lec9-deeplearning.ipynb

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   "source": [
    "# Chapter 9: Introduction to Deep Learning\n",
    "___"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Neural Network"
   ]
  },
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   "source": [
    "### 1.1 Neuron\n",
    "\n",
    "A neuron is the building block of the neural network, which performs a dot product with the input and its weights, adds the bias and applies the non-linear transformation (or **activation function**), in this case the sigmoid $\\sigma(z)=1/(1+e^{-z})$. We will go into more details about different activation functions at the end of this section.\n",
    "\n",
    "<img src=\"images/neuron_model.jpeg\" width=60%>\n",
    "\n",
    "$$\n",
    "\\begin{array}{lcl}\n",
    "z &=& \\mathbf{w}^T \\mathbf{x} \\\\\n",
    "y &=& \\left( 1+\\exp(-z) \\right)^{-1}\n",
    "\\end{array}\n",
    "$$"
   ]
  },
  {
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   "source": [
    "class Neuron(object):\n",
    "    def __init__(self, weights, bias):\n",
    "        self.weights = weights\n",
    "        self.bias = bias\n",
    "        \n",
    "    def forward(self, input):\n",
    "        z = np.sum(self.weights * inputs) + self.bias\n",
    "        return 1.0 / (1.0 + np.exp(-z))"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1.2 Activation functions\n",
    "\n",
    "Every activation function takes a scalar as input and performs a certain mathematical operation on it. In practice, there are several popular activation functions:\n",
    "\n",
    "- **Sigmoid function**: $\\sigma(z) = 1 / (1 + e^{-z})$\n",
    "![](images/sigmoid.jpeg)\n",
    "- Two major drawbacks of Sigmoid function\n",
    "    1. Sigmoids saturate and kill gradients\n",
    "    2. Sigmoid ouputs are not zero-centered.\n",
    "\n",
    "- **tanh function**: $f(z) = 2\\sigma(z) - 1$\n",
    "![](images/tanh.jpeg)\n",
    "\n",
    "- **ReLU (Rectified Linear Unit, 修正线性单元)**: $f(z) = \\max(0, z)$\n",
    "![](images/relu.jpeg)\n",
    "- Pros and cons of ReLU:\n",
    "    * Accelerated convergence of SGD compared to sigmoid/tanh.\n",
    "    * Simpler computation of linear thresholding compared to the expensive exponential for sigmoid/tanh.\n",
    "    * No saturation.\n",
    "    * Fragile to be \"dead\" during training.\n",
    "    \n",
    "- **Leaky ReLU**: $f(z) = \\alpha z \\mathbf{I}(z < 0) + z \\mathbf{I}(z \\ge 0)$\n",
    "    * $\\alpha$ is a very small constant.\n",
    "    * $\\mathbf{I}(\\cdot)$ is an indicator function.\n",
    "    * The result is not always consistent.\n",
    "\n",
    "- **Maxout**: $f(x) = \\max(\\mathbf{w}_1^T \\mathbf{x} + b_1, \\mathbf{w}_2^T \\mathbf{x} + b_2)$\n",
    "    * Extension to ReLU and Leaky-ReLU\n",
    "    * Pros: Linear regime, no saturation, no \"dying\" threat\n",
    "    * Cons: doubling the number of parameters.\n",
    "    \n",
    "#### Take-home messages\n",
    "\n",
    "Use the ReLU non-linearity, be careful with your learning rates and possibly monitor the fraction of \"dead\" units in a network. If this concerns you, give Leaky ReLU or Maxout a try. Never use sigmoid. Try tanh, but expect it to work worse than ReLU/Maxout."
   ]
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   "source": [
    "### 1.3  Neural network architectures\n",
    "\n",
    "A common neural network is a hierarchical strucuture with neurons organized as multiple layers. For regular neural networks, the most common layer type is the **fully-connected (FC, 全连接) layer** in which neurons between two adjacent layers are fully pairwise connected, but neurons within a single layer share no connections. \n",
    "\n",
    "Below are two example Neural Network topologies that use a stack of fully-connected layers:\n",
    "\n",
    "#### Two-layer neural network\n",
    "<img src=\"images/two-layer-neural_net.jpeg\" width=40%>\n",
    "This two-layer neural net has $4 + 2 = 6$ neurons (not counting the inputs),  $[3 \\times 4] + [4 \\times 2] = 20$ weights and $4 + 2 = 6$ biases, for a total of $26$ learnable parameters.\n",
    "\n",
    "#### Three-layer neural network\n",
    "<img src=\"images/three-layer-neural_net.jpeg\" width=50%>\n",
    "This three-layer neural net has $4 + 4 + 1 = 9$ neurons (not counting the inputs),  $[3 \\times 4] + [4 \\times 4] + [4 \\times 1] = 32$ weights and $4 + 4 + 1 = 9$ biases, for a total of $41$ learnable parameters."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.4 Feed-forward computation\n",
    "\n",
    "For the above three-layer neural net:\n",
    "- The input is $\\mathbf{x} \\in \\mathbb{R}^3$\n",
    "- The weights between the input layer and the first hidden layer is $\\mathbf{W}_1 \\in \\mathbb{R}^{4 \\times 3}$; and the biases between these two layers is $\\mathbf{b}_1 \\in \\mathbb{R}^4$ \n",
    "- The weights between the two hidden layers is $\\mathbf{W}_2 \\in \\mathbb{R}^{4 \\times 4}$; and the biases between these two hidden layers is $\\mathbf{b}_2 \\in \\mathbb{R}^4$\n",
    "- The weights between the second hidden layer and the output layer is $\\mathbf{W}_3 \\in \\mathbb{R}^{1 \\times 4}$; and the bias is $\\mathbf{b} \\in \\mathbb{R}$.\n",
    "\n",
    "The output for the first hidden layer is:\n",
    "$$\n",
    "\\mathbf{h}_1 = \\sigma \\left(\\mathbf{W}_1 \\mathbf{x} + \\mathbf{b}_1 \\right)\n",
    "$$\n",
    "The output for the second hidden layer is:\n",
    "$$\n",
    "\\mathbf{h}_2 = \\sigma \\left(\\mathbf{W}_2 \\mathbf{h}_1 + \\mathbf{b}_2 \\right)\n",
    "$$\n",
    "The output for the output layer is:\n",
    "$$\n",
    "\\mathbf{o} = \\sigma \\left(\\mathbf{W}_3 \\mathbf{h}_2 + \\mathbf{b}_3 \\right)\n",
    "$$"
   ]
  },
  {
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   "execution_count": 3,
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 2.3012188]]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "# forward-pass of a 3-layer neural network:\n",
    "f = lambda x: 1.0/(1.0 + np.exp(-x)) # activation function (use sigmoid)\n",
    "x = np.random.randn(3, 1) # random input vector of three numbers (3x1)\n",
    "W1 = np.random.randn(4, 3)\n",
    "b1 = np.random.randn(4, 1)\n",
    "h1 = f(np.dot(W1, x) + b1) # calculate first hidden layer activations (4x1)\n",
    "W2 = np.random.randn(4, 4)\n",
    "b2 = np.random.randn(4, 1)\n",
    "h2 = f(np.dot(W2, h1) + b2) # calculate second hidden layer activations (4x1)\n",
    "W3 = np.random.randn(1, 4)\n",
    "b3 = np.random.randn(1)\n",
    "out = np.dot(W3, h2) + b3 # output neuron (1x1)\n",
    "print out"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1.5 Setting numbers of layers and the sizes\n",
    "\n",
    "If we increase the size (number of hidden nodes) and the number of layers, the **capacity能力** of the neural network also increases.\n",
    "<img src=\"images/layer_sizes.jpeg\" width=80%>\n",
    "\n",
    "However, increasing the size and the number of layer may result in **overfitting 过拟合** (i.e., bad generalization 泛化能力差).\n",
    "\n",
    "However, <font color=\"red\">you should not be using smaller networks to avoid overfitting</font>. Instead, you should use as big of a neural network as your computational budget allows, and pose other **regularization 正则化**  (especially **dropout 神经元失活**) to control overfitting."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.6 Setting up the data and the model\n",
    "\n",
    "- The recommended preprocessing is to conduct the **mean-subtraction** to center the data to zero, and do **normalization** of its scale to [-1, 1] along each feature.\n",
    "- **Initialize the weights** by drawing them from a gaussian distribution with standard deviation of $2/n$ - $\\sqrt{2/n}$, where $n$ is the number of inputs to the neuron.\n",
    "- Use L2 regularization and dropout (the inverted version).\n",
    "- Use batch normalization."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.7 Case study\n",
    "\n",
    "In this section we'll walk through a complete implementation of a toy Neural Network in 2-D space. We'll first implement a simple linear classifier and then extend the code to a 2-layer Neural Network. As we'll see, this extension is surprisingly simple and very few changes are necessary."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### (1) Generating a spiral dataset"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
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   "outputs": [
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7trtaeuPG1bC1NeXcuXP/UDJPVlYWQUFB7N69mxcvXvxrXlkpIIQoMa+8cIrXjBk/CVMT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\nZKDPe0LxQwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6a972a4110>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "N = 100 # number of points per class\n",
    "D = 2 # dimensionality\n",
    "K = 3 # number of classes\n",
    "X = np.zeros((N*K,D)) # data matrix (each row = single example)\n",
    "y = np.zeros(N*K, dtype='uint8') # class labels\n",
    "for j in xrange(K):\n",
    "  ix = range(N*j,N*(j+1))\n",
    "  r = np.linspace(0.0,1,N) # radius\n",
    "  t = np.linspace(j*4,(j+1)*4,N) + np.random.randn(N)*0.2 # theta\n",
    "  X[ix] = np.c_[r*np.sin(t), r*np.cos(t)]\n",
    "  y[ix] = j\n",
    "# visualize the data:\n",
    "plt.scatter(X[:, 0], X[:, 1], c=y, s=40, cmap=plt.cm.Spectral)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For this data set:\n",
    "- $\\mathbf{X} \\in \\mathbb{R}^{300 \\times 2}$;\n",
    "- The weight matrix $\\mathbf{W} \\in \\mathbb{R}^{2 \\times 3}$;\n",
    "- The output score $\\mathbf{Z} = \\mathbf{XW}  + \\mathbf{b} \\in \\mathbb{R}^{300 \\times 3}$\n",
    "- The output probabilities for the $i$-th example belonging to class $k=\\{1, 2, 3\\}$ is: \n",
    "$$\n",
    "p_{ik} = \\frac{\\exp(z_{ik})}{\\sum_{j=1}^3 \\exp(z_{ij})}\n",
    "$$\n",
    "- The label for the $i$-th example is $y_i \\in \\{1, 2, 3\\}$\n",
    "- We use **cross-entropy loss (交叉熵损失)** to evaluate the classifier\n",
    "$$\n",
    "L_i = -\\log(p_{iy_i})\n",
    "$$\n",
    "- And the final average cross-entropy loss with regularization term is:\n",
    "$$\n",
    "L = \\frac{1}{300} \\sum_{i=1}^{300} L_i + \\frac{1}{2} \\lambda \\sum_k \\sum_l w_{k,l}^2\n",
    "$$\n",
    "- Then we can compute the gradient of $L$ with respect to $\\mathbf{Z}$, $\\mathbf{W}$ and $\\mathbf{b}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### (2) Data preprocessing\n",
    "\n",
    "Normally we would want to preprocess the dataset so that each feature has zero-mean and unit standard deviation, but in this case the features are already in a nice range from -1 to 1, so we skip this step."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### (3) Training a softmax linear classifier: gradient descent\n",
    "\n",
    "First we need to initiate the parameters:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# initialize parameters randomly\n",
    "W = 0.01 * np.random.randn(D,K)\n",
    "b = np.zeros((1,K))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With the parameters, compute the class `scores`: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# compute class scores for a linear classifier\n",
    "scores = np.dot(X, W) + b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here the array `scores` will have size $[300 \\times 3]$, where each row gives the class scores corresponding to the 3 classes (blue, red, yellow).\n",
    "\n",
    "Then the `scores` can be easily converted into `probs` - the probability of an example belonging to a given class:\n",
    "$$\n",
    "p_k = \\frac{\\exp(z_{k})}{\\sum_j \\exp(z_j)}, k=1, \\dots, K\n",
    "$$\n",
    "Here $K$ is the number of classes to be classified."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# compute the class prob\n",
    "exp_scores = np.exp(scores)\n",
    "probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True) # [N x K]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using the `probs`, we can compute the cross-entropy loss for example $i$, `corect_logprobs`:\n",
    "$$\n",
    "L_i =  - \\log p_{y_i}\n",
    "$$\n",
    "where $y_i$ is the class label for the $i$-th example.\n",
    "\n",
    "Then we can compute the overall loss to be the average loss plus the $L_2$-regularization loss:\n",
    "$$\n",
    "L = \\frac{1}{n} \\sum_{i=1}^n L_i + \\lambda \\sum_k \\sum_l \\mathbf{W}_{k,l}^2\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Compute the loss average cross-entropy loss and regularization\n",
    "corect_logprobs = -np.log(probs[range(num_examples), y])        # loss for each sample\n",
    "data_loss = np.sum(corect_logprobs)/num_examples                     # model loss\n",
    "reg_loss = 0.5*reg*np.sum(W*W)                                                               # regularization loss\n",
    "loss = data_loss + reg_loss                                                                             # overall loss"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And with the loss function, we can compute the gradient with regard to `scores`:\n",
    "$$\n",
    "\\frac{\\partial L_i}{\\partial z_k} = p_k - 1(y_i = k)\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# compute the gradient on scores\n",
    "dscores = probs\n",
    "dscores[range(num_examples), y] -= 1\n",
    "dscores /= num_examples"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And since $\\mathbf{Z} = \\mathbf{XW} + \\mathbf{b}$, we can easily compute the backpropagated gradient of $\\mathbf{Z}$ with respect to $\\mathbf{W}$ and $\\mathbf{b}$: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# backpropagate the gradient to the parameters  (W, b)\n",
    "dW = np.dot(X.T, dscores)\n",
    "db = np.sum(dscores, axis=0, keepdims=True)\n",
    "dW += reg*W  # plus regularization gradient"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And then we can update $\\mathbf{W}$ and $\\mathbf{b}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Perform a parameter update\n",
    "W += -step_size * dW\n",
    "b += -step_size * db"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's combine the above scripts together:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 0: loss 0.762615\n",
      "Iteration 10: loss 0.762594\n",
      "Iteration 20: loss 0.762578\n",
      "Iteration 30: loss 0.762567\n",
      "Iteration 40: loss 0.762559\n",
      "Iteration 50: loss 0.762553\n",
      "Iteration 60: loss 0.762549\n",
      "Iteration 70: loss 0.762546\n",
      "Iteration 80: loss 0.762543\n",
      "Iteration 90: loss 0.762542\n",
      "Iteration 100: loss 0.762540\n",
      "Iteration 110: loss 0.762540\n",
      "Iteration 120: loss 0.762539\n",
      "Iteration 130: loss 0.762538\n",
      "Iteration 140: loss 0.762538\n",
      "Iteration 150: loss 0.762538\n",
      "Iteration 160: loss 0.762538\n",
      "Iteration 170: loss 0.762537\n",
      "Iteration 180: loss 0.762537\n",
      "Iteration 190: loss 0.762537\n",
      "Iteration 200: loss 0.762537\n",
      "Iteration 210: loss 0.762537\n",
      "Iteration 220: loss 0.762537\n",
      "Iteration 230: loss 0.762537\n",
      "Iteration 240: loss 0.762537\n",
      "Iteration 250: loss 0.762537\n",
      "Iteration 260: loss 0.762537\n",
      "Iteration 270: loss 0.762537\n",
      "Iteration 280: loss 0.762537\n",
      "Iteration 290: loss 0.762537\n",
      "Iteration 300: loss 0.762537\n",
      "Iteration 310: loss 0.762537\n",
      "Iteration 320: loss 0.762537\n",
      "Iteration 330: loss 0.762537\n",
      "Iteration 340: loss 0.762537\n",
      "Iteration 350: loss 0.762537\n",
      "Iteration 360: loss 0.762537\n",
      "Iteration 370: loss 0.762537\n",
      "Iteration 380: loss 0.762537\n",
      "Iteration 390: loss 0.762537\n"
     ]
    }
   ],
   "source": [
    "# some hyperparameters\n",
    "step_size = 1e-0\n",
    "reg = 1e-3               # regularization strength $\\lambda$\n",
    "\n",
    "# gradient descent loop\n",
    "num_examples = X.shape[0]\n",
    "for i in xrange(400):\n",
    "    \n",
    "    # evaluate class scores, [N x K]\n",
    "    scores = np.dot(X, W) + b\n",
    "    \n",
    "    # compute the class prob\n",
    "    exp_scores = np.exp(scores)\n",
    "    probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True) # [N x K]\n",
    "    \n",
    "    \n",
    "    # Compute the loss average cross-entropy loss and regularization\n",
    "    corect_logprobs = -np.log(probs[range(num_examples), y])\n",
    "    data_loss = np.sum(corect_logprobs)/num_examples\n",
    "    reg_loss = 0.5*reg*np.sum(W*W)\n",
    "    loss = data_loss + reg_loss\n",
    "    \n",
    "    if i % 10 == 0:\n",
    "        print \"Iteration %d: loss %f\" %(i, loss)\n",
    "        \n",
    "    # compute the gradient on scores\n",
    "    dscores = probs\n",
    "    dscores[range(num_examples), y] -= 1\n",
    "    dscores /= num_examples\n",
    "    \n",
    "    # backpropagate the gradient to the parameters  (W, b)\n",
    "    dW = np.dot(X.T, dscores)\n",
    "    db = np.sum(dscores, axis=0, keepdims=True)\n",
    "    \n",
    "    dW += reg*W  # plus regularization gradient\n",
    "    \n",
    "    # Perform a parameter update\n",
    "    W += -step_size * dW\n",
    "    b += -step_size * db"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "now we can evaluate the **training set accuracy**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "training accuracy: 0.52\n"
     ]
    }
   ],
   "source": [
    "# evaluate training set accuracy\n",
    "scores = np.dot(X, W) + b\n",
    "predicted_class = np.argmax(scores, axis=1)\n",
    "print 'training accuracy: %.2f' % (np.mean(predicted_class == y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And this is the diagram of classification margins:\n",
    "<img src=\"images/spiral_linear.png\" width=50%>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <font color=\"red\">Exercise 1</font>\n",
    "1. The training accuracy is so low for the above linear classifier - softmax, can you improve it by including more basis functions, like second-order, or even the third-order terms into the model?\n",
    "\n",
    "2. In practice, we often use cross-validation (CV) to test the accuracy of the classifiers. Here, try to use 5-fold cross validation to compute the prediction accuracy of this classifier."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### (4) Training a 2-layer neural network"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iteration 0: loss 1.098724\n",
      "iteration 1000: loss 0.312690\n",
      "iteration 2000: loss 0.256362\n",
      "iteration 3000: loss 0.249350\n",
      "iteration 4000: loss 0.248308\n",
      "iteration 5000: loss 0.248183\n",
      "iteration 6000: loss 0.251487\n",
      "iteration 7000: loss 0.248855\n",
      "iteration 8000: loss 0.246873\n",
      "iteration 9000: loss 0.247030\n"
     ]
    }
   ],
   "source": [
    "# initialize parameters randomly\n",
    "h = 100 # size of hidden layer\n",
    "W = 0.01 * np.random.randn(D,h)\n",
    "b = np.zeros((1,h))\n",
    "W2 = 0.01 * np.random.randn(h,K)\n",
    "b2 = np.zeros((1,K))\n",
    "\n",
    "# some hyperparameters\n",
    "step_size = 1e-0\n",
    "reg = 1e-3 # regularization strength\n",
    "\n",
    "# gradient descent loop\n",
    "num_examples = X.shape[0]\n",
    "for i in xrange(10000):\n",
    "  \n",
    "  # evaluate class scores, [N x K]\n",
    "  hidden_layer = np.maximum(0, np.dot(X, W) + b) # note, ReLU activation\n",
    "  scores = np.dot(hidden_layer, W2) + b2\n",
    "  \n",
    "  # compute the class probabilities\n",
    "  exp_scores = np.exp(scores)\n",
    "  probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True) # [N x K]\n",
    "  \n",
    "  # compute the loss: average cross-entropy loss and regularization\n",
    "  corect_logprobs = -np.log(probs[range(num_examples),y])\n",
    "  data_loss = np.sum(corect_logprobs)/num_examples\n",
    "  reg_loss = 0.5*reg*np.sum(W*W) + 0.5*reg*np.sum(W2*W2)\n",
    "  loss = data_loss + reg_loss\n",
    "  if i % 1000 == 0:\n",
    "    print \"iteration %d: loss %f\" % (i, loss)\n",
    "  \n",
    "  # compute the gradient on scores\n",
    "  dscores = probs\n",
    "  dscores[range(num_examples),y] -= 1\n",
    "  dscores /= num_examples\n",
    "  \n",
    "  # backpropate the gradient to the parameters\n",
    "  # first backprop into parameters W2 and b2\n",
    "  dW2 = np.dot(hidden_layer.T, dscores)\n",
    "  db2 = np.sum(dscores, axis=0, keepdims=True)\n",
    "  # next backprop into hidden layer\n",
    "  dhidden = np.dot(dscores, W2.T)\n",
    "  # backprop the ReLU non-linearity\n",
    "  dhidden[hidden_layer <= 0] = 0\n",
    "  # finally into W,b\n",
    "  dW = np.dot(X.T, dhidden)\n",
    "  db = np.sum(dhidden, axis=0, keepdims=True)\n",
    "  \n",
    "  # add regularization gradient contribution\n",
    "  dW2 += reg * W2\n",
    "  dW += reg * W\n",
    "  \n",
    "  # perform a parameter update\n",
    "  W += -step_size * dW\n",
    "  b += -step_size * db\n",
    "  W2 += -step_size * dW2\n",
    "  b2 += -step_size * db2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "training accuracy: 0.98\n"
     ]
    }
   ],
   "source": [
    "# evaluate training set accuracy\n",
    "hidden_layer = np.maximum(0, np.dot(X, W) + b)\n",
    "scores = np.dot(hidden_layer, W2) + b2\n",
    "predicted_class = np.argmax(scores, axis=1)\n",
    "print 'training accuracy: %.2f' % (np.mean(predicted_class == y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"images/spiral_net.png\" width=60%>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <font color=\"red\">Exercise 2</font>\n",
    "\n",
    "Do you think we need to add more layer to improve the performance, although it seems that it is already very good? To validate the model, you'd better use a 5-fold cross-validation to obtain the prediction accuracy. \n",
    "\n",
    "Or you can change the number of neurons in the hidden layer, see how the accuracy changes."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Convolutional Neural Network (CNN) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "![](images/convnet.jpeg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Recurrent Neural Network (RNN)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": []
  }
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