CalvinTChi/Neural Networks.ipynb

## Neural Networks.ipynb
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   "source": [
    "# Neural Networks\n",
    "### Author: _Calvin Chi_"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introduction\n",
    "Neural network is a powerful supervised machine learning algorithm suited for both regression and classification problems. A unique feature about this algorithm is that it allows for the implicit learning of features without having the user explicitly design them. The neural network is a constructed by several neurons with the following architecture: \n",
    "\n",
    "<img src=\"http://i.imgur.com/yZfR8RH.png\", width=300, height=300>\n",
    "\n",
    "In this representation, $x_{1},...,x_{3}$ are the input values for features $1$ to $3$, and $+1$ is the bias term to increase the model space. The neuron then accepts these input to output $h_{W, b}(x) = f(W^{T}x) = f(\\sum_{i=1}^{3}W_{i}x_{i} + W_{4}b)$, where $W \\in \\mathbb{R}^{4 x 1}$ is a vector of weights for each of the input values and $b$ represents the bias of $+1$. $f()$ is an activation function, typically chosen to be the signmoid function or tanh function. \n",
    "\n",
    "A neural network is built from these individual neurons. An example of a neural network with $3$ layers looks like this:\n",
    "\n",
    "<img src=\"http://i.imgur.com/W8jxjyy.png\", width=400, height=150>\n",
    "\n",
    "With this architure, our weight vector becomes a matrix instead. To go from layer $L_{1}$ to layer $L_{2}$, we would need a $W^{(1)} \\in \\mathbb{R}^{4 x 3}$ matrix, where $4$ is the number of input values $n_{in}$ and $3$ is the number of output values $n_{out}$. The neurons in the second layer are denoted as $a^{(2)}_{j}$, where $j$ denotes the $j\\text{th}$ neuron. According to above diagram, $a^{(2)}_{j}$ is calculated as $f((\\sum_{i=1}^{3} W^{(1)}_{ij}x_{i}) + W^{(1)}_{4j} b^{(1)}_{j})$. To go from layer $2$ to layer $3$, one would use a $W^{(2)} \\in \\mathbb{R}^{4 x 1}$ vector, and the value of $h_{W, b}(x)$ is computed as $f((\\sum_{i=1}^{3} W^{(2)}_{i1}a^{(2)}_{i}) + W^{(2)}_{41} b^{(2)}_{1})$. The logic of these computational steps can be extrapolated to neural networks in general with a number of layers and a given number of neurons per layer. Note that a neural network with more than $3$ layers is often called a deep neural network. "
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    "# Backpropagation\n",
    "Training a neural network involves iterately finding the values of the weight matrices via gradient descent to minimize error, which could be set as the mean square error or cross entropy error. Either way, the cost function is non-convex such that the result from training may represent a local solution instead of a global solution. One way to combat this is to randomly intialize the weight values prior to training. \n",
    "\n",
    "In this tutorial's implementation, we are going to implement neural network with stochastic gradient descent. In stochastic gradient descent, each sample will first be used to compute all $a$ using the current weight values in a procedure called feedforward propagation. Then in backpropagation, these computed $a$ are used to compute the gradients for each individual weight value, and used to update the weight value. \n",
    "\n",
    "Let us illustrate the procedure for deriving the gradients for each layer in matrix notation. Assume that we have $3$ layers with $2$ weight matrices. The middle layer will use the tanh activation function whereas the third layer will use the sigmoid function. We shall also derive the gradients separately for mean-square error and cross entropy error respectively.\n",
    "\n",
    "In our notation, we denote $n_{out}$ to refer to the last layer and $\\alpha$ as our learning rate. The sigmoid function is defiend as:\n",
    "\n",
    "$$g(z) = \\frac{1}{1+e^{-z}}$$\n",
    "\n",
    "Where as the tanh activation function is defined as:\n",
    "\n",
    "$$f(z) = tanh(z)$$\n",
    "\n",
    "In our derivation, we will make use of two mathematical properties:\n",
    "\n",
    "1. $\\frac{dg(z)}{dz} = \\frac{e^{-z}}{(1 + e^{-z})^{2}} = \\frac{1 + e^{-z}}{(1 + e^{-z})^{2}} - \\frac{1}{(1 + e^{-z})^{2}} = \\frac{1}{1 + e^{-z}} - \\frac{1}{(1 + e^{-z})^{2}} = g(z)(1-g(z))\\\\$\n",
    "\n",
    "2. $\\frac{df(z)}{dz} = 1 - tanh^{2}(z)$\n",
    "\n",
    "## Mean Square Error\n",
    "According to our neural network design, mean square error is defined as: \n",
    "\n",
    "$$J = -\\sum_{j=1}^{n_{out}} (y_{j} - g_{j}(a^{(3)}))^{2}$$\n",
    "\n",
    "Where $y_{j}$ in our case represents the $j$th class we are trying to predict. Then our gradient for $W^{(2)}$ would be:\n",
    "\n",
    "$$\\frac{\\partial J}{\\partial W^{(2)}} = \\frac{\\partial J}{\\partial g(a^{(3)})} \\: \\frac{\\partial g(a^{(3)})}{\\partial a^{(3)}} \\: \\frac{\\partial a^{(3)}}{\\partial W^{(2)}} = -a^{(2)}(y - g(a^{(3)}))^{T} diag[g(a^{(3)})(1 - g(a^{(3)}))]$$\n",
    "\n",
    "The gradient for $W^{(1)}$ would be:\n",
    "\n",
    "$$\\frac{\\partial J}{\\partial W^{(1)}} = \\frac{\\partial J}{\\partial g(a^{(3)})} \\: \\frac{\\partial g(a^{(3)})}{\\partial a^{(3)}} \\: \\frac{\\partial a^{(3)}}{\\partial f(a^{(2)})} \\: \\frac{\\partial f(a^{(2)})}{\\partial a^{(2)}} \\: \\frac{\\partial a^{(2)}}{\\partial W^{(1)}} = -x[ diag(1-tanh^{2}(a^{(2)}))(W^{(2)})^{T}diag[g(a^{(3)})(1 - g(a^{(3)}))](y - g(a^{(3)})) ]^{T}$$ \n",
    "\n",
    "Taking care of dimensions in matrix multiplication. The update equations then would be:\n",
    "\n",
    "$$W^{(2)} = W^{(2)} - \\alpha \\frac{\\partial J}{\\partial W^{(2)}}\\\\$$\n",
    "\n",
    "$$W^{(1)} = W^{(1)} - \\alpha \\frac{\\partial J}{\\partial W^{(1)}}$$\n",
    "\n",
    "## Cross Entropy Error\n",
    "According to our neural network design, cross entropy error is defined as: \n",
    "\n",
    "$$J = -\\sum_{j=1}^{n_{out}} [y_{j} ln(g_{j}(a^{(3)}) + (1-y_{j})ln(1-g_{j}(a^{(3)}))]$$\n",
    "\n",
    "The gradient for $W^{(2)}$ would be:\n",
    "\n",
    "$$ \\frac{\\partial J}{\\partial W^{(2)}} = \\frac{\\partial J}{\\partial g(a^{(3)})} \\: \\frac{\\partial g(a^{(3)})}{\\partial a^{(3)}} \\: \\frac{\\partial a^{(3)}}{\\partial W^{(2)}} = a^{(2)}(\\frac{1-y}{1-g(a^{(3)})} - \\frac{y}{g(a^{(3)})})^{T} diag[g(a^{(3)})(1 - g(a^{(3)})]$$\n",
    "\n",
    "And the gradient for $W^{(1)}$ would be:\n",
    "\n",
    "$$ \\frac{\\partial J}{\\partial W^{(1)}} = \\frac{\\partial J}{\\partial g(a^{(3)})} \\: \\frac{\\partial g(a^{(3)})}{\\partial a^{(3)}} \\: \\frac{\\partial a^{(3)}}{\\partial f(a^{(2)})} \\: \\frac{\\partial f(a^{(2)})}{\\partial a^{(2)}} \\: \\frac{\\partial a^{(2)}}{\\partial W^{(1)}} = x [ diag(1 - tanh^{2}(a^{(2)})) (W^{(2)})^{T} diag[g(a^{(3)})(1 - g(a^{(3)}))] (\\frac{1-y}{1 + e^{-a^{(3)}}} - y \\frac{e^{-a^{(3)}}}{1 + e^{-a^{(3)}}}) ]^{T}$$\n",
    "\n",
    "## Implementation\n",
    "Our implementation shall have the following features:\n",
    "\n",
    "+ Learning rate of $\\alpha = \\frac{1}{100 + 10k}$, where $k$ = number of epochs.\n",
    "+ Weight initialization: each weight value is drawn randomly from a $N(0, \\sigma=0.01)$ distribution. \n",
    "+ The neural network could optionally accept pre-trained weights as the starting point for training. \n",
    "\n",
    "The dataset we will be applying our implementation on will be the MNIST digits dataset. Thus, there will be 10 classes for each digit from $0$ to $9$. To train our classifier in recognizing these digits, we will have an input layer of $784$ neurons, a hidden layer of $200$ neurons, and an output layer of $10$ neurons. The number of neurons for the hidden layer can be user-defined, but the number of neurons for the input and output layers are determined by the problem we are working on. In the MNIST dataset, each sample is a $24 x 24$ pixel image, so that translates to $784$ feature values. There are a total of $10$ classes, leading to an output layer with $10$ neurons. The neural network is trained to output a probability for each class, and in our implementation we shall pick the class with the highest probability as our prediction. The usage of our `NeuralNetwork` class will be as follows:\n",
    "\n",
    "```\n",
    "classifier = NeuralNetwork(layers, loss=\"mean_square\", monitor=False, plot=False, weights=[], iterations=[], errors=[], accuracies=[])\n",
    "```\n",
    "\n",
    "+ layers: a list of integers, each representing number of neurons for the layer it corresponds to by index. The integer at index $0$ indicates number of neurons for the input layer and the integer at index $len(layers)-1$ indicates the number of neurons for the output layer. \n",
    "+ loss: string of either \"mean_square\" for mean square loss or \"cross_entropy\" for cross entropy loss.\n",
    "+ monitor: boolean indicating whether to print to screen the iteration number, current loss based on entire training dataset, accuracy in predicting on the training set, and accuracy in predicting on the validation set if appropriate.\n",
    "+ plot: boolean indicating whether to plot a error vs iterations and accuracy vs iterations on the training set at the conclusion of the training process. \n",
    "+ weights, iterations, errors, accuracies: list of weights, list containing iteration counts, list of errors, and list of accuracies from prior training.\n",
    "\n",
    "The usage for training the neural network is as follows:\n",
    "\n",
    "```\n",
    "classifier.train(x, Y, epochs, validation=None, validation_label=None)\n",
    "```\n",
    "\n",
    "+ x: training data in a numpy array \n",
    "+ Y: training labels in a numpy array\n",
    "+ epochs: number of complete passes through the training data\n",
    "+ validation: validation data in a numpy array\n",
    "+ validation_label: validation label in a numpy array\n",
    "\n",
    "Now we implement our `NeuralNetwork` class:"
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    "import numpy as np\n",
    "import sys\n",
    "import pickle\n",
    "import matplotlib.pyplot as plt\n",
    "import collections \n",
    " \n",
    "class NeuralNetwork: \n",
    "    \n",
    "    def __init__(self, layers, loss=\"mean_square\", monitor=False, \n",
    "        plot=False, weights=[], iterations=[], errors=[], accuracies=[]): \n",
    "\n",
    "        def tanh(x): \n",
    "            return np.tanh(x)\n",
    "\n",
    "        def tanh_deriv(x): \n",
    "            return 1.0 - (x)**2\n",
    "\n",
    "        def logistic(x):\n",
    "            return 1 / (1 + np.exp(-x))\n",
    "            \n",
    "        def logistic_deriv(x): \n",
    "            return x * (1 - x)\n",
    "            \n",
    "        def mean_square(x, y): \n",
    "            total = 0\n",
    "            for predict, label in zip(x, y):\n",
    "                total += 0.5*np.sum((label - predict)**2)\n",
    "            return total\n",
    "            \n",
    "        def mean_square_deriv(x, y):\n",
    "            return -(y - x)\n",
    "            \n",
    "        def cross_entropy(x, y):\n",
    "            total = 0\n",
    "            for predict, label in zip(x, y):\n",
    "                total += -(np.sum(label*np.log(predict) + (1-label)*np.log(1-predict)))\n",
    "            return total\n",
    "\n",
    "        def cross_entropy_deriv(x, y):\n",
    "            return ((1-y)/(1-x)-(y/x))\n",
    "\n",
    "        self.output_activation = logistic\n",
    "        self.output_activation_deriv = logistic_deriv\n",
    "        self.hidden_activation = tanh\n",
    "        self.hidden_activation_deriv = tanh_deriv\n",
    "        self.layers = layers\n",
    "        self.weights = weights\n",
    "        self.iterations = iterations\n",
    "        self.errors = errors\n",
    "        self.accuracies = accuracies\n",
    "        self.monitor = monitor\n",
    "        self.plot = plot\n",
    "        self.lossName = loss\n",
    "        if len(self.weights) == 0:\n",
    "            for i in range(1, len(layers)-1):\n",
    "                self.weights.append(np.random.normal(0, 0.01, (layers[i-1]+1, layers[i]+1)))\n",
    "            self.weights.append(np.random.normal(0, 0.01, (layers[i]+1, layers[i+1])))\n",
    "        if loss == \"mean_square\": \n",
    "            self.loss = mean_square\n",
    "            self.loss_deriv = mean_square_deriv\n",
    "        elif loss == \"cross_entropy\": \n",
    "            self.loss = cross_entropy\n",
    "            self.loss_deriv = cross_entropy_deriv\n",
    "\n",
    "    def train(self, x, Y, epochs, validation=None, validation_label=None):\n",
    "        def accuracy(x, y):\n",
    "            correct = 0\n",
    "            for a, b in zip(x, y):\n",
    "                if np.argmax(a) == np.argmax(b): \n",
    "                    correct += 1\n",
    "            return correct/len(x)\n",
    "        if len(self.iterations) != 0 and len(self.errors) != 0 and len(self.accuracies) != 0: \n",
    "            count = self.iterations[-1] + 1\n",
    "        else:\n",
    "            count = 0\n",
    "        bias = np.ones((x.shape[0], 1))\n",
    "        x = np.hstack((x, bias))\n",
    "        for k in range(epochs):\n",
    "            for i in range(len(x)):\n",
    "                a = [x[[i], :]]\n",
    "                y = Y[i]\n",
    "                gradients = collections.deque()\n",
    "                # Feedforward propagation step\n",
    "                for j in range(len(self.weights)):\n",
    "                    if j == len(self.weights) - 1: \n",
    "                        a.append(self.output_activation((np.dot(a[j], self.weights[j]))))\n",
    "                    else: \n",
    "                        a.append(self.hidden_activation((np.dot(a[j], self.weights[j]))))\n",
    "                # Backpropagation step\n",
    "                deltas = collections.deque()\n",
    "                deltas.append(np.dot((self.loss_deriv(a[-1], y)), \n",
    "                    np.diag(self.output_activation_deriv(a[-1])[0])))\n",
    "                for j in range(len(self.weights) - 1, 0, -1):\n",
    "                    delta = np.dot(np.dot(np.diag(self.hidden_activation_deriv(a[j])[0]), \n",
    "                        self.weights[j]), deltas[-1].T).T\n",
    "                    deltas.appendleft(delta)\n",
    "                # Final gradient calculation and weight update\n",
    "                for j in range(len(deltas)-1, -1, -1):\n",
    "                    gradient = np.dot(a[j].T, deltas[j])\n",
    "                    gradients.appendleft(gradient)\n",
    "                for j in range(len(self.weights)): \n",
    "                    self.weights[j] -= 1/float(100+10*k) * gradients[j]\n",
    "                if self.monitor and count % 20000 == 0 or count in [1000, 5000, 10000, 15000]: \n",
    "                    self.iterations.append(count)\n",
    "                    predictions1 = self.predict(x)\n",
    "                    correct1 = accuracy(predictions1, Y)\n",
    "                    self.accuracies.append(correct1)\n",
    "                    error = self.loss(predictions1, Y)\n",
    "                    self.errors.append(error)\n",
    "                    if validation is not None and validation_label is not None:\n",
    "                        predictions2 = self.predict(validation)\n",
    "                        correct2 = accuracy(predictions2, validation_label)\n",
    "                        print(\"# \" + str(count), \"error: \", error, \"|\", \"training: \", \n",
    "                            correct1, \"|\", \"validation: \", correct2)\n",
    "                    else:\n",
    "                        print(\"# \" + str(count), \"error: \", error, \"|\", \"training: \",\n",
    "                             correct1)\n",
    "                if count % 54000 == 0: \n",
    "                    pickle.dump(self.weights, open(\"result/weights.p\", 'wb'))\n",
    "                    pickle.dump(self.iterations, open(\"result/iterations.p\", 'wb'))\n",
    "                    pickle.dump(self.accuracies, open(\"result/accuracies.p\", 'wb'))\n",
    "                    pickle.dump(self.errors, open(\"result/errors.p\", 'wb'))\n",
    "                count += 1\n",
    "        pickle.dump(self.weights, open(\"result/weights.p\", 'wb'))\n",
    "        pickle.dump(self.iterations, open(\"result/iterations.p\", 'wb'))\n",
    "        pickle.dump(self.accuracies, open(\"result/accuracies.p\", 'wb'))\n",
    "        pickle.dump(self.errors, open(\"result/errors.p\", 'wb'))\n",
    "        if self.plot:\n",
    "            plt.plot(self.iterations, self.errors)\n",
    "            plt.xlabel(\"Iterations\")\n",
    "            plt.ylabel(\"Error\")\n",
    "            plt.title(\"Training Error vs Iterations (\" + self.lossName + \")\")\n",
    "            plt.show()\n",
    "            plt.plot(self.iterations, self.accuracies)\n",
    "            plt.xlabel(\"Iterations\")\n",
    "            plt.ylabel(\"Accuracy\")\n",
    "            plt.title(\"Training Accuracy vs Iterations (\" + self.lossName + \")\")\n",
    "            plt.show()\n",
    "\n",
    "    def predict(self, x):\n",
    "        predictions = []\n",
    "        if x.shape[1] != self.layers[0]+1: \n",
    "            bias = np.ones((x.shape[0], 1))\n",
    "            x = np.hstack((x, bias))\n",
    "        for i in range(len(x)): \n",
    "            for l in range(0, len(self.weights)):\n",
    "                if l == len(self.weights)-1: \n",
    "                    a = self.output_activation(np.dot(a, self.weights[l]))\n",
    "                else: \n",
    "                    a = self.hidden_activation(np.dot(x[[i], :], self.weights[l]))\n",
    "            predictions.append(a)\n",
    "        return np.array(predictions)"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "In this tutorial, we are going to train a neural network with mean square error and cross entropy as the loss function respectively, and compare the effectiveness of each loss function at the end. Now let us load our MNIST dataset, visualize it,and pre-process it:"
   ]
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    {
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yxqcWH2oCFTFrUlSl5bUL5BDIp3/M+cjvuR9JEyeYGvRUUwVHHS0hGVLWCAoE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\nHHyzCjn4ZhVy8M0qdBtlANUbvZQbIAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x103402f98>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "from scipy import io\n",
    "from sklearn.preprocessing import scale\n",
    "import matplotlib.pyplot as plt\n",
    "import time\n",
    "\n",
    "datasetDir = \"datasets/MNIST/\"\n",
    "datamat = io.loadmat(datasetDir + 'train.mat')\n",
    "test_images = io.loadmat(datasetDir + 'test.mat')['test_images']\n",
    "train_images = datamat['train_images']\n",
    "labels = datamat['train_labels']\n",
    "train_images = np.transpose(train_images, (2, 0, 1))\n",
    "plt.imshow(train_images[0])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "train_images = np.reshape(train_images, (train_images.shape[0], 784))\n",
    "indices = np.random.permutation(labels.shape[0])\n",
    "train_images = train_images[indices].astype(float)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In data pre-processing it is common to scale the dataset so that each feature(column) are standardized."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "train_images = scale(train_images, axis = 0, copy=False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Conver the label vector to a matrix. The label for each sample is now represented by a $10$-dimensional vector with a $1$ at the index position corresponding to its class and $0$ elsewhere. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "labels = labels[indices]\n",
    "train_labels = [np.zeros((1, 10)) for i in range(labels.shape[0])]\n",
    "for array, label in zip(train_labels, labels): \n",
    "    array[0][label] = 1\n",
    "train_labels = np.array(train_labels)\n",
    "test_images = np.transpose(test_images, (2, 0, 1))\n",
    "test_images = np.reshape(test_images, (test_images.shape[0], 784))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And now we are ready to train our neural network! Let us first split our training dataset so that $9/10$th of it is used for training and $1/10$th of it is used for validation error. Let us train a model using mean square error as our loss function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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IK9P6twNzgLWldVL9dODdEfHx0joR8YCkgcDGiBhdIZao57mame2PJBERVd/k10pdu+lS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\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x12ae29278>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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8bzVJrwSOjYivMcBdiZ1MzMyqqZ1nwJfti6x948imCeUb35jCddel8b6+Pvr6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\nYmZmHXMyMTOzjjmZmJlZx5xMzMysY/8fC3TBvGinwNIAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1083d8d68>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "With 6 epochs, the training time is:  385.72842597961426\n",
      "The validation accuracy is:  0.9631666666666666\n"
     ]
    }
   ],
   "source": [
    "def accuracy(x, y):\n",
    "    correct = 0\n",
    "    for a, b in zip(x, y):\n",
    "        if np.argmax(a) == np.argmax(b): \n",
    "            correct += 1\n",
    "    return correct/len(x)\n",
    "\n",
    "train = train_images[:(9*len(train_images)//10)]\n",
    "train_label = train_labels[:(9*len(train_images)//10)]\n",
    "validation = train_images[(9*len(train_images)//10):len(train_images)]\n",
    "validation_label = train_labels[(9*len(train_images)//10):len(train_images)]\n",
    "NN = NeuralNetwork([784, 200, 10], loss=\"mean_square\", \n",
    "                  plot=True)\n",
    "start = time.time()\n",
    "epochs = 6\n",
    "NN.train(train, train_label, epochs, validation=validation, \n",
    "         validation_label=validation_label)\n",
    "end = time.time()\n",
    "print(\"With \" + str(epochs) + \" epochs, the training time is: \", end-start)\n",
    "predictions = NN.predict(validation)\n",
    "p_correct = accuracy(predictions, validation_label)\n",
    "print(\"The validation accuracy is: \", p_correct) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let us train the model using cross entropy error as our loss function, with the same number of epochs. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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zs+JoxTWjMRHRAxARW4ExKT4O2JhbbnOKjQM25eKbUmyfdSJiD7Bd0uhakvA1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\nzsXIzMxazsXIzMxa7v8DmoiiPo8WAIMAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x12ae0d9e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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+jSObJpRvf3sGV1+d+vv6+ujr6ys1MDOzXtPf309/f/+wL1cRUd7MpanAjIiY\nlodPByIizi6UebjWC2wHvAB8MiKuqptX3HFH8IY3lBaumdmII4mIKP1a2LLPTGYDu0uaRHrQ1nTS\nY4DXiojX1volnQ/8uD6R1Liay8ysmkpNJhGxWtIpwLWk9pmZETFH0slpcpxX/5ZW83MyMTOrplKr\nuYaSpLj33uCAA7odiZlZ7xiuaq6eOtb3mYmZWTX11O7ZycTMrJp6avfsZGJmVk09tXv2jR7NzKqp\np5KJz0zMzKqpp3bPTiZmZtXUU7tnJxMzs2rqqd2zk4mZWTX11O7ZDfBmZtXUU8nEZyZmZtXUU7tn\nJxMzs2rqqd2zk4mZWTX11O7ZbSZmZtXUU8nEZyZmZtXUU7tnJxMzs2rqqd2zk4mZWTWVvnuWNE3S\nXEnzJJ3WYPp7Jd0r6W5Jt0t6c7N5OZmYmVVTqU9alDQKmAccBjxKeib89IiYWyizeUQsz/37A5dG\nxD4N5hVLlwbjx5cWrpnZiDNSnrQ4BZgfEQsiYhUwCziqWKCWSLItgDXNZuYzEzOzaip79zwRWFgY\nXpTHrUfS0ZLmAD8GPtpsZk4mZmbVNKbbAQBExBXAFZIOAf4BeGejcmeeOYMxOeK+vj76+vqGK0Qz\ns57Q399Pf3//sC+37DaTqcCMiJiWh08HIiLObvGe3wIHR8TTdeNjxYpg3LjSwjUzG3FGSpvJbGB3\nSZMkjQWmA1cVC0jardB/EDC2PpHUuJrLzKyaSq3miojVkk4BriUlrpkRMUfSyWlynAccK+lE4CXg\nReADzebnZGJmVk2lVnMNJUmxZk34/lxmZoMwUqq5hpQTiZlZNfVUMjEzs2pyMjEzs445mZiZWcec\nTMzMrGNOJmZm1jEnEzMz65iTiZmZdczJxMzMOuZkYmZmHXMyMTOzjjmZmJlZx5xMzMysY04mZmbW\nMScTMzPrmJOJmZl1rPRkImmapLmS5kk6rcH0D0q6N3c3S9q/7JjMzGxolfqkRUmjgHnAYcCjpGfC\nT4+IuYUyU4E5EbFU0jRgRkRMbTCv6JWnQpqZVcVIedLiFGB+RCyIiFXALOCoYoGIuDUilubBW4GJ\nJcdkZmZDrOxkMhFYWBheROtk8XHgZ6VGZGZmQ25MtwOokfQ24CTgkGZlZsyYsba/r6+Pvr6+0uMy\nM+sl/f399Pf3D/tyy24zmUpqA5mWh08HIiLOrit3AHA5MC0ifttkXm4zMTMbpJHSZjIb2F3SJElj\ngenAVcWWFRnbAAAH9klEQVQCknYhJZIPNUskZmZWbaVWc0XEakmnANeSEtfMiJgj6eQ0Oc4DvgRs\nA3xTkoBVETGlzLjMzGxolVrNNZRczWVmNngjpZrLzMw2Ak4mZmbWMScTMzPrmJOJmZl1zMnEzMw6\n5mRiZmYdczIxM7OOOZmYmVnHnEzMzKxjTiZmZtYxJxMzM+uYk4mZmXXMycTMzDrmZGJmZh1zMjEz\ns46VnkwkTZM0V9I8Sac1mL6XpF9KWiHp82XHY2ZmQ6/UZCJpFHAucASwH3C8pL3rij0FnAr8U5mx\ndFt/f3+3Q+iI4++uXo6/l2OH3o9/uJR9ZjIFmB8RCyJiFTALOKpYICKejIg7gZdLjqWrev0L6fi7\nq5fj7+XYoffjHy5lJ5OJwMLC8KI8zszMRhA3wJuZWccUEeXNXJoKzIiIaXn4dCAi4uwGZb8MPB8R\nX20yr/ICNTMbwSJCZS9jTMnznw3sLmkS8BgwHTi+RfmmKzwcG8PMzF6ZUs9MIF0aDHydVKU2MyLO\nknQy6QzlPEk7AHcAWwJrgGXAvhGxrNTAzMxsyJSeTMzMbOTriQb4gf74OMyx/F7SvZLulnR7Hre1\npGslPSTpGkkTCuXPkDRf0hxJhxfGHyTpvrxOXyuMHytpVn7PryTt0mG8MyUtkXRfYdywxCvpw7n8\nQ5JOHML4vyxpkaS7cjetivFL2knS/0h6UNL9kv5XHt8T279B/Kfm8b2y/cdJui3/Vu9Xapftpe3f\nLP5qbv+IqHRHSni/ASYBmwD3AHt3MZ6Hga3rxp0N/E3uPw04K/fvC9xNapuanNejdjZ4G3Bw7r8a\nOCL3/wXwzdx/HDCrw3gPAQ4E7hvOeIGtgd8CE4Ctav1DFP+Xgc83KLtPleIHXgMcmPu3AB4C9u6V\n7d8i/p7Y/nk+m+fX0cCtpP++9cT2bxF/Jbd/L5yZDPjHx2EmNjyjOwq4IPdfAByd+99L+nBejojf\nA/OBKZJeA2wZEbNzuQsL7ynO6zLgsE6CjYibgWeGMd635/4jgGsjYmlEPAtcC6w9guowfmh8scZR\nVYo/Iv4QEffk/mXAHGAnemT7N4m/9j+xym//HPfy3DuOtJMNemT7t4gfKrj9eyGZVO2PjwH8QtJs\nSR/P43aIiCWQfoDA9nl8feyL87iJpPWoKa7T2vdExGrgWUnbDPE6bF9ivEtzvM3mNVROkXSPpP8s\nVFNUNn5Jk0lnWLdS7vel7Phvy6N6YvtLGiXpbuAPwC/yDrVntn+T+KGC278XkknVvDkiDgKOBP5S\n0ltYd7RQM5RXNQzHJdG9Fu83gddGxIGkH9m/DOG8hzx+SVuQjvo+k4/we+r70iD+ntn+EbEmIl5P\nOiOcImk/emj7N4h/Xyq6/XshmSwGio3QO+VxXRERj+XXJ4ArSNVwS5QucSafUj6eiy8Gdi68vRZ7\ns/HrvUfSaGB8RDw9xKsxHPGW9rlFxBORK3aBb5M+g0rGL2kMaUd8UURcmUf3zPZvFH8vbf+aiHgO\n6CdV1fTM9m8Uf2W3/2AbhIa7IzU81Rrgx5Ia4PfpUiybA1vk/lcBtwCHkxr0TovmDXpjgV1Zv0Gs\n1pgmUoPYtDz+06xrEJtOhw3weT6TgfsLw6XHy/oNeLX+rYYo/tcU+j8HXFzV+En101+tG9cz279J\n/D2x/YHtyI3GwGbAjaQahZ7Y/i3ir+T2H/Yd8ivpSEcTD5EalE7vYhy7kpLZ3cD9tViAbYDrcozX\nFjc6cEb+UOcAhxfGvyHPYz7w9cL4ccClefytwOQOY74YeBRYCTwCnJS/HKXHC3wkj58HnDiE8V8I\n3Jc/iytIdeCVix94M7C68J25K3+Xh+X7UmL8vbL9988x35Pj/T/D+XstMf5Kbn//adHMzDrWC20m\nZmZWcU4mZmbWMScTMzPrmJOJmZl1zMnEzMw65mRiZmYdczKxjYak5/PrJEmtnvj5SuZ9Rt3wzUM5\nf7OqczKxjUntT1W7Ah8czBvzrSZa+d/rLSjikMHM36zXOZnYxugrwCH5wUKfyXdmPSc/iOgeSZ8A\nkHSopBslXQk8mMf9KN8x+v7aXaMlfQXYLM/vojzu+drCJP1TLn+vpA8U5n29pB/mBxldVCh/lqQH\nciznDNtWMevAmG4HYNYFpwN/FRHvBcjJ49mIeKOkscAtkq7NZV8P7BcRj+ThkyLiWUmbArMlXR4R\nZ0j6y0h3k66JPO9jgQMiYn9J2+f33JDLHEi6n9If8jLfBMwFjo6IvfP7x5e1EcyGks9MzNLNOk/M\nz424jXTvpj3ytNsLiQTgs5LuId3HaKdCuWbeDFwCEBGPk+78enBh3o9FuqfRPaQbWi4FXszPqXgf\n8GKH62Y2LJxMzNKdVE+NiNfnbreIuC5Pe2FtIelQ0pPo3hjpWRL3AJsW5tHusmpWFvpXA2MiPaBo\nCum27+8Bfj7otTHrAicT25jUduTPA1sWxl8DfDo/uwNJe0javMH7JwDPRMRKSXsDUwvTXqq9v25Z\nNwHH5XaZVwNvAW5vGmBa7lYR8XPg88AB7a+eWfe4zcQ2JrWrue4D1uRqre9GxNfzY2nvkiTSw5KO\nbvD+nwOfkvQg6fblvypMOw+4T9KdEfGh2rIi4keSpgL3AmuAL0TE45L2aRLbeODK3CYD6XkVZpXn\nW9CbmVnHXM1lZmYdczIxM7OOOZmYmVnHnEzMzKxjTiZmZtYxJxMzM+uYk4mZmXXMycTMzDr2/wHy\n7B6hOyuFGQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x106f6e0b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "With 6 epochs, the training time is:  425.2575750350952\n",
      "The validation accuracy is:  0.9636666666666667\n"
     ]
    }
   ],
   "source": [
    "NN = NeuralNetwork([784, 200, 10], loss=\"cross_entropy\", \n",
    "                  plot=True)\n",
    "start = time.time()\n",
    "epochs = 6\n",
    "NN.train(train, train_label, epochs, validation=validation, \n",
    "         validation_label=validation_label)\n",
    "end = time.time()\n",
    "print(\"With \" + str(epochs) + \" epochs, the training time is: \", end-start)\n",
    "predictions = NN.predict(validation)\n",
    "p_correct = accuracy(predictions, validation_label)\n",
    "print(\"The validation accuracy is: \", p_correct) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}