In our discussions of ordinary differential equations and neural network codes -we will also study the usage of Autograd, see for example https://www.youtube.com/watch?v=fRf4l5qaX1M&ab_channel=AlexSmola in computing gradients for deep learning. For the documentation of Autograd and examples see the lectures slides from week 39 and the Autograd documentation at https://github.com/HIPS/autograd. +we will also study the usage of Autograd, see for example https://www.youtube.com/watch?v=fRf4l5qaX1M&ab_channel=AlexSmola in computing gradients for deep learning. For the documentation of Autograd and examples see the Autograd documentation at https://github.com/HIPS/autograd and the lecture slides from week 40, see https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/week41.html.
@@ -561,11 +438,11 @@ we will also study the usage of Autograd, see for example http://cs231n.stanford.edu/slides/2017/cs231n_2017_lecture4.pdf -This is a reminder from where we ended last week.
+This is a reminder from last week.
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. We let \( \theta \) represent the unknown weights and biases to be adjusted by our equations). The variable \( x \) -represents our activation values \( z \). We have -
-$$ -P(y = 0 \mid \boldsymbol{x}, \boldsymbol{\theta}) = \frac{1}{1 + \exp{(- \boldsymbol{x}})} , -$$ - -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:
- -$$ -\mathcal{C}(\boldsymbol{\theta}) = - \log{P(\mathcal{D} \mid \boldsymbol{\theta})}. -$$ - -See the logistic regression lectures for a full definition of the cost function.
- -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!
- - -As an example of the above, relevant for project 2 as well, let us consider a binary class. As discussed in our logistic regression lectures, we defined a cost function in terms of the parameters \( \beta \) as
-$$ -\mathcal{C}(\boldsymbol{\beta}) = - \sum_{i=1}^n \left(y_i\log{p(y_i \vert x_i,\boldsymbol{\beta})}+(1-y_i)\log{1-p(y_i \vert x_i,\boldsymbol{\beta})}\right), -$$ - -where we had defined the logistic (sigmoid) function
-$$ -p(y_i =1\vert x_i,\boldsymbol{\beta})=\frac{\exp{(\beta_0+\beta_1 x_i)}}{1+\exp{(\beta_0+\beta_1 x_i)}}, -$$ - -and
-$$ -p(y_i =0\vert x_i,\boldsymbol{\beta})=1-p(y_i =1\vert x_i,\boldsymbol{\beta}). -$$ - -The parameters \( \boldsymbol{\beta} \) were defined using a minimization method like gradient descent or Newton-Raphson's method.
- -Now we replace \( x_i \) with the activation \( z_i^l \) for a given layer \( l \) and the outputs as \( y_i=a_i^l=f(z_i^l) \), with \( z_i^l \) now being a function of the weights \( w_{ij}^l \) and biases \( b_i^l \). -We have then -
-$$ -a_i^l = y_i = \frac{\exp{(z_i^l)}}{1+\exp{(z_i^l)}}, -$$ - -with
-$$ -z_i^l = \sum_{j}w_{ij}^l a_j^{l-1}+b_i^l, -$$ - -where the superscript \( l-1 \) indicates that these are the outputs from layer \( l-1 \). -Our cost function at the final layer \( l=L \) is now -
-$$ -\mathcal{C}(\boldsymbol{W}) = - \sum_{i=1}^n \left(t_i\log{a_i^L}+(1-t_i)\log{(1-a_i^L)}\right), -$$ - -where we have defined the targets \( t_i \). The derivatives of the cost function with respect to the output \( a_i^L \) are then easily calculated and we get
-$$ -\frac{\partial \mathcal{C}(\boldsymbol{W})}{\partial a_i^L} = \frac{a_i^L-t_i}{a_i^L(1-a_i^L)}. -$$ - -In case we use another activation function than the logistic one, we need to evaluate other derivatives.
- - -In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation \( z_i^l \), that is we need
-$$ -\frac{\partial f(z_i^l)}{\partial w_{jk}^l} = -\frac{\partial f(z_i^l)}{\partial z_j^l} \frac{\partial z_j^l}{\partial w_{jk}^l}= \frac{\partial f(z_i^l)}{\partial z_j^l}a_k^{l-1}. -$$ - -For the Softmax function we have
-$$ -f(z_i^l) = \frac{\exp{(z_i^l)}}{\sum_{m=1}^K\exp{(z_m^l)}}. -$$ - -Its derivative with respect to \( z_j^l \) gives
-$$ -\frac{\partial f(z_i^l)}{\partial z_j^l}= f(z_i^l)\left(\delta_{ij}-f(z_j^l)\right), -$$ - -which in case of the simply binary model reduces to having \( i=j \).
- - -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 \( 28\times 28 \) 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 design/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 design/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 -design/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. In case of the simple perceptron model we have
- -$$ z = \sum_{i=1}^n w_i a_i ,$$
- -$$ y = f(z) ,$$
- -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. See the discussion below concerning other activation functions.
- - -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:
- -$$ 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: -
- -$$ z_j = \sum_{i=1}^n w_ {ij} a_i+b_j.$$
- -Since each neuron in the output layer is connected to the 50 inputs from the hidden layer we have 50x10 = 500 -weights to the output layer. -
- - -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 \( j \), \( b_j \): -
- -$$ z_j = \sum_{i=1}^n w_ {ij} a_i + b_j.$$
- -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 \( l \): -
- -$$ z_{j}^{l} = \sum_{i=1}^{F} w_{ij}^{l} x_i + b_{j}^{l},$$
- -this is then passed through our activation function
- -$$ a_{j}^{l} = f(z_{j}^{l}) .$$
- -We calculate a weighted sum of inputs (activations in the hidden layer) to each neuron \( j \) in the output layer:
- -$$ z_{j}^{L} = \sum_{i=1}^{H} w_{ij}^{L} a_{i}^{l} + b_{j}^{L}.$$
- -Finally we calculate the output of neuron \( j \) in the output layer using the softmax function:
- -$$ a_{j}^{L} = \frac{\exp{(z_j^{L})}} -{\sum_{c=0}^{C-1} \exp{(z_c^{L})}} .$$ -
- - -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 \( Z^{h} \): -
- -$$ \boldsymbol{z}^{l} = \boldsymbol{X} \boldsymbol{W}^{l} + \boldsymbol{b}^{l} ,$$
- -meaning the same bias (1D array with size equal number of hidden neurons) is added to each input image. -This is then passed through the activation: -
- -$$ \boldsymbol{a}^{l} = f(\boldsymbol{z}^l) .$$
- -This is fed to the output layer:
- -$$ \boldsymbol{z}^{L} = \boldsymbol{a}^{L} \boldsymbol{W}^{L} + \boldsymbol{b}^{L} .$$
- -Finally we receive our output values for each image and each category by passing it through the softmax function:
- -$$ output = softmax (\boldsymbol{z}^{L}) = (n_{inputs}, n_{categories}) .$$
- - - -# setup the feed-forward pass, subscript h = hidden layer
-
-def sigmoid(x):
- return 1/(1 + np.exp(-x))
-
-def feed_forward(X):
- # weighted sum of inputs to the hidden layer
- z_h = np.matmul(X, hidden_weights) + hidden_bias
- # activation in the hidden layer
- a_h = sigmoid(z_h)
-
- # weighted sum of inputs to the output layer
- z_o = np.matmul(a_h, output_weights) + output_bias
- # softmax output
- # axis 0 holds each input and axis 1 the probabilities of each category
- exp_term = np.exp(z_o)
- probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
-
- return probabilities
-
-probabilities = feed_forward(X_train)
-print("probabilities = (n_inputs, n_categories) = " + str(probabilities.shape))
-print("probability that image 0 is in category 0,1,2,...,9 = \n" + str(probabilities[0]))
-print("probabilities sum up to: " + str(probabilities[0].sum()))
-print()
-
-# we obtain a prediction by taking the class with the highest likelihood
-def predict(X):
- probabilities = feed_forward(X)
- return np.argmax(probabilities, axis=1)
-
-predictions = predict(X_train)
-print("predictions = (n_inputs) = " + str(predictions.shape))
-print("prediction for image 0: " + str(predictions[0]))
-print("correct label for image 0: " + str(Y_train[0]))
-
-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. -
- -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. -
- - - -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:
-The various optmization methods, with codes and algorithms, are discussed in our lectures on Gradient descent approaches.
- - -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:
- -$$ \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \mathcal{L}_i(\theta) \quad \rightarrow \quad -\frac{1}{N} \sum_{i=1}^N \mathcal{L}_i(\theta) + \lambda \lvert \lvert \boldsymbol{w} \rvert \rvert_2^2 -= \frac{1}{N} \sum_{i=1}^N \mathcal{L}(\theta) + \lambda \sum_{ij} w_{ij}^2,$$ -
- -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, with \( \boldsymbol{t} \) being our targets, -
- -$$ \delta_L = \boldsymbol{t} - \boldsymbol{y} = (n_{inputs}, n_{categories}) .$$
- -The gradient for the output weights is calculated as
- -$$ \nabla W_{L} = \boldsymbol{a}^T \delta_L = (n_{hidden}, n_{categories}) ,$$
- -where \( \boldsymbol{a} = (n_{inputs}, n_{hidden}) \). 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
- -$$ \nabla \boldsymbol{b}_{L} = \sum_{i=1}^{n_{inputs}} \delta_L = (n_{categories}) .$$
- -The error in the hidden layer is
- -$$ \Delta_h = \delta_L W_{L}^T \circ f'(z_{h}) = \delta_L W_{L}^T \circ a_{h} \circ (1 - a_{h}) = (n_{inputs}, n_{hidden}) ,$$
- -where \( f'(a_{h}) \) is the derivative of the activation in the hidden layer. The matrix products mean -that we are summing up the products for each neuron in the output layer. The symbol \( \circ \) denotes -the Hadamard product, meaning element-wise multiplication. -
- -This again gives us the gradients in the hidden layer:
- -$$ \nabla W_{h} = X^T \delta_h = (n_{features}, n_{hidden}) ,$$
- -$$ \nabla b_{h} = \sum_{i=1}^{n_{inputs}} \delta_h = (n_{hidden}) .$$
- - - -# to categorical turns our integer vector into a onehot representation
-from sklearn.metrics import accuracy_score
-
-# one-hot in numpy
-def to_categorical_numpy(integer_vector):
- n_inputs = len(integer_vector)
- n_categories = np.max(integer_vector) + 1
- onehot_vector = np.zeros((n_inputs, n_categories))
- onehot_vector[range(n_inputs), integer_vector] = 1
-
- return onehot_vector
-
-#Y_train_onehot, Y_test_onehot = to_categorical(Y_train), to_categorical(Y_test)
-Y_train_onehot, Y_test_onehot = to_categorical_numpy(Y_train), to_categorical_numpy(Y_test)
-
-def feed_forward_train(X):
- # weighted sum of inputs to the hidden layer
- z_h = np.matmul(X, hidden_weights) + hidden_bias
- # activation in the hidden layer
- a_h = sigmoid(z_h)
-
- # weighted sum of inputs to the output layer
- z_o = np.matmul(a_h, output_weights) + output_bias
- # softmax output
- # axis 0 holds each input and axis 1 the probabilities of each category
- exp_term = np.exp(z_o)
- probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
-
- # for backpropagation need activations in hidden and output layers
- return a_h, probabilities
-
-def backpropagation(X, Y):
- a_h, probabilities = feed_forward_train(X)
-
- # error in the output layer
- error_output = probabilities - Y
- # error in the hidden layer
- error_hidden = np.matmul(error_output, output_weights.T) * a_h * (1 - a_h)
-
- # gradients for the output layer
- output_weights_gradient = np.matmul(a_h.T, error_output)
- output_bias_gradient = np.sum(error_output, axis=0)
-
- # gradient for the hidden layer
- hidden_weights_gradient = np.matmul(X.T, error_hidden)
- hidden_bias_gradient = np.sum(error_hidden, axis=0)
-
- return output_weights_gradient, output_bias_gradient, hidden_weights_gradient, hidden_bias_gradient
-
-print("Old accuracy on training data: " + str(accuracy_score(predict(X_train), Y_train)))
-
-eta = 0.01
-lmbd = 0.01
-for i in range(1000):
- # calculate gradients
- dWo, dBo, dWh, dBh = backpropagation(X_train, Y_train_onehot)
-
- # regularization term gradients
- dWo += lmbd * output_weights
- dWh += lmbd * hidden_weights
-
- # update weights and biases
- output_weights -= eta * dWo
- output_bias -= eta * dBo
- hidden_weights -= eta * dWh
- hidden_bias -= eta * dBh
-
-print("New accuracy on training data: " + str(accuracy_score(predict(X_train), Y_train)))
-
-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 \( 1 \). -
- -$$ \text{Accuracy} = \frac{\sum_{i=1}^n I(\tilde{y}_i = y_i)}{n} ,$$
- -where \( I \) is the indicator function, \( 1 \) if \( \tilde{y}_i = y_i \) 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()
-
-# visual representation of grid search
-# uses seaborn heatmap, 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 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()
-
-import tensorflow as tf
-from tensorflow.keras.layers import Input
-from tensorflow.keras.models import Sequential #This allows appending layers to existing models
-from tensorflow.keras.layers import Dense #This allows defining the characteristics of a particular layer
-from tensorflow.keras import optimizers #This allows using whichever optimiser we want (sgd,adam,RMSprop)
-from tensorflow.keras import regularizers #This allows using whichever regularizer we want (l1,l2,l1_l2)
-from tensorflow.keras.utils import to_categorical #This allows using categorical cross entropy as the cost function
-import numpy as np
-import matplotlib.pyplot as plt
-import seaborn as sns
-from sklearn.model_selection import train_test_split as splitter
-from sklearn.datasets import load_breast_cancer
-import pickle
-import os
+ import torch
+import torch.nn as nn
+import torch.optim as optim
+import torchvision
+import torchvision.transforms as transforms
+
+# Device configuration: use GPU if available
+device = torch.device("cuda" if torch.cuda.is_available() else "cpu")
+
+# MNIST dataset (downloads if not already present)
+transform = transforms.Compose([
+ transforms.ToTensor(),
+ transforms.Normalize((0.5,), (0.5,)) # normalize to mean=0.5, std=0.5 (approx. [-1,1] pixel range)
+])
+train_dataset = torchvision.datasets.MNIST(root='./data', train=True, download=True, transform=transform)
+test_dataset = torchvision.datasets.MNIST(root='./data', train=False, download=True, transform=transform)
+
+train_loader = torch.utils.data.DataLoader(train_dataset, batch_size=64, shuffle=True)
+test_loader = torch.utils.data.DataLoader(test_dataset, batch_size=64, shuffle=False)
-"""Load breast cancer dataset"""
+class NeuralNet(nn.Module):
+ def __init__(self):
+ super(NeuralNet, self).__init__()
+ self.fc1 = nn.Linear(28*28, 100) # first hidden layer (784 -> 100)
+ self.fc2 = nn.Linear(100, 100) # second hidden layer (100 -> 100)
+ self.fc3 = nn.Linear(100, 10) # output layer (100 -> 10 classes)
+ def forward(self, x):
+ x = x.view(x.size(0), -1) # flatten images into vectors of size 784
+ x = torch.relu(self.fc1(x)) # hidden layer 1 + ReLU activation
+ x = torch.relu(self.fc2(x)) # hidden layer 2 + ReLU activation
+ x = self.fc3(x) # output layer (logits for 10 classes)
+ return x
-np.random.seed(0) #create same seed for random number every time
-
-cancer=load_breast_cancer() #Download breast cancer dataset
-
-inputs=cancer.data #Feature matrix of 569 rows (samples) and 30 columns (parameters)
-outputs=cancer.target #Label array of 569 rows (0 for benign and 1 for malignant)
-labels=cancer.feature_names[0:30]
-
-print('The content of the breast cancer dataset is:') #Print information about the datasets
-print(labels)
-print('-------------------------')
-print("inputs = " + str(inputs.shape))
-print("outputs = " + str(outputs.shape))
-print("labels = "+ str(labels.shape))
-
-x=inputs #Reassign the Feature and Label matrices to other variables
-y=outputs
-
-#%%
-
-# Visualisation of dataset (for correlation analysis)
-
-plt.figure()
-plt.scatter(x[:,0],x[:,2],s=40,c=y,cmap=plt.cm.Spectral)
-plt.xlabel('Mean radius',fontweight='bold')
-plt.ylabel('Mean perimeter',fontweight='bold')
-plt.show()
-
-plt.figure()
-plt.scatter(x[:,5],x[:,6],s=40,c=y, cmap=plt.cm.Spectral)
-plt.xlabel('Mean compactness',fontweight='bold')
-plt.ylabel('Mean concavity',fontweight='bold')
-plt.show()
+model = NeuralNet().to(device)
-plt.figure()
-plt.scatter(x[:,0],x[:,1],s=40,c=y,cmap=plt.cm.Spectral)
-plt.xlabel('Mean radius',fontweight='bold')
-plt.ylabel('Mean texture',fontweight='bold')
-plt.show()
+criterion = nn.CrossEntropyLoss()
+optimizer = optim.SGD(model.parameters(), lr=0.01, weight_decay=1e-4)
-plt.figure()
-plt.scatter(x[:,2],x[:,1],s=40,c=y,cmap=plt.cm.Spectral)
-plt.xlabel('Mean perimeter',fontweight='bold')
-plt.ylabel('Mean compactness',fontweight='bold')
-plt.show()
+num_epochs = 10
+for epoch in range(num_epochs):
+ model.train() # set model to training mode
+ running_loss = 0.0
+ for images, labels in train_loader:
+ # Move data to device (GPU if available, else CPU)
+ images, labels = images.to(device), labels.to(device)
+
+ optimizer.zero_grad() # reset gradients to zero
+ outputs = model(images) # forward pass: compute predictions
+ loss = criterion(outputs, labels) # compute cross-entropy loss
+ loss.backward() # backpropagate to compute gradients
+ optimizer.step() # update weights using SGD step
+
+ running_loss += loss.item()
+ # Compute average loss over all batches in this epoch
+ avg_loss = running_loss / len(train_loader)
+ print(f"Epoch {epoch+1}/{num_epochs}, Loss: {avg_loss:.4f}")
+
+#Evaluation on the Test Set
-# Generate training and testing datasets
-#Select features relevant to classification (texture,perimeter,compactness and symmetery)
-#and add to input matrix
+model.eval() # set model to evaluation mode
+correct = 0
+total = 0
+with torch.no_grad(): # disable gradient calculation for evaluation
+ for images, labels in test_loader:
+ images, labels = images.to(device), labels.to(device)
+ outputs = model(images)
+ _, predicted = torch.max(outputs, dim=1) # class with highest score
+ total += labels.size(0)
+ correct += (predicted == labels).sum().item()
-temp1=np.reshape(x[:,1],(len(x[:,1]),1))
-temp2=np.reshape(x[:,2],(len(x[:,2]),1))
-X=np.hstack((temp1,temp2))
-temp=np.reshape(x[:,5],(len(x[:,5]),1))
-X=np.hstack((X,temp))
-temp=np.reshape(x[:,8],(len(x[:,8]),1))
-X=np.hstack((X,temp))
-
-X_train,X_test,y_train,y_test=splitter(X,y,test_size=0.1) #Split datasets into training and testing
-
-y_train=to_categorical(y_train) #Convert labels to categorical when using categorical cross entropy
-y_test=to_categorical(y_test)
-
-del temp1,temp2,temp
-
-# %%
-
-# Define tunable parameters"
-
-eta=np.logspace(-3,-1,3) #Define vector of learning rates (parameter to SGD optimiser)
-lamda=0.01 #Define hyperparameter
-n_layers=2 #Define number of hidden layers in the model
-n_neuron=np.logspace(0,3,4,dtype=int) #Define number of neurons per layer
-epochs=100 #Number of reiterations over the input data
-batch_size=100 #Number of samples per gradient update
-
-# %%
-
-"""Define function to return Deep Neural Network model"""
-
-def NN_model(inputsize,n_layers,n_neuron,eta,lamda):
- model=Sequential()
- for i in range(n_layers): #Run loop to add hidden layers to the model
- if (i==0): #First layer requires input dimensions
- model.add(Dense(n_neuron,activation='relu',kernel_regularizer=regularizers.l2(lamda),input_dim=inputsize))
- else: #Subsequent layers are capable of automatic shape inferencing
- model.add(Dense(n_neuron,activation='relu',kernel_regularizer=regularizers.l2(lamda)))
- model.add(Dense(2,activation='softmax')) #2 outputs - ordered and disordered (softmax for prob)
- sgd=optimizers.SGD(learning_rate=eta)
- model.compile(loss='categorical_crossentropy',optimizer=sgd,metrics=['accuracy'])
- return model
-
-
-Train_accuracy=np.zeros((len(n_neuron),len(eta))) #Define matrices to store accuracy scores as a function
-Test_accuracy=np.zeros((len(n_neuron),len(eta))) #of learning rate and number of hidden neurons for
-
-for i in range(len(n_neuron)): #run loops over hidden neurons and learning rates to calculate
- for j in range(len(eta)): #accuracy scores
- DNN_model=NN_model(X_train.shape[1],n_layers,n_neuron[i],eta[j],lamda)
- DNN_model.fit(X_train,y_train,epochs=epochs,batch_size=batch_size,verbose=1)
- Train_accuracy[i,j]=DNN_model.evaluate(X_train,y_train)[1]
- Test_accuracy[i,j]=DNN_model.evaluate(X_test,y_test)[1]
-
-
-def plot_data(x,y,data,title=None):
-
- # plot results
- fontsize=16
-
-
- fig = plt.figure()
- ax = fig.add_subplot(111)
- cax = ax.matshow(data, interpolation='nearest', vmin=0, vmax=1)
-
- cbar=fig.colorbar(cax)
- cbar.ax.set_ylabel('accuracy (%)',rotation=90,fontsize=fontsize)
- cbar.set_ticks([0,.2,.4,0.6,0.8,1.0])
- cbar.set_ticklabels(['0%','20%','40%','60%','80%','100%'])
-
- # put text on matrix elements
- for i, x_val in enumerate(np.arange(len(x))):
- for j, y_val in enumerate(np.arange(len(y))):
- c = "${0:.1f}\\%$".format( 100*data[j,i])
- ax.text(x_val, y_val, c, va='center', ha='center')
-
- # convert axis vaues to to string labels
- x=[str(i) for i in x]
- y=[str(i) for i in y]
-
-
- ax.set_xticklabels(['']+x)
- ax.set_yticklabels(['']+y)
-
- ax.set_xlabel('$\\mathrm{learning\\ rate}$',fontsize=fontsize)
- ax.set_ylabel('$\\mathrm{hidden\\ neurons}$',fontsize=fontsize)
- if title is not None:
- ax.set_title(title)
-
- plt.tight_layout()
-
- plt.show()
-
-plot_data(eta,n_neuron,Train_accuracy, 'training')
-plot_data(eta,n_neuron,Test_accuracy, 'testing')
+accuracy = 100 * correct / total
+print(f"Test Accuracy: {accuracy:.2f}%")
import tensorflow as tf
+from tensorflow import keras
+from tensorflow.keras import layers, regularizers
+
+# Check for GPU (TensorFlow will use it automatically if available)
+gpus = tf.config.list_physical_devices('GPU')
+print(f"GPUs available: {gpus}")
+
+# 1) Load and preprocess MNIST
+(x_train, y_train), (x_test, y_test) = keras.datasets.mnist.load_data()
+# Normalize to [0, 1]
+x_train = (x_train.astype("float32") / 255.0)
+x_test = (x_test.astype("float32") / 255.0)
+
+# 2) Build the model: 784 -> 100 -> 100 -> 10
+l2_reg = 1e-4 # L2 regularization strength
+
+model = keras.Sequential([
+ layers.Input(shape=(28, 28)),
+ layers.Flatten(),
+ layers.Dense(100, activation="relu",
+ kernel_regularizer=regularizers.l2(l2_reg)),
+ layers.Dense(100, activation="relu",
+ kernel_regularizer=regularizers.l2(l2_reg)),
+ layers.Dense(10, activation="softmax") # output probabilities for 10 classes
+])
+
+# 3) Compile with SGD + weight decay via L2 regularizers
+model.compile(
+ optimizer=keras.optimizers.SGD(learning_rate=0.01),
+ loss="sparse_categorical_crossentropy",
+ metrics=["accuracy"],
+)
+
+model.summary()
+
+# 4) Train
+history = model.fit(
+ x_train, y_train,
+ epochs=10,
+ batch_size=64,
+ validation_split=0.1, # optional: monitor validation during training
+ verbose=1
+)
+
+# 5) Evaluate on test set
+test_loss, test_acc = model.evaluate(x_test, y_test, verbose=0)
+print(f"Test accuracy: {test_acc:.4f}, Test loss: {test_loss:.4f}")
+
+Here we present a flexible object oriented codebase for a feed forward neural network, along with a demonstration of how @@ -6584,7 +5213,7 @@ $$ -->