@@ -157,12 +237,9 @@ MathJax.Hub.Config({
-
- - -
@@ -184,7 +261,7 @@ MathJax.Hub.Config({
In quantum information theory, it has been shown that one can perform -gate decompositions with the help of neural. In lattice quantum chromodynamics, -DNNs have been used to learn action parameters in regions of parameter -space where PCA fails. +gate decompositions with the help of neural.
-The applications are not limited to the natural sciences. There is a plethora of applications in essentially all disciplines, from the humanities to life science and medicine. +The applications are not limited to the natural sciences. There is a +plethora of applications in essentially all disciplines, from the +humanities to life science and medicine.
@@ -217,7 +296,7 @@ The applications are not limited to the natural sciences. There is a plethora of
-An artificial neural network (NN), is a computational model that -consists of layers of connected neurons, or nodes. It is supposed -to mimic a biological nervous system by letting each neuron interact -with other neurons by sending signals in the form of mathematical -functions between layers. A wide variety of different NNs have been -developed, but most of them consist of an input layer, an output layer -and eventual layers in-between, called hidden layers. All layers can -contain an arbitrary number of nodes, and each connection between two -nodes is associated with a weight variable. +An artificial neural network (ANN), is a computational model that +consists of layers of connected neurons, or nodes or units. We will +refer to these interchangeably as units or nodes, and sometimes as +neurons. + +
+It is supposed to mimic a biological nervous system by letting each +neuron interact with other neurons by sending signals in the form of +mathematical functions between layers. A wide variety of different +ANNs have been developed, but most of them consist of an input layer, +an output layer and eventual layers in-between, called hidden +layers. All layers can contain an arbitrary number of nodes, and each +connection between two nodes is associated with a weight variable.
Neural networks (also called neural nets) are neural-inspired nonlinear models for supervised learning. As we will see, neural nets can be viewed as natural, more powerful extensions of supervised learning methods such as linear and logistic regression and soft-max -methods. +methods we discussed earlier.
@@ -177,7 +261,7 @@ methods.
-The feed-forward neural network (FFNN) was the first and simplest type of NN devised. In this network, -the information moves in only one direction: forward through the layers. +The feed-forward neural network (FFNN) was the first and simplest type +of ANNs that were devised. In this network, the information moves in +only one direction: forward through the layers.
-Nodes are represented by circles, while the arrows display the connections between the nodes, including the -direction of information flow. Additionally, each arrow corresponds to a weight variable, not displayed here. -We observe that each node in a layer is connected to all nodes in the subsequent layer, -making this a so-called fully-connected FFNN. - -
-A different variant of FFNNs are convolutional neural networks (CNNs), which have a connectivity pattern -inspired by the animal visual cortex. Individual neurons in the visual cortex only respond to stimuli from -small sub-regions of the visual field, called a receptive field. This makes the neurons well-suited to exploit the strong -spatially local correlation present in natural images. The response of each neuron can be approximated mathematically -as a convolution operation. - -
-CNNs emulate the behaviour of neurons in the visual cortex by enforcing a local connectivity pattern -between nodes of adjacent layers: Each node -in a convolutional layer is connected only to a subset of the nodes in the previous layer, -in contrast to the fully-connected FFNN. -Often, CNNs -consist of several convolutional layers that learn local features of the input, with a fully-connected layer at the end, -which gathers all the local data and produces the outputs. They have wide applications in image and video recognition +Nodes are represented by circles, while the arrows display the +connections between the nodes, including the direction of information +flow. Additionally, each arrow corresponds to a weight variable +(figure to come). We observe that each node in a layer is connected +to all nodes in the subsequent layer, making this a so-called +fully-connected FFNN.
@@ -186,7 +253,7 @@ which gathers all the local data and produces the outputs. They have wide applic
-So far we have only mentioned NNs where information flows in one direction: forward. Recurrent neural networks on -the other hand, have connections between nodes that form directed cycles. This creates a form of -internal memory which are able to capture information on what has been calculated before; the output is dependent -on the previous computations. Recurrent NNs make use of sequential information by performing the same task for -every element in a sequence, where each element depends on previous elements. An example of such information is -sentences, making recurrent NNs especially well-suited for handwriting and speech recognition. +A different variant of FFNNs are convolutional neural networks +(CNNs), which have a connectivity pattern inspired by the animal +visual cortex. Individual neurons in the visual cortex only respond to +stimuli from small sub-regions of the visual field, called a receptive +field. This makes the neurons well-suited to exploit the strong +spatially local correlation present in natural images. The response of +each neuron can be approximated mathematically as a convolution +operation. (figure to come) + +
+Convolutional neural networks emulate the behaviour of neurons in the +visual cortex by enforcing a local connectivity pattern between +nodes of adjacent layers: Each node in a convolutional layer is +connected only to a subset of the nodes in the previous layer, in +contrast to the fully-connected FFNN. Often, CNNs consist of several +convolutional layers that learn local features of the input, with a +fully-connected layer at the end, which gathers all the local data and +produces the outputs. They have wide applications in image and video +recognition.
@@ -169,7 +262,7 @@ sentences, making recurrent NNs especially well-suited for handwriting and speec
-There are many other kinds of NNs that have been developed. One type that is specifically designed for interpolation -in multidimensional space is the radial basis function (RBF) network. RBFs are typically made up of three layers: -an input layer, a hidden layer with non-linear radial symmetric activation functions and a linear output layer (''linear'' here -means that each node in the output layer has a linear activation function). The layers are normally fully-connected and -there are no cycles, thus RBFs can be viewed as a type of fully-connected FFNN. They are however usually treated as -a separate type of NN due the unusual activation functions. +So far we have only mentioned ANNs where information flows in one +direction: forward. Recurrent neural networks on the other hand, +have connections between nodes that form directed cycles. This +creates a form of internal memory which are able to capture +information on what has been calculated before; the output is +dependent on the previous computations. Recurrent NNs make use of +sequential information by performing the same task for every element +in a sequence, where each element depends on previous elements. An +example of such information is sentences, making recurrent NNs +especially well-suited for handwriting and speech recognition.
@@ -170,7 +254,7 @@ a separate type of NN due the unusual activation functions.
-One uses often so-called fully-connected feed-forward neural networks -with three or more layers (an input layer, one or more hidden layers -and an output layer) consisting of neurons that have non-linear -activation functions. - -
-Such networks are often called multilayer perceptrons (MLPs) +There are many other kinds of ANNs that have been developed. One type +that is specifically designed for interpolation in multidimensional +space is the radial basis function (RBF) network. RBFs are typically +made up of three layers: an input layer, a hidden layer with +non-linear radial symmetric activation functions and a linear output +layer (''linear'' here means that each node in the output layer has a +linear activation function). The layers are normally fully-connected +and there are no cycles, thus RBFs can be viewed as a type of +fully-connected FFNN. They are however usually treated as a separate +type of NN due the unusual activation functions.
@@ -172,7 +255,7 @@ Such networks are often called multilayer perceptrons (MLPs)
-According to the Universal approximation theorem, a feed-forward neural network with just a single hidden layer containing -a finite number of neurons can approximate a continuous multidimensional function to arbitrary accuracy, -assuming the activation function for the hidden layer is a non-constant, bounded and monotonically-increasing continuous function. +One uses often so-called fully-connected feed-forward neural networks +with three or more layers (an input layer, one or more hidden layers +and an output layer) consisting of neurons that have non-linear +activation functions.
-Note that the requirements on the activation function only applies to -the hidden layer, the output nodes are always assumed to be linear, so -as to not restrict the range of output values. - -
-We note that this theorem is only applicable to an NN with one hidden -layer. Therefore, we can easily construct an NN that employs -activation functions which do not satisfy the above requirements, as -long as we have at least one layer with activation functions that -do. Furthermore, although the universal approximation theorem lays -the theoretical foundation for regression with neural networks, it -does not say anything about how things work in practice: A neural -network can still be able to approximate a given function reasonably -well without having the flexibility to fit all other functions. +Such networks are often called multilayer perceptrons (MLPs).
@@ -185,7 +253,7 @@ well without having the flexibility to fit all other functions.
-In an FFNN of such neurons, the inputs \( x_i \) are the outputs of -the neurons in the preceding layer. Furthermore, an MLP is -fully-connected, which means that each neuron receives a weighted sum -of the outputs of all neurons in the previous layer. +According to the Universal approximation theorem, a feed-forward +neural network with just a single hidden layer containing a finite +number of neurons can approximate a continuous multidimensional +function to arbitrary accuracy, assuming the activation function for +the hidden layer is a non-constant, bounded and +monotonically-increasing continuous function. + +
+Note that the requirements on the activation function only applies to +the hidden layer, the output nodes are always assumed to be linear, so +as to not restrict the range of output values.
@@ -178,7 +258,7 @@ of the outputs of all neurons in the previous layer.
-First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( u_i^1 \) of the input coordinates \( x_j \), - +The output \( y \) is produced via the activation function \( f \) $$ -\begin{equation} - u_i^1 = \sum_{j=1}^2 w_{ij}^1 x_j + b_i^1 -\tag{3} -\end{equation} + y = f\left(\sum_{i=1}^n w_ix_i + b_i\right) = f(z), $$ -This value is the argument to the activation function \( f_1 \) of each neuron \( i \), -producing the output \( y_i^1 \) of all neurons in layer 1, - -$$ -\begin{equation} - y_i^1 = f_1(u_i^1) = f_1\left(\sum_{j=1}^2 w_{ij}^1 x_j + b_i^1\right) -\tag{4} -\end{equation} -$$ - -
-where we assume that all nodes in the same layer have identical -activation functions, hence the notation \( f_l \) - -$$ -\begin{equation} - y_i^l = f_l(u_i^l) = f_l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right) -\tag{5} -\end{equation} -$$ - -
-where \( N_l \) is the number of nodes in layer \( l \). When the output of -all the nodes in the first hidden layer are computed, the values of -the subsequent layer can be calculated and so forth until the output -is obtained. +This function receives \( x_i \) as inputs. +Here the activation \( z=\sum_{i=1}^n w_ix_i \). +In an FFNN of such neurons, the inputs \( x_i \) are the outputs of +the neurons in the preceding layer. Furthermore, an MLP is +fully-connected, which means that each neuron receives a weighted sum +of the outputs of all neurons in the previous layer.
@@ -203,7 +259,7 @@ is obtained.
-The output of neuron \( i \) in layer 2 is thus, +First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( z_i^1 \) of the input coordinates \( x_j \), $$ -\begin{align} - y_i^2 &= f_2\left(\sum_{j=1}^3 w_{ij}^2 y_j^1 + b_i^2\right) -\tag{6}\\ - &= f_2\left[\sum_{j=1}^3 w_{ij}^2f_1\left(\sum_{k=1}^2 w_{jk}^1 x_k + b_j^1\right) + b_i^2\right] -\tag{7} -\end{align} +\begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1 +\tag{2} +\end{equation} $$ -where we have substituted \( y_m^1 \) with. Finally, the NN output yields, +
+Here \( b_i \) is the so-called bias which is normally needed in +case of zero activation weights or inputs. How to fix the biases and +the weights will be discussed below. The value of \( z_i^1 \) is the +argument to the activation function \( f_i \) of each node \( i \), The +variable \( M \) stands for all possible inputs to a given node \( i \) in the +first layer. We define the output \( y_i^1 \) of all neurons in layer 1 as $$ -\begin{align} - y_1^3 &= f_3\left(\sum_{j=1}^3 w_{1m}^3 y_j^2 + b_1^3\right) -\tag{8}\\ - &= f_3\left[\sum_{j=1}^3 w_{1j}^3 f_2\left(\sum_{k=1}^3 w_{jk}^2 f_1\left(\sum_{m=1}^2 w_{km}^1 x_m + b_k^1\right) + b_j^2\right) - + b_1^3\right] -\tag{9} -\end{align} +\begin{equation} + y_i^1 = f(z_i^1) = f\left(\sum_{j=1}^M w_{ij}^1 x_j + b_i^1\right) +\tag{3} +\end{equation} $$ +
+where we assume that all nodes in the same layer have identical +activation functions, hence the notation \( f \). In general, we could assume in the more general case that different layers have different activation functions. +In this case we would identify these functions with a superscript \( l \) for the \( l \)-th layer, + +$$ +\begin{equation} + y_i^l = f^l(u_i^l) = f^l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right) +\tag{4} +\end{equation} +$$ + +
+where \( N_l \) is the number of nodes in layer \( l \). When the output of +all the nodes in the first hidden layer are computed, the values of +the subsequent layer can be calculated and so forth until the output +is obtained. +
@@ -189,6 +287,8 @@ $$
-We can generalize this expression to an MLP with \( l \) hidden -layers. The complete functional form is, +The output of neuron \( i \) in layer 2 is thus, $$ \begin{align} -&y^{l+1}_1\! = \!f_{l+1}\!\left[\!\sum_{j=1}^{N_l}\! w_{1j}^3 f_l\!\left(\!\sum_{k=1}^{N_{l-1}}\! w_{jk}^2 f_{l-1}\!\left(\! - \dots \!f_1\!\left(\!\sum_{n=1}^{N_0} \!w_{mn}^1 x_n\! + \!b_m^1\!\right) - \!\dots \!\right) \!+ \!b_k^2\!\right) - \!+ \!b_1^3\!\right] && -\tag{10} + y_i^2 &= f^2\left(\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\right) +\tag{5}\\ + &= f^2\left[\sum_{j=1}^N w_{ij}^2f^1\left(\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\right) + b_i^2\right] +\tag{6} \end{align} $$ -
-which illustrates a basic property of MLPs: The only independent -variables are the input values \( x_n \). +where we have substituted \( y_k^1 \) with the inputs \( x_k \). Finally, the ANN output reads + +$$ +\begin{align} + y_i^3 &= f^3\left(\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\right) +\tag{7}\\ + &= f_3\left[\sum_{j} w_{ij}^3 f^2\left(\sum_{k} w_{jk}^2 f^1\left(\sum_{m} w_{km}^1 x_m + b_k^1\right) + b_j^2\right) + + b_1^3\right] +\tag{8} +\end{align} +$$
@@ -182,6 +268,9 @@ variables are the input values \( x_n \).
-This confirms that an MLP, despite its quite convoluted mathematical -form, is nothing more than an analytic function, specifically a -mapping of real-valued vectors \( \vec{x} \in \mathbb{R}^n \rightarrow -\vec{y} \in \mathbb{R}^m \). In our example, \( n=2 \) and -\( m=1 \). Consequentially, the number of input and output values of the -function we want to fit must be equal to the number of inputs and -outputs of our MLP. - -
-Furthermore, the flexibility and universality of a MLP can be -illustrated by realizing that the expression is essentially a nested -sum of scaled activation functions of the form +We can generalize this expression to an MLP with \( l \) hidden +layers. The complete functional form is, $$ -\begin{equation} - h(x) = c_1 f(c_2 x + c_3) + c_4 -\tag{11} -\end{equation} +\begin{align} +&y^{l+1}_i = f^{l+1}\left[\!\sum_{j=1}^{N_l} w_{ij}^3 f^l\left(\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\left(\dots f^1\left(\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\right)\dots\right)+b_k^2\right)+b_1^3\right] && +\tag{9} +\end{align} $$
-where the parameters \( c_i \) are weights and biases. By adjusting these -parameters, the activation functions can be shifted up and down or -left and right, change slope or be rescaled which is the key to the -flexibility of a neural network. +which illustrates a basic property of MLPs: The only independent +variables are the input values \( x_n \).
@@ -190,6 +258,10 @@ flexibility of a neural network.
-We can introduce a more convenient notation for the activations in a NN. +This confirms that an MLP, despite its quite convoluted mathematical +form, is nothing more than an analytic function, specifically a +mapping of real-valued vectors \( \hat{x} \in \mathbb{R}^n \rightarrow +\hat{y} \in \mathbb{R}^m \).
-Additionally, we can represent the biases and activations -as layer-wise column vectors \( \vec{b}_l \) and \( \vec{y}_l \), so that the \( i \)-th element of each vector -is the bias \( b_i^l \) and activation \( y_i^l \) of node \( i \) in layer \( l \) respectively. +Furthermore, the flexibility and universality of an MLP can be +illustrated by realizing that the expression is essentially a nested +sum of scaled activation functions of the form -
-We have that \( \mathrm{W}_l \) is a \( N_{l-1} \times N_l \) matrix, while \( \vec{b}_l \) and \( \vec{y}_l \) are \( N_l \times 1 \) column vectors. -With this notation, the sum in becomes a matrix-vector multiplication, and we can write -the equation for the activations of hidden layer 2 in $$ \begin{equation} - \vec{y}_2 = f_2(\mathrm{W}_2 \vec{y}_{1} + \vec{b}_{2}) = - f_2\left(\left[\begin{array}{ccc} - w^2_{11} &w^2_{12} &w^2_{13} \\ - w^2_{21} &w^2_{22} &w^2_{23} \\ - w^2_{31} &w^2_{32} &w^2_{33} \\ - \end{array} \right] \cdot - \left[\begin{array}{c} - y^1_1 \\ - y^1_2 \\ - y^1_3 \\ - \end{array}\right] + - \left[\begin{array}{c} - b^2_1 \\ - b^2_2 \\ - b^2_3 \\ - \end{array}\right]\right). -\tag{12} + f(x) = c_1 f(c_2 x + c_3) + c_4 +\tag{10} \end{equation} $$ +
+where the parameters \( c_i \) are weights and biases. By adjusting these +parameters, the activation functions can be shifted up and down or +left and right, change slope or be rescaled which is the key to the +flexibility of a neural network. +
@@ -196,6 +266,11 @@ $$
-The activation of node \( i \) in layer 2 is +We can introduce a more convenient notation for the activations in an A NN. +
+Additionally, we can represent the biases and activations +as layer-wise column vectors \( \hat{b}_l \) and \( \hat{y}_l \), so that the \( i \)-th element of each vector +is the bias \( b_i^l \) and activation \( y_i^l \) of node \( i \) in layer \( l \) respectively. + +
+We have that \( \mathrm{W}_l \) is an \( N_{l-1} \times N_l \) matrix, while \( \hat{b}_l \) and \( \hat{y}_l \) are \( N_l \times 1 \) column vectors. +With this notation, the sum becomes a matrix-vector multiplication, and we can write +the equation for the activations of hidden layer 2 as $$ \begin{equation} - y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) = - f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right). -\tag{13} + \hat{y}_2 = f_2(\mathrm{W}_2 \hat{y}_{1} + \hat{b}_{2}) = + f_2\left(\left[\begin{array}{ccc} + w^2_{11} &w^2_{12} &w^2_{13} \\ + w^2_{21} &w^2_{22} &w^2_{23} \\ + w^2_{31} &w^2_{32} &w^2_{33} \\ + \end{array} \right] \cdot + \left[\begin{array}{c} + y^1_1 \\ + y^1_2 \\ + y^1_3 \\ + \end{array}\right] + + \left[\begin{array}{c} + b^2_1 \\ + b^2_2 \\ + b^2_3 \\ + \end{array}\right]\right). +\tag{11} \end{equation} $$ -
-This is not just a convenient and compact notation, but also a useful -and intuitive way to think about MLPs: The output is calculated by a -series of matrix-vector multiplications and vector additions that are -used as input to the activation functions. For each operation -\( \mathrm{W}_l \vec{y}_{l-1} \) we move forward one layer. -
@@ -179,6 +275,12 @@ used as input to the activation functions. For each operation
-A property that characterizes a neural network, other than its -connectivity, is the choice of activation function(s). As described -in, the following restrictions are imposed on an activation function -for a FFNN to fulfill the universal approximation theorem +The activation of node \( i \) in layer 2 is -
+This is not just a convenient and compact notation, but also a useful +and intuitive way to think about MLPs: The output is calculated by a +series of matrix-vector multiplications and vector additions that are +used as input to the activation functions. For each operation +\( \mathrm{W}_l \hat{y}_{l-1} \) we move forward one layer. + +
-The second requirement excludes all linear functions. Furthermore, in -a MLP with only linear activation functions, each layer simply -performs a linear transformation of its inputs. +A property that characterizes a neural network, other than its +connectivity, is the choice of activation function(s). As described +in, the following restrictions are imposed on an activation function +for a FFNN to fulfill the universal approximation theorem -
-Regardless of the number of layers, the output of the NN will be -nothing but a linear function of the inputs. Thus we need to introduce -some kind of non-linearity to the NN to be able to fit non-linear -functions Typical examples are the logistic Sigmoid +
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs018.html b/doc/pub/NeuralNet/html/._NeuralNet-bs018.html index 168753939..773b51dce 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs018.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs018.html @@ -6,9 +6,9 @@ Automatically generated HTML file from DocOnce source - + -
-The sigmoid function are more biologically plausible because the -output of inactive neurons are zero. Such activation function are -called one-sided. However, it has been shown that the hyperbolic -tangent performs better than the sigmoid for training MLPs. has -become the most popular for deep neural networks +The second requirement excludes all linear functions. Furthermore, in +a MLP with only linear activation functions, each layer simply +performs a linear transformation of its inputs.
+Regardless of the number of layers, the output of the NN will be +nothing but a linear function of the inputs. Thus we need to introduce +some kind of non-linearity to the NN to be able to fit non-linear +functions Typical examples are the logistic Sigmoid - -
"""The sigmoid function (or the logistic curve) is a
-function that takes any real number, z, and outputs a number (0,1).
-It is useful in neural networks for assigning weights on a relative scale.
-The value z is the weighted sum of parameters involved in the learning algorithm."""
+$$
+ f(x) = \frac{1}{1 + e^{-x}},
+$$
-import numpy
-import matplotlib.pyplot as plt
-import math as mt
+and the hyperbolic tangent function
+$$
+ f(x) = \tanh(x)
+$$
-z = numpy.arange(-5, 5, .1)
-sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))
-sigma = sigma_fn(z)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, sigma)
-ax.set_ylim([-0.1, 1.1])
-ax.set_xlim([-5,5])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('sigmoid function')
-
-plt.show()
-
-"""Step Function"""
-z = numpy.arange(-5, 5, .02)
-step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)
-step = step_fn(z)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, step)
-ax.set_ylim([-0.5, 1.5])
-ax.set_xlim([-5,5])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('step function')
-
-plt.show()
-
-"""Sine Function"""
-z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)
-t = numpy.sin(z)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, t)
-ax.set_ylim([-1.0, 1.0])
-ax.set_xlim([-2*mt.pi,2*mt.pi])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('sine function')
-
-plt.show()
-
-"""Plots a graph of the squashing function used by a rectified linear
-unit"""
-z = numpy.arange(-2, 2, .1)
-zero = numpy.zeros(len(z))
-y = numpy.max([zero, z], axis=0)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, y)
-ax.set_ylim([-2.0, 2.0])
-ax.set_xlim([-2.0, 2.0])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('Rectified linear unit')
-
-plt.show()
-
@@ -240,6 +258,15 @@ plt.show()
- + -
+The sigmoid function are more biologically plausible because the +output of inactive neurons are zero. Such activation function are +called one-sided. However, it has been shown that the hyperbolic +tangent performs better than the sigmoid for training MLPs. has +become the most popular for deep neural networks
-
from scipy import optimize
+"""The sigmoid function (or the logistic curve) is a
+function that takes any real number, z, and outputs a number (0,1).
+It is useful in neural networks for assigning weights on a relative scale.
+The value z is the weighted sum of parameters involved in the learning algorithm."""
-class Neural_Network(object):
- def __init__(self, Lambda=0):
- #Define Hyperparameters
- self.inputLayerSize = 2
- self.outputLayerSize = 1
- self.hiddenLayerSize = 3
-
- #Weights (parameters)
- self.W1 = np.random.randn(self.inputLayerSize,self.hiddenLayerSize)
- self.W2 = np.random.randn(self.hiddenLayerSize,self.outputLayerSize)
-
- #Regularization Parameter:
- self.Lambda = Lambda
-
- def forward(self, X):
- #Propogate inputs though network
- self.z2 = np.dot(X, self.W1)
- self.a2 = self.sigmoid(self.z2)
- self.z3 = np.dot(self.a2, self.W2)
- yHat = self.sigmoid(self.z3)
- return yHat
-
- def sigmoid(self, z):
- #Apply sigmoid activation function to scalar, vector, or matrix
- return 1/(1+np.exp(-z))
-
- def sigmoidPrime(self,z):
- #Gradient of sigmoid
- return np.exp(-z)/((1+np.exp(-z))**2)
-
- def costFunction(self, X, y):
- #Compute cost for given X,y, use weights already stored in class.
- self.yHat = self.forward(X)
- J = 0.5*sum((y-self.yHat)**2)/X.shape[0] + (self.Lambda/2)*(np.sum(self.W1**2)+np.sum(self.W2**2))
- return J
-
- def costFunctionPrime(self, X, y):
- #Compute derivative with respect to W and W2 for a given X and y:
- self.yHat = self.forward(X)
-
- delta3 = np.multiply(-(y-self.yHat), self.sigmoidPrime(self.z3))
- #Add gradient of regularization term:
- dJdW2 = np.dot(self.a2.T, delta3)/X.shape[0] + self.Lambda*self.W2
-
- delta2 = np.dot(delta3, self.W2.T)*self.sigmoidPrime(self.z2)
- #Add gradient of regularization term:
- dJdW1 = np.dot(X.T, delta2)/X.shape[0] + self.Lambda*self.W1
-
- return dJdW1, dJdW2
-
- #Helper functions for interacting with other methods/classes
- def getParams(self):
- #Get W1 and W2 Rolled into vector:
- params = np.concatenate((self.W1.ravel(), self.W2.ravel()))
- return params
-
- def setParams(self, params):
- #Set W1 and W2 using single parameter vector:
- W1_start = 0
- W1_end = self.hiddenLayerSize*self.inputLayerSize
- self.W1 = np.reshape(params[W1_start:W1_end], \
- (self.inputLayerSize, self.hiddenLayerSize))
- W2_end = W1_end + self.hiddenLayerSize*self.outputLayerSize
- self.W2 = np.reshape(params[W1_end:W2_end], \
- (self.hiddenLayerSize, self.outputLayerSize))
-
- def computeGradients(self, X, y):
- dJdW1, dJdW2 = self.costFunctionPrime(X, y)
- return np.concatenate((dJdW1.ravel(), dJdW2.ravel()))
-
-
-class trainer(object):
- def __init__(self, N):
- #Make Local reference to network:
- self.N = N
-
- def callbackF(self, params):
- self.N.setParams(params)
- self.J.append(self.N.costFunction(self.X, self.y))
- self.testJ.append(self.N.costFunction(self.testX, self.testY))
-
- def costFunctionWrapper(self, params, X, y):
- self.N.setParams(params)
- cost = self.N.costFunction(X, y)
- grad = self.N.computeGradients(X,y)
- return cost, grad
-
- def train(self, trainX, trainY, testX, testY):
- #Make an internal variable for the callback function:
- self.X = trainX
- self.y = trainY
-
- self.testX = testX
- self.testY = testY
+import numpy
+import matplotlib.pyplot as plt
+import math as mt
- #Make empty list to store training costs:
- self.J = []
- self.testJ = []
-
- params0 = self.N.getParams()
+z = numpy.arange(-5, 5, .1)
+sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))
+sigma = sigma_fn(z)
- options = {'maxiter': 200, 'disp' : True}
- _res = optimize.minimize(self.costFunctionWrapper, params0, jac=True, method='BFGS', \
- args=(trainX, trainY), options=options, callback=self.callbackF)
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, sigma)
+ax.set_ylim([-0.1, 1.1])
+ax.set_xlim([-5,5])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('sigmoid function')
- self.N.setParams(_res.x)
- self.optimizationResults = _res
+plt.show()
+
+"""Step Function"""
+z = numpy.arange(-5, 5, .02)
+step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)
+step = step_fn(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, step)
+ax.set_ylim([-0.5, 1.5])
+ax.set_xlim([-5,5])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('step function')
+
+plt.show()
+
+"""Sine Function"""
+z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)
+t = numpy.sin(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, t)
+ax.set_ylim([-1.0, 1.0])
+ax.set_xlim([-2*mt.pi,2*mt.pi])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('sine function')
+
+plt.show()
+
+"""Plots a graph of the squashing function used by a rectified linear
+unit"""
+z = numpy.arange(-2, 2, .1)
+zero = numpy.zeros(len(z))
+y = numpy.max([zero, z], axis=0)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, y)
+ax.set_ylim([-2.0, 2.0])
+ax.set_xlim([-2.0, 2.0])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('Rectified linear unit')
+
+plt.show()
@@ -270,6 +319,16 @@ MathJax.Hub.Config({
- + + +
+The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of - -
import numpy as np
+
+- A neural network with one or more layers of nodes between the input and the output nodes.
+- The multilayer network structure, or architecture, or topology, consists of an input layer, one or more hidden layers, and one output layer.
+- The input nodes pass values to the first hidden layer, its nodes pass the information on to the second and so on till we reach the output layer.
+
-#sigmoid
-def nonlin(x, deriv=False):
- if (deriv==True):
- return x*(1-x)
- return 1/(1+np.exp(-x))
+As a convention it is normal to call a network with one layer of input units, one layer of hidden
+units and one layer of output units as a two-layer network. A network with two layers of hidden units is called a three-layer network etc etc.
-#input data
-x=np.array([[0,0,1],[0,1,1],[1,0,1],[1,1,1]])
-
-#output data
-y=np.array([0,1,1,0]).T
-
-#seed random numbers to make calculation
-np.random.seed(1)
-
-#initialize weights with mean=0
-syn0=2*np.random.random((3,4))-1
-
-for iter in range(10000):
- #forward propogation
- l0=x
- l1=nonlin(np.dot(l0,syn0))
- l1_error=y-l1
- #multiply error by slope of sigmoid at values of l1
- l1_delta=l1_error*nonlin(l1,True)
- #update weights
- syn0+=np.dot(l0.T, l1_delta)
-
-print("Output after training: ",l1 )
-+For an MLP there is no direct connection between the output nodes/neurons/units and the input nodes/neurons/units. +Hereafter we will call the various entities of a layer for nodes. +There are also no connections within a single layer. - -
import numpy as np
-import random
-class Network(object):
-
- def _init_(self, sizes):
- self.num_layers=len(sizes)
- self.sizes=sizes
- self.biases=[np.random.randn(y,1) for y in sizes[1:]]
- self.weights=[np.random.randn(y,x) for x,y in zip(sizes[:-1], sizes[1:])]
-
-#sizes is the number of neurons in each layer
-#for example, say n_1st_layer=3, n_2nd_layer=3, n_3rd_layer=1, then net=Network([3,3,1])
-
-#The biases and weights are initialized randomly, using Gaussian distributions of mean=0, stdev=1
-#z is a vector (or a np.array)
-
- def feedforward(self,a):
- #returns output w/ 'a' as an input
- for b, w in zip(self.biases, self.weights):
- a=sigmoid(np.dot(w,b)+b)
- return a
-
-#Apply a Stochastic Gradient Descent (SGD) method:
- def SGD(self, training_data, epochs, mini_batch_size, eta, test_data=None):
- """Trains network using batches incorporating SGD. The network will be evaluated against the
- test data after each epoch, with partial progress being printed out (this is useful for tracking,
- but slows the process.)"""
- if test_data: n_test=len(test_data)
- n=len(training_data)
- for j in xrange(epochs):
- random.shuffle(training_data)
- mini_batches=[training_data[k:k+mini_batch_size] for k in xrange(o,n,mini_batch_size)]
- for mini_batch in mini_batches:
- self.update_mini_batch(mini_batch, eta)
- if test_data:
- print ("Epoch {0}: {1}/{2}".format(j, self.evaluate(test_data), n_test))
- else:
- print ("Epoch {0} complete".format(j))
-
-
- def update_mini_batch(self, mini_batch, eta):
- #updates w and b using backpropagation to a single mini batch. eta is the learning rate."
- nabla_b=[np.zeros(b.shape) for b in self.biases]
- nabla_w=[np.zeros(w.shape) for w in self.weights]
- for x,y in mini_batch:
- delta_nabla_b, delta_nabla_w=self.backprop(x,y)
- nabla_b=[nb+dnb for nb, dnb in zip(nabla_b, delta_nabla_b)]
- nabla_w=[nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)]
- self.weights=[w-(eta/len(mini_batch))*nw for w, nw in zip(self.weights, nabla_w)]
- self.biases=[b-(eta/len(mini_batch))*nb for b, nb in zip(self.biases, nabla_b)]
-
- def backprop(self, x, y):
- """Return a tuple ``(nabla_b, nabla_w)`` representing the
- gradient for the cost function C_x. ``nabla_b`` and
- ``nabla_w`` are layer-by-layer lists of numpy arrays, similar
- to ``self.biases`` and ``self.weights``."""
- nabla_b = [np.zeros(b.shape) for b in self.biases]
- nabla_w = [np.zeros(w.shape) for w in self.weights]
- # feedforward
- activation = x
- activations = [x] # list to store all the activations, layer by layer
- zs = [] # list to store all the z vectors, layer by layer
- for b, w in zip(self.biases, self.weights):
- z = np.dot(w, activation)+b
- zs.append(z)
- activation = sigmoid(z)
- activations.append(activation)
- # backward pass
- delta = self.cost_derivative(activations[-1], y) * \
- sigmoid_prime(zs[-1])
- nabla_b[-1] = delta
- nabla_w[-1] = np.dot(delta, activations[-2].transpose())
- # Note that the variable l in the loop below is used a little
- # differently to the notation in Chapter 2 of the book. Here,
- # l = 1 means the last layer of neurons, l = 2 is the
- # second-last layer, and so on. It's a renumbering of the
- # scheme in the book, used here to take advantage of the fact
- # that Python can use negative indices in lists.
- for l in xrange(2, self.num_layers):
- z = zs[-l]
- sp = sigmoid_prime(z)
- delta = np.dot(self.weights[-l+1].transpose(), delta) * sp
- nabla_b[-l] = delta
- nabla_w[-l] = np.dot(delta, activations[-l-1].transpose())
- return (nabla_b, nabla_w)
-
- def evaluate(self, test_data):
- """Return the number of test inputs for which the neural
- network outputs the correct result. Note that the neural
- network's output is assumed to be the index of whichever
- neuron in the final layer has the highest activation."""
- test_results = [(np.argmax(self.feedforward(x)), y)
- for (x, y) in test_data]
- return sum(int(x == y) for (x, y) in test_results)
-
- def cost_derivative(self, output_activations, y):
- """Return the vector of partial derivatives \partial C_x /
- \partial a for the output activations."""
- return (output_activations-y)
-
-
-
-#Functions
-def sigmoid(z):
- return 1.0/(1.0+np.exp(-z))
-
-def sigmoid_prime(z):
- return sigmoid(z)*(1-sigmoid(z))
-
-network=Network()
-+The number of input nodes does not need to equal the number of output +nodes. This applies also to the hidden layers. Each layer may have its +own number of nodes and activation functions. - -
# %load neural-networks-and-deep-learning/src/mnist_loader.py
-"""
-mnist_loader
-~~~~~~~~~~~~
-
-A library to load the MNIST image data. For details of the data
-structures that are returned, see the doc strings for ``load_data``
-and ``load_data_wrapper``. In practice, ``load_data_wrapper`` is the
-function usually called by our neural network code.
-"""
-
-#### Libraries
-# Standard library
-import pickle
-import gzip
-
-# Third-party libraries
-import numpy as np
-
-def load_data():
- """Return the MNIST data as a tuple containing the training data,
- the validation data, and the test data.
-
- The ``training_data`` is returned as a tuple with two entries.
- The first entry contains the actual training images. This is a
- numpy ndarray with 50,000 entries. Each entry is, in turn, a
- numpy ndarray with 784 values, representing the 28 * 28 = 784
- pixels in a single MNIST image.
-
- The second entry in the ``training_data`` tuple is a numpy ndarray
- containing 50,000 entries. Those entries are just the digit
- values (0...9) for the corresponding images contained in the first
- entry of the tuple.
-
- The ``validation_data`` and ``test_data`` are similar, except
- each contains only 10,000 images.
-
- This is a nice data format, but for use in neural networks it's
- helpful to modify the format of the ``training_data`` a little.
- That's done in the wrapper function ``load_data_wrapper()``, see
- below.
- """
- f = gzip.open('../data/mnist.pkl.gz', 'rb')
- training_data, validation_data, test_data = cPickle.load(f)
- f.close()
- return (training_data, validation_data, test_data)
-
-def load_data_wrapper():
- """Return a tuple containing ``(training_data, validation_data,
- test_data)``. Based on ``load_data``, but the format is more
- convenient for use in our implementation of neural networks.
-
- In particular, ``training_data`` is a list containing 50,000
- 2-tuples ``(x, y)``. ``x`` is a 784-dimensional numpy.ndarray
- containing the input image. ``y`` is a 10-dimensional
- numpy.ndarray representing the unit vector corresponding to the
- correct digit for ``x``.
-
- ``validation_data`` and ``test_data`` are lists containing 10,000
- 2-tuples ``(x, y)``. In each case, ``x`` is a 784-dimensional
- numpy.ndarry containing the input image, and ``y`` is the
- corresponding classification, i.e., the digit values (integers)
- corresponding to ``x``.
-
- Obviously, this means we're using slightly different formats for
- the training data and the validation / test data. These formats
- turn out to be the most convenient for use in our neural network
- code."""
- tr_d, va_d, te_d = load_data()
- training_inputs = [np.reshape(x, (784, 1)) for x in tr_d[0]]
- training_results = [vectorized_result(y) for y in tr_d[1]]
- training_data = zip(training_inputs, training_results)
- validation_inputs = [np.reshape(x, (784, 1)) for x in va_d[0]]
- validation_data = zip(validation_inputs, va_d[1])
- test_inputs = [np.reshape(x, (784, 1)) for x in te_d[0]]
- test_data = zip(test_inputs, te_d[1])
- return (training_data, validation_data, test_data)
-
-def vectorized_result(j):
- """Return a 10-dimensional unit vector with a 1.0 in the jth
- position and zeroes elsewhere. This is used to convert a digit
- (0...9) into a corresponding desired output from the neural
- network."""
- e = np.zeros((10, 1))
- e[j] = 1.0
- return e
-
-net=network.Network([784,30,30])
-net.SGD(training_data,30,10,3,test_data=test_data)
-+The hidden layers have their name from the fact that they are not +linked to observables and as we will see below when we define the +so-called activation \( \hat{z} \), we can think of this as a basis +expansion of the original inputs \( \hat{x} \). The difference however +between neural networks and say linear regression is that now these +basis functions (which will correspond to the weights in the network) +are learned from data. This results in an important difference between +neural networks and deep learning approaches on one side and methods +like logistic regression or linear regression and their modifications on the other side. +
+ + + + +
+A neural network with only one layer, what we called the simple +perceptron, is best suited if we have a standard binary model with +clear (linear) boundaries between the outcomes. As such it could +equally well be replaced by standard linear regression or logistic +regression. Networks with one or more hidden layers approximate +systems with more complex boundaries. + +
+As stated earlier, +an important theorem in studies of neural networks, restated without +proof here, is the universal approximation +theorem. + +
+It states that a feed-forward network with a single hidden layer +containing a finite number of neurons can approximate continuous +functions on compact subsets of real functions. The theorem thus +states that simple neural networks can represent a wide variety of +interesting functions when given appropriate parameters. It is the +multilayer feedforward architecture itself which gives neural networks +the potential of being universal approximators. + +
+
+ +
+ + +
+ + + + +
+Note: figures will be inserted later! + +
+As we have seen now in a feed forward network, we can express the final output of our network in terms of basic matrix-vector multiplications. +The unknowwn quantities are our weights \( w_{ij} \) and we need to find an algorithm for changing them so that our errors are as small as possible. +This leads us to the famous back propagation algorithm. + +
+The questions we want to ask are how do changes in the biases and the +weights in our network change the cost function and how can we use the +final output to modify the weights? + +
+To derive these equations let us start with a plain regression problem +and define our cost function as + +$$ +{\cal C}(\hat{W}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2, +$$ + +
+where the $t_i$s are our \( n \) targets (the values we want to +reproduce), while the outputs of the network after having propagated +all inputs \( \hat{x} \) are given by \( y_i \). Below we will demonstrate +how the basic equations arising from the back propagation algorithm +can be modified in order to study classification problems with \( K \) +classes. + +
+
+ +
+ + +
+ + + + +
+With our definition of the targets \( \hat{t} \), the outputs of the +network \( \hat{y} \) and the inputs \( \hat{x} \) we +define now the activation \( z_j^l \) of node/neuron/unit \( j \) of the +\( l \)-th layer as a function of the bias, the weights which add up from +the previous layer \( l-1 \) and the forward passes/outputs +\( \hat{a}^{l-1} \) from the previous layer as + +$$ +z_j^l = \sum_{i=1}^{M_{l-1}}w_{ij}^la_j^{l-1}+b_j^l, +$$ + +
+where \( b_k^l \) are the biases from layer \( l \). Here \( M_{l-1} \) +represents the total number of nodes/neurons/units of layer \( l-1 \). The +figure here illustrates this equation. We can rewrite this in a more +compact form as the matrix-vector products we discussed earlier, + +$$ +\hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l. +$$ + +
+With the activation function \( \hat{z}^l \) we can in turn define the +output of layer \( l \) as \( \hat{a}^l = f(\hat{z}^l) \) where \( f \) is our +activation function. In the examples here we will use the sigmoid +function discussed in our logistic regression lectures and here as +well. We will also use the same activation function \( f \) for all layers +and their nodes. It means we have + +$$ +a_j^l = f(z_j^l) = \frac{1}{1+\exp{-(z_j^l)}}. +$$ + +
+
+ +
+ + +
+ + + + +
+From the definition of the activation \( z_j^l \) we have +$$ +\frac{\partial z_j^l}{\partial w_{ji}^l} = a_i^{l-1}, +$$ + +and +$$ +\frac{\partial z_j^l}{\partial a_i^{l-1}} = w_{ji}^l. +$$ + +
+With our definition of the activation function we have that (note that this function depends only on \( z_j^l \)) +$$ +\frac{\partial a_j^l}{\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)). +$$ + +
+
+ +
+ + +
+ + + + +
+With these definitions we can now compute the derivative of the cost function in terms of the weights. + +
+Let us specialize to the output layer \( l=L \). Our cost function is +$$ +{\cal C}(\hat{W^L}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2, +$$ + +The derivative of this function with respect to the weights is + +$$ +\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}}, +$$ + +The last partial derivative can easily be computed and reads (by applying the chain rule) +$$ +\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1}, +$$ + +
+
+ +
+ + +
+ + + + +
+We have thus +$$ +\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1}, +$$ + +
+Defining +$$ +\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}, +$$ + +and using the Hadamard product of two vectors we can write this as +$$ +\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}L)}. +$$ + +
+This is an important expression. The second term on the right handside +measures how fast the cost is changing as a function of the $j$th +output activation. If, for example, the cost function doesn't depend +much on a particular output node \( j \), then \( \delta_j^L \) will be small, +which is what we would expect. The first term on the right, measures +how fast the activation function \( f \) is changing at a given activation +value \( z_j^L \). + +
+Notice that everything in the above equations is easily computed. In +particular, we compute \( z_j^L \) while computing the behaviour of the +network, and it is only a small additional overhead to compute +\( f'(z^L_j) \). The exact form of the derivative with respect to the +outpuwill, of course, depend on the form of the cost function. +However, provided the cost function is known there should be little +trouble computing + +$$ +\frac{\partial {\cal C}}{\partial (a_j^L)} +$$ + +
+With the definition of \( \delta_j^L \) we have a more compact definition of the derivative of the cost function in terms of the weights, namely +$$ +\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}. +$$ + +
+
+ +
+ + +
+ + + + +
+It is also easy to see that our previous equation can be written as + +$$ +\delta_j^L =\frac{\partial {\cal C}}{\partial z_j^L}= \frac{\partial {\cal C}}{\partial a_j^L}\frac{\partial a_j^L}{\partial z_j^L}, +$$ + +which can also be interpreted as the partial derivative of the cost function with respect to the biases \( b_j^L \), namely +$$ +\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}\frac{\partial b_j^L}{\partial z_j^L}=\frac{\partial {\cal C}}{\partial b_j^L}, +$$ + +That is, the error \( \delta_j^L \) is exactly equal to the rate of change of the cost function as a function of the bias. +
+ +
+ + +
+ + + + +
+We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are +
+ +$$ +\begin{equation} +\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}, +\tag{13} +\end{equation} +$$ + +and +$$ +\begin{equation} +\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}, +\tag{14} +\end{equation} +$$ + +and + +$$ +\begin{equation} +\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}, +\tag{15} +\end{equation} +$$ +
+An interesting consequence of the above equations is that when the +activation \( a_k^{L-1} \) is small, the gradient term, that is the +derivative of the cost function with respect to the weights, will also +tend to be small. We say then that the weight learns slowly, meaning +that it changes slowly when we minimize the weights via say gradient +descent. In this case we say the system learns slowly. + +
+Another interesting feature is that is when the activation function, +represented by the sigmoid function here, is rather flat when towards +its end values \( 0 \) and \( 1 \) (see the above Python codes). In these +cases, the derivatives of the activation function will also be close +to zero, meaning again that the gradients will be small and the +network learns slowly again. + +
+We need a fourth equation and we are set. We are going to propagate +backwards in order to the determine the weights and biases. In order +to do so we need to represent the error in the layer before the final +one \( L-1 \) in terms of the errors in the final output layer. + +
+
+ +
+ + +
+ + + + +
+We have that (replacing \( L \) with a general layer \( l \)) +$$ +\delta_j^l =\frac{\partial {\cal C}}{\partial z_j^l}. +$$ + +We want to express this in terms of the equations for layer \( l+1 \). Using the chain rule and summing over all \( k \) entries we have + +$$ +\delta_j^l =\sum_k \frac{\partial {\cal C}}{\partial z_k^{l+1}}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}=\sum_k \delta_k^{l+1}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}, +$$ + +and recalling that +$$ +z_j^{l+1} = \sum_{i=1}^{M_{l}}w_{ij}^{l+1}a_j^{l}+b_j^{l+1}, +$$ + +we obtain +$$ +\delta_j^l =\sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l), +$$ + +This is our final equation. + +
+We are now ready to set up the algorithm for back propagation and learning the weights and biases. + +
+
+ +
+ + +
+ + + + +
+The four equations provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm. + +
+
+First, we set up the input data \( \hat{x} \) and the activations +\( \hat{z}_1 \) of the input layer and compute the activation function and +the pertinent outputs \( \hat{a}^1 \). +
+
+Secondly, we perform then the feed forward till we reach the output +layer and compute all \( \hat{z}_l \) of the input layer and compute the +activation function and the pertinent outputs \( \hat{a}^l \) for +\( l=2,3,\dots,L \). +
+
+Thereafter we compute the ouput error \( \hat{\delta}^L \) by computing all +$$ +\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}. +$$ +
+
+Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as +$$ +\delta_j^l =\sum_k \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l). +$$ +
+
+Finally, we update the weights and the biases using gradient descent for each \( l=L-1,L-2,dots,2 \) and update the weights and biases according to the rules +$$ +w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1}, +$$ + + +$$ +b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^L}, +$$ +
+The parameter \( \eta \) is the learning parameter discussed in connection with the gradient descent methods. +Here it is convenient to use stochastic radient descent with mini-batches with an outer loop that steps through multiple epochs of training. + +
+
+ +
+ + +
+ + + + +
+We are now gong to develop an example based on the MNIST data +base. This is a classification problem and we need to use our +cross-entropy function we discussed in connection with logistic +regression. The cross-entropy defines our cost function for the +classificaton problems with neural networks. + +
+In binary classification with two classes \( (0, 1) \) we define the +logistic/sigmoid function as the probability that a particular input +is in class \( 0 \) or \( 1 \). This is possible because the logistic +function takes any input from the real numbers and inputs a number +between 0 and 1, and can therefore be interpreted as a probability. It +also has other nice properties, such as a derivative that is simple to +calculate. + +
+For an input \( \boldsymbol{a} \) from the hidden layer, the probability that the input \( \boldsymbol{x} \) +is in class 0 or 1 is just: + +$$ +P(y = 0 \mid \boldsymbol{x}, \boldsymbol{\theta}) = \frac{1}{1 + \exp (- \boldsymbol{a}^T \boldsymbol{w}_{out})} , +$$ + +and +$$ +P(y = 1 \mid \boldsymbol{x}, \boldsymbol{\theta}) = 1 - P(y = 0 \mid \boldsymbol{x}, \boldsymbol{\theta}) , +$$ + +
+where \( y \in \{0, 1\} \) and \( \boldsymbol{\theta} \) represents the weights and biases +of our network. + +
+
+ +
+ + +
+ + + + +
+Our cost function is given as (see the Logistic regression lectures) +$$ +\mathcal{C}(\boldsymbol{\theta}) = - \ln P(\mathcal{D} \mid \boldsymbol{\theta}) = - \sum_{i=1}^n +y_i \ln[P(y_i = 0)] + (1 - y_i) \ln [1 - P(y_i = 0)] = \sum_{i=1}^n \mathcal{L}_i(\boldsymbol{\theta}) . +$$ + +
+This last equality means that we can interpret our cost function as a sum over the loss function +for each point in the dataset \( \mathcal{L}_i(\boldsymbol{\theta}) \). +The negative sign is just so that we can think about our algorithm as minimizing a positive number, rather +than maximizing a negative number. + +
+In multiclass classification it is common to treat each integer label as a so called one-hot vector: + +
+\( y = 5 \quad \rightarrow \quad \boldsymbol{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) , \) and + +
+\( y = 1 \quad \rightarrow \quad \boldsymbol{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) , \) + +
+i.e. a binary bit string of length \( C \), where \( C = 10 \) is the number of classes in the MNIST dataset (numbers from \( 0 \) to \( 9 \)).. + +
+If \( \boldsymbol{x}_i \) is the \( i \)-th input (image), \( y_{ic} \) refers to the \( c \)-th component of the \( i \)-th +output vector \( \boldsymbol{y}_i \). +The probability of \( \boldsymbol{x}_i \) being in class \( c \) will be given by the softmax function: + +$$ +P(y_{ic} = 1 \mid \boldsymbol{x}_i, \boldsymbol{\theta}) = \frac{\exp{((\boldsymbol{a}_i^{hidden})^T \boldsymbol{w}_c)}} +{\sum_{c'=0}^{C-1} \exp{((\boldsymbol{a}_i^{hidden})^T \boldsymbol{w}_{c'})}} , +$$ + +
+which reduces to the logistic function in the binary case. +The likelihood of this \( C \)-class classifier +is now given as: + +$$ +P(\mathcal{D} \mid \boldsymbol{\theta}) = \prod_{i=1}^n \prod_{c=0}^{C-1} [P(y_{ic} = 1)]^{y_{ic}} . +$$ + +Again we take the negative log-likelihood to define our cost function: + +$$ +\mathcal{C}(\boldsymbol{\theta}) = - \ln P(\mathcal{D} \mid \boldsymbol{\theta}) = - \sum_{i=1}^n \sum_{c=0}^{C-1} +y_{ic} \ln[P(y_{ic} = 1)] = \sum_{i=1}^n +\mathcal{L}_i(\boldsymbol{\theta}) . +$$ + +
+The back propagation equations need now only a small change, namely the definition of a new cost function. We are thus ready to use the same equations as before! +We leave it as an exercise in project 2 to derive these equations. + +
+
+ +
+ + +
+ + + + +
+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. + +
+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()
++
+ +
+ + +
+ + + + +
+Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions. + +
+We will reserve \( 80 \% \) of our dataset for training and \( 20 \% \) for testing. + +
+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. + +
+ + +
from sklearn.model_selection import train_test_split
+
+# 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
+
+#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)))
++
+ +
+ + +
+ + + + +
+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: + +$$ z = \sum_{i=1}^n w_i a_i ,$$ + +$$ y = f(z) ,$$ + +
+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). + +
+The simplest activation function for a neuron is the Heaviside function: + +$$ f(z) = +\begin{cases} +1, & z > 0\\ +0, & \text{otherwise} +\end{cases} +$$ + +
+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. + +
+
+ +
+ + +
+ + + + +
+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. + +
+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 \( \boldsymbol{a} \)}) = \frac{\exp{(\boldsymbol{a}^T \boldsymbol{w}_j)}} +{\sum_{c=0}^{9} \exp{(\boldsymbol{a}^T \boldsymbol{w}_c)}} ,$$ + +
+i.e. each neuron \( j \) outputs the probability of being in class \( j \) given an input from the hidden layer \( \boldsymbol{a} \), with \( \boldsymbol{w}_j \) 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: + +$$ z_j = \sum_{i=1}^n w_ {ij} a_i = \boldsymbol{a}^T \boldsymbol{w}_j .$$ + +
+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. + +
+
+ +
+ + +
+ + + + +
+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. + +
+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 \( j \), \( b_j \): + +$$ z_j = \sum_{i=1}^n w_ {ij} a_i + 1\cdot b_j = \boldsymbol{a}^T \boldsymbol{w}_j + b_j .$$ + +
+The bias weights \( \boldsymbol{b} \) 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. +
+ + +
# 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
++
+ +
+ + +
+ + + + +
+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: + +$$ z_{j}^{h} = \sum_{i=1}^{F} w_{ij}^{h} x_i + b_{j}^{h} = \boldsymbol{x}^T \boldsymbol{w}_{j}^{h} + b_{j}^{h} ,$$ + +
+this is then passed through our activation function + +$$ a_{j}^{h} = f(z_{j}^{h}) .$$ + +
+We calculate a weighted sum of inputs (activations in the hidden layer) to each neuron \( j \) in the output layer: + +$$ z_{j}^{o} = \sum_{i=1}^{H} w_{ij}^{o} a_{i}^{h} + b_{j}^{o} = (\boldsymbol{a}^{h})^T \boldsymbol{w}_{j}^{o} + b_{j}^{o} .$$ + +
+Finally we calculate the output of neuron \( j \) in the output layer using the softmax function: + +$$ a_{j}^{o} = \frac{\exp{(z_j^{o})}} +{\sum_{c=0}^{C-1} \exp{(z_c^{o})}} .$$ + +
+
+ +
+ + +
+ + + + +
+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 + +$$ X W^{h} = (n_{inputs}, n_{hidden}),$$ + +
+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 \( Z^{h} \): + +$$ Z^{h} = X W^{h} + B^{h} ,$$ + +
+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: + +$$ A^{h} = f(Z^h) .$$ + +
+This is fed to the output layer: + +$$ Z^{o} = A^{h} W^{o} + B^{o} .$$ + +
+Finally we receive our output values for each image and each category by passing it through the softmax function: + +$$ output = softmax (Z^{o}) = (n_{inputs}, n_{categories}) .$$ + +
+ + +
# setup the feed-forward pass
+
+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]))
++
+ +
+ + +
+ + + + +
+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. + +
+In multiclass classification it is common to treat each integer label as a so called one-hot vector: + +$$ y = 5 \quad \rightarrow \quad \boldsymbol{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$$ + +$$ y = 1 \quad \rightarrow \quad \boldsymbol{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,$$ + +
+i.e. a binary bit string of length \( C \), where \( C = 10 \) is the number of classes in the MNIST dataset. + +
+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 \( \boldsymbol{x}_i \) in the dataset: + +$$ +\begin{align} \mathcal{C}(\boldsymbol{\theta}) &= \sum_{i=1}^N \mathcal{L}_i (\boldsymbol{\theta}) +\tag{16}\\ + &= -\sum_{i=1}^N \sum_{c=0}^{C-1} y_{ic} \log P(y_{ic} = 1 \mid \boldsymbol{x}_i, \boldsymbol{\theta}) . +\tag{17} +\end{align} +$$ + +
+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 \( \boldsymbol{\theta} \) represents the parameters of our network, i.e. all the weights and biases. + +
+A full derivation is given in the appendix at the end. + +
+
+ +
+ + +
+ + + + +
+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 + +$$ \theta_{i+1} = \theta_i - \eta \nabla \mathcal{C}(\theta_i) ,$$ + +
+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: + +$$ \nabla \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) \quad \rightarrow \quad +\frac{1}{M} \sum_{i \in B_k} \nabla \mathcal{L}_i(\theta) ,$$ + +
+i.e. instead of averaging the loss over the entire dataset, we average over a minibatch. + +
+This has two important benefits: + +
+ +
+ + +
+ + + + +
+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. + +
+We will measure the size of the weights using the so called L2-norm, meaning our cost function becomes: + +$$ \nabla \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) \quad \rightarrow \quad +\frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) + \lambda \lvert \lvert \boldsymbol{w} \rvert \rvert_2^2 += \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}(\theta) + \lambda \sum_{ij} w_{ij}^2,$$ + +
+i.e. we sum up all the weights squared. The factor \( \lambda \) is known as a regularization parameter. + +
+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. + +
+
+ +
+ + +
+ + + + +
+To more efficently train our network these equations are implemented using matrix operations. +The error in the output layer is calculated simply as + +$$ \Delta_o = \hat{y} - y = (n_{inputs}, n_{categories}) .$$ + +
+The gradient for the output weights is calculated as + +$$ \nabla W_{o} = A^T \Delta_o = (n_{hidden}, n_{categories}) ,$$ + +
+where \( A = (n_{inputs}, n_{hidden}) \). 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. + +
+The gradient with respect to the output bias is then + +$$ \nabla B_{o} = \sum_{i=1}^{n_{inputs}} \Delta_o = (n_{categories}) .$$ + +
+The error in the hidden layer is + +$$ \Delta_h = \Delta_o W_{o}^T \circ f'(Z_{h}) = \Delta_o W_{o}^T \circ A_{h} \circ (1 - A_{h}) = (n_{inputs}, n_{hidden}) ,$$ + +
+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: + +$$ \nabla W_{h} = X^T \Delta_h = (n_{features}, n_{hidden}) ,$$ + +$$ \nabla B_{h} = \sum_{i=1}^{n_{inputs}} \Delta_h = (n_{hidden}) .$$ + +
+ + +
# to categorical turns our integer vector into a onehot representation
+#from keras.utils import to_categorical
+
+# calculate the accuracy score of our model
+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)))
++
+ +
+ + +
+ + + + +
+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. + +
+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. + +
+
+ +
+ + +
+ + + + +
+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. + +
+ + +
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,
+
+ ):
+ self.X_data_full = X_data
+ self.Y_data_full = Y_data
+
+ 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()
++
+ +
+ + +
+ + + + +
+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. + +
+ + +
epochs = 100
+batch_size = 100
+
+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))
++
+ +
+ + +
+ + + + +
+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). + +
+ + +
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)
+
+# 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()
++ + +
# optional
+# visual representation of grid search
+# uses seaborn heatmap, I believe you can also do this with matplotlib imshow
+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_numpy[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()
++
+ +
+ + +
+ + + + +
+scikit-learn is a machine learning library for Python. It 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. + +
+ + +
from sklearn.neural_network import MLPClassifier
+
+# store models for later use
+DNN_scikit = 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 = MLPClassifier(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',
+ alpha=lmbd, learning_rate_init=eta, max_iter=epochs)
+ dnn.fit(X_train, Y_train)
+
+ DNN_scikit[i][j] = dnn
+
+ print("Learning rate = ", eta)
+ print("Lambda = ", lmbd)
+ print("Accuracy score on test set: ", dnn.score(X_test, Y_test))
+ 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_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()
++
+ +
+ + +@@ -157,12 +237,9 @@ MathJax.Hub.Config({
-
- - -
@@ -184,7 +261,7 @@ MathJax.Hub.Config({
@@ -148,11 +148,8 @@ MathJax.Hub.Config({
-
- -
-In natural science, DNNs and CNNs have already found numerous applications. In -statistical physics, they have been applied to detect phase -transitions in 2D Ising and Potts models, lattice gauge theories, and -different phases of polymers, or solving the Navier-Stokes equation in weather forecasting. -Deep learning has also found interesting applications in quantum -physics. Various quantum phase transitions can be detected and studied -using DNNs and CNNs, +In natural science, DNNs and CNNs have already found numerous +applications. In statistical physics, they have been applied to detect +phase transitions in 2D Ising and Potts models, lattice gauge +theories, and different phases of polymers, or solving the +Navier-Stokes equation in weather forecasting. Deep learning has also +found interesting applications in quantum physics. Various quantum +phase transitions can be detected and studied using DNNs and CNNs, topological phases, and even non-equilibrium many-body localization. Representing quantum states as DNNs quantum state -tomography are among some of the impressive -achievements to reveal the potential of DNNs to facilitate the study -of quantum systems. +tomography are among some of the impressive achievements to reveal the +potential of DNNs to facilitate the study of quantum systems.
In quantum information theory, it has been shown that one can perform -gate decompositions with the help of neural. In lattice quantum chromodynamics, -DNNs have been used to learn action parameters in regions of parameter -space where PCA fails. +gate decompositions with the help of neural.
-The applications are not limited to the natural sciences. There is a plethora of applications in essentially all disciplines, from the humanities to life science and medicine. +The applications are not limited to the natural sciences. There is a +plethora of applications in essentially all disciplines, from the +humanities to life science and medicine. @@ -246,22 +242,26 @@ The applications are not limited to the natural sciences. There is a plethora of
-An artificial neural network (NN), is a computational model that -consists of layers of connected neurons, or nodes. It is supposed -to mimic a biological nervous system by letting each neuron interact -with other neurons by sending signals in the form of mathematical -functions between layers. A wide variety of different NNs have been -developed, but most of them consist of an input layer, an output layer -and eventual layers in-between, called hidden layers. All layers can -contain an arbitrary number of nodes, and each connection between two -nodes is associated with a weight variable. +An artificial neural network (ANN), is a computational model that +consists of layers of connected neurons, or nodes or units. We will +refer to these interchangeably as units or nodes, and sometimes as +neurons. + +
+It is supposed to mimic a biological nervous system by letting each +neuron interact with other neurons by sending signals in the form of +mathematical functions between layers. A wide variety of different +ANNs have been developed, but most of them consist of an input layer, +an output layer and eventual layers in-between, called hidden +layers. All layers can contain an arbitrary number of nodes, and each +connection between two nodes is associated with a weight variable.
Neural networks (also called neural nets) are neural-inspired nonlinear models for supervised learning. As we will see, neural nets can be viewed as natural, more powerful extensions of supervised learning methods such as linear and logistic regression and soft-max -methods. +methods we discussed earlier. @@ -269,61 +269,82 @@ methods.
-The feed-forward neural network (FFNN) was the first and simplest type of NN devised. In this network, -the information moves in only one direction: forward through the layers. +The feed-forward neural network (FFNN) was the first and simplest type +of ANNs that were devised. In this network, the information moves in +only one direction: forward through the layers.
-Nodes are represented by circles, while the arrows display the connections between the nodes, including the -direction of information flow. Additionally, each arrow corresponds to a weight variable, not displayed here. -We observe that each node in a layer is connected to all nodes in the subsequent layer, -making this a so-called fully-connected FFNN. - -
-A different variant of FFNNs are convolutional neural networks (CNNs), which have a connectivity pattern -inspired by the animal visual cortex. Individual neurons in the visual cortex only respond to stimuli from -small sub-regions of the visual field, called a receptive field. This makes the neurons well-suited to exploit the strong -spatially local correlation present in natural images. The response of each neuron can be approximated mathematically -as a convolution operation. - -
-CNNs emulate the behaviour of neurons in the visual cortex by enforcing a local connectivity pattern
-between nodes of adjacent layers: Each node
-in a convolutional layer is connected only to a subset of the nodes in the previous layer,
-in contrast to the fully-connected FFNN.
-Often, CNNs
-consist of several convolutional layers that learn local features of the input, with a fully-connected layer at the end,
-which gathers all the local data and produces the outputs. They have wide applications in image and video recognition
+Nodes are represented by circles, while the arrows display the
+connections between the nodes, including the direction of information
+flow. Additionally, each arrow corresponds to a weight variable
+(figure to come). We observe that each node in a layer is connected
+to all nodes in the subsequent layer, making this a so-called
+fully-connected FFNN.
-So far we have only mentioned NNs where information flows in one direction: forward. Recurrent neural networks on
-the other hand, have connections between nodes that form directed cycles. This creates a form of
-internal memory which are able to capture information on what has been calculated before; the output is dependent
-on the previous computations. Recurrent NNs make use of sequential information by performing the same task for
-every element in a sequence, where each element depends on previous elements. An example of such information is
-sentences, making recurrent NNs especially well-suited for handwriting and speech recognition.
+A different variant of FFNNs are convolutional neural networks
+(CNNs), which have a connectivity pattern inspired by the animal
+visual cortex. Individual neurons in the visual cortex only respond to
+stimuli from small sub-regions of the visual field, called a receptive
+field. This makes the neurons well-suited to exploit the strong
+spatially local correlation present in natural images. The response of
+each neuron can be approximated mathematically as a convolution
+operation. (figure to come)
+
+
+Convolutional neural networks emulate the behaviour of neurons in the
+visual cortex by enforcing a local connectivity pattern between
+nodes of adjacent layers: Each node in a convolutional layer is
+connected only to a subset of the nodes in the previous layer, in
+contrast to the fully-connected FFNN. Often, CNNs consist of several
+convolutional layers that learn local features of the input, with a
+fully-connected layer at the end, which gathers all the local data and
+produces the outputs. They have wide applications in image and video
+recognition.
-There are many other kinds of NNs that have been developed. One type that is specifically designed for interpolation
-in multidimensional space is the radial basis function (RBF) network. RBFs are typically made up of three layers:
-an input layer, a hidden layer with non-linear radial symmetric activation functions and a linear output layer (''linear'' here
-means that each node in the output layer has a linear activation function). The layers are normally fully-connected and
-there are no cycles, thus RBFs can be viewed as a type of fully-connected FFNN. They are however usually treated as
-a separate type of NN due the unusual activation functions.
+So far we have only mentioned ANNs where information flows in one
+direction: forward. Recurrent neural networks on the other hand,
+have connections between nodes that form directed cycles. This
+creates a form of internal memory which are able to capture
+information on what has been calculated before; the output is
+dependent on the previous computations. Recurrent NNs make use of
+sequential information by performing the same task for every element
+in a sequence, where each element depends on previous elements. An
+example of such information is sentences, making recurrent NNs
+especially well-suited for handwriting and speech recognition.
+There are many other kinds of ANNs that have been developed. One type
+that is specifically designed for interpolation in multidimensional
+space is the radial basis function (RBF) network. RBFs are typically
+made up of three layers: an input layer, a hidden layer with
+non-linear radial symmetric activation functions and a linear output
+layer (''linear'' here means that each node in the output layer has a
+linear activation function). The layers are normally fully-connected
+and there are no cycles, thus RBFs can be viewed as a type of
+fully-connected FFNN. They are however usually treated as a separate
+type of NN due the unusual activation functions.
+
One uses often so-called fully-connected feed-forward neural networks
@@ -332,49 +353,41 @@ and an output layer) consisting of neurons that have non-linear
activation functions.
-Such networks are often called multilayer perceptrons (MLPs)
+Such networks are often called multilayer perceptrons (MLPs).
-According to the Universal approximation theorem, a feed-forward neural network with just a single hidden layer containing
-a finite number of neurons can approximate a continuous multidimensional function to arbitrary accuracy,
-assuming the activation function for the hidden layer is a non-constant, bounded and monotonically-increasing continuous function.
+According to the Universal approximation theorem, a feed-forward
+neural network with just a single hidden layer containing a finite
+number of neurons can approximate a continuous multidimensional
+function to arbitrary accuracy, assuming the activation function for
+the hidden layer is a non-constant, bounded and
+monotonically-increasing continuous function.
Note that the requirements on the activation function only applies to
the hidden layer, the output nodes are always assumed to be linear, so
as to not restrict the range of output values.
-
-
-We note that this theorem is only applicable to an NN with one hidden
-layer. Therefore, we can easily construct an NN that employs
-activation functions which do not satisfy the above requirements, as
-long as we have at least one layer with activation functions that
-do. Furthermore, although the universal approximation theorem lays
-the theoretical foundation for regression with neural networks, it
-does not say anything about how things work in practice: A neural
-network can still be able to approximate a given function reasonably
-well without having the flexibility to fit all other functions.
+The output \( y \) is produced via the activation function \( f \)
+
-First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( u_i^1 \) of the input coordinates \( x_j \),
+First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( z_i^1 \) of the input coordinates \( x_j \),
+Here \( b_i \) is the so-called bias which is normally needed in
+case of zero activation weights or inputs. How to fix the biases and
+the weights will be discussed below. The value of \( z_i^1 \) is the
+argument to the activation function \( f_i \) of each node \( i \), The
+variable \( M \) stands for all possible inputs to a given node \( i \) in the
+first layer. We define the output \( y_i^1 \) of all neurons in layer 1 as
where we assume that all nodes in the same layer have identical
-activation functions, hence the notation \( f_l \)
+activation functions, hence the notation \( f \). In general, we could assume in the more general case that different layers have different activation functions.
+In this case we would identify these functions with a superscript \( l \) for the \( l \)-th layer,
The output of neuron \( i \) in layer 2 is thus,
@@ -439,24 +457,24 @@ The output of neuron \( i \) in layer 2 is thus,
We can generalize this expression to an MLP with \( l \) hidden
@@ -473,11 +491,8 @@ layers. The complete functional form is,
This confirms that an MLP, despite its quite convoluted mathematical
form, is nothing more than an analytic function, specifically a
-mapping of real-valued vectors \( \vec{x} \in \mathbb{R}^n \rightarrow
-\vec{y} \in \mathbb{R}^m \). In our example, \( n=2 \) and
-\( m=1 \). Consequentially, the number of input and output values of the
-function we want to fit must be equal to the number of inputs and
-outputs of our MLP.
+mapping of real-valued vectors \( \hat{x} \in \mathbb{R}^n \rightarrow
+\hat{y} \in \mathbb{R}^m \).
-Furthermore, the flexibility and universality of a MLP can be
+Furthermore, the flexibility and universality of an MLP can be
illustrated by realizing that the expression is essentially a nested
sum of scaled activation functions of the form
-We can introduce a more convenient notation for the activations in a NN.
+We can introduce a more convenient notation for the activations in an A NN.
Additionally, we can represent the biases and activations
-as layer-wise column vectors \( \vec{b}_l \) and \( \vec{y}_l \), so that the \( i \)-th element of each vector
+as layer-wise column vectors \( \hat{b}_l \) and \( \hat{y}_l \), so that the \( i \)-th element of each vector
is the bias \( b_i^l \) and activation \( y_i^l \) of node \( i \) in layer \( l \) respectively.
-We have that \( \mathrm{W}_l \) is a \( N_{l-1} \times N_l \) matrix, while \( \vec{b}_l \) and \( \vec{y}_l \) are \( N_l \times 1 \) column vectors.
-With this notation, the sum in becomes a matrix-vector multiplication, and we can write
-the equation for the activations of hidden layer 2 in
+We have that \( \mathrm{W}_l \) is an \( N_{l-1} \times N_l \) matrix, while \( \hat{b}_l \) and \( \hat{y}_l \) are \( N_l \times 1 \) column vectors.
+With this notation, the sum becomes a matrix-vector multiplication, and we can write
+the equation for the activations of hidden layer 2 as
The activation of node \( i \) in layer 2 is
@@ -576,7 +588,7 @@ $$
\begin{equation}
y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) =
f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right).
-\tag{13}
+\tag{12}
\end{equation}
$$
A property that characterizes a neural network, other than its
@@ -615,7 +627,7 @@ for a FFNN to fulfill the universal approximation theorem
The second requirement excludes all linear functions. Furthermore, in
@@ -630,20 +642,14 @@ functions Typical examples are the logistic Sigmoid
The sigmoid function are more biologically plausible because the
@@ -739,368 +745,1596 @@ plt.show()
+The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of
+
+
+
+As a convention it is normal to call a network with one layer of input units, one layer of hidden
+units and one layer of output units as a two-layer network. A network with two layers of hidden units is called a three-layer network etc etc.
+
+
+For an MLP there is no direct connection between the output nodes/neurons/units and the input nodes/neurons/units.
+Hereafter we will call the various entities of a layer for nodes.
+There are also no connections within a single layer.
+
+
+The number of input nodes does not need to equal the number of output
+nodes. This applies also to the hidden layers. Each layer may have its
+own number of nodes and activation functions.
+
+
+The hidden layers have their name from the fact that they are not
+linked to observables and as we will see below when we define the
+so-called activation \( \hat{z} \), we can think of this as a basis
+expansion of the original inputs \( \hat{x} \). The difference however
+between neural networks and say linear regression is that now these
+basis functions (which will correspond to the weights in the network)
+are learned from data. This results in an important difference between
+neural networks and deep learning approaches on one side and methods
+like logistic regression or linear regression and their modifications on the other side.
+
+A neural network with only one layer, what we called the simple
+perceptron, is best suited if we have a standard binary model with
+clear (linear) boundaries between the outcomes. As such it could
+equally well be replaced by standard linear regression or logistic
+regression. Networks with one or more hidden layers approximate
+systems with more complex boundaries.
+
+
+As stated earlier,
+an important theorem in studies of neural networks, restated without
+proof here, is the universal approximation
+theorem.
+
+
+It states that a feed-forward network with a single hidden layer
+containing a finite number of neurons can approximate continuous
+functions on compact subsets of real functions. The theorem thus
+states that simple neural networks can represent a wide variety of
+interesting functions when given appropriate parameters. It is the
+multilayer feedforward architecture itself which gives neural networks
+the potential of being universal approximators.
+
+Note: figures will be inserted later!
+
+
+As we have seen now in a feed forward network, we can express the final output of our network in terms of basic matrix-vector multiplications.
+The unknowwn quantities are our weights \( w_{ij} \) and we need to find an algorithm for changing them so that our errors are as small as possible.
+This leads us to the famous back propagation algorithm.
+
+
+The questions we want to ask are how do changes in the biases and the
+weights in our network change the cost function and how can we use the
+final output to modify the weights?
+
+
+To derive these equations let us start with a plain regression problem
+and define our cost function as
+
+
+where the $t_i$s are our \( n \) targets (the values we want to
+reproduce), while the outputs of the network after having propagated
+all inputs \( \hat{x} \) are given by \( y_i \). Below we will demonstrate
+how the basic equations arising from the back propagation algorithm
+can be modified in order to study classification problems with \( K \)
+classes.
+
+With our definition of the targets \( \hat{t} \), the outputs of the
+network \( \hat{y} \) and the inputs \( \hat{x} \) we
+define now the activation \( z_j^l \) of node/neuron/unit \( j \) of the
+\( l \)-th layer as a function of the bias, the weights which add up from
+the previous layer \( l-1 \) and the forward passes/outputs
+\( \hat{a}^{l-1} \) from the previous layer as
+
+
+where \( b_k^l \) are the biases from layer \( l \). Here \( M_{l-1} \)
+represents the total number of nodes/neurons/units of layer \( l-1 \). The
+figure here illustrates this equation. We can rewrite this in a more
+compact form as the matrix-vector products we discussed earlier,
+
+
+With the activation function \( \hat{z}^l \) we can in turn define the
+output of layer \( l \) as \( \hat{a}^l = f(\hat{z}^l) \) where \( f \) is our
+activation function. In the examples here we will use the sigmoid
+function discussed in our logistic regression lectures and here as
+well. We will also use the same activation function \( f \) for all layers
+and their nodes. It means we have
+
+
+From the definition of the activation \( z_j^l \) we have
+
+With our definition of the activation function we have that (note that this function depends only on \( z_j^l \))
+
+With these definitions we can now compute the derivative of the cost function in terms of the weights.
+
+
+Let us specialize to the output layer \( l=L \). Our cost function is
+
+We have thus
+
+Defining
+
+This is an important expression. The second term on the right handside
+measures how fast the cost is changing as a function of the $j$th
+output activation. If, for example, the cost function doesn't depend
+much on a particular output node \( j \), then \( \delta_j^L \) will be small,
+which is what we would expect. The first term on the right, measures
+how fast the activation function \( f \) is changing at a given activation
+value \( z_j^L \).
+
+
+Notice that everything in the above equations is easily computed. In
+particular, we compute \( z_j^L \) while computing the behaviour of the
+network, and it is only a small additional overhead to compute
+\( f'(z^L_j) \). The exact form of the derivative with respect to the
+outpuwill, of course, depend on the form of the cost function.
+However, provided the cost function is known there should be little
+trouble computing
+
+
+With the definition of \( \delta_j^L \) we have a more compact definition of the derivative of the cost function in terms of the weights, namely
+
+It is also easy to see that our previous equation can be written as
+
+
+We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are
+
+An interesting consequence of the above equations is that when the
+activation \( a_k^{L-1} \) is small, the gradient term, that is the
+derivative of the cost function with respect to the weights, will also
+tend to be small. We say then that the weight learns slowly, meaning
+that it changes slowly when we minimize the weights via say gradient
+descent. In this case we say the system learns slowly.
+
+
+Another interesting feature is that is when the activation function,
+represented by the sigmoid function here, is rather flat when towards
+its end values \( 0 \) and \( 1 \) (see the above Python codes). In these
+cases, the derivatives of the activation function will also be close
+to zero, meaning again that the gradients will be small and the
+network learns slowly again.
+
+
+We need a fourth equation and we are set. We are going to propagate
+backwards in order to the determine the weights and biases. In order
+to do so we need to represent the error in the layer before the final
+one \( L-1 \) in terms of the errors in the final output layer.
+
+We have that (replacing \( L \) with a general layer \( l \))
+
+We are now ready to set up the algorithm for back propagation and learning the weights and biases.
+
+The four equations provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.
+
+
+
+First, we set up the input data \( \hat{x} \) and the activations
+\( \hat{z}_1 \) of the input layer and compute the activation function and
+the pertinent outputs \( \hat{a}^1 \).
+
+
+Secondly, we perform then the feed forward till we reach the output
+layer and compute all \( \hat{z}_l \) of the input layer and compute the
+activation function and the pertinent outputs \( \hat{a}^l \) for
+\( l=2,3,\dots,L \).
+
+
+Thereafter we compute the ouput error \( \hat{\delta}^L \) by computing all
+
+
+Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as
+
+
+Finally, we update the weights and the biases using gradient descent for each \( l=L-1,L-2,dots,2 \) and update the weights and biases according to the rules
+
+The parameter \( \eta \) is the learning parameter discussed in connection with the gradient descent methods.
+Here it is convenient to use stochastic radient descent with mini-batches with an outer loop that steps through multiple epochs of training.
+
+We are now gong to develop an example based on the MNIST data
+base. This is a classification problem and we need to use our
+cross-entropy function we discussed in connection with logistic
+regression. The cross-entropy defines our cost function for the
+classificaton problems with neural networks.
+
+
+In binary classification with two classes \( (0, 1) \) we define the
+logistic/sigmoid function as the probability that a particular input
+is in class \( 0 \) or \( 1 \). This is possible because the logistic
+function takes any input from the real numbers and inputs a number
+between 0 and 1, and can therefore be interpreted as a probability. It
+also has other nice properties, such as a derivative that is simple to
+calculate.
+
+
+For an input \( \boldsymbol{a} \) from the hidden layer, the probability that the input \( \boldsymbol{x} \)
+is in class 0 or 1 is just:
+
+
+where \( y \in \{0, 1\} \) and \( \boldsymbol{\theta} \) represents the weights and biases
+of our network.
+
+Our cost function is given as (see the Logistic regression lectures)
+
+This last equality means that we can interpret our cost function as a sum over the loss function
+for each point in the dataset \( \mathcal{L}_i(\boldsymbol{\theta}) \).
+The negative sign is just so that we can think about our algorithm as minimizing a positive number, rather
+than maximizing a negative number.
+
+
+In multiclass classification it is common to treat each integer label as a so called one-hot vector:
+
+
+\( y = 5 \quad \rightarrow \quad \boldsymbol{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) , \) and
+
+
+\( y = 1 \quad \rightarrow \quad \boldsymbol{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) , \)
+
+
+i.e. a binary bit string of length \( C \), where \( C = 10 \) is the number of classes in the MNIST dataset (numbers from \( 0 \) to \( 9 \))..
+
+
+If \( \boldsymbol{x}_i \) is the \( i \)-th input (image), \( y_{ic} \) refers to the \( c \)-th component of the \( i \)-th
+output vector \( \boldsymbol{y}_i \).
+The probability of \( \boldsymbol{x}_i \) being in class \( c \) will be given by the softmax function:
+
+
+which reduces to the logistic function in the binary case.
+The likelihood of this \( C \)-class classifier
+is now given as:
+
+
+The back propagation equations need now only a small change, namely the definition of a new cost function. We are thus ready to use the same equations as before!
+We leave it as an exercise in project 2 to derive these equations.
+
+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.
+
+
+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:
+
+
+and the targets would be:
+
+
+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.
-
+Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions.
+
+
+We will reserve \( 80 \% \) of our dataset for training and \( 20 \% \) for testing.
+
+
+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.
+
-
+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:
+
+
+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).
+
+
+The simplest activation function for a neuron is the Heaviside function:
+
+
+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 \( \sigma(x) \):
+
+
+which is inspired by probability theory (see logistic regression) and was most commonly used until about 2011.
+
+
+Since each input image has 8x8 = 64 pixels or features, we have an input layer of 64 neurons.
+
+
+
+We will use 50 neurons in the hidden layer receiving input from the neurons in the input layer.
+Since each neuron in the hidden layer is connected to the 64 inputs we have 64x50 = 3200 weights to the hidden layer.
+
+
+
+If we were building a binary classifier, it would be sufficient with a single neuron in the output layer,
+which could output 0 or 1 according to the Heaviside function. This would be an example of a hard classifier, meaning it outputs the class of the input directly. However, if we are dealing with noisy data it is often beneficial to use a soft classifier, which outputs the probability of being in class 0 or 1.
+
+
+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.
+
+
+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:
+
+
+i.e. each neuron \( j \) outputs the probability of being in class \( j \) given an input from the hidden layer \( \boldsymbol{a} \), with \( \boldsymbol{w}_j \) 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:
+
+
+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.
+
+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.
+
+
+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 \( j \), \( b_j \):
+
+
+The bias weights \( \boldsymbol{b} \) 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.
-
+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:
+
+
+this is then passed through our activation function
+
+
+We calculate a weighted sum of inputs (activations in the hidden layer) to each neuron \( j \) in the output layer:
+
+
+Finally we calculate the output of neuron \( j \) in the output layer using the softmax function:
+
+
+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
+
+
+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 \( Z^{h} \):
+
+
+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:
+
+
+This is fed to the output layer:
+
+
+Finally we receive our output values for each image and each category by passing it through the softmax function:
+
+
+
+
+
+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.
+
+
+In multiclass classification it is common to treat each integer label as a so called one-hot vector:
+
+
+i.e. a binary bit string of length \( C \), where \( C = 10 \) is the number of classes in the MNIST dataset.
+
+
+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 \( \boldsymbol{x}_i \) in the dataset:
+
+
+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 \( \boldsymbol{\theta} \) represents the parameters of our network, i.e. all the weights and biases.
+
+
+A full derivation is given in the appendix at the end.
+
+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
+
+
+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:
+
+
+i.e. instead of averaging the loss over the entire dataset, we average over a minibatch.
+
+
+This has two important benefits:
+
+
+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.
+
+
+We will measure the size of the weights using the so called L2-norm, meaning our cost function becomes:
+
+
+i.e. we sum up all the weights squared. The factor \( \lambda \) is known as a regularization parameter.
+
+
+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.
+
+To more efficently train our network these equations are implemented using matrix operations.
+The error in the output layer is calculated simply as
+
+
+The gradient for the output weights is calculated as
+
+
+where \( A = (n_{inputs}, n_{hidden}) \). 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.
+
+
+The gradient with respect to the output bias is then
+
+
+The error in the hidden layer is
+
+
+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:
+
+
+
+
+
+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.
+
+
+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.
+
+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.
+
+
+
+
+
+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 \( 1 \).
+
+
+where \( I \) is the indicator function, \( 1 \) if \( \hat{y}_i = y_i \) and \( 0 \) otherwise.
+
+
+
+
+
+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).
+
+
+
+
+
-
+
+
+
+scikit-learn is a machine learning library for Python. It 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.
+
+
+
+
+
+
+
+
@@ -101,11 +177,8 @@ MathJax.Hub.Config({
-
-
-
In quantum information theory, it has been shown that one can perform
-gate decompositions with the help of neural. In lattice quantum chromodynamics,
-DNNs have been used to learn action parameters in regions of parameter
-space where PCA fails.
+gate decompositions with the help of neural.
-The applications are not limited to the natural sciences. There is a plethora of applications in essentially all disciplines, from the humanities to life science and medicine.
+The applications are not limited to the natural sciences. There is a
+plethora of applications in essentially all disciplines, from the
+humanities to life science and medicine.
-An artificial neural network (NN), is a computational model that
-consists of layers of connected neurons, or nodes. It is supposed
-to mimic a biological nervous system by letting each neuron interact
-with other neurons by sending signals in the form of mathematical
-functions between layers. A wide variety of different NNs have been
-developed, but most of them consist of an input layer, an output layer
-and eventual layers in-between, called hidden layers. All layers can
-contain an arbitrary number of nodes, and each connection between two
-nodes is associated with a weight variable.
+An artificial neural network (ANN), is a computational model that
+consists of layers of connected neurons, or nodes or units. We will
+refer to these interchangeably as units or nodes, and sometimes as
+neurons.
+
+
+It is supposed to mimic a biological nervous system by letting each
+neuron interact with other neurons by sending signals in the form of
+mathematical functions between layers. A wide variety of different
+ANNs have been developed, but most of them consist of an input layer,
+an output layer and eventual layers in-between, called hidden
+layers. All layers can contain an arbitrary number of nodes, and each
+connection between two nodes is associated with a weight variable.
Neural networks (also called neural nets) are neural-inspired
nonlinear models for supervised learning. As we will see, neural nets
can be viewed as natural, more powerful extensions of supervised
learning methods such as linear and logistic regression and soft-max
-methods.
+methods we discussed earlier.
-The feed-forward neural network (FFNN) was the first and simplest type of NN devised. In this network,
-the information moves in only one direction: forward through the layers.
+The feed-forward neural network (FFNN) was the first and simplest type
+of ANNs that were devised. In this network, the information moves in
+only one direction: forward through the layers.
-Nodes are represented by circles, while the arrows display the connections between the nodes, including the
-direction of information flow. Additionally, each arrow corresponds to a weight variable, not displayed here.
-We observe that each node in a layer is connected to all nodes in the subsequent layer,
-making this a so-called fully-connected FFNN.
-
-
-A different variant of FFNNs are convolutional neural networks (CNNs), which have a connectivity pattern
-inspired by the animal visual cortex. Individual neurons in the visual cortex only respond to stimuli from
-small sub-regions of the visual field, called a receptive field. This makes the neurons well-suited to exploit the strong
-spatially local correlation present in natural images. The response of each neuron can be approximated mathematically
-as a convolution operation.
-
-
-CNNs emulate the behaviour of neurons in the visual cortex by enforcing a local connectivity pattern
-between nodes of adjacent layers: Each node
-in a convolutional layer is connected only to a subset of the nodes in the previous layer,
-in contrast to the fully-connected FFNN.
-Often, CNNs
-consist of several convolutional layers that learn local features of the input, with a fully-connected layer at the end,
-which gathers all the local data and produces the outputs. They have wide applications in image and video recognition
+Nodes are represented by circles, while the arrows display the
+connections between the nodes, including the direction of information
+flow. Additionally, each arrow corresponds to a weight variable
+(figure to come). We observe that each node in a layer is connected
+to all nodes in the subsequent layer, making this a so-called
+fully-connected FFNN.
-So far we have only mentioned NNs where information flows in one direction: forward. Recurrent neural networks on
-the other hand, have connections between nodes that form directed cycles. This creates a form of
-internal memory which are able to capture information on what has been calculated before; the output is dependent
-on the previous computations. Recurrent NNs make use of sequential information by performing the same task for
-every element in a sequence, where each element depends on previous elements. An example of such information is
-sentences, making recurrent NNs especially well-suited for handwriting and speech recognition.
+A different variant of FFNNs are convolutional neural networks
+(CNNs), which have a connectivity pattern inspired by the animal
+visual cortex. Individual neurons in the visual cortex only respond to
+stimuli from small sub-regions of the visual field, called a receptive
+field. This makes the neurons well-suited to exploit the strong
+spatially local correlation present in natural images. The response of
+each neuron can be approximated mathematically as a convolution
+operation. (figure to come)
+
+
+Convolutional neural networks emulate the behaviour of neurons in the
+visual cortex by enforcing a local connectivity pattern between
+nodes of adjacent layers: Each node in a convolutional layer is
+connected only to a subset of the nodes in the previous layer, in
+contrast to the fully-connected FFNN. Often, CNNs consist of several
+convolutional layers that learn local features of the input, with a
+fully-connected layer at the end, which gathers all the local data and
+produces the outputs. They have wide applications in image and video
+recognition.
-There are many other kinds of NNs that have been developed. One type that is specifically designed for interpolation
-in multidimensional space is the radial basis function (RBF) network. RBFs are typically made up of three layers:
-an input layer, a hidden layer with non-linear radial symmetric activation functions and a linear output layer (''linear'' here
-means that each node in the output layer has a linear activation function). The layers are normally fully-connected and
-there are no cycles, thus RBFs can be viewed as a type of fully-connected FFNN. They are however usually treated as
-a separate type of NN due the unusual activation functions.
+So far we have only mentioned ANNs where information flows in one
+direction: forward. Recurrent neural networks on the other hand,
+have connections between nodes that form directed cycles. This
+creates a form of internal memory which are able to capture
+information on what has been calculated before; the output is
+dependent on the previous computations. Recurrent NNs make use of
+sequential information by performing the same task for every element
+in a sequence, where each element depends on previous elements. An
+example of such information is sentences, making recurrent NNs
+especially well-suited for handwriting and speech recognition.
+There are many other kinds of ANNs that have been developed. One type
+that is specifically designed for interpolation in multidimensional
+space is the radial basis function (RBF) network. RBFs are typically
+made up of three layers: an input layer, a hidden layer with
+non-linear radial symmetric activation functions and a linear output
+layer (''linear'' here means that each node in the output layer has a
+linear activation function). The layers are normally fully-connected
+and there are no cycles, thus RBFs can be viewed as a type of
+fully-connected FFNN. They are however usually treated as a separate
+type of NN due the unusual activation functions.
+
+
+
One uses often so-called fully-connected feed-forward neural networks
@@ -276,47 +373,39 @@ and an output layer) consisting of neurons that have non-linear
activation functions.
-Such networks are often called multilayer perceptrons (MLPs)
+Such networks are often called multilayer perceptrons (MLPs).
-According to the Universal approximation theorem, a feed-forward neural network with just a single hidden layer containing
-a finite number of neurons can approximate a continuous multidimensional function to arbitrary accuracy,
-assuming the activation function for the hidden layer is a non-constant, bounded and monotonically-increasing continuous function.
+According to the Universal approximation theorem, a feed-forward
+neural network with just a single hidden layer containing a finite
+number of neurons can approximate a continuous multidimensional
+function to arbitrary accuracy, assuming the activation function for
+the hidden layer is a non-constant, bounded and
+monotonically-increasing continuous function.
Note that the requirements on the activation function only applies to
the hidden layer, the output nodes are always assumed to be linear, so
as to not restrict the range of output values.
-
-We note that this theorem is only applicable to an NN with one hidden
-layer. Therefore, we can easily construct an NN that employs
-activation functions which do not satisfy the above requirements, as
-long as we have at least one layer with activation functions that
-do. Furthermore, although the universal approximation theorem lays
-the theoretical foundation for regression with neural networks, it
-does not say anything about how things work in practice: A neural
-network can still be able to approximate a given function reasonably
-well without having the flexibility to fit all other functions.
-
+The output \( y \) is produced via the activation function \( f \)
+$$
+ y = f\left(\sum_{i=1}^n w_ix_i + b_i\right) = f(z),
+$$
+
+This function receives \( x_i \) as inputs.
+Here the activation \( z=\sum_{i=1}^n w_ix_i \).
In an FFNN of such neurons, the inputs \( x_i \) are the outputs of
the neurons in the preceding layer. Furthermore, an MLP is
fully-connected, which means that each neuron receives a weighted sum
@@ -325,35 +414,40 @@ of the outputs of all neurons in the previous layer.
-First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( u_i^1 \) of the input coordinates \( x_j \),
+First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( z_i^1 \) of the input coordinates \( x_j \),
$$
-\begin{equation}
- u_i^1 = \sum_{j=1}^2 w_{ij}^1 x_j + b_i^1
+\begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1
\label{_auto1}
\end{equation}
$$
-This value is the argument to the activation function \( f_1 \) of each neuron \( i \),
-producing the output \( y_i^1 \) of all neurons in layer 1,
+
+Here \( b_i \) is the so-called bias which is normally needed in
+case of zero activation weights or inputs. How to fix the biases and
+the weights will be discussed below. The value of \( z_i^1 \) is the
+argument to the activation function \( f_i \) of each node \( i \), The
+variable \( M \) stands for all possible inputs to a given node \( i \) in the
+first layer. We define the output \( y_i^1 \) of all neurons in layer 1 as
$$
\begin{equation}
- y_i^1 = f_1(u_i^1) = f_1\left(\sum_{j=1}^2 w_{ij}^1 x_j + b_i^1\right)
+ y_i^1 = f(z_i^1) = f\left(\sum_{j=1}^M w_{ij}^1 x_j + b_i^1\right)
\label{outputLayer1}
\end{equation}
$$
where we assume that all nodes in the same layer have identical
-activation functions, hence the notation \( f_l \)
+activation functions, hence the notation \( f \). In general, we could assume in the more general case that different layers have different activation functions.
+In this case we would identify these functions with a superscript \( l \) for the \( l \)-th layer,
$$
\begin{equation}
- y_i^l = f_l(u_i^l) = f_l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
+ y_i^l = f^l(u_i^l) = f^l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
\label{generalLayer}
\end{equation}
$$
@@ -367,27 +461,27 @@ is obtained.
The output of neuron \( i \) in layer 2 is thus,
$$
\begin{align}
- y_i^2 &= f_2\left(\sum_{j=1}^3 w_{ij}^2 y_j^1 + b_i^2\right)
+ y_i^2 &= f^2\left(\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\right)
\label{_auto2}\\
- &= f_2\left[\sum_{j=1}^3 w_{ij}^2f_1\left(\sum_{k=1}^2 w_{jk}^1 x_k + b_j^1\right) + b_i^2\right]
+ &= f^2\left[\sum_{j=1}^N w_{ij}^2f^1\left(\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\right) + b_i^2\right]
\label{outputLayer2}
\end{align}
$$
-where we have substituted \( y_m^1 \) with. Finally, the NN output yields,
+where we have substituted \( y_k^1 \) with the inputs \( x_k \). Finally, the ANN output reads
$$
\begin{align}
- y_1^3 &= f_3\left(\sum_{j=1}^3 w_{1m}^3 y_j^2 + b_1^3\right)
+ y_i^3 &= f^3\left(\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\right)
\label{_auto3}\\
- &= f_3\left[\sum_{j=1}^3 w_{1j}^3 f_2\left(\sum_{k=1}^3 w_{jk}^2 f_1\left(\sum_{m=1}^2 w_{km}^1 x_m + b_k^1\right) + b_j^2\right)
+ &= f_3\left[\sum_{j} w_{ij}^3 f^2\left(\sum_{k} w_{jk}^2 f^1\left(\sum_{m} w_{km}^1 x_m + b_k^1\right) + b_j^2\right)
+ b_1^3\right]
\label{_auto4}
\end{align}
@@ -396,7 +490,7 @@ $$
We can generalize this expression to an MLP with \( l \) hidden
@@ -404,10 +498,7 @@ layers. The complete functional form is,
$$
\begin{align}
-&y^{l+1}_1\! = \!f_{l+1}\!\left[\!\sum_{j=1}^{N_l}\! w_{1j}^3 f_l\!\left(\!\sum_{k=1}^{N_{l-1}}\! w_{jk}^2 f_{l-1}\!\left(\!
- \dots \!f_1\!\left(\!\sum_{n=1}^{N_0} \!w_{mn}^1 x_n\! + \!b_m^1\!\right)
- \!\dots \!\right) \!+ \!b_k^2\!\right)
- \!+ \!b_1^3\!\right] &&
+&y^{l+1}_i = f^{l+1}\left[\!\sum_{j=1}^{N_l} w_{ij}^3 f^l\left(\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\left(\dots f^1\left(\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\right)\dots\right)+b_k^2\right)+b_1^3\right] &&
\label{completeNN}
\end{align}
$$
@@ -419,25 +510,22 @@ variables are the input values \( x_n \).
This confirms that an MLP, despite its quite convoluted mathematical
form, is nothing more than an analytic function, specifically a
-mapping of real-valued vectors \( \vec{x} \in \mathbb{R}^n \rightarrow
-\vec{y} \in \mathbb{R}^m \). In our example, \( n=2 \) and
-\( m=1 \). Consequentially, the number of input and output values of the
-function we want to fit must be equal to the number of inputs and
-outputs of our MLP.
+mapping of real-valued vectors \( \hat{x} \in \mathbb{R}^n \rightarrow
+\hat{y} \in \mathbb{R}^m \).
-Furthermore, the flexibility and universality of a MLP can be
+Furthermore, the flexibility and universality of an MLP can be
illustrated by realizing that the expression is essentially a nested
sum of scaled activation functions of the form
$$
\begin{equation}
- h(x) = c_1 f(c_2 x + c_3) + c_4
+ f(x) = c_1 f(c_2 x + c_3) + c_4
\label{_auto5}
\end{equation}
$$
@@ -451,23 +539,23 @@ flexibility of a neural network.
-We can introduce a more convenient notation for the activations in a NN.
+We can introduce a more convenient notation for the activations in an A NN.
Additionally, we can represent the biases and activations
-as layer-wise column vectors \( \vec{b}_l \) and \( \vec{y}_l \), so that the \( i \)-th element of each vector
+as layer-wise column vectors \( \hat{b}_l \) and \( \hat{y}_l \), so that the \( i \)-th element of each vector
is the bias \( b_i^l \) and activation \( y_i^l \) of node \( i \) in layer \( l \) respectively.
-We have that \( \mathrm{W}_l \) is a \( N_{l-1} \times N_l \) matrix, while \( \vec{b}_l \) and \( \vec{y}_l \) are \( N_l \times 1 \) column vectors.
-With this notation, the sum in becomes a matrix-vector multiplication, and we can write
-the equation for the activations of hidden layer 2 in
+We have that \( \mathrm{W}_l \) is an \( N_{l-1} \times N_l \) matrix, while \( \hat{b}_l \) and \( \hat{y}_l \) are \( N_l \times 1 \) column vectors.
+With this notation, the sum becomes a matrix-vector multiplication, and we can write
+the equation for the activations of hidden layer 2 as
$$
\begin{equation}
- \vec{y}_2 = f_2(\mathrm{W}_2 \vec{y}_{1} + \vec{b}_{2}) =
+ \hat{y}_2 = f_2(\mathrm{W}_2 \hat{y}_{1} + \hat{b}_{2}) =
f_2\left(\left[\begin{array}{ccc}
w^2_{11} &w^2_{12} &w^2_{13} \\
w^2_{21} &w^2_{22} &w^2_{23} \\
@@ -490,7 +578,7 @@ $$
The activation of node \( i \) in layer 2 is
@@ -508,12 +596,12 @@ This is not just a convenient and compact notation, but also a useful
and intuitive way to think about MLPs: The output is calculated by a
series of matrix-vector multiplications and vector additions that are
used as input to the activation functions. For each operation
-\( \mathrm{W}_l \vec{y}_{l-1} \) we move forward one layer.
+\( \mathrm{W}_l \hat{y}_{l-1} \) we move forward one layer.
A property that characterizes a neural network, other than its
@@ -530,7 +618,7 @@ for a FFNN to fulfill the universal approximation theorem
The second requirement excludes all linear functions. Furthermore, in
@@ -544,24 +632,18 @@ some kind of non-linearity to the NN to be able to fit non-linear
functions Typical examples are the logistic Sigmoid
$$
-\begin{equation}
f(x) = \frac{1}{1 + e^{-x}},
-\label{sigmoidActivationFunction}
-\end{equation}
$$
and the hyperbolic tangent function
$$
-\begin{equation}
f(x) = \tanh(x)
-\label{tanhActivationFunction}
-\end{equation}
$$
The sigmoid function are more biologically plausible because the
@@ -646,369 +728,1455 @@ ax.set_title('Rectified linear unit'
plt.show()
-
+
+The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of
-
-
+For an MLP there is no direct connection between the output nodes/neurons/units and the input nodes/neurons/units.
+Hereafter we will call the various entities of a layer for nodes.
+There are also no connections within a single layer.
- options = {'maxiter': 200, 'disp' : True}
- _res = optimize.minimize(self.costFunctionWrapper, params0, jac=True, method='BFGS', \
- args=(trainX, trainY), options=options, callback=self.callbackF)
+
+The number of input nodes does not need to equal the number of output
+nodes. This applies also to the hidden layers. Each layer may have its
+own number of nodes and activation functions.
+
+
+The hidden layers have their name from the fact that they are not
+linked to observables and as we will see below when we define the
+so-called activation \( \hat{z} \), we can think of this as a basis
+expansion of the original inputs \( \hat{x} \). The difference however
+between neural networks and say linear regression is that now these
+basis functions (which will correspond to the weights in the network)
+are learned from data. This results in an important difference between
+neural networks and deep learning approaches on one side and methods
+like logistic regression or linear regression and their modifications on the other side.
+
+
+
+A neural network with only one layer, what we called the simple
+perceptron, is best suited if we have a standard binary model with
+clear (linear) boundaries between the outcomes. As such it could
+equally well be replaced by standard linear regression or logistic
+regression. Networks with one or more hidden layers approximate
+systems with more complex boundaries.
+
+
+As stated earlier,
+an important theorem in studies of neural networks, restated without
+proof here, is the universal approximation
+theorem.
+
+
+It states that a feed-forward network with a single hidden layer
+containing a finite number of neurons can approximate continuous
+functions on compact subsets of real functions. The theorem thus
+states that simple neural networks can represent a wide variety of
+interesting functions when given appropriate parameters. It is the
+multilayer feedforward architecture itself which gives neural networks
+the potential of being universal approximators.
+
+
+
+Note: figures will be inserted later!
+
+
+As we have seen now in a feed forward network, we can express the final output of our network in terms of basic matrix-vector multiplications.
+The unknowwn quantities are our weights \( w_{ij} \) and we need to find an algorithm for changing them so that our errors are as small as possible.
+This leads us to the famous back propagation algorithm.
+
+
+The questions we want to ask are how do changes in the biases and the
+weights in our network change the cost function and how can we use the
+final output to modify the weights?
+
+
+To derive these equations let us start with a plain regression problem
+and define our cost function as
+
+$$
+{\cal C}(\hat{W}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2,
+$$
+
+
+where the $t_i$s are our \( n \) targets (the values we want to
+reproduce), while the outputs of the network after having propagated
+all inputs \( \hat{x} \) are given by \( y_i \). Below we will demonstrate
+how the basic equations arising from the back propagation algorithm
+can be modified in order to study classification problems with \( K \)
+classes.
+
+
+
+With our definition of the targets \( \hat{t} \), the outputs of the
+network \( \hat{y} \) and the inputs \( \hat{x} \) we
+define now the activation \( z_j^l \) of node/neuron/unit \( j \) of the
+\( l \)-th layer as a function of the bias, the weights which add up from
+the previous layer \( l-1 \) and the forward passes/outputs
+\( \hat{a}^{l-1} \) from the previous layer as
+
+$$
+z_j^l = \sum_{i=1}^{M_{l-1}}w_{ij}^la_j^{l-1}+b_j^l,
+$$
+
+
+where \( b_k^l \) are the biases from layer \( l \). Here \( M_{l-1} \)
+represents the total number of nodes/neurons/units of layer \( l-1 \). The
+figure here illustrates this equation. We can rewrite this in a more
+compact form as the matrix-vector products we discussed earlier,
+
+$$
+\hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l.
+$$
+
+
+With the activation function \( \hat{z}^l \) we can in turn define the
+output of layer \( l \) as \( \hat{a}^l = f(\hat{z}^l) \) where \( f \) is our
+activation function. In the examples here we will use the sigmoid
+function discussed in our logistic regression lectures and here as
+well. We will also use the same activation function \( f \) for all layers
+and their nodes. It means we have
+
+$$
+a_j^l = f(z_j^l) = \frac{1}{1+\exp{-(z_j^l)}}.
+$$
+
+
+
+From the definition of the activation \( z_j^l \) we have
+$$
+\frac{\partial z_j^l}{\partial w_{ji}^l} = a_i^{l-1},
+$$
+
+and
+$$
+\frac{\partial z_j^l}{\partial a_i^{l-1}} = w_{ji}^l.
+$$
+
+
+With our definition of the activation function we have that (note that this function depends only on \( z_j^l \))
+$$
+\frac{\partial a_j^l}{\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)).
+$$
+
+
+
+With these definitions we can now compute the derivative of the cost function in terms of the weights.
+
+
+Let us specialize to the output layer \( l=L \). Our cost function is
+$$
+{\cal C}(\hat{W^L}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2,
+$$
+
+The derivative of this function with respect to the weights is
+
+$$
+\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}},
+$$
+
+The last partial derivative can easily be computed and reads (by applying the chain rule)
+$$
+\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1},
+$$
+
+
+
+We have thus
+$$
+\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1},
+$$
+
+
+Defining
+$$
+\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
+$$
+
+and using the Hadamard product of two vectors we can write this as
+$$
+\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}L)}.
+$$
+
+
+This is an important expression. The second term on the right handside
+measures how fast the cost is changing as a function of the $j$th
+output activation. If, for example, the cost function doesn't depend
+much on a particular output node \( j \), then \( \delta_j^L \) will be small,
+which is what we would expect. The first term on the right, measures
+how fast the activation function \( f \) is changing at a given activation
+value \( z_j^L \).
+
+
+Notice that everything in the above equations is easily computed. In
+particular, we compute \( z_j^L \) while computing the behaviour of the
+network, and it is only a small additional overhead to compute
+\( f'(z^L_j) \). The exact form of the derivative with respect to the
+outpuwill, of course, depend on the form of the cost function.
+However, provided the cost function is known there should be little
+trouble computing
+
+$$
+\frac{\partial {\cal C}}{\partial (a_j^L)}
+$$
+
+
+With the definition of \( \delta_j^L \) we have a more compact definition of the derivative of the cost function in terms of the weights, namely
+$$
+\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}.
+$$
+
+
+
+It is also easy to see that our previous equation can be written as
+
+$$
+\delta_j^L =\frac{\partial {\cal C}}{\partial z_j^L}= \frac{\partial {\cal C}}{\partial a_j^L}\frac{\partial a_j^L}{\partial z_j^L},
+$$
+
+which can also be interpreted as the partial derivative of the cost function with respect to the biases \( b_j^L \), namely
+$$
+\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}\frac{\partial b_j^L}{\partial z_j^L}=\frac{\partial {\cal C}}{\partial b_j^L},
+$$
+
+That is, the error \( \delta_j^L \) is exactly equal to the rate of change of the cost function as a function of the bias.
+
+We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are
+
+
+$$
+\begin{equation}
+\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1},
+\label{_auto8}
+\end{equation}
+$$
+
+and
+$$
+\begin{equation}
+\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
+\label{_auto9}
+\end{equation}
+$$
+
+and
+
+$$
+\begin{equation}
+\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L},
+\label{_auto10}
+\end{equation}
+$$
+
+An interesting consequence of the above equations is that when the
+activation \( a_k^{L-1} \) is small, the gradient term, that is the
+derivative of the cost function with respect to the weights, will also
+tend to be small. We say then that the weight learns slowly, meaning
+that it changes slowly when we minimize the weights via say gradient
+descent. In this case we say the system learns slowly.
+
+
+Another interesting feature is that is when the activation function,
+represented by the sigmoid function here, is rather flat when towards
+its end values \( 0 \) and \( 1 \) (see the above Python codes). In these
+cases, the derivatives of the activation function will also be close
+to zero, meaning again that the gradients will be small and the
+network learns slowly again.
+
+
+We need a fourth equation and we are set. We are going to propagate
+backwards in order to the determine the weights and biases. In order
+to do so we need to represent the error in the layer before the final
+one \( L-1 \) in terms of the errors in the final output layer.
+
+
+
+We have that (replacing \( L \) with a general layer \( l \))
+$$
+\delta_j^l =\frac{\partial {\cal C}}{\partial z_j^l}.
+$$
+
+We want to express this in terms of the equations for layer \( l+1 \). Using the chain rule and summing over all \( k \) entries we have
+
+$$
+\delta_j^l =\sum_k \frac{\partial {\cal C}}{\partial z_k^{l+1}}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}=\sum_k \delta_k^{l+1}\frac{\partial z_k^{l+1}}{\partial z_j^{l}},
+$$
+
+and recalling that
+$$
+z_j^{l+1} = \sum_{i=1}^{M_{l}}w_{ij}^{l+1}a_j^{l}+b_j^{l+1},
+$$
+
+we obtain
+$$
+\delta_j^l =\sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l),
+$$
+
+This is our final equation.
+
+
+We are now ready to set up the algorithm for back propagation and learning the weights and biases.
+
+
+
+The four equations provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.
+
+
+
+First, we set up the input data \( \hat{x} \) and the activations
+\( \hat{z}_1 \) of the input layer and compute the activation function and
+the pertinent outputs \( \hat{a}^1 \).
+
+
+Secondly, we perform then the feed forward till we reach the output
+layer and compute all \( \hat{z}_l \) of the input layer and compute the
+activation function and the pertinent outputs \( \hat{a}^l \) for
+\( l=2,3,\dots,L \).
+
+
+Thereafter we compute the ouput error \( \hat{\delta}^L \) by computing all
+$$
+\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
+$$
+
+
+Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as
+$$
+\delta_j^l =\sum_k \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).
+$$
+
+
+Finally, we update the weights and the biases using gradient descent for each \( l=L-1,L-2,dots,2 \) and update the weights and biases according to the rules
+$$
+w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1},
+$$
+
+
+$$
+b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^L},
+$$
+
+The parameter \( \eta \) is the learning parameter discussed in connection with the gradient descent methods.
+Here it is convenient to use stochastic radient descent with mini-batches with an outer loop that steps through multiple epochs of training.
- self.N.setParams(_res.x)
- self.optimizationResults = _res
-
-
+We are now gong to develop an example based on the MNIST data
+base. This is a classification problem and we need to use our
+cross-entropy function we discussed in connection with logistic
+regression. The cross-entropy defines our cost function for the
+classificaton problems with neural networks.
+
+
+In binary classification with two classes \( (0, 1) \) we define the
+logistic/sigmoid function as the probability that a particular input
+is in class \( 0 \) or \( 1 \). This is possible because the logistic
+function takes any input from the real numbers and inputs a number
+between 0 and 1, and can therefore be interpreted as a probability. It
+also has other nice properties, such as a derivative that is simple to
+calculate.
+
+
+For an input \( \boldsymbol{a} \) from the hidden layer, the probability that the input \( \boldsymbol{x} \)
+is in class 0 or 1 is just:
+
+$$
+P(y = 0 \mid \boldsymbol{x}, \boldsymbol{\theta}) = \frac{1}{1 + \exp (- \boldsymbol{a}^T \boldsymbol{w}_{out})} ,
+$$
+
+and
+$$
+P(y = 1 \mid \boldsymbol{x}, \boldsymbol{\theta}) = 1 - P(y = 0 \mid \boldsymbol{x}, \boldsymbol{\theta}) ,
+$$
+
+
+where \( y \in \{0, 1\} \) and \( \boldsymbol{\theta} \) represents the weights and biases
+of our network.
+
+
+
+Our cost function is given as (see the Logistic regression lectures)
+$$
+\mathcal{C}(\boldsymbol{\theta}) = - \ln P(\mathcal{D} \mid \boldsymbol{\theta}) = - \sum_{i=1}^n
+y_i \ln[P(y_i = 0)] + (1 - y_i) \ln [1 - P(y_i = 0)] = \sum_{i=1}^n \mathcal{L}_i(\boldsymbol{\theta}) .
+$$
+
+
+This last equality means that we can interpret our cost function as a sum over the loss function
+for each point in the dataset \( \mathcal{L}_i(\boldsymbol{\theta}) \).
+The negative sign is just so that we can think about our algorithm as minimizing a positive number, rather
+than maximizing a negative number.
+
+
+In multiclass classification it is common to treat each integer label as a so called one-hot vector:
+
+
+\( y = 5 \quad \rightarrow \quad \boldsymbol{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) , \) and
+
+
+\( y = 1 \quad \rightarrow \quad \boldsymbol{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) , \)
+
+
+i.e. a binary bit string of length \( C \), where \( C = 10 \) is the number of classes in the MNIST dataset (numbers from \( 0 \) to \( 9 \))..
+
+
+If \( \boldsymbol{x}_i \) is the \( i \)-th input (image), \( y_{ic} \) refers to the \( c \)-th component of the \( i \)-th
+output vector \( \boldsymbol{y}_i \).
+The probability of \( \boldsymbol{x}_i \) being in class \( c \) will be given by the softmax function:
+
+$$
+P(y_{ic} = 1 \mid \boldsymbol{x}_i, \boldsymbol{\theta}) = \frac{\exp{((\boldsymbol{a}_i^{hidden})^T \boldsymbol{w}_c)}}
+{\sum_{c'=0}^{C-1} \exp{((\boldsymbol{a}_i^{hidden})^T \boldsymbol{w}_{c'})}} ,
+$$
+
+
+which reduces to the logistic function in the binary case.
+The likelihood of this \( C \)-class classifier
+is now given as:
+
+$$
+P(\mathcal{D} \mid \boldsymbol{\theta}) = \prod_{i=1}^n \prod_{c=0}^{C-1} [P(y_{ic} = 1)]^{y_{ic}} .
+$$
+
+Again we take the negative log-likelihood to define our cost function:
+
+$$
+\mathcal{C}(\boldsymbol{\theta}) = - \ln P(\mathcal{D} \mid \boldsymbol{\theta}) = - \sum_{i=1}^n \sum_{c=0}^{C-1}
+y_{ic} \ln[P(y_{ic} = 1)] = \sum_{i=1}^n
+\mathcal{L}_i(\boldsymbol{\theta}) .
+$$
+
+
+The back propagation equations need now only a small change, namely the definition of a new cost function. We are thus ready to use the same equations as before!
+We leave it as an exercise in project 2 to derive these equations.
+
+
+
+
+
+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.
+
+
+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.
+
-
-
-
-
-
-
-
+
+Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions.
+
+
+We will reserve \( 80 \% \) of our dataset for training and \( 20 \% \) for testing.
+
+
+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.
+
+
+
+
+
+
+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:
+
+$$ z = \sum_{i=1}^n w_i a_i ,$$
+
+$$ y = f(z) ,$$
+
+
+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).
+
+
+The simplest activation function for a neuron is the Heaviside function:
+
+$$ f(z) =
+\begin{cases}
+1, & z > 0\\
+0, & \text{otherwise}
+\end{cases}
+$$
+
+
+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.
+
+
+
+
+
+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.
+
+
+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 \( \boldsymbol{a} \)}) = \frac{\exp{(\boldsymbol{a}^T \boldsymbol{w}_j)}}
+{\sum_{c=0}^{9} \exp{(\boldsymbol{a}^T \boldsymbol{w}_c)}} ,$$
+
+
+i.e. each neuron \( j \) outputs the probability of being in class \( j \) given an input from the hidden layer \( \boldsymbol{a} \), with \( \boldsymbol{w}_j \) 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:
+
+$$ z_j = \sum_{i=1}^n w_ {ij} a_i = \boldsymbol{a}^T \boldsymbol{w}_j .$$
+
+
+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.
+
+
+
+
+
+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.
+
+
+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 \( j \), \( b_j \):
+
+$$ z_j = \sum_{i=1}^n w_ {ij} a_i + 1\cdot b_j = \boldsymbol{a}^T \boldsymbol{w}_j + b_j .$$
+
+
+The bias weights \( \boldsymbol{b} \) 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.
+
+
+
+
+
+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:
+
+$$ z_{j}^{h} = \sum_{i=1}^{F} w_{ij}^{h} x_i + b_{j}^{h} = \boldsymbol{x}^T \boldsymbol{w}_{j}^{h} + b_{j}^{h} ,$$
+
+
+this is then passed through our activation function
+
+$$ a_{j}^{h} = f(z_{j}^{h}) .$$
+
+
+We calculate a weighted sum of inputs (activations in the hidden layer) to each neuron \( j \) in the output layer:
+
+$$ z_{j}^{o} = \sum_{i=1}^{H} w_{ij}^{o} a_{i}^{h} + b_{j}^{o} = (\boldsymbol{a}^{h})^T \boldsymbol{w}_{j}^{o} + b_{j}^{o} .$$
+
+
+Finally we calculate the output of neuron \( j \) in the output layer using the softmax function:
+
+$$ a_{j}^{o} = \frac{\exp{(z_j^{o})}}
+{\sum_{c=0}^{C-1} \exp{(z_c^{o})}} .$$
+
+
+
+
+
+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
+
+$$ X W^{h} = (n_{inputs}, n_{hidden}),$$
+
+
+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 \( Z^{h} \):
+
+$$ Z^{h} = X W^{h} + B^{h} ,$$
+
+
+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:
+
+$$ A^{h} = f(Z^h) .$$
+
+
+This is fed to the output layer:
+
+$$ Z^{o} = A^{h} W^{o} + B^{o} .$$
+
+
+Finally we receive our output values for each image and each category by passing it through the softmax function:
+
+$$ output = softmax (Z^{o}) = (n_{inputs}, n_{categories}) .$$
+
+
+
+
+
+
+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.
+
+
+In multiclass classification it is common to treat each integer label as a so called one-hot vector:
+
+$$ y = 5 \quad \rightarrow \quad \boldsymbol{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$$
+
+$$ y = 1 \quad \rightarrow \quad \boldsymbol{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,$$
+
+
+i.e. a binary bit string of length \( C \), where \( C = 10 \) is the number of classes in the MNIST dataset.
+
+
+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 \( \boldsymbol{x}_i \) in the dataset:
+
+$$
+\begin{align} \mathcal{C}(\boldsymbol{\theta}) &= \sum_{i=1}^N \mathcal{L}_i (\boldsymbol{\theta})
+\label{_auto11}\\
+ &= -\sum_{i=1}^N \sum_{c=0}^{C-1} y_{ic} \log P(y_{ic} = 1 \mid \boldsymbol{x}_i, \boldsymbol{\theta}) .
+\label{_auto12}
+\end{align}
+$$
+
+
+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 \( \boldsymbol{\theta} \) represents the parameters of our network, i.e. all the weights and biases.
+
+
+A full derivation is given in the appendix at the end.
+
+
+
+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
+
+$$ \theta_{i+1} = \theta_i - \eta \nabla \mathcal{C}(\theta_i) ,$$
+
+
+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:
+
+$$ \nabla \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) \quad \rightarrow \quad
+\frac{1}{M} \sum_{i \in B_k} \nabla \mathcal{L}_i(\theta) ,$$
+
+
+i.e. instead of averaging the loss over the entire dataset, we average over a minibatch.
+
+
+This has two important benefits:
+
+
+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.
+
+
+We will measure the size of the weights using the so called L2-norm, meaning our cost function becomes:
+
+$$ \nabla \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) \quad \rightarrow \quad
+\frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) + \lambda \lvert \lvert \boldsymbol{w} \rvert \rvert_2^2
+= \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}(\theta) + \lambda \sum_{ij} w_{ij}^2,$$
+
+
+i.e. we sum up all the weights squared. The factor \( \lambda \) is known as a regularization parameter.
+
+
+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.
+
+
+
+To more efficently train our network these equations are implemented using matrix operations.
+The error in the output layer is calculated simply as
+
+$$ \Delta_o = \hat{y} - y = (n_{inputs}, n_{categories}) .$$
+
+
+The gradient for the output weights is calculated as
+
+$$ \nabla W_{o} = A^T \Delta_o = (n_{hidden}, n_{categories}) ,$$
+
+
+where \( A = (n_{inputs}, n_{hidden}) \). 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.
+
+
+The gradient with respect to the output bias is then
+
+$$ \nabla B_{o} = \sum_{i=1}^{n_{inputs}} \Delta_o = (n_{categories}) .$$
+
+
+The error in the hidden layer is
+
+$$ \Delta_h = \Delta_o W_{o}^T \circ f'(Z_{h}) = \Delta_o W_{o}^T \circ A_{h} \circ (1 - A_{h}) = (n_{inputs}, n_{hidden}) ,$$
+
+
+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:
+
+$$ \nabla W_{h} = X^T \Delta_h = (n_{features}, n_{hidden}) ,$$
+
+$$ \nabla B_{h} = \sum_{i=1}^{n_{inputs}} \Delta_h = (n_{hidden}) .$$
+
+
+
+
+
+
+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.
+
+
+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.
+
+
+
+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.
+
+
+
+
+
+
+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.
+
+
+
+
+
+
+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).
+
+
+
+
+
+
+
+
+
+
+
+
+scikit-learn is a machine learning library for Python. It 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.
+
+
+
+
+
+
+
+
+
diff --git a/doc/pub/NeuralNet/html/NeuralNet.html b/doc/pub/NeuralNet/html/NeuralNet.html
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+++ b/doc/pub/NeuralNet/html/NeuralNet.html
@@ -6,9 +6,9 @@ Automatically generated HTML file from DocOnce source
@@ -106,11 +182,8 @@ MathJax.Hub.Config({
-
-
-
In quantum information theory, it has been shown that one can perform
-gate decompositions with the help of neural. In lattice quantum chromodynamics,
-DNNs have been used to learn action parameters in regions of parameter
-space where PCA fails.
+gate decompositions with the help of neural.
-The applications are not limited to the natural sciences. There is a plethora of applications in essentially all disciplines, from the humanities to life science and medicine.
+The applications are not limited to the natural sciences. There is a
+plethora of applications in essentially all disciplines, from the
+humanities to life science and medicine.
-An artificial neural network (NN), is a computational model that
-consists of layers of connected neurons, or nodes. It is supposed
-to mimic a biological nervous system by letting each neuron interact
-with other neurons by sending signals in the form of mathematical
-functions between layers. A wide variety of different NNs have been
-developed, but most of them consist of an input layer, an output layer
-and eventual layers in-between, called hidden layers. All layers can
-contain an arbitrary number of nodes, and each connection between two
-nodes is associated with a weight variable.
+An artificial neural network (ANN), is a computational model that
+consists of layers of connected neurons, or nodes or units. We will
+refer to these interchangeably as units or nodes, and sometimes as
+neurons.
+
+
+It is supposed to mimic a biological nervous system by letting each
+neuron interact with other neurons by sending signals in the form of
+mathematical functions between layers. A wide variety of different
+ANNs have been developed, but most of them consist of an input layer,
+an output layer and eventual layers in-between, called hidden
+layers. All layers can contain an arbitrary number of nodes, and each
+connection between two nodes is associated with a weight variable.
Neural networks (also called neural nets) are neural-inspired
nonlinear models for supervised learning. As we will see, neural nets
can be viewed as natural, more powerful extensions of supervised
learning methods such as linear and logistic regression and soft-max
-methods.
+methods we discussed earlier.
-The feed-forward neural network (FFNN) was the first and simplest type of NN devised. In this network,
-the information moves in only one direction: forward through the layers.
+The feed-forward neural network (FFNN) was the first and simplest type
+of ANNs that were devised. In this network, the information moves in
+only one direction: forward through the layers.
-Nodes are represented by circles, while the arrows display the connections between the nodes, including the
-direction of information flow. Additionally, each arrow corresponds to a weight variable, not displayed here.
-We observe that each node in a layer is connected to all nodes in the subsequent layer,
-making this a so-called fully-connected FFNN.
-
-
-A different variant of FFNNs are convolutional neural networks (CNNs), which have a connectivity pattern
-inspired by the animal visual cortex. Individual neurons in the visual cortex only respond to stimuli from
-small sub-regions of the visual field, called a receptive field. This makes the neurons well-suited to exploit the strong
-spatially local correlation present in natural images. The response of each neuron can be approximated mathematically
-as a convolution operation.
-
-
-CNNs emulate the behaviour of neurons in the visual cortex by enforcing a local connectivity pattern
-between nodes of adjacent layers: Each node
-in a convolutional layer is connected only to a subset of the nodes in the previous layer,
-in contrast to the fully-connected FFNN.
-Often, CNNs
-consist of several convolutional layers that learn local features of the input, with a fully-connected layer at the end,
-which gathers all the local data and produces the outputs. They have wide applications in image and video recognition
+Nodes are represented by circles, while the arrows display the
+connections between the nodes, including the direction of information
+flow. Additionally, each arrow corresponds to a weight variable
+(figure to come). We observe that each node in a layer is connected
+to all nodes in the subsequent layer, making this a so-called
+fully-connected FFNN.
-So far we have only mentioned NNs where information flows in one direction: forward. Recurrent neural networks on
-the other hand, have connections between nodes that form directed cycles. This creates a form of
-internal memory which are able to capture information on what has been calculated before; the output is dependent
-on the previous computations. Recurrent NNs make use of sequential information by performing the same task for
-every element in a sequence, where each element depends on previous elements. An example of such information is
-sentences, making recurrent NNs especially well-suited for handwriting and speech recognition.
+A different variant of FFNNs are convolutional neural networks
+(CNNs), which have a connectivity pattern inspired by the animal
+visual cortex. Individual neurons in the visual cortex only respond to
+stimuli from small sub-regions of the visual field, called a receptive
+field. This makes the neurons well-suited to exploit the strong
+spatially local correlation present in natural images. The response of
+each neuron can be approximated mathematically as a convolution
+operation. (figure to come)
+
+
+Convolutional neural networks emulate the behaviour of neurons in the
+visual cortex by enforcing a local connectivity pattern between
+nodes of adjacent layers: Each node in a convolutional layer is
+connected only to a subset of the nodes in the previous layer, in
+contrast to the fully-connected FFNN. Often, CNNs consist of several
+convolutional layers that learn local features of the input, with a
+fully-connected layer at the end, which gathers all the local data and
+produces the outputs. They have wide applications in image and video
+recognition.
-There are many other kinds of NNs that have been developed. One type that is specifically designed for interpolation
-in multidimensional space is the radial basis function (RBF) network. RBFs are typically made up of three layers:
-an input layer, a hidden layer with non-linear radial symmetric activation functions and a linear output layer (''linear'' here
-means that each node in the output layer has a linear activation function). The layers are normally fully-connected and
-there are no cycles, thus RBFs can be viewed as a type of fully-connected FFNN. They are however usually treated as
-a separate type of NN due the unusual activation functions.
+So far we have only mentioned ANNs where information flows in one
+direction: forward. Recurrent neural networks on the other hand,
+have connections between nodes that form directed cycles. This
+creates a form of internal memory which are able to capture
+information on what has been calculated before; the output is
+dependent on the previous computations. Recurrent NNs make use of
+sequential information by performing the same task for every element
+in a sequence, where each element depends on previous elements. An
+example of such information is sentences, making recurrent NNs
+especially well-suited for handwriting and speech recognition.
+There are many other kinds of ANNs that have been developed. One type
+that is specifically designed for interpolation in multidimensional
+space is the radial basis function (RBF) network. RBFs are typically
+made up of three layers: an input layer, a hidden layer with
+non-linear radial symmetric activation functions and a linear output
+layer (''linear'' here means that each node in the output layer has a
+linear activation function). The layers are normally fully-connected
+and there are no cycles, thus RBFs can be viewed as a type of
+fully-connected FFNN. They are however usually treated as a separate
+type of NN due the unusual activation functions.
+
+
+
One uses often so-called fully-connected feed-forward neural networks
@@ -281,47 +378,39 @@ and an output layer) consisting of neurons that have non-linear
activation functions.
-Such networks are often called multilayer perceptrons (MLPs)
+Such networks are often called multilayer perceptrons (MLPs).
-According to the Universal approximation theorem, a feed-forward neural network with just a single hidden layer containing
-a finite number of neurons can approximate a continuous multidimensional function to arbitrary accuracy,
-assuming the activation function for the hidden layer is a non-constant, bounded and monotonically-increasing continuous function.
+According to the Universal approximation theorem, a feed-forward
+neural network with just a single hidden layer containing a finite
+number of neurons can approximate a continuous multidimensional
+function to arbitrary accuracy, assuming the activation function for
+the hidden layer is a non-constant, bounded and
+monotonically-increasing continuous function.
Note that the requirements on the activation function only applies to
the hidden layer, the output nodes are always assumed to be linear, so
as to not restrict the range of output values.
-
-We note that this theorem is only applicable to an NN with one hidden
-layer. Therefore, we can easily construct an NN that employs
-activation functions which do not satisfy the above requirements, as
-long as we have at least one layer with activation functions that
-do. Furthermore, although the universal approximation theorem lays
-the theoretical foundation for regression with neural networks, it
-does not say anything about how things work in practice: A neural
-network can still be able to approximate a given function reasonably
-well without having the flexibility to fit all other functions.
-
+The output \( y \) is produced via the activation function \( f \)
+$$
+ y = f\left(\sum_{i=1}^n w_ix_i + b_i\right) = f(z),
+$$
+
+This function receives \( x_i \) as inputs.
+Here the activation \( z=\sum_{i=1}^n w_ix_i \).
In an FFNN of such neurons, the inputs \( x_i \) are the outputs of
the neurons in the preceding layer. Furthermore, an MLP is
fully-connected, which means that each neuron receives a weighted sum
@@ -330,35 +419,40 @@ of the outputs of all neurons in the previous layer.
-First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( u_i^1 \) of the input coordinates \( x_j \),
+First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( z_i^1 \) of the input coordinates \( x_j \),
$$
-\begin{equation}
- u_i^1 = \sum_{j=1}^2 w_{ij}^1 x_j + b_i^1
+\begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1
\label{_auto1}
\end{equation}
$$
-This value is the argument to the activation function \( f_1 \) of each neuron \( i \),
-producing the output \( y_i^1 \) of all neurons in layer 1,
+
+Here \( b_i \) is the so-called bias which is normally needed in
+case of zero activation weights or inputs. How to fix the biases and
+the weights will be discussed below. The value of \( z_i^1 \) is the
+argument to the activation function \( f_i \) of each node \( i \), The
+variable \( M \) stands for all possible inputs to a given node \( i \) in the
+first layer. We define the output \( y_i^1 \) of all neurons in layer 1 as
$$
\begin{equation}
- y_i^1 = f_1(u_i^1) = f_1\left(\sum_{j=1}^2 w_{ij}^1 x_j + b_i^1\right)
+ y_i^1 = f(z_i^1) = f\left(\sum_{j=1}^M w_{ij}^1 x_j + b_i^1\right)
\label{outputLayer1}
\end{equation}
$$
where we assume that all nodes in the same layer have identical
-activation functions, hence the notation \( f_l \)
+activation functions, hence the notation \( f \). In general, we could assume in the more general case that different layers have different activation functions.
+In this case we would identify these functions with a superscript \( l \) for the \( l \)-th layer,
$$
\begin{equation}
- y_i^l = f_l(u_i^l) = f_l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
+ y_i^l = f^l(u_i^l) = f^l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
\label{generalLayer}
\end{equation}
$$
@@ -372,27 +466,27 @@ is obtained.
The output of neuron \( i \) in layer 2 is thus,
$$
\begin{align}
- y_i^2 &= f_2\left(\sum_{j=1}^3 w_{ij}^2 y_j^1 + b_i^2\right)
+ y_i^2 &= f^2\left(\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\right)
\label{_auto2}\\
- &= f_2\left[\sum_{j=1}^3 w_{ij}^2f_1\left(\sum_{k=1}^2 w_{jk}^1 x_k + b_j^1\right) + b_i^2\right]
+ &= f^2\left[\sum_{j=1}^N w_{ij}^2f^1\left(\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\right) + b_i^2\right]
\label{outputLayer2}
\end{align}
$$
-where we have substituted \( y_m^1 \) with. Finally, the NN output yields,
+where we have substituted \( y_k^1 \) with the inputs \( x_k \). Finally, the ANN output reads
$$
\begin{align}
- y_1^3 &= f_3\left(\sum_{j=1}^3 w_{1m}^3 y_j^2 + b_1^3\right)
+ y_i^3 &= f^3\left(\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\right)
\label{_auto3}\\
- &= f_3\left[\sum_{j=1}^3 w_{1j}^3 f_2\left(\sum_{k=1}^3 w_{jk}^2 f_1\left(\sum_{m=1}^2 w_{km}^1 x_m + b_k^1\right) + b_j^2\right)
+ &= f_3\left[\sum_{j} w_{ij}^3 f^2\left(\sum_{k} w_{jk}^2 f^1\left(\sum_{m} w_{km}^1 x_m + b_k^1\right) + b_j^2\right)
+ b_1^3\right]
\label{_auto4}
\end{align}
@@ -401,7 +495,7 @@ $$
We can generalize this expression to an MLP with \( l \) hidden
@@ -409,10 +503,7 @@ layers. The complete functional form is,
$$
\begin{align}
-&y^{l+1}_1\! = \!f_{l+1}\!\left[\!\sum_{j=1}^{N_l}\! w_{1j}^3 f_l\!\left(\!\sum_{k=1}^{N_{l-1}}\! w_{jk}^2 f_{l-1}\!\left(\!
- \dots \!f_1\!\left(\!\sum_{n=1}^{N_0} \!w_{mn}^1 x_n\! + \!b_m^1\!\right)
- \!\dots \!\right) \!+ \!b_k^2\!\right)
- \!+ \!b_1^3\!\right] &&
+&y^{l+1}_i = f^{l+1}\left[\!\sum_{j=1}^{N_l} w_{ij}^3 f^l\left(\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\left(\dots f^1\left(\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\right)\dots\right)+b_k^2\right)+b_1^3\right] &&
\label{completeNN}
\end{align}
$$
@@ -424,25 +515,22 @@ variables are the input values \( x_n \).
This confirms that an MLP, despite its quite convoluted mathematical
form, is nothing more than an analytic function, specifically a
-mapping of real-valued vectors \( \vec{x} \in \mathbb{R}^n \rightarrow
-\vec{y} \in \mathbb{R}^m \). In our example, \( n=2 \) and
-\( m=1 \). Consequentially, the number of input and output values of the
-function we want to fit must be equal to the number of inputs and
-outputs of our MLP.
+mapping of real-valued vectors \( \hat{x} \in \mathbb{R}^n \rightarrow
+\hat{y} \in \mathbb{R}^m \).
-Furthermore, the flexibility and universality of a MLP can be
+Furthermore, the flexibility and universality of an MLP can be
illustrated by realizing that the expression is essentially a nested
sum of scaled activation functions of the form
$$
\begin{equation}
- h(x) = c_1 f(c_2 x + c_3) + c_4
+ f(x) = c_1 f(c_2 x + c_3) + c_4
\label{_auto5}
\end{equation}
$$
@@ -456,23 +544,23 @@ flexibility of a neural network.
-We can introduce a more convenient notation for the activations in a NN.
+We can introduce a more convenient notation for the activations in an A NN.
Additionally, we can represent the biases and activations
-as layer-wise column vectors \( \vec{b}_l \) and \( \vec{y}_l \), so that the \( i \)-th element of each vector
+as layer-wise column vectors \( \hat{b}_l \) and \( \hat{y}_l \), so that the \( i \)-th element of each vector
is the bias \( b_i^l \) and activation \( y_i^l \) of node \( i \) in layer \( l \) respectively.
-We have that \( \mathrm{W}_l \) is a \( N_{l-1} \times N_l \) matrix, while \( \vec{b}_l \) and \( \vec{y}_l \) are \( N_l \times 1 \) column vectors.
-With this notation, the sum in becomes a matrix-vector multiplication, and we can write
-the equation for the activations of hidden layer 2 in
+We have that \( \mathrm{W}_l \) is an \( N_{l-1} \times N_l \) matrix, while \( \hat{b}_l \) and \( \hat{y}_l \) are \( N_l \times 1 \) column vectors.
+With this notation, the sum becomes a matrix-vector multiplication, and we can write
+the equation for the activations of hidden layer 2 as
$$
\begin{equation}
- \vec{y}_2 = f_2(\mathrm{W}_2 \vec{y}_{1} + \vec{b}_{2}) =
+ \hat{y}_2 = f_2(\mathrm{W}_2 \hat{y}_{1} + \hat{b}_{2}) =
f_2\left(\left[\begin{array}{ccc}
w^2_{11} &w^2_{12} &w^2_{13} \\
w^2_{21} &w^2_{22} &w^2_{23} \\
@@ -495,7 +583,7 @@ $$
The activation of node \( i \) in layer 2 is
@@ -513,12 +601,12 @@ This is not just a convenient and compact notation, but also a useful
and intuitive way to think about MLPs: The output is calculated by a
series of matrix-vector multiplications and vector additions that are
used as input to the activation functions. For each operation
-\( \mathrm{W}_l \vec{y}_{l-1} \) we move forward one layer.
+\( \mathrm{W}_l \hat{y}_{l-1} \) we move forward one layer.
A property that characterizes a neural network, other than its
@@ -535,7 +623,7 @@ for a FFNN to fulfill the universal approximation theorem
The second requirement excludes all linear functions. Furthermore, in
@@ -549,24 +637,18 @@ some kind of non-linearity to the NN to be able to fit non-linear
functions Typical examples are the logistic Sigmoid
$$
-\begin{equation}
f(x) = \frac{1}{1 + e^{-x}},
-\label{sigmoidActivationFunction}
-\end{equation}
$$
and the hyperbolic tangent function
$$
-\begin{equation}
f(x) = \tanh(x)
-\label{tanhActivationFunction}
-\end{equation}
$$
The sigmoid function are more biologically plausible because the
@@ -651,369 +733,1455 @@ ax.set_title(&#
plt.show()
-
+
+The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of
-
-
+For an MLP there is no direct connection between the output nodes/neurons/units and the input nodes/neurons/units.
+Hereafter we will call the various entities of a layer for nodes.
+There are also no connections within a single layer.
- options = {'maxiter': 200, 'disp' : True}
- _res = optimize.minimize(self.costFunctionWrapper, params0, jac=True, method='BFGS', \
- args=(trainX, trainY), options=options, callback=self.callbackF)
+
+The number of input nodes does not need to equal the number of output
+nodes. This applies also to the hidden layers. Each layer may have its
+own number of nodes and activation functions.
+
+
+The hidden layers have their name from the fact that they are not
+linked to observables and as we will see below when we define the
+so-called activation \( \hat{z} \), we can think of this as a basis
+expansion of the original inputs \( \hat{x} \). The difference however
+between neural networks and say linear regression is that now these
+basis functions (which will correspond to the weights in the network)
+are learned from data. This results in an important difference between
+neural networks and deep learning approaches on one side and methods
+like logistic regression or linear regression and their modifications on the other side.
+
+
+
+A neural network with only one layer, what we called the simple
+perceptron, is best suited if we have a standard binary model with
+clear (linear) boundaries between the outcomes. As such it could
+equally well be replaced by standard linear regression or logistic
+regression. Networks with one or more hidden layers approximate
+systems with more complex boundaries.
+
+
+As stated earlier,
+an important theorem in studies of neural networks, restated without
+proof here, is the universal approximation
+theorem.
+
+
+It states that a feed-forward network with a single hidden layer
+containing a finite number of neurons can approximate continuous
+functions on compact subsets of real functions. The theorem thus
+states that simple neural networks can represent a wide variety of
+interesting functions when given appropriate parameters. It is the
+multilayer feedforward architecture itself which gives neural networks
+the potential of being universal approximators.
+
+
+
+Note: figures will be inserted later!
+
+
+As we have seen now in a feed forward network, we can express the final output of our network in terms of basic matrix-vector multiplications.
+The unknowwn quantities are our weights \( w_{ij} \) and we need to find an algorithm for changing them so that our errors are as small as possible.
+This leads us to the famous back propagation algorithm.
+
+
+The questions we want to ask are how do changes in the biases and the
+weights in our network change the cost function and how can we use the
+final output to modify the weights?
+
+
+To derive these equations let us start with a plain regression problem
+and define our cost function as
+
+$$
+{\cal C}(\hat{W}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2,
+$$
+
+
+where the $t_i$s are our \( n \) targets (the values we want to
+reproduce), while the outputs of the network after having propagated
+all inputs \( \hat{x} \) are given by \( y_i \). Below we will demonstrate
+how the basic equations arising from the back propagation algorithm
+can be modified in order to study classification problems with \( K \)
+classes.
+
+
+
+With our definition of the targets \( \hat{t} \), the outputs of the
+network \( \hat{y} \) and the inputs \( \hat{x} \) we
+define now the activation \( z_j^l \) of node/neuron/unit \( j \) of the
+\( l \)-th layer as a function of the bias, the weights which add up from
+the previous layer \( l-1 \) and the forward passes/outputs
+\( \hat{a}^{l-1} \) from the previous layer as
+
+$$
+z_j^l = \sum_{i=1}^{M_{l-1}}w_{ij}^la_j^{l-1}+b_j^l,
+$$
+
+
+where \( b_k^l \) are the biases from layer \( l \). Here \( M_{l-1} \)
+represents the total number of nodes/neurons/units of layer \( l-1 \). The
+figure here illustrates this equation. We can rewrite this in a more
+compact form as the matrix-vector products we discussed earlier,
+
+$$
+\hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l.
+$$
+
+
+With the activation function \( \hat{z}^l \) we can in turn define the
+output of layer \( l \) as \( \hat{a}^l = f(\hat{z}^l) \) where \( f \) is our
+activation function. In the examples here we will use the sigmoid
+function discussed in our logistic regression lectures and here as
+well. We will also use the same activation function \( f \) for all layers
+and their nodes. It means we have
+
+$$
+a_j^l = f(z_j^l) = \frac{1}{1+\exp{-(z_j^l)}}.
+$$
+
+
+
+From the definition of the activation \( z_j^l \) we have
+$$
+\frac{\partial z_j^l}{\partial w_{ji}^l} = a_i^{l-1},
+$$
+
+and
+$$
+\frac{\partial z_j^l}{\partial a_i^{l-1}} = w_{ji}^l.
+$$
+
+
+With our definition of the activation function we have that (note that this function depends only on \( z_j^l \))
+$$
+\frac{\partial a_j^l}{\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)).
+$$
+
+
+
+With these definitions we can now compute the derivative of the cost function in terms of the weights.
+
+
+Let us specialize to the output layer \( l=L \). Our cost function is
+$$
+{\cal C}(\hat{W^L}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2,
+$$
+
+The derivative of this function with respect to the weights is
+
+$$
+\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}},
+$$
+
+The last partial derivative can easily be computed and reads (by applying the chain rule)
+$$
+\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1},
+$$
+
+
+
+We have thus
+$$
+\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1},
+$$
+
+
+Defining
+$$
+\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
+$$
+
+and using the Hadamard product of two vectors we can write this as
+$$
+\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}L)}.
+$$
+
+
+This is an important expression. The second term on the right handside
+measures how fast the cost is changing as a function of the $j$th
+output activation. If, for example, the cost function doesn't depend
+much on a particular output node \( j \), then \( \delta_j^L \) will be small,
+which is what we would expect. The first term on the right, measures
+how fast the activation function \( f \) is changing at a given activation
+value \( z_j^L \).
+
+
+Notice that everything in the above equations is easily computed. In
+particular, we compute \( z_j^L \) while computing the behaviour of the
+network, and it is only a small additional overhead to compute
+\( f'(z^L_j) \). The exact form of the derivative with respect to the
+outpuwill, of course, depend on the form of the cost function.
+However, provided the cost function is known there should be little
+trouble computing
+
+$$
+\frac{\partial {\cal C}}{\partial (a_j^L)}
+$$
+
+
+With the definition of \( \delta_j^L \) we have a more compact definition of the derivative of the cost function in terms of the weights, namely
+$$
+\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}.
+$$
+
+
+
+It is also easy to see that our previous equation can be written as
+
+$$
+\delta_j^L =\frac{\partial {\cal C}}{\partial z_j^L}= \frac{\partial {\cal C}}{\partial a_j^L}\frac{\partial a_j^L}{\partial z_j^L},
+$$
+
+which can also be interpreted as the partial derivative of the cost function with respect to the biases \( b_j^L \), namely
+$$
+\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}\frac{\partial b_j^L}{\partial z_j^L}=\frac{\partial {\cal C}}{\partial b_j^L},
+$$
+
+That is, the error \( \delta_j^L \) is exactly equal to the rate of change of the cost function as a function of the bias.
+
+We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are
+
+
+$$
+\begin{equation}
+\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1},
+\label{_auto8}
+\end{equation}
+$$
+
+and
+$$
+\begin{equation}
+\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
+\label{_auto9}
+\end{equation}
+$$
+
+and
+
+$$
+\begin{equation}
+\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L},
+\label{_auto10}
+\end{equation}
+$$
+
+An interesting consequence of the above equations is that when the
+activation \( a_k^{L-1} \) is small, the gradient term, that is the
+derivative of the cost function with respect to the weights, will also
+tend to be small. We say then that the weight learns slowly, meaning
+that it changes slowly when we minimize the weights via say gradient
+descent. In this case we say the system learns slowly.
+
+
+Another interesting feature is that is when the activation function,
+represented by the sigmoid function here, is rather flat when towards
+its end values \( 0 \) and \( 1 \) (see the above Python codes). In these
+cases, the derivatives of the activation function will also be close
+to zero, meaning again that the gradients will be small and the
+network learns slowly again.
+
+
+We need a fourth equation and we are set. We are going to propagate
+backwards in order to the determine the weights and biases. In order
+to do so we need to represent the error in the layer before the final
+one \( L-1 \) in terms of the errors in the final output layer.
+
+
+
+We have that (replacing \( L \) with a general layer \( l \))
+$$
+\delta_j^l =\frac{\partial {\cal C}}{\partial z_j^l}.
+$$
+
+We want to express this in terms of the equations for layer \( l+1 \). Using the chain rule and summing over all \( k \) entries we have
+
+$$
+\delta_j^l =\sum_k \frac{\partial {\cal C}}{\partial z_k^{l+1}}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}=\sum_k \delta_k^{l+1}\frac{\partial z_k^{l+1}}{\partial z_j^{l}},
+$$
+
+and recalling that
+$$
+z_j^{l+1} = \sum_{i=1}^{M_{l}}w_{ij}^{l+1}a_j^{l}+b_j^{l+1},
+$$
+
+we obtain
+$$
+\delta_j^l =\sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l),
+$$
+
+This is our final equation.
+
+
+We are now ready to set up the algorithm for back propagation and learning the weights and biases.
+
+
+
+The four equations provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.
+
+
+
+First, we set up the input data \( \hat{x} \) and the activations
+\( \hat{z}_1 \) of the input layer and compute the activation function and
+the pertinent outputs \( \hat{a}^1 \).
+
+
+Secondly, we perform then the feed forward till we reach the output
+layer and compute all \( \hat{z}_l \) of the input layer and compute the
+activation function and the pertinent outputs \( \hat{a}^l \) for
+\( l=2,3,\dots,L \).
+
+
+Thereafter we compute the ouput error \( \hat{\delta}^L \) by computing all
+$$
+\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
+$$
+
+
+Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as
+$$
+\delta_j^l =\sum_k \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).
+$$
+
+
+Finally, we update the weights and the biases using gradient descent for each \( l=L-1,L-2,dots,2 \) and update the weights and biases according to the rules
+$$
+w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1},
+$$
+
+
+$$
+b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^L},
+$$
+
+The parameter \( \eta \) is the learning parameter discussed in connection with the gradient descent methods.
+Here it is convenient to use stochastic radient descent with mini-batches with an outer loop that steps through multiple epochs of training.
- self.N.setParams(_res.x)
- self.optimizationResults = _res
-
-
+We are now gong to develop an example based on the MNIST data
+base. This is a classification problem and we need to use our
+cross-entropy function we discussed in connection with logistic
+regression. The cross-entropy defines our cost function for the
+classificaton problems with neural networks.
+
+
+In binary classification with two classes \( (0, 1) \) we define the
+logistic/sigmoid function as the probability that a particular input
+is in class \( 0 \) or \( 1 \). This is possible because the logistic
+function takes any input from the real numbers and inputs a number
+between 0 and 1, and can therefore be interpreted as a probability. It
+also has other nice properties, such as a derivative that is simple to
+calculate.
+
+
+For an input \( \boldsymbol{a} \) from the hidden layer, the probability that the input \( \boldsymbol{x} \)
+is in class 0 or 1 is just:
+
+$$
+P(y = 0 \mid \boldsymbol{x}, \boldsymbol{\theta}) = \frac{1}{1 + \exp (- \boldsymbol{a}^T \boldsymbol{w}_{out})} ,
+$$
+
+and
+$$
+P(y = 1 \mid \boldsymbol{x}, \boldsymbol{\theta}) = 1 - P(y = 0 \mid \boldsymbol{x}, \boldsymbol{\theta}) ,
+$$
+
+
+where \( y \in \{0, 1\} \) and \( \boldsymbol{\theta} \) represents the weights and biases
+of our network.
+
+
+
+Our cost function is given as (see the Logistic regression lectures)
+$$
+\mathcal{C}(\boldsymbol{\theta}) = - \ln P(\mathcal{D} \mid \boldsymbol{\theta}) = - \sum_{i=1}^n
+y_i \ln[P(y_i = 0)] + (1 - y_i) \ln [1 - P(y_i = 0)] = \sum_{i=1}^n \mathcal{L}_i(\boldsymbol{\theta}) .
+$$
+
+
+This last equality means that we can interpret our cost function as a sum over the loss function
+for each point in the dataset \( \mathcal{L}_i(\boldsymbol{\theta}) \).
+The negative sign is just so that we can think about our algorithm as minimizing a positive number, rather
+than maximizing a negative number.
+
+
+In multiclass classification it is common to treat each integer label as a so called one-hot vector:
+
+
+\( y = 5 \quad \rightarrow \quad \boldsymbol{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) , \) and
+
+
+\( y = 1 \quad \rightarrow \quad \boldsymbol{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) , \)
+
+
+i.e. a binary bit string of length \( C \), where \( C = 10 \) is the number of classes in the MNIST dataset (numbers from \( 0 \) to \( 9 \))..
+
+
+If \( \boldsymbol{x}_i \) is the \( i \)-th input (image), \( y_{ic} \) refers to the \( c \)-th component of the \( i \)-th
+output vector \( \boldsymbol{y}_i \).
+The probability of \( \boldsymbol{x}_i \) being in class \( c \) will be given by the softmax function:
+
+$$
+P(y_{ic} = 1 \mid \boldsymbol{x}_i, \boldsymbol{\theta}) = \frac{\exp{((\boldsymbol{a}_i^{hidden})^T \boldsymbol{w}_c)}}
+{\sum_{c'=0}^{C-1} \exp{((\boldsymbol{a}_i^{hidden})^T \boldsymbol{w}_{c'})}} ,
+$$
+
+
+which reduces to the logistic function in the binary case.
+The likelihood of this \( C \)-class classifier
+is now given as:
+
+$$
+P(\mathcal{D} \mid \boldsymbol{\theta}) = \prod_{i=1}^n \prod_{c=0}^{C-1} [P(y_{ic} = 1)]^{y_{ic}} .
+$$
+
+Again we take the negative log-likelihood to define our cost function:
+
+$$
+\mathcal{C}(\boldsymbol{\theta}) = - \ln P(\mathcal{D} \mid \boldsymbol{\theta}) = - \sum_{i=1}^n \sum_{c=0}^{C-1}
+y_{ic} \ln[P(y_{ic} = 1)] = \sum_{i=1}^n
+\mathcal{L}_i(\boldsymbol{\theta}) .
+$$
+
+
+The back propagation equations need now only a small change, namely the definition of a new cost function. We are thus ready to use the same equations as before!
+We leave it as an exercise in project 2 to derive these equations.
+
+
+
+
+
+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.
+
+
+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.
+
-
-
-
-
-
-
-
+
+Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions.
+
+
+We will reserve \( 80 \% \) of our dataset for training and \( 20 \% \) for testing.
+
+
+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.
+
+
+
+
+
+
+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:
+
+$$ z = \sum_{i=1}^n w_i a_i ,$$
+
+$$ y = f(z) ,$$
+
+
+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).
+
+
+The simplest activation function for a neuron is the Heaviside function:
+
+$$ f(z) =
+\begin{cases}
+1, & z > 0\\
+0, & \text{otherwise}
+\end{cases}
+$$
+
+
+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.
+
+
+
+
+
+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.
+
+
+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 \( \boldsymbol{a} \)}) = \frac{\exp{(\boldsymbol{a}^T \boldsymbol{w}_j)}}
+{\sum_{c=0}^{9} \exp{(\boldsymbol{a}^T \boldsymbol{w}_c)}} ,$$
+
+
+i.e. each neuron \( j \) outputs the probability of being in class \( j \) given an input from the hidden layer \( \boldsymbol{a} \), with \( \boldsymbol{w}_j \) 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:
+
+$$ z_j = \sum_{i=1}^n w_ {ij} a_i = \boldsymbol{a}^T \boldsymbol{w}_j .$$
+
+
+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.
+
+
+
+
+
+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.
+
+
+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 \( j \), \( b_j \):
+
+$$ z_j = \sum_{i=1}^n w_ {ij} a_i + 1\cdot b_j = \boldsymbol{a}^T \boldsymbol{w}_j + b_j .$$
+
+
+The bias weights \( \boldsymbol{b} \) 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.
+
+
+
+
+
+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:
+
+$$ z_{j}^{h} = \sum_{i=1}^{F} w_{ij}^{h} x_i + b_{j}^{h} = \boldsymbol{x}^T \boldsymbol{w}_{j}^{h} + b_{j}^{h} ,$$
+
+
+this is then passed through our activation function
+
+$$ a_{j}^{h} = f(z_{j}^{h}) .$$
+
+
+We calculate a weighted sum of inputs (activations in the hidden layer) to each neuron \( j \) in the output layer:
+
+$$ z_{j}^{o} = \sum_{i=1}^{H} w_{ij}^{o} a_{i}^{h} + b_{j}^{o} = (\boldsymbol{a}^{h})^T \boldsymbol{w}_{j}^{o} + b_{j}^{o} .$$
+
+
+Finally we calculate the output of neuron \( j \) in the output layer using the softmax function:
+
+$$ a_{j}^{o} = \frac{\exp{(z_j^{o})}}
+{\sum_{c=0}^{C-1} \exp{(z_c^{o})}} .$$
+
+
+
+
+
+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
+
+$$ X W^{h} = (n_{inputs}, n_{hidden}),$$
+
+
+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 \( Z^{h} \):
+
+$$ Z^{h} = X W^{h} + B^{h} ,$$
+
+
+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:
+
+$$ A^{h} = f(Z^h) .$$
+
+
+This is fed to the output layer:
+
+$$ Z^{o} = A^{h} W^{o} + B^{o} .$$
+
+
+Finally we receive our output values for each image and each category by passing it through the softmax function:
+
+$$ output = softmax (Z^{o}) = (n_{inputs}, n_{categories}) .$$
+
+
+
+
+
+
+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.
+
+
+In multiclass classification it is common to treat each integer label as a so called one-hot vector:
+
+$$ y = 5 \quad \rightarrow \quad \boldsymbol{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$$
+
+$$ y = 1 \quad \rightarrow \quad \boldsymbol{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,$$
+
+
+i.e. a binary bit string of length \( C \), where \( C = 10 \) is the number of classes in the MNIST dataset.
+
+
+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 \( \boldsymbol{x}_i \) in the dataset:
+
+$$
+\begin{align} \mathcal{C}(\boldsymbol{\theta}) &= \sum_{i=1}^N \mathcal{L}_i (\boldsymbol{\theta})
+\label{_auto11}\\
+ &= -\sum_{i=1}^N \sum_{c=0}^{C-1} y_{ic} \log P(y_{ic} = 1 \mid \boldsymbol{x}_i, \boldsymbol{\theta}) .
+\label{_auto12}
+\end{align}
+$$
+
+
+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 \( \boldsymbol{\theta} \) represents the parameters of our network, i.e. all the weights and biases.
+
+
+A full derivation is given in the appendix at the end.
+
+
+
+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
+
+$$ \theta_{i+1} = \theta_i - \eta \nabla \mathcal{C}(\theta_i) ,$$
+
+
+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:
+
+$$ \nabla \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) \quad \rightarrow \quad
+\frac{1}{M} \sum_{i \in B_k} \nabla \mathcal{L}_i(\theta) ,$$
+
+
+i.e. instead of averaging the loss over the entire dataset, we average over a minibatch.
+
+
+This has two important benefits:
+
+
+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.
+
+
+We will measure the size of the weights using the so called L2-norm, meaning our cost function becomes:
+
+$$ \nabla \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) \quad \rightarrow \quad
+\frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) + \lambda \lvert \lvert \boldsymbol{w} \rvert \rvert_2^2
+= \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}(\theta) + \lambda \sum_{ij} w_{ij}^2,$$
+
+
+i.e. we sum up all the weights squared. The factor \( \lambda \) is known as a regularization parameter.
+
+
+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.
+
+
+
+To more efficently train our network these equations are implemented using matrix operations.
+The error in the output layer is calculated simply as
+
+$$ \Delta_o = \hat{y} - y = (n_{inputs}, n_{categories}) .$$
+
+
+The gradient for the output weights is calculated as
+
+$$ \nabla W_{o} = A^T \Delta_o = (n_{hidden}, n_{categories}) ,$$
+
+
+where \( A = (n_{inputs}, n_{hidden}) \). 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.
+
+
+The gradient with respect to the output bias is then
+
+$$ \nabla B_{o} = \sum_{i=1}^{n_{inputs}} \Delta_o = (n_{categories}) .$$
+
+
+The error in the hidden layer is
+
+$$ \Delta_h = \Delta_o W_{o}^T \circ f'(Z_{h}) = \Delta_o W_{o}^T \circ A_{h} \circ (1 - A_{h}) = (n_{inputs}, n_{hidden}) ,$$
+
+
+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:
+
+$$ \nabla W_{h} = X^T \Delta_h = (n_{features}, n_{hidden}) ,$$
+
+$$ \nabla B_{h} = \sum_{i=1}^{n_{inputs}} \Delta_h = (n_{hidden}) .$$
+
+
+
+
+
+
+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.
+
+
+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.
+
+
+
+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.
+
+
+
+
+
+
+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.
+
+
+
+
+
+
+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).
+
+
+
+
+
+
+
+
+
+
+
+
+scikit-learn is a machine learning library for Python. It 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.
+
+
+
+
+
+
+
+
+
diff --git a/doc/pub/NeuralNet/ipynb/NeuralNet.ipynb b/doc/pub/NeuralNet/ipynb/NeuralNet.ipynb
index da8ec380c..c56f955ed 100644
--- a/doc/pub/NeuralNet/ipynb/NeuralNet.ipynb
+++ b/doc/pub/NeuralNet/ipynb/NeuralNet.ipynb
@@ -4,19 +4,18 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "\n",
- "# Data Analysis and Machine Learning: Elements of machine learning\n",
+ "\n",
+ "# Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning\n",
"\n",
" \n",
"**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n",
"\n",
- "Date: **Sep 28, 2018**\n",
+ "Date: **Oct 2, 2018**\n",
"\n",
"Copyright 1999-2018, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n",
"\n",
"\n",
"\n",
- "\n",
"\n",
"## Neural networks\n",
"\n",
@@ -78,88 +77,108 @@
"\n",
"4. neural networks for unsupervised learning such as Deep Boltzmann Machines.\n",
"\n",
- "In natural science, DNNs and CNNs have already found numerous applications. In\n",
- "statistical physics, they have been applied to detect phase\n",
- "transitions in 2D Ising and Potts models, lattice gauge theories, and\n",
- "different phases of polymers, or solving the Navier-Stokes equation in weather forecasting.\n",
- "Deep learning has also found interesting applications in quantum\n",
- "physics. Various quantum phase transitions can be detected and studied\n",
- "using DNNs and CNNs,\n",
+ "In natural science, DNNs and CNNs have already found numerous\n",
+ "applications. In statistical physics, they have been applied to detect\n",
+ "phase transitions in 2D Ising and Potts models, lattice gauge\n",
+ "theories, and different phases of polymers, or solving the\n",
+ "Navier-Stokes equation in weather forecasting. Deep learning has also\n",
+ "found interesting applications in quantum physics. Various quantum\n",
+ "phase transitions can be detected and studied using DNNs and CNNs,\n",
"topological phases, and even non-equilibrium many-body\n",
"localization. Representing quantum states as DNNs quantum state\n",
- "tomography are among some of the impressive\n",
- "achievements to reveal the potential of DNNs to facilitate the study\n",
- "of quantum systems.\n",
+ "tomography are among some of the impressive achievements to reveal the\n",
+ "potential of DNNs to facilitate the study of quantum systems.\n",
"\n",
"In quantum information theory, it has been shown that one can perform\n",
- "gate decompositions with the help of neural. In lattice quantum chromodynamics,\n",
- "DNNs have been used to learn action parameters in regions of parameter\n",
- "space where PCA fails. \n",
+ "gate decompositions with the help of neural. \n",
"\n",
- "The applications are not limited to the natural sciences. There is a plethora of applications in essentially all disciplines, from the humanities to life science and medicine.\n",
+ "The applications are not limited to the natural sciences. There is a\n",
+ "plethora of applications in essentially all disciplines, from the\n",
+ "humanities to life science and medicine.\n",
"\n",
"## Neural network types\n",
"\n",
- "An artificial neural network (NN), is a computational model that\n",
- "consists of layers of connected neurons, or *nodes*. It is supposed\n",
- "to mimic a biological nervous system by letting each neuron interact\n",
- "with other neurons by sending signals in the form of mathematical\n",
- "functions between layers. A wide variety of different NNs have been\n",
- "developed, but most of them consist of an input layer, an output layer\n",
- "and eventual layers in-between, called *hidden layers*. All layers can\n",
- "contain an arbitrary number of nodes, and each connection between two\n",
- "nodes is associated with a weight variable.\n",
+ "An artificial neural network (ANN), is a computational model that\n",
+ "consists of layers of connected neurons, or nodes or units. We will\n",
+ "refer to these interchangeably as units or nodes, and sometimes as\n",
+ "neurons.\n",
+ "\n",
+ "It is supposed to mimic a biological nervous system by letting each\n",
+ "neuron interact with other neurons by sending signals in the form of\n",
+ "mathematical functions between layers. A wide variety of different\n",
+ "ANNs have been developed, but most of them consist of an input layer,\n",
+ "an output layer and eventual layers in-between, called *hidden\n",
+ "layers*. All layers can contain an arbitrary number of nodes, and each\n",
+ "connection between two nodes is associated with a weight variable.\n",
"\n",
"Neural networks (also called neural nets) are neural-inspired\n",
"nonlinear models for supervised learning. As we will see, neural nets\n",
"can be viewed as natural, more powerful extensions of supervised\n",
"learning methods such as linear and logistic regression and soft-max\n",
- "methods.\n",
+ "methods we discussed earlier.\n",
"\n",
"\n",
"## Feed-forward neural networks\n",
"\n",
- "The feed-forward neural network (FFNN) was the first and simplest type of NN devised. In this network, \n",
- "the information moves in only one direction: forward through the layers.\n",
+ "The feed-forward neural network (FFNN) was the first and simplest type\n",
+ "of ANNs that were devised. In this network, the information moves in\n",
+ "only one direction: forward through the layers.\n",
"\n",
- "Nodes are represented by circles, while the arrows display the connections between the nodes, including the \n",
- "direction of information flow. Additionally, each arrow corresponds to a weight variable, not displayed here. \n",
- "We observe that each node in a layer is connected to *all* nodes in the subsequent layer, \n",
- "making this a so-called *fully-connected* FFNN. \n",
+ "Nodes are represented by circles, while the arrows display the\n",
+ "connections between the nodes, including the direction of information\n",
+ "flow. Additionally, each arrow corresponds to a weight variable\n",
+ "(figure to come). We observe that each node in a layer is connected\n",
+ "to *all* nodes in the subsequent layer, making this a so-called\n",
+ "*fully-connected* FFNN.\n",
"\n",
"\n",
"\n",
- "A different variant of FFNNs are *convolutional neural networks* (CNNs), which have a connectivity pattern\n",
- "inspired by the animal visual cortex. Individual neurons in the visual cortex only respond to stimuli from\n",
- "small sub-regions of the visual field, called a receptive field. This makes the neurons well-suited to exploit the strong\n",
- "spatially local correlation present in natural images. The response of each neuron can be approximated mathematically \n",
- "as a convolution operation. \n",
+ "## Convolutional Neural Network\n",
"\n",
- "CNNs emulate the behaviour of neurons in the visual cortex by enforcing a *local* connectivity pattern\n",
- "between nodes of adjacent layers: Each node\n",
- "in a convolutional layer is connected only to a subset of the nodes in the previous layer, \n",
- "in contrast to the fully-connected FFNN.\n",
- "Often, CNNs \n",
- "consist of several convolutional layers that learn local features of the input, with a fully-connected layer at the end, \n",
- "which gathers all the local data and produces the outputs. They have wide applications in image and video recognition\n",
+ "A different variant of FFNNs are *convolutional neural networks*\n",
+ "(CNNs), which have a connectivity pattern inspired by the animal\n",
+ "visual cortex. Individual neurons in the visual cortex only respond to\n",
+ "stimuli from small sub-regions of the visual field, called a receptive\n",
+ "field. This makes the neurons well-suited to exploit the strong\n",
+ "spatially local correlation present in natural images. The response of\n",
+ "each neuron can be approximated mathematically as a convolution\n",
+ "operation. (figure to come)\n",
+ "\n",
+ "Convolutional neural networks emulate the behaviour of neurons in the\n",
+ "visual cortex by enforcing a *local* connectivity pattern between\n",
+ "nodes of adjacent layers: Each node in a convolutional layer is\n",
+ "connected only to a subset of the nodes in the previous layer, in\n",
+ "contrast to the fully-connected FFNN. Often, CNNs consist of several\n",
+ "convolutional layers that learn local features of the input, with a\n",
+ "fully-connected layer at the end, which gathers all the local data and\n",
+ "produces the outputs. They have wide applications in image and video\n",
+ "recognition.\n",
"\n",
"## Recurrent neural networks\n",
"\n",
- "So far we have only mentioned NNs where information flows in one direction: forward. *Recurrent neural networks* on\n",
- "the other hand, have connections between nodes that form directed *cycles*. This creates a form of \n",
- "internal memory which are able to capture information on what has been calculated before; the output is dependent \n",
- "on the previous computations. Recurrent NNs make use of sequential information by performing the same task for \n",
- "every element in a sequence, where each element depends on previous elements. An example of such information is \n",
- "sentences, making recurrent NNs especially well-suited for handwriting and speech recognition.\n",
+ "So far we have only mentioned ANNs where information flows in one\n",
+ "direction: forward. *Recurrent neural networks* on the other hand,\n",
+ "have connections between nodes that form directed *cycles*. This\n",
+ "creates a form of internal memory which are able to capture\n",
+ "information on what has been calculated before; the output is\n",
+ "dependent on the previous computations. Recurrent NNs make use of\n",
+ "sequential information by performing the same task for every element\n",
+ "in a sequence, where each element depends on previous elements. An\n",
+ "example of such information is sentences, making recurrent NNs\n",
+ "especially well-suited for handwriting and speech recognition.\n",
"\n",
"## Other types of networks\n",
"\n",
- "There are many other kinds of NNs that have been developed. One type that is specifically designed for interpolation\n",
- "in multidimensional space is the radial basis function (RBF) network. RBFs are typically made up of three layers: \n",
- "an input layer, a hidden layer with non-linear radial symmetric activation functions and a linear output layer (''linear'' here\n",
- "means that each node in the output layer has a linear activation function). The layers are normally fully-connected and \n",
- "there are no cycles, thus RBFs can be viewed as a type of fully-connected FFNN. They are however usually treated as\n",
- "a separate type of NN due the unusual activation functions.\n",
+ "There are many other kinds of ANNs that have been developed. One type\n",
+ "that is specifically designed for interpolation in multidimensional\n",
+ "space is the radial basis function (RBF) network. RBFs are typically\n",
+ "made up of three layers: an input layer, a hidden layer with\n",
+ "non-linear radial symmetric activation functions and a linear output\n",
+ "layer (''linear'' here means that each node in the output layer has a\n",
+ "linear activation function). The layers are normally fully-connected\n",
+ "and there are no cycles, thus RBFs can be viewed as a type of\n",
+ "fully-connected FFNN. They are however usually treated as a separate\n",
+ "type of NN due the unusual activation functions.\n",
"\n",
"## Multilayer perceptrons\n",
"\n",
@@ -168,45 +187,33 @@
"and an output layer) consisting of neurons that have non-linear\n",
"activation functions.\n",
"\n",
- "Such networks are often called *multilayer perceptrons* (MLPs)\n",
+ "Such networks are often called *multilayer perceptrons* (MLPs).\n",
"\n",
"## Why multilayer perceptrons?\n",
"\n",
- "According to the *Universal approximation theorem*, a feed-forward neural network with just a single hidden layer containing \n",
- "a finite number of neurons can approximate a continuous multidimensional function to arbitrary accuracy, \n",
- "assuming the activation function for the hidden layer is a **non-constant, bounded and monotonically-increasing continuous function**.\n",
+ "According to the *Universal approximation theorem*, a feed-forward\n",
+ "neural network with just a single hidden layer containing a finite\n",
+ "number of neurons can approximate a continuous multidimensional\n",
+ "function to arbitrary accuracy, assuming the activation function for\n",
+ "the hidden layer is a **non-constant, bounded and\n",
+ "monotonically-increasing continuous function**.\n",
"\n",
"Note that the requirements on the activation function only applies to\n",
"the hidden layer, the output nodes are always assumed to be linear, so\n",
"as to not restrict the range of output values.\n",
"\n",
- "We note that this theorem is only applicable to an NN with *one* hidden\n",
- "layer. Therefore, we can easily construct an NN that employs\n",
- "activation functions which do not satisfy the above requirements, as\n",
- "long as we have at least one layer with activation functions that\n",
- "*do*. Furthermore, although the universal approximation theorem lays\n",
- "the theoretical foundation for regression with neural networks, it\n",
- "does not say anything about how things work in practice: A neural\n",
- "network can still be able to approximate a given function reasonably\n",
- "well without having the flexibility to fit *all other* functions.\n",
"\n",
+ "## Mathematical model\n",
"\n",
- "\n",
- "## Mathematical model"
+ "The output $y$ is produced via the activation function $f$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
- "\n",
- "Recurrent neural networks
+Convolutional Neural Network
Other types of networks
+Recurrent neural networks
Multilayer perceptrons
+Other types of networks
+
+Multilayer perceptrons
Why multilayer perceptrons?
+Why multilayer perceptrons?
Mathematical model
-
-
-$$
-\begin{equation}
- y = f\left(\sum_{i=1}^n w_ix_i + b_i\right) = f(u)
-\tag{2}
-\end{equation}
-$$
-
+Mathematical model
+$$
+ y = f\left(\sum_{i=1}^n w_ix_i + b_i\right) = f(z),
+$$
+
+
+This function receives \( x_i \) as inputs.
+Here the activation \( z=\sum_{i=1}^n w_ix_i \).
In an FFNN of such neurons, the inputs \( x_i \) are the outputs of
the neurons in the preceding layer. Furthermore, an MLP is
fully-connected, which means that each neuron receives a weighted sum
@@ -383,41 +396,46 @@ of the outputs of all neurons in the previous layer.
Mathematical model
+Mathematical model
$$
-\begin{equation}
- u_i^1 = \sum_{j=1}^2 w_{ij}^1 x_j + b_i^1
-\tag{3}
+\begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1
+\tag{2}
\end{equation}
$$
-This value is the argument to the activation function \( f_1 \) of each neuron \( i \),
-producing the output \( y_i^1 \) of all neurons in layer 1,
+
$$
\begin{equation}
- y_i^1 = f_1(u_i^1) = f_1\left(\sum_{j=1}^2 w_{ij}^1 x_j + b_i^1\right)
-\tag{4}
+ y_i^1 = f(z_i^1) = f\left(\sum_{j=1}^M w_{ij}^1 x_j + b_i^1\right)
+\tag{3}
\end{equation}
$$
$$
\begin{equation}
- y_i^l = f_l(u_i^l) = f_l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
-\tag{5}
+ y_i^l = f^l(u_i^l) = f^l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
+\tag{4}
\end{equation}
$$
@@ -431,7 +449,7 @@ is obtained.
Mathematical model
+Mathematical model
$$
\begin{align}
- y_i^2 &= f_2\left(\sum_{j=1}^3 w_{ij}^2 y_j^1 + b_i^2\right)
-\tag{6}\\
- &= f_2\left[\sum_{j=1}^3 w_{ij}^2f_1\left(\sum_{k=1}^2 w_{jk}^1 x_k + b_j^1\right) + b_i^2\right]
-\tag{7}
+ y_i^2 &= f^2\left(\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\right)
+\tag{5}\\
+ &= f^2\left[\sum_{j=1}^N w_{ij}^2f^1\left(\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\right) + b_i^2\right]
+\tag{6}
\end{align}
$$
-where we have substituted \( y_m^1 \) with. Finally, the NN output yields,
+where we have substituted \( y_k^1 \) with the inputs \( x_k \). Finally, the ANN output reads
$$
\begin{align}
- y_1^3 &= f_3\left(\sum_{j=1}^3 w_{1m}^3 y_j^2 + b_1^3\right)
-\tag{8}\\
- &= f_3\left[\sum_{j=1}^3 w_{1j}^3 f_2\left(\sum_{k=1}^3 w_{jk}^2 f_1\left(\sum_{m=1}^2 w_{km}^1 x_m + b_k^1\right) + b_j^2\right)
+ y_i^3 &= f^3\left(\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\right)
+\tag{7}\\
+ &= f_3\left[\sum_{j} w_{ij}^3 f^2\left(\sum_{k} w_{jk}^2 f^1\left(\sum_{m} w_{km}^1 x_m + b_k^1\right) + b_j^2\right)
+ b_1^3\right]
-\tag{9}
+\tag{8}
\end{align}
$$
@@ -464,7 +482,7 @@ $$
Mathematical model
+Mathematical model
$$
\begin{align}
-&y^{l+1}_1\! = \!f_{l+1}\!\left[\!\sum_{j=1}^{N_l}\! w_{1j}^3 f_l\!\left(\!\sum_{k=1}^{N_{l-1}}\! w_{jk}^2 f_{l-1}\!\left(\!
- \dots \!f_1\!\left(\!\sum_{n=1}^{N_0} \!w_{mn}^1 x_n\! + \!b_m^1\!\right)
- \!\dots \!\right) \!+ \!b_k^2\!\right)
- \!+ \!b_1^3\!\right] &&
-\tag{10}
+&y^{l+1}_i = f^{l+1}\left[\!\sum_{j=1}^{N_l} w_{ij}^3 f^l\left(\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\left(\dots f^1\left(\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\right)\dots\right)+b_k^2\right)+b_1^3\right] &&
+\tag{9}
\end{align}
$$
@@ -489,27 +504,24 @@ variables are the input values \( x_n \).
Mathematical model
+Mathematical model
$$
\begin{equation}
- h(x) = c_1 f(c_2 x + c_3) + c_4
-\tag{11}
+ f(x) = c_1 f(c_2 x + c_3) + c_4
+\tag{10}
\end{equation}
$$
@@ -524,24 +536,24 @@ flexibility of a neural network.
Matrix-vector notation
+Matrix-vector notation
$$
\begin{equation}
- \vec{y}_2 = f_2(\mathrm{W}_2 \vec{y}_{1} + \vec{b}_{2}) =
+ \hat{y}_2 = f_2(\mathrm{W}_2 \hat{y}_{1} + \hat{b}_{2}) =
f_2\left(\left[\begin{array}{ccc}
w^2_{11} &w^2_{12} &w^2_{13} \\
w^2_{21} &w^2_{22} &w^2_{23} \\
@@ -557,7 +569,7 @@ $$
b^2_2 \\
b^2_3 \\
\end{array}\right]\right).
-\tag{12}
+\tag{11}
\end{equation}
$$
@@ -566,7 +578,7 @@ $$
Matrix-vector notation and activation
+Matrix-vector notation and activation
@@ -586,13 +598,13 @@ This is not just a convenient and compact notation, but also a useful
and intuitive way to think about MLPs: The output is calculated by a
series of matrix-vector multiplications and vector additions that are
used as input to the activation functions. For each operation
-\( \mathrm{W}_l \vec{y}_{l-1} \) we move forward one layer.
+\( \mathrm{W}_l \hat{y}_{l-1} \) we move forward one layer.
Activation functions
+Activation functions
Activation functions, Logistic and Hyperbolic ones
+Activation functions, Logistic and Hyperbolic ones
$$
-\begin{equation}
f(x) = \frac{1}{1 + e^{-x}},
-\tag{14}
-\end{equation}
$$
and the hyperbolic tangent function
$$
-\begin{equation}
f(x) = \tanh(x)
-\tag{15}
-\end{equation}
$$
Relevance
+Relevance
Setting up a Multi-layer perceptron model
+The multilayer perceptron (MLP)
+
+
+
+From one to many layers, the universal approximation theorem
+
+Deriving the back propagation code for a multilayer perceptron model
+
+
+$$
+{\cal C}(\hat{W}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2,
+$$
+
+
+Definitions
+
+
+$$
+z_j^l = \sum_{i=1}^{M_{l-1}}w_{ij}^la_j^{l-1}+b_j^l,
+$$
+
+
+
+$$
+\hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l.
+$$
+
+
+
+$$
+a_j^l = f(z_j^l) = \frac{1}{1+\exp{-(z_j^l)}}.
+$$
+
+Derivatives and the chain rule
+
+
+$$
+\frac{\partial z_j^l}{\partial w_{ji}^l} = a_i^{l-1},
+$$
+
+
+and
+
+$$
+\frac{\partial z_j^l}{\partial a_i^{l-1}} = w_{ji}^l.
+$$
+
+
+
+$$
+\frac{\partial a_j^l}{\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)).
+$$
+
+Derivative of the cost function
+
+
+$$
+{\cal C}(\hat{W^L}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2,
+$$
+
+
+The derivative of this function with respect to the weights is
+
+
+$$
+\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}},
+$$
+
+
+The last partial derivative can easily be computed and reads (by applying the chain rule)
+
+$$
+\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1},
+$$
+
+Bringing it together, first back propagation equation
+
+
+$$
+\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1},
+$$
+
+
+
+$$
+\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
+$$
+
+
+and using the Hadamard product of two vectors we can write this as
+
+$$
+\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}L)}.
+$$
+
+
+
+$$
+\frac{\partial {\cal C}}{\partial (a_j^L)}
+$$
+
+
+
+$$
+\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}.
+$$
+
+Derivatives in terms of \( z_j^L \)
+
+
+$$
+\delta_j^L =\frac{\partial {\cal C}}{\partial z_j^L}= \frac{\partial {\cal C}}{\partial a_j^L}\frac{\partial a_j^L}{\partial z_j^L},
+$$
+
+
+which can also be interpreted as the partial derivative of the cost function with respect to the biases \( b_j^L \), namely
+
+$$
+\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}\frac{\partial b_j^L}{\partial z_j^L}=\frac{\partial {\cal C}}{\partial b_j^L},
+$$
+
+
+That is, the error \( \delta_j^L \) is exactly equal to the rate of change of the cost function as a function of the bias.
+Bringing it together
+
+
+$$
+\begin{equation}
+\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1},
+\tag{13}
+\end{equation}
+$$
+
+
+and
+
+$$
+\begin{equation}
+\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
+\tag{14}
+\end{equation}
+$$
+
+
+and
+
+
+$$
+\begin{equation}
+\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L},
+\tag{15}
+\end{equation}
+$$
+
+Final back propagating equation
+
+
+$$
+\delta_j^l =\frac{\partial {\cal C}}{\partial z_j^l}.
+$$
+
+
+We want to express this in terms of the equations for layer \( l+1 \). Using the chain rule and summing over all \( k \) entries we have
+
+
+$$
+\delta_j^l =\sum_k \frac{\partial {\cal C}}{\partial z_k^{l+1}}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}=\sum_k \delta_k^{l+1}\frac{\partial z_k^{l+1}}{\partial z_j^{l}},
+$$
+
+
+and recalling that
+
+$$
+z_j^{l+1} = \sum_{i=1}^{M_{l}}w_{ij}^{l+1}a_j^{l}+b_j^{l+1},
+$$
+
+
+we obtain
+
+$$
+\delta_j^l =\sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l),
+$$
+
+
+This is our final equation.
+
+Setting up the Back propagation algorithm
+
+
+$$
+\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
+$$
+
+
+$$
+\delta_j^l =\sum_k \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).
+$$
+
+
+$$
+w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1},
+$$
+
+
+
+$$
+b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^L},
+$$
+
+Setting up a Multi-layer perceptron model for classification
+
+
+$$
+P(y = 0 \mid \boldsymbol{x}, \boldsymbol{\theta}) = \frac{1}{1 + \exp (- \boldsymbol{a}^T \boldsymbol{w}_{out})} ,
+$$
+
+
+and
+
+$$
+P(y = 1 \mid \boldsymbol{x}, \boldsymbol{\theta}) = 1 - P(y = 0 \mid \boldsymbol{x}, \boldsymbol{\theta}) ,
+$$
+
+
+Defining the cost function
+
+
+$$
+\mathcal{C}(\boldsymbol{\theta}) = - \ln P(\mathcal{D} \mid \boldsymbol{\theta}) = - \sum_{i=1}^n
+y_i \ln[P(y_i = 0)] + (1 - y_i) \ln [1 - P(y_i = 0)] = \sum_{i=1}^n \mathcal{L}_i(\boldsymbol{\theta}) .
+$$
+
+
+
+$$
+P(y_{ic} = 1 \mid \boldsymbol{x}_i, \boldsymbol{\theta}) = \frac{\exp{((\boldsymbol{a}_i^{hidden})^T \boldsymbol{w}_c)}}
+{\sum_{c'=0}^{C-1} \exp{((\boldsymbol{a}_i^{hidden})^T \boldsymbol{w}_{c'})}} ,
+$$
+
+
+
+$$
+P(\mathcal{D} \mid \boldsymbol{\theta}) = \prod_{i=1}^n \prod_{c=0}^{C-1} [P(y_{ic} = 1)]^{y_{ic}} .
+$$
+
+
+Again we take the negative log-likelihood to define our cost function:
+
+
+$$
+\mathcal{C}(\boldsymbol{\theta}) = - \ln P(\mathcal{D} \mid \boldsymbol{\theta}) = - \sum_{i=1}^n \sum_{c=0}^{C-1}
+y_{ic} \ln[P(y_{ic} = 1)] = \sum_{i=1}^n
+\mathcal{L}_i(\boldsymbol{\theta}) .
+$$
+
+
+Developing a code for doing neural networks with back propagation
+
+
+
+Collect and pre-process data
+
+
+$$ X = \begin{bmatrix}
+1.85 & 81\\
+1.71 & 65\\
+1.95 & 103\\
+1.55 & 42\\
+1.63 & 56
+\end{bmatrix} ,$$
+
+
+
+$$ Y = (23.7, 22.2, 27.1, 17.5, 21.1) $$
+
+
+from scipy import optimize
+
# import necessary packages
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn import datasets
-class Neural_Network(object):
- def __init__(self, Lambda=0):
- #Define Hyperparameters
- self.inputLayerSize = 2
- self.outputLayerSize = 1
- self.hiddenLayerSize = 3
-
- #Weights (parameters)
- self.W1 = np.random.randn(self.inputLayerSize,self.hiddenLayerSize)
- self.W2 = np.random.randn(self.hiddenLayerSize,self.outputLayerSize)
-
- #Regularization Parameter:
- self.Lambda = Lambda
-
- def forward(self, X):
- #Propogate inputs though network
- self.z2 = np.dot(X, self.W1)
- self.a2 = self.sigmoid(self.z2)
- self.z3 = np.dot(self.a2, self.W2)
- yHat = self.sigmoid(self.z3)
- return yHat
-
- def sigmoid(self, z):
- #Apply sigmoid activation function to scalar, vector, or matrix
- return 1/(1+np.exp(-z))
-
- def sigmoidPrime(self,z):
- #Gradient of sigmoid
- return np.exp(-z)/((1+np.exp(-z))**2)
-
- def costFunction(self, X, y):
- #Compute cost for given X,y, use weights already stored in class.
- self.yHat = self.forward(X)
- J = 0.5*sum((y-self.yHat)**2)/X.shape[0] + (self.Lambda/2)*(np.sum(self.W1**2)+np.sum(self.W2**2))
- return J
-
- def costFunctionPrime(self, X, y):
- #Compute derivative with respect to W and W2 for a given X and y:
- self.yHat = self.forward(X)
-
- delta3 = np.multiply(-(y-self.yHat), self.sigmoidPrime(self.z3))
- #Add gradient of regularization term:
- dJdW2 = np.dot(self.a2.T, delta3)/X.shape[0] + self.Lambda*self.W2
-
- delta2 = np.dot(delta3, self.W2.T)*self.sigmoidPrime(self.z2)
- #Add gradient of regularization term:
- dJdW1 = np.dot(X.T, delta2)/X.shape[0] + self.Lambda*self.W1
-
- return dJdW1, dJdW2
-
- #Helper functions for interacting with other methods/classes
- def getParams(self):
- #Get W1 and W2 Rolled into vector:
- params = np.concatenate((self.W1.ravel(), self.W2.ravel()))
- return params
-
- def setParams(self, params):
- #Set W1 and W2 using single parameter vector:
- W1_start = 0
- W1_end = self.hiddenLayerSize*self.inputLayerSize
- self.W1 = np.reshape(params[W1_start:W1_end], \
- (self.inputLayerSize, self.hiddenLayerSize))
- W2_end = W1_end + self.hiddenLayerSize*self.outputLayerSize
- self.W2 = np.reshape(params[W1_end:W2_end], \
- (self.hiddenLayerSize, self.outputLayerSize))
-
- def computeGradients(self, X, y):
- dJdW1, dJdW2 = self.costFunctionPrime(X, y)
- return np.concatenate((dJdW1.ravel(), dJdW2.ravel()))
-
-
-class trainer(object):
- def __init__(self, N):
- #Make Local reference to network:
- self.N = N
-
- def callbackF(self, params):
- self.N.setParams(params)
- self.J.append(self.N.costFunction(self.X, self.y))
- self.testJ.append(self.N.costFunction(self.testX, self.testY))
-
- def costFunctionWrapper(self, params, X, y):
- self.N.setParams(params)
- cost = self.N.costFunction(X, y)
- grad = self.N.computeGradients(X,y)
- return cost, grad
-
- def train(self, trainX, trainY, testX, testY):
- #Make an internal variable for the callback function:
- self.X = trainX
- self.y = trainY
-
- self.testX = testX
- self.testY = testY
- #Make empty list to store training costs:
- self.J = []
- self.testJ = []
-
- params0 = self.N.getParams()
+# ensure the same random numbers appear every time
+np.random.seed(0)
- options = {'maxiter': 200, 'disp' : True}
- _res = optimize.minimize(self.costFunctionWrapper, params0, jac=True, method='BFGS', \
- args=(trainX, trainY), options=options, callback=self.callbackF)
+# display images in notebook
+%matplotlib inline
+plt.rcParams['figure.figsize'] = (12,12)
- self.N.setParams(_res.x)
- self.optimizationResults = _res
+
+# 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()
Two-layer Neural Network
+Train and test datasets
+
+import numpy as np
+
from sklearn.model_selection import train_test_split
-#sigmoid
-def nonlin(x, deriv=False):
- if (deriv==True):
- return x*(1-x)
- return 1/(1+np.exp(-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)
-#input data
-x=np.array([[0,0,1],[0,1,1],[1,0,1],[1,1,1]])
-
-#output data
-y=np.array([0,1,1,0]).T
-
-#seed random numbers to make calculation
-np.random.seed(1)
-
-#initialize weights with mean=0
-syn0=2*np.random.random((3,4))-1
-
-for iter in range(10000):
- #forward propogation
- l0=x
- l1=nonlin(np.dot(l0,syn0))
- l1_error=y-l1
- #multiply error by slope of sigmoid at values of l1
- l1_delta=l1_error*nonlin(l1,True)
- #update weights
- syn0+=np.dot(l0.T, l1_delta)
+# 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()
-print("Output after training: ",l1 )
+ 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
+
+#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)))
Define model and architecture
+
+
+$$ z = \sum_{i=1}^n w_i a_i ,$$
+
+
+
+$$ y = f(z) ,$$
+
+
+
+$$ f(z) =
+\begin{cases}
+1, & z > 0\\
+0, & \text{otherwise}
+\end{cases}
+$$
+
+
+
+$$ f(x) = \sigma(x) = \frac{1}{1 + e^{-x}} ,$$
+
+
+Layers
+
+
+
+
+
+
+
+
+$$ P(\text{class \( j \)} \mid \text{input \( \boldsymbol{a} \)}) = \frac{\exp{(\boldsymbol{a}^T \boldsymbol{w}_j)}}
+{\sum_{c=0}^{9} \exp{(\boldsymbol{a}^T \boldsymbol{w}_c)}} ,$$
+
+
+
+$$ z_j = \sum_{i=1}^n w_ {ij} a_i = \boldsymbol{a}^T \boldsymbol{w}_j .$$
+
+
+Weights and biases
+
+
+$$ z_j = \sum_{i=1}^n w_ {ij} a_i + 1\cdot b_j = \boldsymbol{a}^T \boldsymbol{w}_j + b_j .$$
+
+
+import numpy as np
-import random
-class Network(object):
+
# building our neural network
+
+n_inputs, n_features = X_train.shape
+n_hidden_neurons = 50
+n_categories = 10
+
+# we make the weights normally distributed using numpy.random.randn
+
+# weights and bias in the hidden layer
+hidden_weights = np.random.randn(n_features, n_hidden_neurons)
+hidden_bias = np.zeros(n_hidden_neurons) + 0.01
+
+# weights and bias in the output layer
+output_weights = np.random.randn(n_hidden_neurons, n_categories)
+output_bias = np.zeros(n_categories) + 0.01
+
Feed-forward pass
+
+
+$$ z_{j}^{h} = \sum_{i=1}^{F} w_{ij}^{h} x_i + b_{j}^{h} = \boldsymbol{x}^T \boldsymbol{w}_{j}^{h} + b_{j}^{h} ,$$
+
+
+
+$$ a_{j}^{h} = f(z_{j}^{h}) .$$
+
+
+
+$$ z_{j}^{o} = \sum_{i=1}^{H} w_{ij}^{o} a_{i}^{h} + b_{j}^{o} = (\boldsymbol{a}^{h})^T \boldsymbol{w}_{j}^{o} + b_{j}^{o} .$$
+
+
+
+$$ a_{j}^{o} = \frac{\exp{(z_j^{o})}}
+{\sum_{c=0}^{C-1} \exp{(z_c^{o})}} .$$
+
+Matrix multiplication
+
+
+$$ X W^{h} = (n_{inputs}, n_{hidden}),$$
+
+
+
+$$ Z^{h} = X W^{h} + B^{h} ,$$
+
+
+
+$$ A^{h} = f(Z^h) .$$
+
+
+
+$$ Z^{o} = A^{h} W^{o} + B^{o} .$$
+
+
+
+$$ output = softmax (Z^{o}) = (n_{inputs}, n_{categories}) .$$
+
+
+# setup the feed-forward pass
+
+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)
- def _init_(self, sizes):
- self.num_layers=len(sizes)
- self.sizes=sizes
- self.biases=[np.random.randn(y,1) for y in sizes[1:]]
- self.weights=[np.random.randn(y,x) for x,y in zip(sizes[:-1], sizes[1:])]
-
-#sizes is the number of neurons in each layer
-#for example, say n_1st_layer=3, n_2nd_layer=3, n_3rd_layer=1, then net=Network([3,3,1])
-
-#The biases and weights are initialized randomly, using Gaussian distributions of mean=0, stdev=1
-#z is a vector (or a np.array)
-
- def feedforward(self,a):
- #returns output w/ 'a' as an input
- for b, w in zip(self.biases, self.weights):
- a=sigmoid(np.dot(w,b)+b)
- return a
+ # 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)
-#Apply a Stochastic Gradient Descent (SGD) method:
- def SGD(self, training_data, epochs, mini_batch_size, eta, test_data=None):
- """Trains network using batches incorporating SGD. The network will be evaluated against the
- test data after each epoch, with partial progress being printed out (this is useful for tracking,
- but slows the process.)"""
- if test_data: n_test=len(test_data)
- n=len(training_data)
- for j in xrange(epochs):
- random.shuffle(training_data)
- mini_batches=[training_data[k:k+mini_batch_size] for k in xrange(o,n,mini_batch_size)]
- for mini_batch in mini_batches:
- self.update_mini_batch(mini_batch, eta)
- if test_data:
- print ("Epoch {0}: {1}/{2}".format(j, self.evaluate(test_data), n_test))
- else:
- print ("Epoch {0} complete".format(j))
-
+ 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
+
+
+$$ y = 5 \quad \rightarrow \quad \boldsymbol{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$$
+
+
+
+$$ y = 1 \quad \rightarrow \quad \boldsymbol{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,$$
+
+
+
+$$
+\begin{align} \mathcal{C}(\boldsymbol{\theta}) &= \sum_{i=1}^N \mathcal{L}_i (\boldsymbol{\theta})
+\tag{16}\\
+ &= -\sum_{i=1}^N \sum_{c=0}^{C-1} y_{ic} \log P(y_{ic} = 1 \mid \boldsymbol{x}_i, \boldsymbol{\theta}) .
+\tag{17}
+\end{align}
+$$
+
+
+Optimizing the cost function
+
+
+$$ \theta_{i+1} = \theta_i - \eta \nabla \mathcal{C}(\theta_i) ,$$
+
+
+
+$$ \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) ,$$
+
+
+
+
+Regularization
+
+
+$$ \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 \boldsymbol{w} \rvert \rvert_2^2
+= \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}(\theta) + \lambda \sum_{ij} w_{ij}^2,$$
+
+
+Matrix multiplication
+
+
+$$ \Delta_o = \hat{y} - y = (n_{inputs}, n_{categories}) .$$
+
+
+
+$$ \nabla W_{o} = A^T \Delta_o = (n_{hidden}, n_{categories}) ,$$
+
+
+
+$$ \nabla B_{o} = \sum_{i=1}^{n_{inputs}} \Delta_o = (n_{categories}) .$$
+
+
+
+$$ \Delta_h = \Delta_o W_{o}^T \circ f'(Z_{h}) = \Delta_o W_{o}^T \circ A_{h} \circ (1 - A_{h}) = (n_{inputs}, n_{hidden}) ,$$
+
+
+
+$$ \nabla W_{h} = X^T \Delta_h = (n_{features}, n_{hidden}) ,$$
+
+
+
+$$ \nabla B_{h} = \sum_{i=1}^{n_{inputs}} \Delta_h = (n_{hidden}) .$$
+
+
+# to categorical turns our integer vector into a onehot representation
+#from keras.utils import to_categorical
+
+# calculate the accuracy score of our model
+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
+
+Full object-oriented implementation
+
+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,
+
+ ):
+ self.X_data_full = X_data
+ self.Y_data_full = Y_data
+
+ 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
- def update_mini_batch(self, mini_batch, eta):
- #updates w and b using backpropagation to a single mini batch. eta is the learning rate."
- nabla_b=[np.zeros(b.shape) for b in self.biases]
- nabla_w=[np.zeros(w.shape) for w in self.weights]
- for x,y in mini_batch:
- delta_nabla_b, delta_nabla_w=self.backprop(x,y)
- nabla_b=[nb+dnb for nb, dnb in zip(nabla_b, delta_nabla_b)]
- nabla_w=[nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)]
- self.weights=[w-(eta/len(mini_batch))*nw for w, nw in zip(self.weights, nabla_w)]
- self.biases=[b-(eta/len(mini_batch))*nb for b, nb in zip(self.biases, nabla_b)]
-
- def backprop(self, x, y):
- """Return a tuple ``(nabla_b, nabla_w)`` representing the
- gradient for the cost function C_x. ``nabla_b`` and
- ``nabla_w`` are layer-by-layer lists of numpy arrays, similar
- to ``self.biases`` and ``self.weights``."""
- nabla_b = [np.zeros(b.shape) for b in self.biases]
- nabla_w = [np.zeros(w.shape) for w in self.weights]
- # feedforward
- activation = x
- activations = [x] # list to store all the activations, layer by layer
- zs = [] # list to store all the z vectors, layer by layer
- for b, w in zip(self.biases, self.weights):
- z = np.dot(w, activation)+b
- zs.append(z)
- activation = sigmoid(z)
- activations.append(activation)
- # backward pass
- delta = self.cost_derivative(activations[-1], y) * \
- sigmoid_prime(zs[-1])
- nabla_b[-1] = delta
- nabla_w[-1] = np.dot(delta, activations[-2].transpose())
- # Note that the variable l in the loop below is used a little
- # differently to the notation in Chapter 2 of the book. Here,
- # l = 1 means the last layer of neurons, l = 2 is the
- # second-last layer, and so on. It's a renumbering of the
- # scheme in the book, used here to take advantage of the fact
- # that Python can use negative indices in lists.
- for l in xrange(2, self.num_layers):
- z = zs[-l]
- sp = sigmoid_prime(z)
- delta = np.dot(self.weights[-l+1].transpose(), delta) * sp
- nabla_b[-l] = delta
- nabla_w[-l] = np.dot(delta, activations[-l-1].transpose())
- return (nabla_b, nabla_w)
+ exp_term = np.exp(z_o)
+ probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+ return probabilities
- def evaluate(self, test_data):
- """Return the number of test inputs for which the neural
- network outputs the correct result. Note that the neural
- network's output is assumed to be the index of whichever
- neuron in the final layer has the highest activation."""
- test_results = [(np.argmax(self.feedforward(x)), y)
- for (x, y) in test_data]
- return sum(int(x == y) for (x, y) in test_results)
+ 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)
- def cost_derivative(self, output_activations, y):
- """Return the vector of partial derivatives \partial C_x /
- \partial a for the output activations."""
- return (output_activations-y)
-
-
-
-#Functions
-def sigmoid(z):
- return 1.0/(1.0+np.exp(-z))
+ self.output_weights_gradient = np.matmul(self.a_h.T, error_output)
+ self.output_bias_gradient = np.sum(error_output, axis=0)
-def sigmoid_prime(z):
- return sigmoid(z)*(1-sigmoid(z))
+ self.hidden_weights_gradient = np.matmul(self.X_data.T, error_hidden)
+ self.hidden_bias_gradient = np.sum(error_hidden, axis=0)
-network=Network()
+ 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()
+
Evaluate model performance on test data
+
+
+$$ \text{Accuracy} = \frac{\sum_{i=1}^n I(\hat{y}_i = y_i)}{n} ,$$
+
+
+epochs = 100
+batch_size = 100
+
+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))
+
Adjust hyperparameters (if necessary, network architecture
+
+eta_vals = np.logspace(-5, 1, 7)
+lmbd_vals = np.logspace(-5, 1, 7)
# %load neural-networks-and-deep-learning/src/mnist_loader.py
-"""
-mnist_loader
-~~~~~~~~~~~~
+
# store the models for later use
+DNN_numpy = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
-A library to load the MNIST image data. For details of the data
-structures that are returned, see the doc strings for ``load_data``
-and ``load_data_wrapper``. In practice, ``load_data_wrapper`` is the
-function usually called by our neural network code.
-"""
-
-#### Libraries
-# Standard library
-import pickle
-import gzip
-
-# Third-party libraries
-import numpy as np
-
-def load_data():
- """Return the MNIST data as a tuple containing the training data,
- the validation data, and the test data.
-
- The ``training_data`` is returned as a tuple with two entries.
- The first entry contains the actual training images. This is a
- numpy ndarray with 50,000 entries. Each entry is, in turn, a
- numpy ndarray with 784 values, representing the 28 * 28 = 784
- pixels in a single MNIST image.
-
- The second entry in the ``training_data`` tuple is a numpy ndarray
- containing 50,000 entries. Those entries are just the digit
- values (0...9) for the corresponding images contained in the first
- entry of the tuple.
-
- The ``validation_data`` and ``test_data`` are similar, except
- each contains only 10,000 images.
-
- This is a nice data format, but for use in neural networks it's
- helpful to modify the format of the ``training_data`` a little.
- That's done in the wrapper function ``load_data_wrapper()``, see
- below.
- """
- f = gzip.open('../data/mnist.pkl.gz', 'rb')
- training_data, validation_data, test_data = cPickle.load(f)
- f.close()
- return (training_data, validation_data, test_data)
-
-def load_data_wrapper():
- """Return a tuple containing ``(training_data, validation_data,
- test_data)``. Based on ``load_data``, but the format is more
- convenient for use in our implementation of neural networks.
-
- In particular, ``training_data`` is a list containing 50,000
- 2-tuples ``(x, y)``. ``x`` is a 784-dimensional numpy.ndarray
- containing the input image. ``y`` is a 10-dimensional
- numpy.ndarray representing the unit vector corresponding to the
- correct digit for ``x``.
-
- ``validation_data`` and ``test_data`` are lists containing 10,000
- 2-tuples ``(x, y)``. In each case, ``x`` is a 784-dimensional
- numpy.ndarry containing the input image, and ``y`` is the
- corresponding classification, i.e., the digit values (integers)
- corresponding to ``x``.
-
- Obviously, this means we're using slightly different formats for
- the training data and the validation / test data. These formats
- turn out to be the most convenient for use in our neural network
- code."""
- tr_d, va_d, te_d = load_data()
- training_inputs = [np.reshape(x, (784, 1)) for x in tr_d[0]]
- training_results = [vectorized_result(y) for y in tr_d[1]]
- training_data = zip(training_inputs, training_results)
- validation_inputs = [np.reshape(x, (784, 1)) for x in va_d[0]]
- validation_data = zip(validation_inputs, va_d[1])
- test_inputs = [np.reshape(x, (784, 1)) for x in te_d[0]]
- test_data = zip(test_inputs, te_d[1])
- return (training_data, validation_data, test_data)
-
-def vectorized_result(j):
- """Return a 10-dimensional unit vector with a 1.0 in the jth
- position and zeroes elsewhere. This is used to convert a digit
- (0...9) into a corresponding desired output from the neural
- network."""
- e = np.zeros((10, 1))
- e[j] = 1.0
- return e
-
-net=network.Network([784,30,30])
-net.SGD(training_data,30,10,3,test_data=test_data)
+# 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()
# optional
+# visual representation of grid search
+# uses seaborn heatmap, I believe you can also do this with matplotlib imshow
+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_numpy[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()
+
scikit-learn implementation
+
+from sklearn.neural_network import MLPClassifier
+
+# store models for later use
+DNN_scikit = 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 = MLPClassifier(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',
+ alpha=lmbd, learning_rate_init=eta, max_iter=epochs)
+ dnn.fit(X_train, Y_train)
+
+ DNN_scikit[i][j] = dnn
+
+ print("Learning rate = ", eta)
+ print("Lambda = ", lmbd)
+ print("Accuracy score on test set: ", dnn.score(X_test, Y_test))
+ 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_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()
+
And then with Tensorflow
Data Analysis and Machine Learning: Elements of machine learning
Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
Sep 28, 2018
Oct 2, 2018
-
@@ -162,27 +235,26 @@ categories:
@@ -190,22 +262,26 @@ The applications are not limited to the natural sciences. There is a plethora of
Neural network types
@@ -213,61 +289,82 @@ methods.
Feed-forward neural networks
-Recurrent neural networks
+Convolutional Neural Network
-Other types of networks
+Recurrent neural networks
-Multilayer perceptrons
+Other types of networks
+
+
+
+Multilayer perceptrons
-Why multilayer perceptrons?
+Why multilayer perceptrons?
-Mathematical model
-
-$$
-\begin{equation}
- y = f\left(\sum_{i=1}^n w_ix_i + b_i\right) = f(u)
-\label{artificialNeuron2}
-\end{equation}
-$$
+Mathematical model
-Mathematical model
+Mathematical model
-Mathematical model
+Mathematical model
-Mathematical model
+Mathematical model
-Mathematical model
+Mathematical model
-Matrix-vector notation
+Matrix-vector notation
-Matrix-vector notation and activation
+Matrix-vector notation and activation
-Activation functions
+Activation functions
-Activation functions, Logistic and Hyperbolic ones
+Activation functions, Logistic and Hyperbolic ones
-Relevance
+Relevance
-Setting up a Multi-layer perceptron model
+The multilayer perceptron (MLP)
from scipy import optimize
+
+
-class Neural_Network(object):
- def __init__(self, Lambda=0):
- #Define Hyperparameters
- self.inputLayerSize = 2
- self.outputLayerSize = 1
- self.hiddenLayerSize = 3
-
- #Weights (parameters)
- self.W1 = np.random.randn(self.inputLayerSize,self.hiddenLayerSize)
- self.W2 = np.random.randn(self.hiddenLayerSize,self.outputLayerSize)
-
- #Regularization Parameter:
- self.Lambda = Lambda
-
- def forward(self, X):
- #Propogate inputs though network
- self.z2 = np.dot(X, self.W1)
- self.a2 = self.sigmoid(self.z2)
- self.z3 = np.dot(self.a2, self.W2)
- yHat = self.sigmoid(self.z3)
- return yHat
-
- def sigmoid(self, z):
- #Apply sigmoid activation function to scalar, vector, or matrix
- return 1/(1+np.exp(-z))
-
- def sigmoidPrime(self,z):
- #Gradient of sigmoid
- return np.exp(-z)/((1+np.exp(-z))**2)
-
- def costFunction(self, X, y):
- #Compute cost for given X,y, use weights already stored in class.
- self.yHat = self.forward(X)
- J = 0.5*sum((y-self.yHat)**2)/X.shape[0] + (self.Lambda/2)*(np.sum(self.W1**2)+np.sum(self.W2**2))
- return J
-
- def costFunctionPrime(self, X, y):
- #Compute derivative with respect to W and W2 for a given X and y:
- self.yHat = self.forward(X)
-
- delta3 = np.multiply(-(y-self.yHat), self.sigmoidPrime(self.z3))
- #Add gradient of regularization term:
- dJdW2 = np.dot(self.a2.T, delta3)/X.shape[0] + self.Lambda*self.W2
-
- delta2 = np.dot(delta3, self.W2.T)*self.sigmoidPrime(self.z2)
- #Add gradient of regularization term:
- dJdW1 = np.dot(X.T, delta2)/X.shape[0] + self.Lambda*self.W1
-
- return dJdW1, dJdW2
-
- #Helper functions for interacting with other methods/classes
- def getParams(self):
- #Get W1 and W2 Rolled into vector:
- params = np.concatenate((self.W1.ravel(), self.W2.ravel()))
- return params
-
- def setParams(self, params):
- #Set W1 and W2 using single parameter vector:
- W1_start = 0
- W1_end = self.hiddenLayerSize*self.inputLayerSize
- self.W1 = np.reshape(params[W1_start:W1_end], \
- (self.inputLayerSize, self.hiddenLayerSize))
- W2_end = W1_end + self.hiddenLayerSize*self.outputLayerSize
- self.W2 = np.reshape(params[W1_end:W2_end], \
- (self.hiddenLayerSize, self.outputLayerSize))
-
- def computeGradients(self, X, y):
- dJdW1, dJdW2 = self.costFunctionPrime(X, y)
- return np.concatenate((dJdW1.ravel(), dJdW2.ravel()))
-
-
-class trainer(object):
- def __init__(self, N):
- #Make Local reference to network:
- self.N = N
-
- def callbackF(self, params):
- self.N.setParams(params)
- self.J.append(self.N.costFunction(self.X, self.y))
- self.testJ.append(self.N.costFunction(self.testX, self.testY))
-
- def costFunctionWrapper(self, params, X, y):
- self.N.setParams(params)
- cost = self.N.costFunction(X, y)
- grad = self.N.computeGradients(X,y)
- return cost, grad
-
- def train(self, trainX, trainY, testX, testY):
- #Make an internal variable for the callback function:
- self.X = trainX
- self.y = trainY
-
- self.testX = testX
- self.testY = testY
+As a convention it is normal to call a network with one layer of input units, one layer of hidden
+units and one layer of output units as a two-layer network. A network with two layers of hidden units is called a three-layer network etc etc.
- #Make empty list to store training costs:
- self.J = []
- self.testJ = []
-
- params0 = self.N.getParams()
+
+
+From one to many layers, the universal approximation theorem
+
+
+
+Deriving the back propagation code for a multilayer perceptron model
+
+
+
+Definitions
+
+
+
+Derivatives and the chain rule
+
+
+
+Derivative of the cost function
+
+
+
+Bringing it together, first back propagation equation
+
+
+
+Derivatives in terms of \( z_j^L \)
+
+
+
+Bringing it together
+
+
+
+Final back propagating equation
+
+
+
+Setting up the Back propagation algorithm
+
+Two-layer Neural Network
+Setting up a Multi-layer perceptron model for classification
+
+
+
+Defining the cost function
+
+Developing a code for doing neural networks with back propagation
+
+
+
+
+
+
+Collect and pre-process data
+
+import numpy as np
-
-#sigmoid
-def nonlin(x, deriv=False):
- if (deriv==True):
- return x*(1-x)
- return 1/(1+np.exp(-x))
-
-#input data
-x=np.array([[0,0,1],[0,1,1],[1,0,1],[1,1,1]])
-
-#output data
-y=np.array([0,1,1,0]).T
-
-#seed random numbers to make calculation
-np.random.seed(1)
-
-#initialize weights with mean=0
-syn0=2*np.random.random((3,4))-1
-
-for iter in range(10000):
- #forward propogation
- l0=x
- l1=nonlin(np.dot(l0,syn0))
- l1_error=y-l1
- #multiply error by slope of sigmoid at values of l1
- l1_delta=l1_error*nonlin(l1,True)
- #update weights
- syn0+=np.dot(l0.T, l1_delta)
-
-print("Output after training: ",l1 )
-
import numpy as np
-import random
-class Network(object):
-
- def _init_(self, sizes):
- self.num_layers=len(sizes)
- self.sizes=sizes
- self.biases=[np.random.randn(y,1) for y in sizes[1:]]
- self.weights=[np.random.randn(y,x) for x,y in zip(sizes[:-1], sizes[1:])]
-
-#sizes is the number of neurons in each layer
-#for example, say n_1st_layer=3, n_2nd_layer=3, n_3rd_layer=1, then net=Network([3,3,1])
-
-#The biases and weights are initialized randomly, using Gaussian distributions of mean=0, stdev=1
-#z is a vector (or a np.array)
-
- def feedforward(self,a):
- #returns output w/ 'a' as an input
- for b, w in zip(self.biases, self.weights):
- a=sigmoid(np.dot(w,b)+b)
- return a
-
-#Apply a Stochastic Gradient Descent (SGD) method:
- def SGD(self, training_data, epochs, mini_batch_size, eta, test_data=None):
- """Trains network using batches incorporating SGD. The network will be evaluated against the
- test data after each epoch, with partial progress being printed out (this is useful for tracking,
- but slows the process.)"""
- if test_data: n_test=len(test_data)
- n=len(training_data)
- for j in xrange(epochs):
- random.shuffle(training_data)
- mini_batches=[training_data[k:k+mini_batch_size] for k in xrange(o,n,mini_batch_size)]
- for mini_batch in mini_batches:
- self.update_mini_batch(mini_batch, eta)
- if test_data:
- print ("Epoch {0}: {1}/{2}".format(j, self.evaluate(test_data), n_test))
- else:
- print ("Epoch {0} complete".format(j))
-
-
- def update_mini_batch(self, mini_batch, eta):
- #updates w and b using backpropagation to a single mini batch. eta is the learning rate."
- nabla_b=[np.zeros(b.shape) for b in self.biases]
- nabla_w=[np.zeros(w.shape) for w in self.weights]
- for x,y in mini_batch:
- delta_nabla_b, delta_nabla_w=self.backprop(x,y)
- nabla_b=[nb+dnb for nb, dnb in zip(nabla_b, delta_nabla_b)]
- nabla_w=[nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)]
- self.weights=[w-(eta/len(mini_batch))*nw for w, nw in zip(self.weights, nabla_w)]
- self.biases=[b-(eta/len(mini_batch))*nb for b, nb in zip(self.biases, nabla_b)]
-
- def backprop(self, x, y):
- """Return a tuple ``(nabla_b, nabla_w)`` representing the
- gradient for the cost function C_x. ``nabla_b`` and
- ``nabla_w`` are layer-by-layer lists of numpy arrays, similar
- to ``self.biases`` and ``self.weights``."""
- nabla_b = [np.zeros(b.shape) for b in self.biases]
- nabla_w = [np.zeros(w.shape) for w in self.weights]
- # feedforward
- activation = x
- activations = [x] # list to store all the activations, layer by layer
- zs = [] # list to store all the z vectors, layer by layer
- for b, w in zip(self.biases, self.weights):
- z = np.dot(w, activation)+b
- zs.append(z)
- activation = sigmoid(z)
- activations.append(activation)
- # backward pass
- delta = self.cost_derivative(activations[-1], y) * \
- sigmoid_prime(zs[-1])
- nabla_b[-1] = delta
- nabla_w[-1] = np.dot(delta, activations[-2].transpose())
- # Note that the variable l in the loop below is used a little
- # differently to the notation in Chapter 2 of the book. Here,
- # l = 1 means the last layer of neurons, l = 2 is the
- # second-last layer, and so on. It's a renumbering of the
- # scheme in the book, used here to take advantage of the fact
- # that Python can use negative indices in lists.
- for l in xrange(2, self.num_layers):
- z = zs[-l]
- sp = sigmoid_prime(z)
- delta = np.dot(self.weights[-l+1].transpose(), delta) * sp
- nabla_b[-l] = delta
- nabla_w[-l] = np.dot(delta, activations[-l-1].transpose())
- return (nabla_b, nabla_w)
-
- def evaluate(self, test_data):
- """Return the number of test inputs for which the neural
- network outputs the correct result. Note that the neural
- network's output is assumed to be the index of whichever
- neuron in the final layer has the highest activation."""
- test_results = [(np.argmax(self.feedforward(x)), y)
- for (x, y) in test_data]
- return sum(int(x == y) for (x, y) in test_results)
-
- def cost_derivative(self, output_activations, y):
- """Return the vector of partial derivatives \partial C_x /
- \partial a for the output activations."""
- return (output_activations-y)
-
-
-
-#Functions
-def sigmoid(z):
- return 1.0/(1.0+np.exp(-z))
-
-def sigmoid_prime(z):
- return sigmoid(z)*(1-sigmoid(z))
-
-network=Network()
-
# %load neural-networks-and-deep-learning/src/mnist_loader.py
-"""
-mnist_loader
-~~~~~~~~~~~~
-
-A library to load the MNIST image data. For details of the data
-structures that are returned, see the doc strings for ``load_data``
-and ``load_data_wrapper``. In practice, ``load_data_wrapper`` is the
-function usually called by our neural network code.
-"""
-
-#### Libraries
-# Standard library
-import pickle
-import gzip
-
-# Third-party libraries
+
# import necessary packages
import numpy as np
+import matplotlib.pyplot as plt
+from sklearn import datasets
-def load_data():
- """Return the MNIST data as a tuple containing the training data,
- the validation data, and the test data.
- The ``training_data`` is returned as a tuple with two entries.
- The first entry contains the actual training images. This is a
- numpy ndarray with 50,000 entries. Each entry is, in turn, a
- numpy ndarray with 784 values, representing the 28 * 28 = 784
- pixels in a single MNIST image.
+# ensure the same random numbers appear every time
+np.random.seed(0)
- The second entry in the ``training_data`` tuple is a numpy ndarray
- containing 50,000 entries. Those entries are just the digit
- values (0...9) for the corresponding images contained in the first
- entry of the tuple.
+# display images in notebook
+%matplotlib inline
+plt.rcParams['figure.figsize'] = (12,12)
- The ``validation_data`` and ``test_data`` are similar, except
- each contains only 10,000 images.
- This is a nice data format, but for use in neural networks it's
- helpful to modify the format of the ``training_data`` a little.
- That's done in the wrapper function ``load_data_wrapper()``, see
- below.
- """
- f = gzip.open('../data/mnist.pkl.gz', 'rb')
- training_data, validation_data, test_data = cPickle.load(f)
- f.close()
- return (training_data, validation_data, test_data)
+# download MNIST dataset
+digits = datasets.load_digits()
-def load_data_wrapper():
- """Return a tuple containing ``(training_data, validation_data,
- test_data)``. Based on ``load_data``, but the format is more
- convenient for use in our implementation of neural networks.
+# define inputs and labels
+inputs = digits.images
+labels = digits.target
- In particular, ``training_data`` is a list containing 50,000
- 2-tuples ``(x, y)``. ``x`` is a 784-dimensional numpy.ndarray
- containing the input image. ``y`` is a 10-dimensional
- numpy.ndarray representing the unit vector corresponding to the
- correct digit for ``x``.
+print("inputs = (n_inputs, pixel_width, pixel_height) = " + str(inputs.shape))
+print("labels = (n_inputs) = " + str(labels.shape))
- ``validation_data`` and ``test_data`` are lists containing 10,000
- 2-tuples ``(x, y)``. In each case, ``x`` is a 784-dimensional
- numpy.ndarry containing the input image, and ``y`` is the
- corresponding classification, i.e., the digit values (integers)
- corresponding to ``x``.
- Obviously, this means we're using slightly different formats for
- the training data and the validation / test data. These formats
- turn out to be the most convenient for use in our neural network
- code."""
- tr_d, va_d, te_d = load_data()
- training_inputs = [np.reshape(x, (784, 1)) for x in tr_d[0]]
- training_results = [vectorized_result(y) for y in tr_d[1]]
- training_data = zip(training_inputs, training_results)
- validation_inputs = [np.reshape(x, (784, 1)) for x in va_d[0]]
- validation_data = zip(validation_inputs, va_d[1])
- test_inputs = [np.reshape(x, (784, 1)) for x in te_d[0]]
- test_data = zip(test_inputs, te_d[1])
- return (training_data, validation_data, test_data)
+# 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))
-def vectorized_result(j):
- """Return a 10-dimensional unit vector with a 1.0 in the jth
- position and zeroes elsewhere. This is used to convert a digit
- (0...9) into a corresponding desired output from the neural
- network."""
- e = np.zeros((10, 1))
- e[j] = 1.0
- return e
-net=network.Network([784,30,30])
-net.SGD(training_data,30,10,3,test_data=test_data)
+# 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()
+
+Train and test datasets
+
+from sklearn.model_selection import train_test_split
+
+# 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
+
+#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)))
+
+
+Define model and architecture
+
+Layers
+
+
+
+
+Since each input image has 8x8 = 64 pixels or features, we have an input layer of 64 neurons.
+
+
+
+
+We will use 50 neurons in the hidden layer receiving input from the neurons in the input layer.
+Since each neuron in the hidden layer is connected to the 64 inputs we have 64x50 = 3200 weights to the hidden layer.
+
+
+
+
+If we were building a binary classifier, it would be sufficient with a single neuron in the output layer,
+which could output 0 or 1 according to the Heaviside function. This would be an example of a hard classifier, meaning it outputs the class of the input directly. However, if we are dealing with noisy data it is often beneficial to use a soft classifier, which outputs the probability of being in class 0 or 1.
+
+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
+
+
+Feed-forward pass
+
+Matrix multiplication
+
+# setup the feed-forward pass
+
+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
+
+
+
+Optimizing the cost function
+
+
+
+
+
+
+Regularization
+
+
+
+Matrix multiplication
+
+# to categorical turns our integer vector into a onehot representation
+#from keras.utils import to_categorical
+
+# calculate the accuracy score of our model
+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
+
+
+
+Full object-oriented implementation
+
+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,
+
+ ):
+ self.X_data_full = X_data
+ self.Y_data_full = Y_data
+
+ 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()
+
+
+Evaluate model performance on test data
+
+epochs = 100
+batch_size = 100
+
+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))
+
+
+Adjust hyperparameters (if necessary, network architecture
+
+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)
+
+# 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()
+
# optional
+# visual representation of grid search
+# uses seaborn heatmap, I believe you can also do this with matplotlib imshow
+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_numpy[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()
+
+
+scikit-learn implementation
+
+from sklearn.neural_network import MLPClassifier
+
+# store models for later use
+DNN_scikit = 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 = MLPClassifier(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',
+ alpha=lmbd, learning_rate_init=eta, max_iter=epochs)
+ dnn.fit(X_train, Y_train)
+
+ DNN_scikit[i][j] = dnn
+
+ print("Learning rate = ", eta)
+ print("Lambda = ", lmbd)
+ print("Accuracy score on test set: ", dnn.score(X_test, Y_test))
+ 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_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()
+
+
+And then with Tensorflow
+
Data Analysis and Machine Learning: Elements of machine learning
Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
Sep 28, 2018
Oct 2, 2018
-
@@ -167,27 +240,26 @@ categories:
@@ -195,22 +267,26 @@ The applications are not limited to the natural sciences. There is a plethora of
Neural network types
@@ -218,61 +294,82 @@ methods.
Feed-forward neural networks
-Recurrent neural networks
+Convolutional Neural Network
-Other types of networks
+Recurrent neural networks
-Multilayer perceptrons
+Other types of networks
+
+
+
+Multilayer perceptrons
-Why multilayer perceptrons?
+Why multilayer perceptrons?
-Mathematical model
-
-$$
-\begin{equation}
- y = f\left(\sum_{i=1}^n w_ix_i + b_i\right) = f(u)
-\label{artificialNeuron2}
-\end{equation}
-$$
+Mathematical model
-Mathematical model
+Mathematical model
-Mathematical model
+Mathematical model
-Mathematical model
+Mathematical model
-Mathematical model
+Mathematical model
-Matrix-vector notation
+Matrix-vector notation
-Matrix-vector notation and activation
+Matrix-vector notation and activation
-Activation functions
+Activation functions
-Activation functions, Logistic and Hyperbolic ones
+Activation functions, Logistic and Hyperbolic ones
-Relevance
+Relevance
-Setting up a Multi-layer perceptron model
+The multilayer perceptron (MLP)
from scipy import optimize
+
+
-class Neural_Network(object):
- def __init__(self, Lambda=0):
- #Define Hyperparameters
- self.inputLayerSize = 2
- self.outputLayerSize = 1
- self.hiddenLayerSize = 3
-
- #Weights (parameters)
- self.W1 = np.random.randn(self.inputLayerSize,self.hiddenLayerSize)
- self.W2 = np.random.randn(self.hiddenLayerSize,self.outputLayerSize)
-
- #Regularization Parameter:
- self.Lambda = Lambda
-
- def forward(self, X):
- #Propogate inputs though network
- self.z2 = np.dot(X, self.W1)
- self.a2 = self.sigmoid(self.z2)
- self.z3 = np.dot(self.a2, self.W2)
- yHat = self.sigmoid(self.z3)
- return yHat
-
- def sigmoid(self, z):
- #Apply sigmoid activation function to scalar, vector, or matrix
- return 1/(1+np.exp(-z))
-
- def sigmoidPrime(self,z):
- #Gradient of sigmoid
- return np.exp(-z)/((1+np.exp(-z))**2)
-
- def costFunction(self, X, y):
- #Compute cost for given X,y, use weights already stored in class.
- self.yHat = self.forward(X)
- J = 0.5*sum((y-self.yHat)**2)/X.shape[0] + (self.Lambda/2)*(np.sum(self.W1**2)+np.sum(self.W2**2))
- return J
-
- def costFunctionPrime(self, X, y):
- #Compute derivative with respect to W and W2 for a given X and y:
- self.yHat = self.forward(X)
-
- delta3 = np.multiply(-(y-self.yHat), self.sigmoidPrime(self.z3))
- #Add gradient of regularization term:
- dJdW2 = np.dot(self.a2.T, delta3)/X.shape[0] + self.Lambda*self.W2
-
- delta2 = np.dot(delta3, self.W2.T)*self.sigmoidPrime(self.z2)
- #Add gradient of regularization term:
- dJdW1 = np.dot(X.T, delta2)/X.shape[0] + self.Lambda*self.W1
-
- return dJdW1, dJdW2
-
- #Helper functions for interacting with other methods/classes
- def getParams(self):
- #Get W1 and W2 Rolled into vector:
- params = np.concatenate((self.W1.ravel(), self.W2.ravel()))
- return params
-
- def setParams(self, params):
- #Set W1 and W2 using single parameter vector:
- W1_start = 0
- W1_end = self.hiddenLayerSize*self.inputLayerSize
- self.W1 = np.reshape(params[W1_start:W1_end], \
- (self.inputLayerSize, self.hiddenLayerSize))
- W2_end = W1_end + self.hiddenLayerSize*self.outputLayerSize
- self.W2 = np.reshape(params[W1_end:W2_end], \
- (self.hiddenLayerSize, self.outputLayerSize))
-
- def computeGradients(self, X, y):
- dJdW1, dJdW2 = self.costFunctionPrime(X, y)
- return np.concatenate((dJdW1.ravel(), dJdW2.ravel()))
-
-
-class trainer(object):
- def __init__(self, N):
- #Make Local reference to network:
- self.N = N
-
- def callbackF(self, params):
- self.N.setParams(params)
- self.J.append(self.N.costFunction(self.X, self.y))
- self.testJ.append(self.N.costFunction(self.testX, self.testY))
-
- def costFunctionWrapper(self, params, X, y):
- self.N.setParams(params)
- cost = self.N.costFunction(X, y)
- grad = self.N.computeGradients(X,y)
- return cost, grad
-
- def train(self, trainX, trainY, testX, testY):
- #Make an internal variable for the callback function:
- self.X = trainX
- self.y = trainY
-
- self.testX = testX
- self.testY = testY
+As a convention it is normal to call a network with one layer of input units, one layer of hidden
+units and one layer of output units as a two-layer network. A network with two layers of hidden units is called a three-layer network etc etc.
- #Make empty list to store training costs:
- self.J = []
- self.testJ = []
-
- params0 = self.N.getParams()
+
+
+From one to many layers, the universal approximation theorem
+
+
+
+Deriving the back propagation code for a multilayer perceptron model
+
+
+
+Definitions
+
+
+
+Derivatives and the chain rule
+
+
+
+Derivative of the cost function
+
+
+
+Bringing it together, first back propagation equation
+
+
+
+Derivatives in terms of \( z_j^L \)
+
+
+
+Bringing it together
+
+
+
+Final back propagating equation
+
+
+
+Setting up the Back propagation algorithm
+
+Two-layer Neural Network
+Setting up a Multi-layer perceptron model for classification
+
+
+
+Defining the cost function
+
+Developing a code for doing neural networks with back propagation
+
+
+
+
+
+
+Collect and pre-process data
+
+import numpy as np
-
-#sigmoid
-def nonlin(x, deriv=False):
- if (deriv==True):
- return x*(1-x)
- return 1/(1+np.exp(-x))
-
-#input data
-x=np.array([[0,0,1],[0,1,1],[1,0,1],[1,1,1]])
-
-#output data
-y=np.array([0,1,1,0]).T
-
-#seed random numbers to make calculation
-np.random.seed(1)
-
-#initialize weights with mean=0
-syn0=2*np.random.random((3,4))-1
-
-for iter in range(10000):
- #forward propogation
- l0=x
- l1=nonlin(np.dot(l0,syn0))
- l1_error=y-l1
- #multiply error by slope of sigmoid at values of l1
- l1_delta=l1_error*nonlin(l1,True)
- #update weights
- syn0+=np.dot(l0.T, l1_delta)
-
-print("Output after training: ",l1 )
-
import numpy as np
-import random
-class Network(object):
-
- def _init_(self, sizes):
- self.num_layers=len(sizes)
- self.sizes=sizes
- self.biases=[np.random.randn(y,1) for y in sizes[1:]]
- self.weights=[np.random.randn(y,x) for x,y in zip(sizes[:-1], sizes[1:])]
-
-#sizes is the number of neurons in each layer
-#for example, say n_1st_layer=3, n_2nd_layer=3, n_3rd_layer=1, then net=Network([3,3,1])
-
-#The biases and weights are initialized randomly, using Gaussian distributions of mean=0, stdev=1
-#z is a vector (or a np.array)
-
- def feedforward(self,a):
- #returns output w/ 'a' as an input
- for b, w in zip(self.biases, self.weights):
- a=sigmoid(np.dot(w,b)+b)
- return a
-
-#Apply a Stochastic Gradient Descent (SGD) method:
- def SGD(self, training_data, epochs, mini_batch_size, eta, test_data=None):
- """Trains network using batches incorporating SGD. The network will be evaluated against the
- test data after each epoch, with partial progress being printed out (this is useful for tracking,
- but slows the process.)"""
- if test_data: n_test=len(test_data)
- n=len(training_data)
- for j in xrange(epochs):
- random.shuffle(training_data)
- mini_batches=[training_data[k:k+mini_batch_size] for k in xrange(o,n,mini_batch_size)]
- for mini_batch in mini_batches:
- self.update_mini_batch(mini_batch, eta)
- if test_data:
- print ("Epoch {0}: {1}/{2}".format(j, self.evaluate(test_data), n_test))
- else:
- print ("Epoch {0} complete".format(j))
-
-
- def update_mini_batch(self, mini_batch, eta):
- #updates w and b using backpropagation to a single mini batch. eta is the learning rate."
- nabla_b=[np.zeros(b.shape) for b in self.biases]
- nabla_w=[np.zeros(w.shape) for w in self.weights]
- for x,y in mini_batch:
- delta_nabla_b, delta_nabla_w=self.backprop(x,y)
- nabla_b=[nb+dnb for nb, dnb in zip(nabla_b, delta_nabla_b)]
- nabla_w=[nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)]
- self.weights=[w-(eta/len(mini_batch))*nw for w, nw in zip(self.weights, nabla_w)]
- self.biases=[b-(eta/len(mini_batch))*nb for b, nb in zip(self.biases, nabla_b)]
-
- def backprop(self, x, y):
- """Return a tuple ``(nabla_b, nabla_w)`` representing the
- gradient for the cost function C_x. ``nabla_b`` and
- ``nabla_w`` are layer-by-layer lists of numpy arrays, similar
- to ``self.biases`` and ``self.weights``."""
- nabla_b = [np.zeros(b.shape) for b in self.biases]
- nabla_w = [np.zeros(w.shape) for w in self.weights]
- # feedforward
- activation = x
- activations = [x] # list to store all the activations, layer by layer
- zs = [] # list to store all the z vectors, layer by layer
- for b, w in zip(self.biases, self.weights):
- z = np.dot(w, activation)+b
- zs.append(z)
- activation = sigmoid(z)
- activations.append(activation)
- # backward pass
- delta = self.cost_derivative(activations[-1], y) * \
- sigmoid_prime(zs[-1])
- nabla_b[-1] = delta
- nabla_w[-1] = np.dot(delta, activations[-2].transpose())
- # Note that the variable l in the loop below is used a little
- # differently to the notation in Chapter 2 of the book. Here,
- # l = 1 means the last layer of neurons, l = 2 is the
- # second-last layer, and so on. It's a renumbering of the
- # scheme in the book, used here to take advantage of the fact
- # that Python can use negative indices in lists.
- for l in xrange(2, self.num_layers):
- z = zs[-l]
- sp = sigmoid_prime(z)
- delta = np.dot(self.weights[-l+1].transpose(), delta) * sp
- nabla_b[-l] = delta
- nabla_w[-l] = np.dot(delta, activations[-l-1].transpose())
- return (nabla_b, nabla_w)
-
- def evaluate(self, test_data):
- """Return the number of test inputs for which the neural
- network outputs the correct result. Note that the neural
- network's output is assumed to be the index of whichever
- neuron in the final layer has the highest activation."""
- test_results = [(np.argmax(self.feedforward(x)), y)
- for (x, y) in test_data]
- return sum(int(x == y) for (x, y) in test_results)
-
- def cost_derivative(self, output_activations, y):
- """Return the vector of partial derivatives \partial C_x /
- \partial a for the output activations."""
- return (output_activations-y)
-
-
-
-#Functions
-def sigmoid(z):
- return 1.0/(1.0+np.exp(-z))
-
-def sigmoid_prime(z):
- return sigmoid(z)*(1-sigmoid(z))
-
-network=Network()
-
# %load neural-networks-and-deep-learning/src/mnist_loader.py
-"""
-mnist_loader
-~~~~~~~~~~~~
-
-A library to load the MNIST image data. For details of the data
-structures that are returned, see the doc strings for ``load_data``
-and ``load_data_wrapper``. In practice, ``load_data_wrapper`` is the
-function usually called by our neural network code.
-"""
-
-#### Libraries
-# Standard library
-import pickle
-import gzip
-
-# Third-party libraries
+
# import necessary packages
import numpy as np
+import matplotlib.pyplot as plt
+from sklearn import datasets
-def load_data():
- """Return the MNIST data as a tuple containing the training data,
- the validation data, and the test data.
- The ``training_data`` is returned as a tuple with two entries.
- The first entry contains the actual training images. This is a
- numpy ndarray with 50,000 entries. Each entry is, in turn, a
- numpy ndarray with 784 values, representing the 28 * 28 = 784
- pixels in a single MNIST image.
+# ensure the same random numbers appear every time
+np.random.seed(0)
- The second entry in the ``training_data`` tuple is a numpy ndarray
- containing 50,000 entries. Those entries are just the digit
- values (0...9) for the corresponding images contained in the first
- entry of the tuple.
+# display images in notebook
+%matplotlib inline
+plt.rcParams['figure.figsize'] = (12,12)
- The ``validation_data`` and ``test_data`` are similar, except
- each contains only 10,000 images.
- This is a nice data format, but for use in neural networks it's
- helpful to modify the format of the ``training_data`` a little.
- That's done in the wrapper function ``load_data_wrapper()``, see
- below.
- """
- f = gzip.open('../data/mnist.pkl.gz', 'rb')
- training_data, validation_data, test_data = cPickle.load(f)
- f.close()
- return (training_data, validation_data, test_data)
+# download MNIST dataset
+digits = datasets.load_digits()
-def load_data_wrapper():
- """Return a tuple containing ``(training_data, validation_data,
- test_data)``. Based on ``load_data``, but the format is more
- convenient for use in our implementation of neural networks.
+# define inputs and labels
+inputs = digits.images
+labels = digits.target
- In particular, ``training_data`` is a list containing 50,000
- 2-tuples ``(x, y)``. ``x`` is a 784-dimensional numpy.ndarray
- containing the input image. ``y`` is a 10-dimensional
- numpy.ndarray representing the unit vector corresponding to the
- correct digit for ``x``.
+print("inputs = (n_inputs, pixel_width, pixel_height) = " + str(inputs.shape))
+print("labels = (n_inputs) = " + str(labels.shape))
- ``validation_data`` and ``test_data`` are lists containing 10,000
- 2-tuples ``(x, y)``. In each case, ``x`` is a 784-dimensional
- numpy.ndarry containing the input image, and ``y`` is the
- corresponding classification, i.e., the digit values (integers)
- corresponding to ``x``.
- Obviously, this means we're using slightly different formats for
- the training data and the validation / test data. These formats
- turn out to be the most convenient for use in our neural network
- code."""
- tr_d, va_d, te_d = load_data()
- training_inputs = [np.reshape(x, (784, 1)) for x in tr_d[0]]
- training_results = [vectorized_result(y) for y in tr_d[1]]
- training_data = zip(training_inputs, training_results)
- validation_inputs = [np.reshape(x, (784, 1)) for x in va_d[0]]
- validation_data = zip(validation_inputs, va_d[1])
- test_inputs = [np.reshape(x, (784, 1)) for x in te_d[0]]
- test_data = zip(test_inputs, te_d[1])
- return (training_data, validation_data, test_data)
+# 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))
-def vectorized_result(j):
- """Return a 10-dimensional unit vector with a 1.0 in the jth
- position and zeroes elsewhere. This is used to convert a digit
- (0...9) into a corresponding desired output from the neural
- network."""
- e = np.zeros((10, 1))
- e[j] = 1.0
- return e
-net=network.Network([784,30,30])
-net.SGD(training_data,30,10,3,test_data=test_data)
+# 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()
+
+Train and test datasets
+
+from sklearn.model_selection import train_test_split
+
+# 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
+
+#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)))
+
+
+Define model and architecture
+
+Layers
+
+
+
+
+Since each input image has 8x8 = 64 pixels or features, we have an input layer of 64 neurons.
+
+
+
+
+We will use 50 neurons in the hidden layer receiving input from the neurons in the input layer.
+Since each neuron in the hidden layer is connected to the 64 inputs we have 64x50 = 3200 weights to the hidden layer.
+
+
+
+
+If we were building a binary classifier, it would be sufficient with a single neuron in the output layer,
+which could output 0 or 1 according to the Heaviside function. This would be an example of a hard classifier, meaning it outputs the class of the input directly. However, if we are dealing with noisy data it is often beneficial to use a soft classifier, which outputs the probability of being in class 0 or 1.
+
+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
+
+
+Feed-forward pass
+
+Matrix multiplication
+
+# setup the feed-forward pass
+
+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
+
+
+
+Optimizing the cost function
+
+
+
+
+
+
+Regularization
+
+
+
+Matrix multiplication
+
+# to categorical turns our integer vector into a onehot representation
+#from keras.utils import to_categorical
+
+# calculate the accuracy score of our model
+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
+
+
+
+Full object-oriented implementation
+
+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,
+
+ ):
+ self.X_data_full = X_data
+ self.Y_data_full = Y_data
+
+ 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()
+
+
+Evaluate model performance on test data
+
+epochs = 100
+batch_size = 100
+
+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))
+
+
+Adjust hyperparameters (if necessary, network architecture
+
+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)
+
+# 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()
+
# optional
+# visual representation of grid search
+# uses seaborn heatmap, I believe you can also do this with matplotlib imshow
+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_numpy[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()
+
+
+scikit-learn implementation
+
+from sklearn.neural_network import MLPClassifier
+
+# store models for later use
+DNN_scikit = 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 = MLPClassifier(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',
+ alpha=lmbd, learning_rate_init=eta, max_iter=epochs)
+ dnn.fit(X_train, Y_train)
+
+ DNN_scikit[i][j] = dnn
+
+ print("Learning rate = ", eta)
+ print("Lambda = ", lmbd)
+ print("Accuracy score on test set: ", dnn.score(X_test, Y_test))
+ 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_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()
+
+
+And then with Tensorflow
+