrics import accuracy_score
-
-# one-hot in numpy
-def to_categorical_numpy(integer_vector):
- n_inputs = len(integer_vector)
- n_categories = np.max(integer_vector) + 1
- onehot_vector = np.zeros((n_inputs, n_categories))
- onehot_vector[range(n_inputs), integer_vector] = 1
-
- return onehot_vector
-
-#Y_train_onehot, Y_test_onehot = to_categorical(Y_train), to_categorical(Y_test)
-Y_train_onehot, Y_test_onehot = to_categorical_numpy(Y_train), to_categorical_numpy(Y_test)
-
-def feed_forward_train(X):
- # weighted sum of inputs to the hidden layer
- z_h = np.matmul(X, hidden_weights) + hidden_bias
- # activation in the hidden layer
- a_h = sigmoid(z_h)
-
- # weighted sum of inputs to the output layer
- z_o = np.matmul(a_h, output_weights) + output_bias
- # softmax output
- # axis 0 holds each input and axis 1 the probabilities of each category
- exp_term = np.exp(z_o)
- probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
-
- # for backpropagation need activations in hidden and output layers
- return a_h, probabilities
-
-def backpropagation(X, Y):
- a_h, probabilities = feed_forward_train(X)
-
- # error in the output layer
- error_output = probabilities - Y
- # error in the hidden layer
- error_hidden = np.matmul(error_output, output_weights.T) * a_h * (1 - a_h)
-
- # gradients for the output layer
- output_weights_gradient = np.matmul(a_h.T, error_output)
- output_bias_gradient = np.sum(error_output, axis=0)
-
- # gradient for the hidden layer
- hidden_weights_gradient = np.matmul(X.T, error_hidden)
- hidden_bias_gradient = np.sum(error_hidden, axis=0)
-
- return output_weights_gradient, output_bias_gradient, hidden_weights_gradient, hidden_bias_gradient
-
-print("Old accuracy on training data: " + str(accuracy_score(predict(X_train), Y_train)))
-
-eta = 0.01
-lmbd = 0.01
-for i in range(1000):
- # calculate gradients
- dWo, dBo, dWh, dBh = backpropagation(X_train, Y_train_onehot)
-
- # regularization term gradients
- dWo += lmbd * output_weights
- dWh += lmbd * hidden_weights
-
- # update weights and biases
- output_weights -= eta * dWo
- output_bias -= eta * dBo
- hidden_weights -= eta * dWh
- hidden_bias -= eta * dBh
-
-print("New accuracy on training data: " + str(accuracy_score(predict(X_train), Y_train)))
-
Improving performance
+Matrix multiplication
# to categorical turns our integer vector into a onehot representation
+from sklearn.metrics import accuracy_score
+
+# one-hot in numpy
+def to_categorical_numpy(integer_vector):
+ n_inputs = len(integer_vector)
+ n_categories = np.max(integer_vector) + 1
+ onehot_vector = np.zeros((n_inputs, n_categories))
+ onehot_vector[range(n_inputs), integer_vector] = 1
+
+ return onehot_vector
+
+#Y_train_onehot, Y_test_onehot = to_categorical(Y_train), to_categorical(Y_test)
+Y_train_onehot, Y_test_onehot = to_categorical_numpy(Y_train), to_categorical_numpy(Y_test)
+
+def feed_forward_train(X):
+ # weighted sum of inputs to the hidden layer
+ z_h = np.matmul(X, hidden_weights) + hidden_bias
+ # activation in the hidden layer
+ a_h = sigmoid(z_h)
+
+ # weighted sum of inputs to the output layer
+ z_o = np.matmul(a_h, output_weights) + output_bias
+ # softmax output
+ # axis 0 holds each input and axis 1 the probabilities of each category
+ exp_term = np.exp(z_o)
+ probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+
+ # for backpropagation need activations in hidden and output layers
+ return a_h, probabilities
+
+def backpropagation(X, Y):
+ a_h, probabilities = feed_forward_train(X)
+
+ # error in the output layer
+ error_output = probabilities - Y
+ # error in the hidden layer
+ error_hidden = np.matmul(error_output, output_weights.T) * a_h * (1 - a_h)
+
+ # gradients for the output layer
+ output_weights_gradient = np.matmul(a_h.T, error_output)
+ output_bias_gradient = np.sum(error_output, axis=0)
+
+ # gradient for the hidden layer
+ hidden_weights_gradient = np.matmul(X.T, error_hidden)
+ hidden_bias_gradient = np.sum(error_hidden, axis=0)
+
+ return output_weights_gradient, output_bias_gradient, hidden_weights_gradient, hidden_bias_gradient
+
+print("Old accuracy on training data: " + str(accuracy_score(predict(X_train), Y_train)))
+
+eta = 0.01
+lmbd = 0.01
+for i in range(1000):
+ # calculate gradients
+ dWo, dBo, dWh, dBh = backpropagation(X_train, Y_train_onehot)
+
+ # regularization term gradients
+ dWo += lmbd * output_weights
+ dWh += lmbd * hidden_weights
+
+ # update weights and biases
+ output_weights -= eta * dWo
+ output_bias -= eta * dBo
+ hidden_weights -= eta * dWh
+ hidden_bias -= eta * dBh
+
+print("New accuracy on training data: " + str(accuracy_score(predict(X_train), Y_train)))
+
Full object-oriented implementation
+Improving performance
class NeuralNetwork:
- def __init__(
- self,
- X_data,
- Y_data,
- n_hidden_neurons=50,
- n_categories=10,
- epochs=10,
- batch_size=100,
- eta=0.1,
- lmbd=0.0,
+
Evaluate model performance on test data
+Full object-oriented implementation
epochs = 100
-batch_size = 100
+
class NeuralNetwork:
+ def __init__(
+ self,
+ X_data,
+ Y_data,
+ n_hidden_neurons=50,
+ n_categories=10,
+ epochs=10,
+ batch_size=100,
+ eta=0.1,
+ lmbd=0.0,
-dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size,
- n_hidden_neurons=n_hidden_neurons, n_categories=n_categories)
-dnn.train()
-test_predict = dnn.predict(X_test)
+ ):
+ self.X_data_full = X_data
+ self.Y_data_full = Y_data
-# accuracy score from scikit library
-print("Accuracy score on test set: ", accuracy_score(Y_test, test_predict))
+ self.n_inputs = X_data.shape[0]
+ self.n_features = X_data.shape[1]
+ self.n_hidden_neurons = n_hidden_neurons
+ self.n_categories = n_categories
-# equivalent in numpy
-def accuracy_score_numpy(Y_test, Y_pred):
- return np.sum(Y_test == Y_pred) / len(Y_test)
+ self.epochs = epochs
+ self.batch_size = batch_size
+ self.iterations = self.n_inputs // self.batch_size
+ self.eta = eta
+ self.lmbd = lmbd
-#print("Accuracy score on test set: ", accuracy_score_numpy(Y_test, test_predict))
+ self.create_biases_and_weights()
+
+ def create_biases_and_weights(self):
+ self.hidden_weights = np.random.randn(self.n_features, self.n_hidden_neurons)
+ self.hidden_bias = np.zeros(self.n_hidden_neurons) + 0.01
+
+ self.output_weights = np.random.randn(self.n_hidden_neurons, self.n_categories)
+ self.output_bias = np.zeros(self.n_categories) + 0.01
+
+ def feed_forward(self):
+ # feed-forward for training
+ self.z_h = np.matmul(self.X_data, self.hidden_weights) + self.hidden_bias
+ self.a_h = sigmoid(self.z_h)
+
+ self.z_o = np.matmul(self.a_h, self.output_weights) + self.output_bias
+
+ exp_term = np.exp(self.z_o)
+ self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+
+ def feed_forward_out(self, X):
+ # feed-forward for output
+ z_h = np.matmul(X, self.hidden_weights) + self.hidden_bias
+ a_h = sigmoid(z_h)
+
+ z_o = np.matmul(a_h, self.output_weights) + self.output_bias
+
+ exp_term = np.exp(z_o)
+ probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+ return probabilities
+
+ def backpropagation(self):
+ error_output = self.probabilities - self.Y_data
+ error_hidden = np.matmul(error_output, self.output_weights.T) * self.a_h * (1 - self.a_h)
+
+ self.output_weights_gradient = np.matmul(self.a_h.T, error_output)
+ self.output_bias_gradient = np.sum(error_output, axis=0)
+
+ self.hidden_weights_gradient = np.matmul(self.X_data.T, error_hidden)
+ self.hidden_bias_gradient = np.sum(error_hidden, axis=0)
+
+ if self.lmbd > 0.0:
+ self.output_weights_gradient += self.lmbd * self.output_weights
+ self.hidden_weights_gradient += self.lmbd * self.hidden_weights
+
+ self.output_weights -= self.eta * self.output_weights_gradient
+ self.output_bias -= self.eta * self.output_bias_gradient
+ self.hidden_weights -= self.eta * self.hidden_weights_gradient
+ self.hidden_bias -= self.eta * self.hidden_bias_gradient
+
+ def predict(self, X):
+ probabilities = self.feed_forward_out(X)
+ return np.argmax(probabilities, axis=1)
+
+ def predict_probabilities(self, X):
+ probabilities = self.feed_forward_out(X)
+ return probabilities
+
+ def train(self):
+ data_indices = np.arange(self.n_inputs)
+
+ for i in range(self.epochs):
+ for j in range(self.iterations):
+ # pick datapoints with replacement
+ chosen_datapoints = np.random.choice(
+ data_indices, size=self.batch_size, replace=False
+ )
+
+ # minibatch training data
+ self.X_data = self.X_data_full[chosen_datapoints]
+ self.Y_data = self.Y_data_full[chosen_datapoints]
+
+ self.feed_forward()
+ self.backpropagation()
Adjust hyperparameters
+Evaluate model performance on test data
eta_vals = np.logspace(-5, 1, 7)
-lmbd_vals = np.logspace(-5, 1, 7)
-# store the models for later use
-DNN_numpy = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+
epochs = 100
+batch_size = 100
-# grid search
-for i, eta in enumerate(eta_vals):
- for j, lmbd in enumerate(lmbd_vals):
- dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size,
- n_hidden_neurons=n_hidden_neurons, n_categories=n_categories)
- dnn.train()
-
- DNN_numpy[i][j] = dnn
-
- test_predict = dnn.predict(X_test)
-
- print("Learning rate = ", eta)
- print("Lambda = ", lmbd)
- print("Accuracy score on test set: ", accuracy_score(Y_test, test_predict))
- print()
+dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size,
+ n_hidden_neurons=n_hidden_neurons, n_categories=n_categories)
+dnn.train()
+test_predict = dnn.predict(X_test)
+
+# accuracy score from scikit library
+print("Accuracy score on test set: ", accuracy_score(Y_test, test_predict))
+
+# equivalent in numpy
+def accuracy_score_numpy(Y_test, Y_pred):
+ return np.sum(Y_test == Y_pred) / len(Y_test)
+
+#print("Accuracy score on test set: ", accuracy_score_numpy(Y_test, test_predict))
Visualization
+Adjust hyperparameters
+
+# visual representation of grid search
-# uses seaborn heatmap, you can also do this with matplotlib imshow
-import seaborn as sns
+
eta_vals = np.logspace(-5, 1, 7)
+lmbd_vals = np.logspace(-5, 1, 7)
+# store the models for later use
+DNN_numpy = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
-sns.set()
-
-train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-
-for i in range(len(eta_vals)):
- for j in range(len(lmbd_vals)):
- dnn = DNN_numpy[i][j]
+# grid search
+for i, eta in enumerate(eta_vals):
+ for j, lmbd in enumerate(lmbd_vals):
+ dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size,
+ n_hidden_neurons=n_hidden_neurons, n_categories=n_categories)
+ dnn.train()
- train_pred = dnn.predict(X_train)
- test_pred = dnn.predict(X_test)
-
- train_accuracy[i][j] = accuracy_score(Y_train, train_pred)
- test_accuracy[i][j] = accuracy_score(Y_test, test_pred)
-
+ DNN_numpy[i][j] = dnn
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Training Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
-
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Test Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
+ test_predict = dnn.predict(X_test)
+
+ print("Learning rate = ", eta)
+ print("Lambda = ", lmbd)
+ print("Accuracy score on test set: ", accuracy_score(Y_test, test_predict))
+ print()
scikit-learn implementation
-
-Visualization
from sklearn.neural_network import MLPClassifier
-# store models for later use
-DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+
# visual representation of grid search
+# uses seaborn heatmap, you can also do this with matplotlib imshow
+import seaborn as sns
-for i, eta in enumerate(eta_vals):
- for j, lmbd in enumerate(lmbd_vals):
- dnn = MLPClassifier(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',
- alpha=lmbd, learning_rate_init=eta, max_iter=epochs)
- dnn.fit(X_train, Y_train)
+sns.set()
+
+train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+
+for i in range(len(eta_vals)):
+ for j in range(len(lmbd_vals)):
+ dnn = DNN_numpy[i][j]
- DNN_scikit[i][j] = dnn
+ train_pred = dnn.predict(X_train)
+ test_pred = dnn.predict(X_test)
+
+ train_accuracy[i][j] = accuracy_score(Y_train, train_pred)
+ test_accuracy[i][j] = accuracy_score(Y_test, test_pred)
+
- print("Learning rate = ", eta)
- print("Lambda = ", lmbd)
- print("Accuracy score on test set: ", dnn.score(X_test, Y_test))
- print()
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Training Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Test Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
Visualization
+scikit-learn implementation
+
+# optional
-# visual representation of grid search
-# uses seaborn heatmap, could probably do this in matplotlib
-import seaborn as sns
+
from sklearn.neural_network import MLPClassifier
+# store models for later use
+DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
-sns.set()
-
-train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-
-for i in range(len(eta_vals)):
- for j in range(len(lmbd_vals)):
- dnn = DNN_scikit[i][j]
+for i, eta in enumerate(eta_vals):
+ for j, lmbd in enumerate(lmbd_vals):
+ dnn = MLPClassifier(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',
+ alpha=lmbd, learning_rate_init=eta, max_iter=epochs)
+ dnn.fit(X_train, Y_train)
- train_pred = dnn.predict(X_train)
- test_pred = dnn.predict(X_test)
-
- train_accuracy[i][j] = accuracy_score(Y_train, train_pred)
- test_accuracy[i][j] = accuracy_score(Y_test, test_pred)
-
+ DNN_scikit[i][j] = dnn
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Training Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
-
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Test Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
+ print("Learning rate = ", eta)
+ print("Lambda = ", lmbd)
+ print("Accuracy score on test set: ", dnn.score(X_test, Y_test))
+ print()
Building neural networks in Tensorflow and Keras
-
+Visualization
# optional
+# visual representation of grid search
+# uses seaborn heatmap, could probably do this in matplotlib
+import seaborn as sns
+sns.set()
+
+train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+
+for i in range(len(eta_vals)):
+ for j in range(len(lmbd_vals)):
+ dnn = DNN_scikit[i][j]
+
+ train_pred = dnn.predict(X_train)
+ test_pred = dnn.predict(X_test)
+
+ train_accuracy[i][j] = accuracy_score(Y_train, train_pred)
+ test_accuracy[i][j] = accuracy_score(Y_test, test_pred)
+
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Training Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Test Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
Tensorflow
+Building neural networks in Tensorflow and Keras
pip3 install tensorflow
-
conda install tensorflow
-
Collect and pre-process data
+Tensorflow
+# import necessary packages
-import numpy as np
-import matplotlib.pyplot as plt
-from sklearn import datasets
-
-
-# ensure the same random numbers appear every time
-np.random.seed(0)
-
-# display images in notebook
-%matplotlib inline
-plt.rcParams['figure.figsize'] = (12,12)
-
-
-# download MNIST dataset
-digits = datasets.load_digits()
-
-# define inputs and labels
-inputs = digits.images
-labels = digits.target
-
-print("inputs = (n_inputs, pixel_width, pixel_height) = " + str(inputs.shape))
-print("labels = (n_inputs) = " + str(labels.shape))
-
-
-# flatten the image
-# the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64
-n_inputs = len(inputs)
-inputs = inputs.reshape(n_inputs, -1)
-print("X = (n_inputs, n_features) = " + str(inputs.shape))
-
-
-# choose some random images to display
-indices = np.arange(n_inputs)
-random_indices = np.random.choice(indices, size=5)
-
-for i, image in enumerate(digits.images[random_indices]):
- plt.subplot(1, 5, i+1)
- plt.axis('off')
- plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
- plt.title("Label: %d" % digits.target[random_indices[i]])
-plt.show()
+
pip3 install tensorflow
from keras.utils import to_categorical
-from sklearn.model_selection import train_test_split
-
-# one-hot representation of labels
-labels = to_categorical(labels)
-
-# split into train and test data
-train_size = 0.8
-test_size = 1 - train_size
-X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
- test_size=test_size)
+
conda install tensorflow
Using TensorFlow backend
-
-
-
+Collect and pre-process data
import tensorflow as tf
+
# import necessary packages
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn import datasets
-class NeuralNetworkTensorflow:
- def __init__(
- self,
- X_train,
- Y_train,
- X_test,
- Y_test,
- n_neurons_layer1=100,
- n_neurons_layer2=50,
- n_categories=2,
- epochs=10,
- batch_size=100,
- eta=0.1,
- lmbd=0.0,
- ):
-
- # keep track of number of steps
- self.global_step = tf.Variable(0, dtype=tf.int32, trainable=False, name='global_step')
-
- self.X_train = X_train
- self.Y_train = Y_train
- self.X_test = X_test
- self.Y_test = Y_test
-
- self.n_inputs = X_train.shape[0]
- self.n_features = X_train.shape[1]
- self.n_neurons_layer1 = n_neurons_layer1
- self.n_neurons_layer2 = n_neurons_layer2
- self.n_categories = n_categories
-
- self.epochs = epochs
- self.batch_size = batch_size
- self.iterations = self.n_inputs // self.batch_size
- self.eta = eta
- self.lmbd = lmbd
-
- # build network piece by piece
- # name scopes (with) are used to enforce creation of new variables
- # https://www.tensorflow.org/guide/variables
- self.create_placeholders()
- self.create_DNN()
- self.create_loss()
- self.create_optimiser()
- self.create_accuracy()
-
- def create_placeholders(self):
- # placeholders are fine here, but "Datasets" are the preferred method
- # of streaming data into a model
- with tf.name_scope('data'):
- self.X = tf.placeholder(tf.float32, shape=(None, self.n_features), name='X_data')
- self.Y = tf.placeholder(tf.float32, shape=(None, self.n_categories), name='Y_data')
-
- def create_DNN(self):
- with tf.name_scope('DNN'):
- # the weights are stored to calculate regularization loss later
-
- # Fully connected layer 1
- self.W_fc1 = self.weight_variable([self.n_features, self.n_neurons_layer1], name='fc1', dtype=tf.float32)
- b_fc1 = self.bias_variable([self.n_neurons_layer1], name='fc1', dtype=tf.float32)
- a_fc1 = tf.nn.sigmoid(tf.matmul(self.X, self.W_fc1) + b_fc1)
-
- # Fully connected layer 2
- self.W_fc2 = self.weight_variable([self.n_neurons_layer1, self.n_neurons_layer2], name='fc2', dtype=tf.float32)
- b_fc2 = self.bias_variable([self.n_neurons_layer2], name='fc2', dtype=tf.float32)
- a_fc2 = tf.nn.sigmoid(tf.matmul(a_fc1, self.W_fc2) + b_fc2)
-
- # Output layer
- self.W_out = self.weight_variable([self.n_neurons_layer2, self.n_categories], name='out', dtype=tf.float32)
- b_out = self.bias_variable([self.n_categories], name='out', dtype=tf.float32)
- self.z_out = tf.matmul(a_fc2, self.W_out) + b_out
-
- def create_loss(self):
- with tf.name_scope('loss'):
- softmax_loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits_v2(labels=self.Y, logits=self.z_out))
-
- regularizer_loss_fc1 = tf.nn.l2_loss(self.W_fc1)
- regularizer_loss_fc2 = tf.nn.l2_loss(self.W_fc2)
- regularizer_loss_out = tf.nn.l2_loss(self.W_out)
- regularizer_loss = self.lmbd*(regularizer_loss_fc1 + regularizer_loss_fc2 + regularizer_loss_out)
-
- self.loss = softmax_loss + regularizer_loss
- def create_accuracy(self):
- with tf.name_scope('accuracy'):
- probabilities = tf.nn.softmax(self.z_out)
- predictions = tf.argmax(probabilities, axis=1)
- labels = tf.argmax(self.Y, axis=1)
-
- correct_predictions = tf.equal(predictions, labels)
- correct_predictions = tf.cast(correct_predictions, tf.float32)
- self.accuracy = tf.reduce_mean(correct_predictions)
-
- def create_optimiser(self):
- with tf.name_scope('optimizer'):
- self.optimizer = tf.train.GradientDescentOptimizer(learning_rate=self.eta).minimize(self.loss, global_step=self.global_step)
-
- def weight_variable(self, shape, name='', dtype=tf.float32):
- initial = tf.truncated_normal(shape, stddev=0.1)
- return tf.Variable(initial, name=name, dtype=dtype)
-
- def bias_variable(self, shape, name='', dtype=tf.float32):
- initial = tf.constant(0.1, shape=shape)
- return tf.Variable(initial, name=name, dtype=dtype)
-
- def fit(self):
- data_indices = np.arange(self.n_inputs)
+# ensure the same random numbers appear every time
+np.random.seed(0)
- with tf.Session() as sess:
- sess.run(tf.global_variables_initializer())
- for i in range(self.epochs):
- for j in range(self.iterations):
- chosen_datapoints = np.random.choice(data_indices, size=self.batch_size, replace=False)
- batch_X, batch_Y = self.X_train[chosen_datapoints], self.Y_train[chosen_datapoints]
-
- sess.run([DNN.loss, DNN.optimizer],
- feed_dict={DNN.X: batch_X,
- DNN.Y: batch_Y})
- accuracy = sess.run(DNN.accuracy,
- feed_dict={DNN.X: batch_X,
- DNN.Y: batch_Y})
- step = sess.run(DNN.global_step)
-
- self.train_loss, self.train_accuracy = sess.run([DNN.loss, DNN.accuracy],
- feed_dict={DNN.X: self.X_train,
- DNN.Y: self.Y_train})
-
- self.test_loss, self.test_accuracy = sess.run([DNN.loss, DNN.accuracy],
- feed_dict={DNN.X: self.X_test,
- DNN.Y: self.Y_test})
+# display images in notebook
+%matplotlib inline
+plt.rcParams['figure.figsize'] = (12,12)
+
+
+# download MNIST dataset
+digits = datasets.load_digits()
+
+# define inputs and labels
+inputs = digits.images
+labels = digits.target
+
+print("inputs = (n_inputs, pixel_width, pixel_height) = " + str(inputs.shape))
+print("labels = (n_inputs) = " + str(labels.shape))
+
+
+# flatten the image
+# the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64
+n_inputs = len(inputs)
+inputs = inputs.reshape(n_inputs, -1)
+print("X = (n_inputs, n_features) = " + str(inputs.shape))
+
+
+# choose some random images to display
+indices = np.arange(n_inputs)
+random_indices = np.random.choice(indices, size=5)
+
+for i, image in enumerate(digits.images[random_indices]):
+ plt.subplot(1, 5, i+1)
+ plt.axis('off')
+ plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
+ plt.title("Label: %d" % digits.target[random_indices[i]])
+plt.show()
+
from keras.utils import to_categorical
+from sklearn.model_selection import train_test_split
+
+# one-hot representation of labels
+labels = to_categorical(labels)
+
+# split into train and test data
+train_size = 0.8
+test_size = 1 - train_size
+X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
+ test_size=test_size)
Optimizing and using gradient descent
+Using TensorFlow backend
+
+
+
epochs = 100
-batch_size = 100
-n_neurons_layer1 = 100
-n_neurons_layer2 = 50
-n_categories = 10
-eta_vals = np.logspace(-5, 1, 7)
-lmbd_vals = np.logspace(-5, 1, 7)
-
import tensorflow as tf
-
-
DNN_tf = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+class NeuralNetworkTensorflow:
+ def __init__(
+ self,
+ X_train,
+ Y_train,
+ X_test,
+ Y_test,
+ n_neurons_layer1=100,
+ n_neurons_layer2=50,
+ n_categories=2,
+ epochs=10,
+ batch_size=100,
+ eta=0.1,
+ lmbd=0.0,
+ ):
-for i, eta in enumerate(eta_vals):
- for j, lmbd in enumerate(lmbd_vals):
- DNN = NeuralNetworkTensorflow(X_train, Y_train, X_test, Y_test,
- n_neurons_layer1, n_neurons_layer2, n_categories,
- epochs=epochs, batch_size=batch_size, eta=eta, lmbd=lmbd)
- DNN.fit()
+ # keep track of number of steps
+ self.global_step = tf.Variable(0, dtype=tf.int32, trainable=False, name='global_step')
- DNN_tf[i][j] = DNN
+ self.X_train = X_train
+ self.Y_train = Y_train
+ self.X_test = X_test
+ self.Y_test = Y_test
- print("Learning rate = ", eta)
- print("Lambda = ", lmbd)
- print("Test accuracy: %.3f" % DNN.test_accuracy)
- print()
-
# optional
-# visual representation of grid search
-# uses seaborn heatmap, could probably do this in matplotlib
-import seaborn as sns
-
-sns.set()
-
-train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-
-for i in range(len(eta_vals)):
- for j in range(len(lmbd_vals)):
- DNN = DNN_tf[i][j]
-
- train_accuracy[i][j] = DNN.train_accuracy
- test_accuracy[i][j] = DNN.test_accuracy
-
+ self.n_inputs = X_train.shape[0]
+ self.n_features = X_train.shape[1]
+ self.n_neurons_layer1 = n_neurons_layer1
+ self.n_neurons_layer2 = n_neurons_layer2
+ self.n_categories = n_categories
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Training Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
+ self.epochs = epochs
+ self.batch_size = batch_size
+ self.iterations = self.n_inputs // self.batch_size
+ self.eta = eta
+ self.lmbd = lmbd
+
+ # build network piece by piece
+ # name scopes (with) are used to enforce creation of new variables
+ # https://www.tensorflow.org/guide/variables
+ self.create_placeholders()
+ self.create_DNN()
+ self.create_loss()
+ self.create_optimiser()
+ self.create_accuracy()
+
+ def create_placeholders(self):
+ # placeholders are fine here, but "Datasets" are the preferred method
+ # of streaming data into a model
+ with tf.name_scope('data'):
+ self.X = tf.placeholder(tf.float32, shape=(None, self.n_features), name='X_data')
+ self.Y = tf.placeholder(tf.float32, shape=(None, self.n_categories), name='Y_data')
+
+ def create_DNN(self):
+ with tf.name_scope('DNN'):
+ # the weights are stored to calculate regularization loss later
+
+ # Fully connected layer 1
+ self.W_fc1 = self.weight_variable([self.n_features, self.n_neurons_layer1], name='fc1', dtype=tf.float32)
+ b_fc1 = self.bias_variable([self.n_neurons_layer1], name='fc1', dtype=tf.float32)
+ a_fc1 = tf.nn.sigmoid(tf.matmul(self.X, self.W_fc1) + b_fc1)
+
+ # Fully connected layer 2
+ self.W_fc2 = self.weight_variable([self.n_neurons_layer1, self.n_neurons_layer2], name='fc2', dtype=tf.float32)
+ b_fc2 = self.bias_variable([self.n_neurons_layer2], name='fc2', dtype=tf.float32)
+ a_fc2 = tf.nn.sigmoid(tf.matmul(a_fc1, self.W_fc2) + b_fc2)
+
+ # Output layer
+ self.W_out = self.weight_variable([self.n_neurons_layer2, self.n_categories], name='out', dtype=tf.float32)
+ b_out = self.bias_variable([self.n_categories], name='out', dtype=tf.float32)
+ self.z_out = tf.matmul(a_fc2, self.W_out) + b_out
+
+ def create_loss(self):
+ with tf.name_scope('loss'):
+ softmax_loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits_v2(labels=self.Y, logits=self.z_out))
+
+ regularizer_loss_fc1 = tf.nn.l2_loss(self.W_fc1)
+ regularizer_loss_fc2 = tf.nn.l2_loss(self.W_fc2)
+ regularizer_loss_out = tf.nn.l2_loss(self.W_out)
+ regularizer_loss = self.lmbd*(regularizer_loss_fc1 + regularizer_loss_fc2 + regularizer_loss_out)
+
+ self.loss = softmax_loss + regularizer_loss
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Test Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
-
# optional
-# we can use log files to visualize our graph in Tensorboard
-writer = tf.summary.FileWriter('logs/')
-writer.add_graph(tf.get_default_graph())
+ with tf.Session() as sess:
+ sess.run(tf.global_variables_initializer())
+ for i in range(self.epochs):
+ for j in range(self.iterations):
+ chosen_datapoints = np.random.choice(data_indices, size=self.batch_size, replace=False)
+ batch_X, batch_Y = self.X_train[chosen_datapoints], self.Y_train[chosen_datapoints]
+
+ sess.run([DNN.loss, DNN.optimizer],
+ feed_dict={DNN.X: batch_X,
+ DNN.Y: batch_Y})
+ accuracy = sess.run(DNN.accuracy,
+ feed_dict={DNN.X: batch_X,
+ DNN.Y: batch_Y})
+ step = sess.run(DNN.global_step)
+
+ self.train_loss, self.train_accuracy = sess.run([DNN.loss, DNN.accuracy],
+ feed_dict={DNN.X: self.X_train,
+ DNN.Y: self.Y_train})
+
+ self.test_loss, self.test_accuracy = sess.run([DNN.loss, DNN.accuracy],
+ feed_dict={DNN.X: self.X_test,
+ DNN.Y: self.Y_test})
Using Keras
-
-conda install keras
-
Optimizing and using gradient descent
pip3 install keras
-
from keras.models import Sequential
-from keras.layers import Dense
-from keras.regularizers import l2
-from keras.optimizers import SGD
-
-def create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories, eta, lmbd):
- model = Sequential()
- model.add(Dense(n_neurons_layer1, activation='sigmoid', kernel_regularizer=l2(lmbd)))
- model.add(Dense(n_neurons_layer2, activation='sigmoid', kernel_regularizer=l2(lmbd)))
- model.add(Dense(n_categories, activation='softmax'))
-
- sgd = SGD(lr=eta)
- model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])
-
- return model
+
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs059.html b/doc/pub/NeuralNet/html/._NeuralNet-bs059.html
index 13d0e4d5e..28d76760e 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs059.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs059.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
'___sec30'),
('Defining the cost function', 2, None, '___sec31'),
('Example: binary classification problem', 2, None, '___sec32'),
+ ('The Softmax function', 2, None, '___sec33'),
('Developing a code for doing neural networks with back '
'propagation',
2,
None,
- '___sec33'),
- ('Collect and pre-process data', 2, None, '___sec34'),
- ('Train and test datasets', 2, None, '___sec35'),
- ('Define model and architecture', 2, None, '___sec36'),
- ('Layers', 2, None, '___sec37'),
- ('Weights and biases', 2, None, '___sec38'),
- ('Feed-forward pass', 2, None, '___sec39'),
- ('Matrix multiplications', 2, None, '___sec40'),
- ('Choose cost function and optimizer', 2, None, '___sec41'),
- ('Optimizing the cost function', 2, None, '___sec42'),
- ('Regularization', 2, None, '___sec43'),
- ('Matrix multiplication', 2, None, '___sec44'),
- ('Improving performance', 2, None, '___sec45'),
- ('Full object-oriented implementation', 2, None, '___sec46'),
- ('Evaluate model performance on test data', 2, None, '___sec47'),
- ('Adjust hyperparameters', 2, None, '___sec48'),
- ('Visualization', 2, None, '___sec49'),
- ('scikit-learn implementation', 2, None, '___sec50'),
- ('Visualization', 2, None, '___sec51'),
+ '___sec34'),
+ ('Collect and pre-process data', 2, None, '___sec35'),
+ ('Train and test datasets', 2, None, '___sec36'),
+ ('Define model and architecture', 2, None, '___sec37'),
+ ('Layers', 2, None, '___sec38'),
+ ('Weights and biases', 2, None, '___sec39'),
+ ('Feed-forward pass', 2, None, '___sec40'),
+ ('Matrix multiplications', 2, None, '___sec41'),
+ ('Choose cost function and optimizer', 2, None, '___sec42'),
+ ('Optimizing the cost function', 2, None, '___sec43'),
+ ('Regularization', 2, None, '___sec44'),
+ ('Matrix multiplication', 2, None, '___sec45'),
+ ('Improving performance', 2, None, '___sec46'),
+ ('Full object-oriented implementation', 2, None, '___sec47'),
+ ('Evaluate model performance on test data', 2, None, '___sec48'),
+ ('Adjust hyperparameters', 2, None, '___sec49'),
+ ('Visualization', 2, None, '___sec50'),
+ ('scikit-learn implementation', 2, None, '___sec51'),
+ ('Visualization', 2, None, '___sec52'),
('Building neural networks in Tensorflow and Keras',
2,
None,
- '___sec52'),
- ('Tensorflow', 2, None, '___sec53'),
- ('Collect and pre-process data', 2, None, '___sec54'),
- ('Using TensorFlow backend', 2, None, '___sec55'),
- ('Optimizing and using gradient descent', 2, None, '___sec56'),
- ('Using Keras', 2, None, '___sec57'),
- ('Which activation function should I use?', 2, None, '___sec58'),
+ '___sec53'),
+ ('Tensorflow', 2, None, '___sec54'),
+ ('Collect and pre-process data', 2, None, '___sec55'),
+ ('Using TensorFlow backend', 2, None, '___sec56'),
+ ('Optimizing and using gradient descent', 2, None, '___sec57'),
+ ('Using Keras', 2, None, '___sec58'),
+ ('Which activation function should I use?', 2, None, '___sec59'),
('Is the Logistic activation function (Sigmoid) our choice?',
2,
None,
- '___sec59'),
- ('The derivative of the Logistic funtion', 2, None, '___sec60'),
- ('The RELU function family', 2, None, '___sec61'),
- ('Which activation function should we use?', 2, None, '___sec62'),
+ '___sec60'),
+ ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+ ('The RELU function family', 2, None, '___sec62'),
+ ('Which activation function should we use?', 2, None, '___sec63'),
('A top-down perspective on Neural networks',
2,
None,
- '___sec63'),
+ '___sec64'),
('Limitations of supervised learning with deep networks',
2,
None,
- '___sec64')]}
+ '___sec65')]}
end of tocinfo -->
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
epochs = 100
+batch_size = 100
+n_neurons_layer1 = 100
+n_neurons_layer2 = 50
+n_categories = 10
+eta_vals = np.logspace(-5, 1, 7)
+lmbd_vals = np.logspace(-5, 1, 7)
DNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+
DNN_tf = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
for i, eta in enumerate(eta_vals):
for j, lmbd in enumerate(lmbd_vals):
- DNN = create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories,
- eta=eta, lmbd=lmbd)
- DNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
- scores = DNN.evaluate(X_test, Y_test)
+ DNN = NeuralNetworkTensorflow(X_train, Y_train, X_test, Y_test,
+ n_neurons_layer1, n_neurons_layer2, n_categories,
+ epochs=epochs, batch_size=batch_size, eta=eta, lmbd=lmbd)
+ DNN.fit()
- DNN_keras[i][j] = DNN
+ DNN_tf[i][j] = DNN
print("Learning rate = ", eta)
print("Lambda = ", lmbd)
- print("Test accuracy: %.3f" % scores[1])
+ print("Test accuracy: %.3f" % DNN.test_accuracy)
print()
# optional
+# we can use log files to visualize our graph in Tensorboard
+writer = tf.summary.FileWriter('logs/')
+writer.add_graph(tf.get_default_graph())
+
Which activation function should I use?
+Using Keras
conda install keras
+
pip3 install keras
+
from keras.models import Sequential
+from keras.layers import Dense
+from keras.regularizers import l2
+from keras.optimizers import SGD
+
+def create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories, eta, lmbd):
+ model = Sequential()
+ model.add(Dense(n_neurons_layer1, activation='sigmoid', kernel_regularizer=l2(lmbd)))
+ model.add(Dense(n_neurons_layer2, activation='sigmoid', kernel_regularizer=l2(lmbd)))
+ model.add(Dense(n_categories, activation='softmax'))
+
+ sgd = SGD(lr=eta)
+ model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])
+
+ return model
+
DNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+
+for i, eta in enumerate(eta_vals):
+ for j, lmbd in enumerate(lmbd_vals):
+ DNN = create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories,
+ eta=eta, lmbd=lmbd)
+ DNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
+ scores = DNN.evaluate(X_test, Y_test)
+
+ DNN_keras[i][j] = DNN
+
+ print("Learning rate = ", eta)
+ print("Lambda = ", lmbd)
+ print("Test accuracy: %.3f" % scores[1])
+ print()
+
# optional
+# visual representation of grid search
+# uses seaborn heatmap, could probably do this in matplotlib
+import seaborn as sns
+
+sns.set()
+
+train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+
+for i in range(len(eta_vals)):
+ for j in range(len(lmbd_vals)):
+ DNN = DNN_keras[i][j]
+
+ train_accuracy[i][j] = DNN.evaluate(X_train, Y_train)[1]
+ test_accuracy[i][j] = DNN.evaluate(X_test, Y_test)[1]
+
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Training Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Test Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
Is the Logistic activation function (Sigmoid) our choice?
+Which activation function should I use?
The derivative of the Logistic funtion
+Is the Logistic activation function (Sigmoid) our choice?
The RELU function family
+The derivative of the Logistic funtion
Which activation function should we use?
+The RELU function family
A top-down perspective on Neural networks
+Which activation function should we use?
-
-
-If the validation and test sets are drawn from the same distributions,
-then a good performance on the validation set should lead to similarly
-good performance on the test set.
+In general it seems that the ELU activation function is better than
+the leaky ReLU function (and its variants), which is better than
+ReLU. ReLU performs better than \( \tanh \) which in turn performs better
+than the logistic function.
Limitations of supervised learning with deep networks
+A top-down perspective on Neural networks
-
-Some of these remarks are particular to DNNs, others are shared by all supervised learning methods. This motivates the use of unsupervised methods which in part circumnavigate these problems.
+If the validation and test sets are drawn from the same distributions,
+then a good performance on the validation set should lead to similarly
+good performance on the test set.
+
@@ -295,6 +314,8 @@ Some of these remarks are particular to DNNs, others are shared by all supervise
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs066.html b/doc/pub/NeuralNet/html/._NeuralNet-bs066.html
new file mode 100644
index 000000000..e1b566889
--- /dev/null
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs066.html
@@ -0,0 +1,323 @@
+
+
+
+
+
+
+
+Limitations of supervised learning with deep networks
+
+
+
+
+Some of these remarks are particular to DNNs, others are shared by all supervised learning methods. This motivates the use of unsupervised methods which in part circumnavigate these problems.
+
+Developing a code for doing neural networks with back propagation
+The Softmax function
+In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation \( z_i^l \), that is we need
+
+$$
+\frac{\partial f(z_i^l)}{\partial w_{jk}^l} =
+\frac{\partial f(z_i^l)}{\partial z_j^l} \frac{\partial z_j^l}{\partial w_{jk}^l}= \frac{\partial f(z_i^l)}{\partial z_j^l}a_k^{-1}.
+$$
+
+
+For the Softmax function we have
+
+$$
+f(z_i^l) = \frac{\exp{(z_i^l)}}{\sum_{k=1}^K\exp{(z_k^l}}.
+$$
+
+
+Its derivative with respect to \( z_j^l \) gives
+
+$$
+\frac{\partial f(z_i^l)}{\partial z_j^l}= f(z_i^l)\left(\delta_{ij}-f(z_i^l)\right),
+$$
+
+
+which in case of the simply binary model reduces to having \( i=j \).
+Developing a code for doing neural networks with back propagation
Collect and pre-process data
+Collect and pre-process data
Train and test datasets
+Train and test datasets
Define model and architecture
+Define model and architecture
Layers
+Layers
Weights and biases
+Weights and biases
Feed-forward pass
+Feed-forward pass
Matrix multiplications
+Matrix multiplications
Choose cost function and optimizer
+Choose cost function and optimizer
Optimizing the cost function
+Optimizing the cost function
Regularization
+Regularization
Matrix multiplication
+Matrix multiplication
Improving performance
+Improving performance
Full object-oriented implementation
Evaluate model performance on test data
+Evaluate model performance on test data
Adjust hyperparameters
+Adjust hyperparameters
Visualization
+Visualization
scikit-learn implementation
+scikit-learn implementation
Visualization
+Visualization
Building neural networks in Tensorflow and Keras
+Building neural networks in Tensorflow and Keras
Tensorflow
+Tensorflow
Collect and pre-process data
+Collect and pre-process data
Using TensorFlow backend
+Using TensorFlow backend
Optimizing and using gradient descent
+Optimizing and using gradient descent
Using Keras
+Using Keras
Which activation function should I use?
+Which activation function should I use?
Is the Logistic activation function (Sigmoid) our choice?
+Is the Logistic activation function (Sigmoid) our choice?
The derivative of the Logistic funtion
+The derivative of the Logistic funtion
The RELU function family
+The RELU function family
Which activation function should we use?
+Which activation function should we use?
A top-down perspective on Neural networks
+A top-down perspective on Neural networks
Limitations of supervised learning with deep networks
+Limitations of supervised learning with deep networks
+
+The Softmax function
+In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation \( z_i^l \), that is we need
+$$
+\frac{\partial f(z_i^l)}{\partial w_{jk}^l} =
+\frac{\partial f(z_i^l)}{\partial z_j^l} \frac{\partial z_j^l}{\partial w_{jk}^l}= \frac{\partial f(z_i^l)}{\partial z_j^l}a_k^{-1}.
+$$
+
+For the Softmax function we have
+$$
+f(z_i^l) = \frac{\exp{(z_i^l)}}{\sum_{k=1}^K\exp{(z_k^l}}.
+$$
+
+Its derivative with respect to \( z_j^l \) gives
+$$
+\frac{\partial f(z_i^l)}{\partial z_j^l}= f(z_i^l)\left(\delta_{ij}-f(z_i^l)\right),
+$$
+
+which in case of the simply binary model reduces to having \( i=j \).
+
Developing a code for doing neural networks with back propagation
+Developing a code for doing neural networks with back propagation
-Collect and pre-process data
+Collect and pre-process data
-Train and test datasets
+Train and test datasets
-Define model and architecture
+Define model and architecture
Layers
+Layers
Weights and biases
+Weights and biases
-Feed-forward pass
+Feed-forward pass
Matrix multiplications
+Matrix multiplications
-Choose cost function and optimizer
+Choose cost function and optimizer
-Optimizing the cost function
+Optimizing the cost function
Regularization
+Regularization
-Matrix multiplication
+Matrix multiplication
-Improving performance
+Improving performance
Full object-oriented implementation
-Evaluate model performance on test data
+Evaluate model performance on test data
-Adjust hyperparameters
+Adjust hyperparameters
-Visualization
+Visualization
-scikit-learn implementation
+scikit-learn implementation
-Visualization
+Visualization
-Building neural networks in Tensorflow and Keras
+Building neural networks in Tensorflow and Keras
-Tensorflow
+Tensorflow
-Collect and pre-process data
+Collect and pre-process data
-Using TensorFlow backend
+Using TensorFlow backend
-Optimizing and using gradient descent
+Optimizing and using gradient descent
-Using Keras
+Using Keras
Which activation function should I use?
+Which activation function should I use?
Is the Logistic activation function (Sigmoid) our choice?
+Is the Logistic activation function (Sigmoid) our choice?
-The derivative of the Logistic funtion
+The derivative of the Logistic funtion
-The RELU function family
+The RELU function family
-Which activation function should we use?
+Which activation function should we use?
A top-down perspective on Neural networks
+A top-down perspective on Neural networks
-Limitations of supervised learning with deep networks
+Limitations of supervised learning with deep networks
+
+The Softmax function
+In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation \( z_i^l \), that is we need
+$$
+\frac{\partial f(z_i^l)}{\partial w_{jk}^l} =
+\frac{\partial f(z_i^l)}{\partial z_j^l} \frac{\partial z_j^l}{\partial w_{jk}^l}= \frac{\partial f(z_i^l)}{\partial z_j^l}a_k^{-1}.
+$$
+
+For the Softmax function we have
+$$
+f(z_i^l) = \frac{\exp{(z_i^l)}}{\sum_{k=1}^K\exp{(z_k^l}}.
+$$
+
+Its derivative with respect to \( z_j^l \) gives
+$$
+\frac{\partial f(z_i^l)}{\partial z_j^l}= f(z_i^l)\left(\delta_{ij}-f(z_i^l)\right),
+$$
+
+which in case of the simply binary model reduces to having \( i=j \).
+
Developing a code for doing neural networks with back propagation
+Developing a code for doing neural networks with back propagation
-Collect and pre-process data
+Collect and pre-process data
-Train and test datasets
+Train and test datasets
-Define model and architecture
+Define model and architecture
Layers
+Layers
Weights and biases
+Weights and biases
-Feed-forward pass
+Feed-forward pass
Matrix multiplications
+Matrix multiplications
-Choose cost function and optimizer
+Choose cost function and optimizer
-Optimizing the cost function
+Optimizing the cost function
Regularization
+Regularization
-Matrix multiplication
+Matrix multiplication
-Improving performance
+Improving performance
Full object-oriented implementation
-Evaluate model performance on test data
+Evaluate model performance on test data
Adjust hyperparameters
-Visualization
+Visualization
-scikit-learn implementation
+scikit-learn implementation
-Visualization
+Visualization
-Building neural networks in Tensorflow and Keras
+Building neural networks in Tensorflow and Keras
-Tensorflow
+Tensorflow
-Collect and pre-process data
+Collect and pre-process data
-Using TensorFlow backend
+Using TensorFlow backend
-Optimizing and using gradient descent
+Optimizing and using gradient descent
Using Keras
Which activation function should I use?
+Which activation function should I use?
Is the Logistic activation function (Sigmoid) our choice?
+Is the Logistic activation function (Sigmoid) our choice?
-The derivative of the Logistic funtion
+The derivative of the Logistic funtion
-The RELU function family
+The RELU function family
-Which activation function should we use?
+Which activation function should we use?
A top-down perspective on Neural networks
+A top-down perspective on Neural networks
-Limitations of supervised learning with deep networks
+Limitations of supervised learning with deep networks