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000000000..e2dbae3dc Binary files /dev/null and b/doc/LectureNotes/_build/html/_images/clustering_24_0.png differ diff --git a/doc/LectureNotes/_build/html/_sources/chapter10.ipynb b/doc/LectureNotes/_build/html/_sources/chapter10.ipynb index cc65cfe33..fb9797ca3 100644 --- a/doc/LectureNotes/_build/html/_sources/chapter10.ipynb +++ b/doc/LectureNotes/_build/html/_sources/chapter10.ipynb @@ -2,7 +2,21 @@ "cells": [ { "cell_type": "markdown", - "metadata": {}, + "id": "7e3af66b", + "metadata": { + "editable": true + }, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "id": "64b4469b", + "metadata": { + "editable": true + }, "source": [ "# Building a Feed Forward Neural Network\n", "\n", @@ -27,7 +41,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a27e39fb", + "metadata": { + "editable": true + }, "source": [ "$$\n", "P(y = 0 \\mid \\hat{x}, \\hat{\\theta}) = \\frac{1}{1 + \\exp{(- \\hat{x}})} ,\n", @@ -36,14 +53,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6332f4a5", + "metadata": { + "editable": true + }, "source": [ "and" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "5d2a5fe5", + "metadata": { + "editable": true + }, "source": [ "$$\n", "P(y = 1 \\mid \\hat{x}, \\hat{\\theta}) = 1 - P(y = 0 \\mid \\hat{x}, \\hat{\\theta}) ,\n", @@ -52,13 +75,22 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "62bf3ae6", + "metadata": { + "editable": true + }, "source": [ "where $y \\in \\{0, 1\\}$ and $\\hat{\\theta}$ represents the weights and biases\n", - "of our network.\n", - "\n", - "\n", - "\n", + "of our network." + ] + }, + { + "cell_type": "markdown", + "id": "5fdb1150", + "metadata": { + "editable": true + }, + "source": [ "## Defining the cost function\n", "\n", "Our cost function is given as (see the Logistic regression lectures)" @@ -66,7 +98,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "64cfe441", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathcal{C}(\\hat{\\theta}) = - \\ln P(\\mathcal{D} \\mid \\hat{\\theta}) = - \\sum_{i=1}^n\n", @@ -76,7 +111,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "dbd78c70", + "metadata": { + "editable": true + }, "source": [ "This last equality means that we can interpret our *cost* function as a sum over the *loss* function\n", "for each point in the dataset $\\mathcal{L}_i(\\hat{\\theta})$. \n", @@ -87,10 +125,8 @@ "\n", "$y = 5 \\quad \\rightarrow \\quad \\hat{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$ and\n", "\n", - "\n", "$y = 1 \\quad \\rightarrow \\quad \\hat{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,$ \n", "\n", - "\n", "i.e. a binary bit string of length $C$, where $C = 10$ is the number of classes in the MNIST dataset (numbers from $0$ to $9$).. \n", "\n", "If $\\hat{x}_i$ is the $i$-th input (image), $y_{ic}$ refers to the $c$-th component of the $i$-th\n", @@ -100,7 +136,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "67834a87", + "metadata": { + "editable": true + }, "source": [ "$$\n", "P(y_{ic} = 1 \\mid \\hat{x}_i, \\hat{\\theta}) = \\frac{\\exp{((\\hat{a}_i^{hidden})^T \\hat{w}_c)}}\n", @@ -110,7 +149,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0521db23", + "metadata": { + "editable": true + }, "source": [ "which reduces to the logistic function in the binary case. \n", "The likelihood of this $C$-class classifier\n", @@ -119,7 +161,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "12a92031", + "metadata": { + "editable": true + }, "source": [ "$$\n", "P(\\mathcal{D} \\mid \\hat{\\theta}) = \\prod_{i=1}^n \\prod_{c=0}^{C-1} [P(y_{ic} = 1)]^{y_{ic}} .\n", @@ -128,14 +173,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e03ffaa9", + "metadata": { + "editable": true + }, "source": [ "Again we take the negative log-likelihood to define our cost function:" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "cfab4fa3", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathcal{C}(\\hat{\\theta}) = - \\log{P(\\mathcal{D} \\mid \\hat{\\theta})}.\n", @@ -144,13 +195,23 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "864840d8", + "metadata": { + "editable": true + }, "source": [ "See the logistic regression lectures for a full definition of the cost function.\n", "\n", - "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!\n", - "\n", - "\n", + "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!" + ] + }, + { + "cell_type": "markdown", + "id": "496bdba3", + "metadata": { + "editable": true + }, + "source": [ "### Example: binary classification problem\n", "\n", "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" @@ -158,7 +219,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "8646c18f", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathcal{C}(\\hat{\\beta}) = - \\sum_{i=1}^n \\left(y_i\\log{p(y_i \\vert x_i,\\hat{\\beta})}+(1-y_i)\\log{1-p(y_i \\vert x_i,\\hat{\\beta})}\\right),\n", @@ -167,14 +231,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "2dbfda71", + "metadata": { + "editable": true + }, "source": [ "where we had defined the logistic (sigmoid) function" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "5cd0a375", + "metadata": { + "editable": true + }, "source": [ "$$\n", "p(y_i =1\\vert x_i,\\hat{\\beta})=\\frac{\\exp{(\\beta_0+\\beta_1 x_i)}}{1+\\exp{(\\beta_0+\\beta_1 x_i)}},\n", @@ -183,14 +253,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "020de545", + "metadata": { + "editable": true + }, "source": [ "and" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "2efde2f5", + "metadata": { + "editable": true + }, "source": [ "$$\n", "p(y_i =0\\vert x_i,\\hat{\\beta})=1-p(y_i =1\\vert x_i,\\hat{\\beta}).\n", @@ -199,7 +275,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "bf9a596d", + "metadata": { + "editable": true + }, "source": [ "The parameters $\\hat{\\beta}$ were defined using a minimization method like gradient descent or Newton-Raphson's method. \n", "\n", @@ -209,7 +288,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0b392cc5", + "metadata": { + "editable": true + }, "source": [ "$$\n", "a_i^l = y_i = \\frac{\\exp{(z_i^l)}}{1+\\exp{(z_i^l)}},\n", @@ -218,14 +300,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5169434f", + "metadata": { + "editable": true + }, "source": [ "with" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "dcc8ed0a", + "metadata": { + "editable": true + }, "source": [ "$$\n", "z_i^l = \\sum_{j}w_{ij}^l a_j^{l-1}+b_i^l,\n", @@ -234,7 +322,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "4010628a", + "metadata": { + "editable": true + }, "source": [ "where the superscript $l-1$ indicates that these are the outputs from layer $l-1$.\n", "Our cost function at the final layer $l=L$ is now" @@ -242,7 +333,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "7bf89431", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathcal{C}(\\hat{W}) = - \\sum_{i=1}^n \\left(t_i\\log{a_i^L}+(1-t_i)\\log{(1-a_i^L)}\\right),\n", @@ -251,14 +345,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "79bb56ee", + "metadata": { + "editable": true + }, "source": [ "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" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "a27a06b5", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial \\mathcal{C}(\\hat{W})}{\\partial a_i^L} = \\frac{a_i^L-t_i}{a_i^L(1-a_i^L)}.\n", @@ -267,12 +367,21 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "cd6acfaf", + "metadata": { + "editable": true + }, + "source": [ + "In case we use another activation function than the logistic one, we need to evaluate other derivatives." + ] + }, + { + "cell_type": "markdown", + "id": "b9e59acd", + "metadata": { + "editable": true + }, "source": [ - "In case we use another activation function than the logistic one, we need to evaluate other derivatives. \n", - "\n", - "\n", - "\n", "### The Softmax function\n", "\n", "In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation $z_i^l$, that is we need" @@ -280,7 +389,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "eceaf7eb", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial f(z_i^l)}{\\partial w_{jk}^l} =\n", @@ -290,14 +402,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "bbc77522", + "metadata": { + "editable": true + }, "source": [ "For the Softmax function we have" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "88d4c18b", + "metadata": { + "editable": true + }, "source": [ "$$\n", "f(z_i^l) = \\frac{\\exp{(z_i^l)}}{\\sum_{m=1}^K\\exp{(z_m^l)}}.\n", @@ -306,14 +424,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b47f4092", + "metadata": { + "editable": true + }, "source": [ "Its derivative with respect to $z_j^l$ gives" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "65ec4336", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial f(z_i^l)}{\\partial z_j^l}= f(z_i^l)\\left(\\delta_{ij}-f(z_j^l)\\right),\n", @@ -322,14 +446,23 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "38f54e19", + "metadata": { + "editable": true + }, + "source": [ + "which in case of the simply binary model reduces to having $i=j$." + ] + }, + { + "cell_type": "markdown", + "id": "c0d8bc5e", + "metadata": { + "editable": true + }, "source": [ - "which in case of the simply binary model reduces to having $i=j$. \n", - "\n", - "\n", "## Developing a code for doing neural networks with back propagation\n", "\n", - "\n", "One can identify a set of key steps when using neural networks to solve supervised learning problems: \n", "\n", "1. Collect and pre-process data \n", @@ -342,8 +475,16 @@ "\n", "5. Evaluate model performance on test data \n", "\n", - "6. Adjust hyperparameters (if necessary, network architecture)\n", - "\n", + "6. Adjust hyperparameters (if necessary, network architecture)" + ] + }, + { + "cell_type": "markdown", + "id": "e3810222", + "metadata": { + "editable": true + }, + "source": [ "### Collect and pre-process data\n", "\n", "Here we will be using the MNIST dataset, which is readily available through the **scikit-learn**\n", @@ -388,7 +529,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 1, + "id": "5e7205ac", "metadata": { "collapsed": false, "editable": true @@ -443,7 +585,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "78721970", + "metadata": { + "editable": true + }, "source": [ "### Train and test datasets\n", "\n", @@ -460,7 +605,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 2, + "id": "dcc066a0", "metadata": { "collapsed": false, "editable": true @@ -498,7 +644,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "95cf3f87", + "metadata": { + "editable": true + }, "source": [ "### Define model and architecture\n", "\n", @@ -534,8 +683,16 @@ "\n", "$$ f(x) = \\sigma(x) = \\frac{1}{1 + e^{-x}} ,$$\n", "\n", - "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.\n", - "\n", + "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." + ] + }, + { + "cell_type": "markdown", + "id": "25c0cbe6", + "metadata": { + "editable": true + }, + "source": [ "### Layers\n", "\n", "* Input \n", @@ -568,7 +725,6 @@ "Since each neuron in the output layer is connected to the 50 inputs from the hidden layer we have 50x10 = 500\n", "weights to the output layer.\n", "\n", - "\n", "Typically weights are initialized with small values distributed around zero, drawn from a uniform\n", "or normal distribution. Setting all weights to zero means all neurons give the same output, making the network useless. \n", "\n", @@ -582,7 +738,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 3, + "id": "9f415ea2", "metadata": { "collapsed": false, "editable": true @@ -608,7 +765,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "afbff46b", + "metadata": { + "editable": true + }, "source": [ "### Feed-forward pass\n", "\n", @@ -630,7 +790,6 @@ "$$ a_{j}^{L} = \\frac{\\exp{(z_j^{L})}}\n", "{\\sum_{c=0}^{C-1} \\exp{(z_c^{L})}} .$$ \n", "\n", - "\n", "Since our data has the dimensions $X = (n_{inputs}, n_{features})$ and our weights to the hidden\n", "layer have the dimensions \n", "$W_{hidden} = (n_{features}, n_{hidden})$,\n", @@ -660,7 +819,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 4, + "id": "e05b2147", "metadata": { "collapsed": false, "editable": true @@ -706,7 +866,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6c8100ba", + "metadata": { + "editable": true + }, "source": [ "### Choose cost function and optimizer\n", "\n", @@ -719,10 +882,8 @@ "\n", "$$ y = 5 \\quad \\rightarrow \\quad \\hat{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$$ \n", "\n", - "\n", "$$ y = 1 \\quad \\rightarrow \\quad \\hat{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,$$ \n", "\n", - "\n", "i.e. a binary bit string of length $C$, where $C = 10$ is the number of classes in the MNIST dataset. \n", "\n", "Let $y_{ic}$ denote the $c$-th component of the $i$-th one-hot vector. \n", @@ -731,10 +892,16 @@ "In the one-hot representation only one of the terms in the loss function is non-zero, namely the\n", "probability of the correct category $c'$ \n", "(i.e. the category $c'$ such that $y_{ic'} = 1$). This means that the cross entropy loss only punishes you for how wrong\n", - "you got the correct label. The probability of category $c$ is given by the softmax function. The vector $\\hat{\\theta}$ represents the parameters of our network, i.e. all the weights and biases. \n", - "\n", - "\n", - "\n", + "you got the correct label. The probability of category $c$ is given by the softmax function. The vector $\\hat{\\theta}$ represents the parameters of our network, i.e. all the weights and biases." + ] + }, + { + "cell_type": "markdown", + "id": "720d19fd", + "metadata": { + "editable": true + }, + "source": [ "### Optimizing the cost function\n", "\n", "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\n", @@ -763,9 +930,16 @@ "\n", "2. It significantly speeds up the calculation, since we do not have to use the entire dataset to calculate the gradient. \n", "\n", - "The various optmization methods, with codes and algorithms, are discussed in our lectures on [Gradient descent approaches](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html).\n", - "\n", - "\n", + "The various optmization methods, with codes and algorithms, are discussed in our lectures on [Gradient descent approaches](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html)." + ] + }, + { + "cell_type": "markdown", + "id": "d0b1f84a", + "metadata": { + "editable": true + }, + "source": [ "### Regularization\n", "\n", "It is common to add an extra term to the cost function, proportional\n", @@ -783,7 +957,6 @@ "\n", "i.e. we sum up all the weights squared. The factor $\\lambda$ is known as a regularization parameter.\n", "\n", - "\n", "In order to train the model, we need to calculate the derivative of\n", "the cost function with respect to every bias and weight in the\n", "network. In total our network has $(64 + 1)\\times 50=3250$ weights in\n", @@ -791,9 +964,16 @@ "layer ($+1$ for the bias), and the gradient must be calculated for\n", "every parameter. We use the *backpropagation* algorithm discussed\n", "above. This is a clever use of the chain rule that allows us to\n", - "calculate the gradient efficently. \n", - "\n", - "\n", + "calculate the gradient efficently." + ] + }, + { + "cell_type": "markdown", + "id": "96417f62", + "metadata": { + "editable": true + }, + "source": [ "### Matrix multiplication\n", "\n", "To more efficently train our network these equations are implemented using matrix operations. \n", @@ -829,7 +1009,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 5, + "id": "c692ab5f", "metadata": { "collapsed": false, "editable": true @@ -908,7 +1089,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9d0ad486", + "metadata": { + "editable": true + }, "source": [ "## Improving performance\n", "\n", @@ -923,14 +1107,14 @@ "If this does not improve network performance, you may want to consider altering the network architecture, adding more neurons or hidden layers. \n", "Andrew Ng goes through some of these considerations in this [video](https://youtu.be/F1ka6a13S9I). You can find a summary of the video [here](https://kevinzakka.github.io/2016/09/26/applying-deep-learning/). \n", "\n", - "\n", "It is very natural to think of the network as an object, with specific instances of the network\n", "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." ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 6, + "id": "84088dcb", "metadata": { "collapsed": false, "editable": true @@ -1040,7 +1224,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "61dae78a", + "metadata": { + "editable": true + }, "source": [ "## Evaluate model performance on test data\n", "\n", @@ -1055,7 +1242,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 7, + "id": "999a01b6", "metadata": { "collapsed": false, "editable": true @@ -1082,7 +1270,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "12b8be76", + "metadata": { + "editable": true + }, "source": [ "## Adjust hyperparameters\n", "\n", @@ -1092,7 +1283,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 8, + "id": "f022ba3c", "metadata": { "collapsed": false, "editable": true @@ -1123,14 +1315,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "118ee392", + "metadata": { + "editable": true + }, "source": [ "## Visualization" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 9, + "id": "01cd7236", "metadata": { "collapsed": false, "editable": true @@ -1174,7 +1370,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "1cef52ca", + "metadata": { + "editable": true + }, "source": [ "## scikit-learn implementation\n", "\n", @@ -1193,7 +1392,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 10, + "id": "08650740", "metadata": { "collapsed": false, "editable": true @@ -1220,14 +1420,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9a5819b0", + "metadata": { + "editable": true + }, "source": [ "## Visualization" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 11, + "id": "c76390e1", "metadata": { "collapsed": false, "editable": true @@ -1272,7 +1476,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6b973f8c", + "metadata": { + "editable": true + }, "source": [ "## Building neural networks in Tensorflow and Keras\n", "\n", @@ -1284,7 +1491,6 @@ "clear how to build one using an arbitrary number of hidden layers, using data structures such as Python lists or\n", "NumPy arrays.\n", "\n", - "\n", "Tensorflow is an open source library machine learning library\n", "developed by the Google Brain team for internal use. It was released\n", "under the Apache 2.0 open source license in November 9, 2015.\n", @@ -1307,32 +1513,15 @@ "Then we will build (effectively) the same graph in Keras, to see just\n", "how simple solving a machine learning problem can be.\n", "\n", - "To install tensorflow on Unix/Linux systems, use pip as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "pip3 install tensorflow" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ + "To install tensorflow on Unix/Linux systems, use pip as **pip3 install tensorflow**\n", "and/or if you use **anaconda**, just write (or install from the graphical user interface)\n", "(current release of CPU-only TensorFlow)" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 12, + "id": "cb71108d", "metadata": { "collapsed": false, "editable": true @@ -1345,14 +1534,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "8cf89ee7", + "metadata": { + "editable": true + }, "source": [ "To install the current release of GPU TensorFlow" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 13, + "id": "ace3d31e", "metadata": { "collapsed": false, "editable": true @@ -1365,7 +1558,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f5d7454d", + "metadata": { + "editable": true + }, "source": [ "Keras is a high level [neural network](https://en.wikipedia.org/wiki/Application_programming_interface)\n", "that supports Tensorflow, CTNK and Theano as backends. \n", @@ -1374,7 +1570,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 14, + "id": "024068e3", "metadata": { "collapsed": false, "editable": true @@ -1386,19 +1583,22 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e36b74d0", + "metadata": { + "editable": true + }, "source": [ "You can look up the [instructions here](https://keras.io/) for more information.\n", "\n", "We will to a large extent use **keras** in this course. \n", "\n", - "\n", "Let us look again at the MINST data set." ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 15, + "id": "42dbd743", "metadata": { "collapsed": false, "editable": true @@ -1452,7 +1652,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 16, + "id": "eba6a34d", "metadata": { "collapsed": false, "editable": true @@ -1480,7 +1681,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 17, + "id": "da80e8b6", "metadata": { "collapsed": false, "editable": true @@ -1509,7 +1711,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 18, + "id": "87273539", "metadata": { "collapsed": false, "editable": true @@ -1535,7 +1738,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 19, + "id": "8cd653a0", "metadata": { "collapsed": false, "editable": true @@ -1577,14 +1781,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "31009a5d", + "metadata": { + "editable": true + }, "source": [ "## The Breast Cancer Data, now with Keras" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 20, + "id": "991bf288", "metadata": { "collapsed": false, "editable": true @@ -1761,7 +1969,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d197746e", + "metadata": { + "editable": true + }, "source": [ "## Fine-tuning neural network hyperparameters\n", "\n", @@ -1779,7 +1990,6 @@ "training a neural network on a large dataset takes a lot of time, you\n", "will only be able to explore a tiny part of the hyperparameter space.\n", "\n", - "\n", "* You can use randomized search.\n", "\n", "* Or use tools like [Oscar](http://oscar.calldesk.ai/), which implements more complex algorithms to help you find a good set of hyperparameters quickly. \n", @@ -1795,12 +2005,16 @@ "as large image classification or speech recognition, typically require networks with dozens of layers\n", "and they need a huge amount\n", "of training data. However, you will rarely have to train such networks from scratch: it is much more\n", - "common to reuse parts of a pretrained state-of-the-art network that performs a similar task.\n", - "\n", - "\n", - "\n", - "\n", - "\n", + "common to reuse parts of a pretrained state-of-the-art network that performs a similar task." + ] + }, + { + "cell_type": "markdown", + "id": "3614fe93", + "metadata": { + "editable": true + }, + "source": [ "## Which activation function should I use?\n", "\n", "The Back propagation algorithm we derived above works by going from\n", @@ -1809,7 +2023,6 @@ "function with regards to each parameter in the network, it uses these\n", "gradients to update each parameter with a Gradient Descent (GD) step.\n", "\n", - "\n", "Unfortunately for us, the gradients often get smaller and smaller as the\n", "algorithm progresses down to the first hidden layers. As a result, the\n", "GD update leaves the lower layer connection weights\n", @@ -1825,9 +2038,6 @@ "neural networks suffer from unstable gradients, different layers may\n", "learn at widely different speeds\n", "\n", - "\n", - "\n", - "\n", "Although this unfortunate behavior has been empirically observed for\n", "quite a while (it was one of the reasons why deep neural networks were\n", "mostly abandoned for a long time), it is only around 2010 that\n", @@ -1850,8 +2060,6 @@ "(the hyperbolic tangent function has a mean of 0 and behaves slightly\n", "better than the logistic function in deep networks).\n", "\n", - "\n", - "\n", "Looking at the logistic activation function, when inputs become large\n", "(negative or positive), the function saturates at 0 or 1, with a\n", "derivative extremely close to 0. Thus when backpropagation kicks in,\n", @@ -1870,8 +2078,6 @@ "its inputs, and we also need the gradients to have equal variance\n", "before and after flowing through a layer in the reverse direction.\n", "\n", - "\n", - "\n", "One of the insights in the 2010 paper by Glorot and Bengio was that\n", "the vanishing/exploding gradients problems were in part due to a poor\n", "choice of activation function. Until then most people had assumed that\n", @@ -1880,9 +2086,16 @@ "that other activation functions behave much better in deep neural\n", "networks, in particular the ReLU activation function, mostly because\n", "it does not saturate for positive values (and also because it is quite\n", - "fast to compute).\n", - "\n", - "\n", + "fast to compute)." + ] + }, + { + "cell_type": "markdown", + "id": "0816093d", + "metadata": { + "editable": true + }, + "source": [ "## The RELU function family\n", "\n", "The ReLU activation function suffers from a problem known as the dying\n", @@ -1903,7 +2116,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d979c7db", + "metadata": { + "editable": true + }, "source": [ "$$\n", "ELU(z) = \\left\\{\\begin{array}{cc} \\alpha\\left( \\exp{(z)}-1\\right) & z < 0,\\\\ z & z \\ge 0.\\end{array}\\right.\n", @@ -1912,7 +2128,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "eadc7692", + "metadata": { + "editable": true + }, "source": [ "In general it seems that the ELU activation function is better than\n", "the leaky ReLU function (and its variants), which is better than\n", @@ -1927,8 +2146,6 @@ "spare time and computing power, you can use cross-validation or\n", "bootstrap to evaluate other activation functions.\n", "\n", - "\n", - "\n", "In most cases you can use the ReLU activation function in the hidden layers (or one of its variants).\n", "\n", "It is a bit faster to compute than other activation functions, and the gradient descent optimization does in general not get stuck.\n", @@ -1937,8 +2154,16 @@ "\n", "* For classification the softmax activation function is generally a good choice for classification tasks (when the classes are mutually exclusive).\n", "\n", - "* For regression tasks, you can simply use no activation function at all.\n", - "\n", + "* For regression tasks, you can simply use no activation function at all." + ] + }, + { + "cell_type": "markdown", + "id": "a78e9091", + "metadata": { + "editable": true + }, + "source": [ "## Batch Normalization\n", "\n", "Batch Normalization\n", @@ -1951,8 +2176,16 @@ "learn the optimal scale and mean of the inputs for each layer.\n", "In order to zero-center and normalize the inputs, the algorithm needs to estimate the inputs’ mean and\n", "standard deviation. It does so by evaluating the mean and standard deviation of the inputs over the current\n", - "mini-batch, from this the name batch normalization.\n", - "\n", + "mini-batch, from this the name batch normalization." + ] + }, + { + "cell_type": "markdown", + "id": "04e309c6", + "metadata": { + "editable": true + }, + "source": [ "## Dropout\n", "\n", "It is a fairly simple algorithm: at every training step, every neuron (including the input neurons but\n", @@ -1961,8 +2194,16 @@ "\n", "The\n", "hyperparameter $p$ is called the dropout rate, and it is typically set to 50%. After training, the neurons are not dropped anymore.\n", - " It is viewed as one of the most popular regularization techniques.\n", - "\n", + " It is viewed as one of the most popular regularization techniques." + ] + }, + { + "cell_type": "markdown", + "id": "f9eb2036", + "metadata": { + "editable": true + }, + "source": [ "## Gradient Clipping\n", "\n", "A popular technique to lessen the exploding gradients problem is to simply clip the gradients during\n", @@ -1972,12 +2213,18 @@ "This technique is called Gradient Clipping.\n", "\n", "In general however, Batch\n", - "Normalization is preferred.\n", - "\n", - "\n", + "Normalization is preferred." + ] + }, + { + "cell_type": "markdown", + "id": "95301882", + "metadata": { + "editable": true + }, + "source": [ "## A top-down perspective on Neural networks\n", "\n", - "\n", "The first thing we would like to do is divide the data into two or three\n", "parts. A training set, a validation or dev (development) set, and a\n", "test set. The test set is the data on which we want to make\n", @@ -1988,7 +2235,6 @@ "do not use any of the test data to train the algorithm. This is a\n", "cardinal sin in ML. Then:\n", "\n", - "\n", "* Estimate optimal error rate\n", "\n", "* Minimize underfitting (bias) on training data set.\n", @@ -2012,9 +2258,16 @@ "the test data. The difference between the performance of the algorithm\n", "on these two validation sets quantifies the train-test mismatch. This\n", "can serve as another important diagnostic when using DNNs for\n", - "supervised learning.\n", - "\n", - "\n", + "supervised learning." + ] + }, + { + "cell_type": "markdown", + "id": "e1da8c6e", + "metadata": { + "editable": true + }, + "source": [ "## Limitations of supervised learning with deep networks\n", "\n", "Like all statistical methods, supervised learning using neural\n", @@ -2027,8 +2280,6 @@ "\n", "Here we list some of the important limitations of supervised neural network based models. \n", "\n", - "\n", - "\n", "* **Need labeled data**. All supervised learning methods, DNNs for supervised learning require labeled data. Often, labeled data is harder to acquire than unlabeled data (e.g. one must pay for human experts to label images).\n", "\n", "* **Supervised neural networks are extremely data intensive.** DNNs are data hungry. They perform best when data is plentiful. This is doubly so for supervised methods where the data must also be labeled. The utility of DNNs is extremely limited if data is hard to acquire or the datasets are small (hundreds to a few thousand samples). In this case, the performance of other methods that utilize hand-engineered features can exceed that of DNNs.\n", @@ -2043,5 +2294,5 @@ ], "metadata": {}, "nbformat": 4, - "nbformat_minor": 4 + "nbformat_minor": 5 } diff --git a/doc/LectureNotes/_build/html/_sources/chapter11.ipynb b/doc/LectureNotes/_build/html/_sources/chapter11.ipynb index c03e4a368..00bf35796 100644 --- a/doc/LectureNotes/_build/html/_sources/chapter11.ipynb +++ b/doc/LectureNotes/_build/html/_sources/chapter11.ipynb @@ -2,7 +2,21 @@ "cells": [ { "cell_type": "markdown", - "metadata": {}, + "id": "7660a7d5", + "metadata": { + "editable": true + }, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "id": "7174820b", + "metadata": { + "editable": true + }, "source": [ "# Solving Differential Equations with Deep Learning\n", "\n", @@ -10,7 +24,6 @@ "approximate any function at a single hidden layer along with one input\n", "and output layer to any given precision. \n", "\n", - "\n", "An ordinary differential equation (ODE) is an equation involving functions having one variable.\n", "\n", "In general, an ordinary differential equation looks like" @@ -18,7 +31,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ac838a12", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -32,7 +48,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ccee4286", + "metadata": { + "editable": true + }, "source": [ "where $g(x)$ is the function to find, and $g^{(n)}(x)$ is the $n$-th derivative of $g(x)$.\n", "\n", @@ -42,14 +61,15 @@ "Along with ([1](#ode)), some additional conditions of the function $g(x)$ are typically given\n", "for the solution to be unique.\n", "\n", - "\n", - "\n", "Let the trial solution $g_t(x)$ be" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "39d7dabd", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -64,7 +84,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "74f33170", + "metadata": { + "editable": true + }, "source": [ "where $h_1(x)$ is a function that makes $g_t(x)$ satisfy a given set\n", "of conditions, $N(x,P)$ a neural network with weights and biases\n", @@ -77,11 +100,8 @@ "\n", "But what about the network $N(x,P)$?\n", "\n", - "\n", "As described previously, an optimization method could be used to minimize the parameters of a neural network, that being its weights and biases, through backward propagation.\n", "\n", - "\n", - "\n", "For the minimization to be defined, we need to have a cost function at hand to minimize.\n", "\n", "It is given that $f\\left(x, \\, g(x), \\, g'(x), \\, g''(x), \\, \\dots \\, , \\, g^{(n)}(x)\\right)$ should be equal to zero in ([1](#ode)).\n", @@ -92,7 +112,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "2565ab16", + "metadata": { + "editable": true + }, "source": [ "$$\n", "C\\left(x, P\\right) = \\big(f\\left(x, \\, g(x), \\, g'(x), \\, g''(x), \\, \\dots \\, , \\, g^{(n)}(x)\\right)\\big)^2\n", @@ -101,7 +124,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "83d4ee25", + "metadata": { + "editable": true + }, "source": [ "If $N$ inputs are given as a vector $\\boldsymbol{x}$ with elements $x_i$ for $i = 1,\\dots,N$,\n", "the cost function becomes" @@ -109,7 +135,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a7a3a709", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -123,20 +152,28 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d3ca8026", + "metadata": { + "editable": true + }, "source": [ "The neural net should then find the parameters $P$ that minimizes the cost function in\n", "([3](#cost)) for a set of $N$ training samples $x_i$.\n", "\n", - "\n", - "\n", "To perform the minimization using gradient descent, the gradient of $C\\left(\\boldsymbol{x}, P\\right)$ is needed.\n", "It might happen so that finding an analytical expression of the gradient of $C(\\boldsymbol{x}, P)$ from ([3](#cost)) gets too messy, depending on which cost function one desires to use.\n", "\n", "Luckily, there exists libraries that makes the job for us through automatic differentiation.\n", - "Automatic differentiation is a method of finding the derivatives numerically with very high precision.\n", - "\n", - "\n", + "Automatic differentiation is a method of finding the derivatives numerically with very high precision." + ] + }, + { + "cell_type": "markdown", + "id": "9bde4bd4", + "metadata": { + "editable": true + }, + "source": [ "### Example: Exponential decay\n", "\n", "An exponential decay of a quantity $g(x)$ is described by the equation" @@ -144,7 +181,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e74337c3", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -158,7 +198,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "273549a4", + "metadata": { + "editable": true + }, "source": [ "with $g(0) = g_0$ for some chosen initial value $g_0$.\n", "\n", @@ -167,7 +210,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "db3c6623", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -182,18 +228,22 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "8547e3c8", + "metadata": { + "editable": true + }, "source": [ "Having an analytical solution at hand, it is possible to use it to compare how well a neural network finds a solution of ([4](#solve_expdec)).\n", "\n", - "\n", - "\n", "The program will use a neural network to solve" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "48341fc6", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -207,19 +257,24 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d25b1e02", + "metadata": { + "editable": true + }, "source": [ "where $g(0) = g_0$ with $\\gamma$ and $g_0$ being some chosen values.\n", "\n", "In this example, $\\gamma = 2$ and $g_0 = 10$.\n", "\n", - "\n", "To begin with, a trial solution $g_t(t)$ must be chosen. A general trial solution for ordinary differential equations could be" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "7c5dd91e", + "metadata": { + "editable": true + }, "source": [ "$$\n", "g_t(x, P) = h_1(x) + h_2(x, N(x, P))\n", @@ -228,12 +283,13 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "24223062", + "metadata": { + "editable": true + }, "source": [ "with $h_1(x)$ ensuring that $g_t(x)$ satisfies some conditions and $h_2(x,N(x, P))$ an expression involving $x$ and the output from the neural network $N(x,P)$ with $P $ being the collection of the weights and biases for each layer. For now, it is assumed that the network consists of one input layer, one hidden layer, and one output layer.\n", "\n", - "\n", - "\n", "In this network, there are no weights and bias at the input layer, so $P = \\{ P_{\\text{hidden}}, P_{\\text{output}} \\}$.\n", "If there are $N_{\\text{hidden} }$ neurons in the hidden layer, then $P_{\\text{hidden}}$ is a $N_{\\text{hidden} } \\times (1 + N_{\\text{input}})$ matrix, given that there are $N_{\\text{input}}$ neurons in the input layer.\n", "\n", @@ -247,7 +303,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ebf04383", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -261,7 +320,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ea4a0013", + "metadata": { + "editable": true + }, "source": [ "### Reformulating the problem\n", "\n", @@ -277,7 +339,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "2351b84f", + "metadata": { + "editable": true + }, "source": [ "$$\n", "g_t(x, P) = g_0 + x \\cdot N(x, P)\n", @@ -286,14 +351,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f6cc00e4", + "metadata": { + "editable": true + }, "source": [ "has been chosen such that it already solves the condition $g(0) = g_0$. What remains, is to find $P$ such that" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "84e066a1", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -307,11 +378,13 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "feff9ab3", + "metadata": { + "editable": true + }, "source": [ "is fulfilled as *best as possible*.\n", "\n", - "\n", "The left hand side and right hand side of ([8](#nnmin)) must be computed separately, and then the neural network must choose weights and biases, contained in $P$, such that the sides are equal as best as possible.\n", "This means that the absolute or squared difference between the sides must be as close to zero, ideally equal to zero.\n", "In this case, the difference squared shows to be an appropriate measurement of how erroneous the trial solution is with respect to $P$ of the neural network.\n", @@ -321,7 +394,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0c6f0e79", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\min_{P}\\Big\\{ \\big(g_t'(x, P) - ( -\\gamma g_t(x, P) \\big)^2 \\Big\\}\n", @@ -330,7 +406,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9302a1dd", + "metadata": { + "editable": true + }, "source": [ "(the notation $\\min_{P}\\{ f(x, P) \\}$ means that we desire to find $P$ that yields the minimum of $f(x, P)$)\n", "\n", @@ -339,7 +418,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f7f204bb", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\min_{P_{\\text{hidden} }, \\ P_{\\text{output} }}\\Big\\{ \\big(g_t'(x, \\{ P_{\\text{hidden} }, P_{\\text{output} }\\}) - ( -\\gamma g_t(x, \\{ P_{\\text{hidden} }, P_{\\text{output} }\\}) \\big)^2 \\Big\\}\n", @@ -348,18 +430,22 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "8d61d75f", + "metadata": { + "editable": true + }, "source": [ "for an input value $x$.\n", "\n", - "\n", - "\n", "If the neural network evaluates $g_t(x, P)$ at more values for $x$, say $N$ values $x_i$ for $i = 1, \\dots, N$, then the *total* error to minimize becomes" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "10d3aec9", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -373,14 +459,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5e296388", + "metadata": { + "editable": true + }, "source": [ "Letting $\\boldsymbol{x}$ be a vector with elements $x_i$ and $C(\\boldsymbol{x}, P) = \\frac{1}{N} \\sum_i \\big(g_t'(x_i, P) - ( -\\gamma g_t(x_i, P) \\big)^2$ denote the cost function, the minimization problem that our network must solve, becomes" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "fe010d79", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\min_{P} C(\\boldsymbol{x}, P)\n", @@ -389,7 +481,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "add715f9", + "metadata": { + "editable": true + }, "source": [ "In terms of $P_{\\text{hidden} }$ and $P_{\\text{output} }$, this could also be expressed as\n", "\n", @@ -397,20 +492,21 @@ "\\min_{P_{\\text{hidden} }, \\ P_{\\text{output} }} C(\\boldsymbol{x}, \\{P_{\\text{hidden} }, P_{\\text{output} }\\})\n", "$$\n", "\n", - "\n", "For simplicity, it is assumed that the input is an array $\\boldsymbol{x} = (x_1, \\dots, x_N)$ with $N$ elements. It is at these points the neural network should find $P$ such that it fulfills ([9](#min)).\n", "\n", "First, the neural network must feed forward the inputs.\n", "This means that $\\boldsymbol{x}s$ must be passed through an input layer, a hidden layer and a output layer. The input layer in this case, does not need to process the data any further.\n", "The input layer will consist of $N_{\\text{input} }$ neurons, passing its element to each neuron in the hidden layer. The number of neurons in the hidden layer will be $N_{\\text{hidden} }$.\n", "\n", - "\n", "For the $i$-th in the hidden layer with weight $w_i^{\\text{hidden} }$ and bias $b_i^{\\text{hidden} }$, the weighting from the $j$-th neuron at the input layer is:" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "cb9b22eb", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{aligned}\n", @@ -429,14 +525,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "49f3c493", + "metadata": { + "editable": true + }, "source": [ "The result after weighting the inputs at the $i$-th hidden neuron can be written as a vector:" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "a352bd61", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{aligned}\n", @@ -456,7 +558,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e5636ad7", + "metadata": { + "editable": true + }, "source": [ "The vector $\\boldsymbol{p}_{i, \\text{hidden}}^T$ constitutes each row in $P_{\\text{hidden} }$, which contains the weights for the neural network to minimize according to ([9](#min)).\n", "\n", @@ -467,7 +572,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "68bb8b9c", + "metadata": { + "editable": true + }, "source": [ "$$\n", "f(z) = \\frac{1}{1 + \\exp{(-z)}}\n", @@ -476,7 +584,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ff6e6536", + "metadata": { + "editable": true + }, "source": [ "It is possible to use other activations functions for the hidden layer also.\n", "\n", @@ -494,14 +605,15 @@ "and biases $b_i^{\\text{output}}$. In this case,\n", "it is assumes that the number of neurons in the output layer is one.\n", "\n", - "\n", - "\n", "The procedure of weighting the output neuron $j$ in the hidden layer to the $i$-th neuron in the output layer is similar as for the hidden layer described previously." ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "6fb4c55b", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{aligned}\n", @@ -519,14 +631,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "09c31d8d", + "metadata": { + "editable": true + }, "source": [ "Expressing $z_{1,j}^{\\text{output}}$ as a vector gives the following way of weighting the inputs from the hidden layer:" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "f81fe361", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{z}_{1}^{\\text{output}} =\n", @@ -542,11 +660,13 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "c309a4ce", + "metadata": { + "editable": true + }, "source": [ "In this case we seek a continuous range of values since we are approximating a function. This means that after computing $\\boldsymbol{z}_{1}^{\\text{output}}$ the neural network has finished its feed forward step, and $\\boldsymbol{z}_{1}^{\\text{output}}$ is the final output of the network.\n", "\n", - "\n", "The next step is to decide how the parameters should be changed such that they minimize the cost function.\n", "\n", "The chosen cost function for this problem is" @@ -554,7 +674,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ea47ae29", + "metadata": { + "editable": true + }, "source": [ "$$\n", "C(\\boldsymbol{x}, P) = \\frac{1}{N} \\sum_i \\big(g_t'(x_i, P) - ( -\\gamma g_t(x_i, P) \\big)^2\n", @@ -563,12 +686,23 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9dd83767", + "metadata": { + "editable": true + }, "source": [ "In order to minimize the cost function, an optimization method must be chosen.\n", "\n", - "Here, gradient descent with a constant step size has been chosen.\n", - "\n", + "Here, gradient descent with a constant step size has been chosen." + ] + }, + { + "cell_type": "markdown", + "id": "531a7b4f", + "metadata": { + "editable": true + }, + "source": [ "### Gradient descent\n", "\n", "The idea of the gradient descent algorithm is to update parameters in\n", @@ -581,7 +715,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e10c204a", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{\\omega}_{\\text{new} } = \\boldsymbol{\\omega} - \\lambda \\nabla_{\\boldsymbol{\\omega}} C(\\boldsymbol{x}, \\boldsymbol{\\omega})\n", @@ -590,7 +727,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9f4fa48e", + "metadata": { + "editable": true + }, "source": [ "for a number of iterations or until $ \\big|\\big| \\boldsymbol{\\omega}_{\\text{new} } - \\boldsymbol{\\omega} \\big|\\big|$ becomes smaller than some given tolerance.\n", "\n", @@ -609,7 +749,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "4652fc79", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{aligned}\n", @@ -621,14 +764,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "41ae0626", + "metadata": { + "editable": true + }, "source": [ "### The code for solving the ODE" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 1, + "id": "af7c165f", "metadata": { "collapsed": false, "editable": true @@ -785,7 +932,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "875a0c0b", + "metadata": { + "editable": true + }, "source": [ "## The network with one input layer, specified number of hidden layers, and one output layer\n", "\n", @@ -796,7 +946,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 2, + "id": "00684827", "metadata": { "collapsed": false, "editable": true @@ -965,7 +1116,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5420c490", + "metadata": { + "editable": true + }, "source": [ "### Example: Population growth\n", "\n", @@ -975,7 +1129,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "1436ed3c", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -989,7 +1146,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a590bdb6", + "metadata": { + "editable": true + }, "source": [ "where $g(t)$ is the population density at time $t$, $\\alpha > 0$ the growth rate and $A > 0$ is the maximum population number in the environment.\n", "Also, at $t = 0$ the population has the size $g(0) = g_0$, where $g_0$ is some chosen constant.\n", @@ -999,15 +1159,16 @@ "using a library like TensorFlow is recommended.\n", "Here, we stay with a more simple approach and implement for comparison, the simple forward Euler method.\n", "\n", - "\n", - "\n", "Here, we will model a population $g(t)$ in an environment having carrying capacity $A$.\n", "The population follows the model" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "48d788d6", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -1021,13 +1182,15 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "169c6c25", + "metadata": { + "editable": true + }, "source": [ "where $g(0) = g_0$.\n", "\n", "In this example, we let $\\alpha = 2$, $A = 1$, and $g_0 = 1.2$.\n", "\n", - "\n", "We will get a slightly different trial solution, as the boundary conditions are different\n", "compared to the case for exponential decay.\n", "\n", @@ -1045,14 +1208,13 @@ "g(t) = \\frac{Ag_0}{g_0 + (A - g_0)\\exp(-\\alpha A t)}\n", "$$\n", "\n", - "\n", - "\n", "The network will be the similar as for the exponential decay example, but with some small modifications for our problem." ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 3, + "id": "6d2e33bf", "metadata": { "collapsed": false, "editable": true @@ -1226,7 +1388,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e31ec549", + "metadata": { + "editable": true + }, "source": [ "## Using forward Euler to solve the ODE\n", "\n", @@ -1243,7 +1408,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "2e9ee105", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{aligned}\n", @@ -1255,7 +1423,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e876da16", + "metadata": { + "editable": true + }, "source": [ "along with the condition that $g(0) = g_0$.\n", "\n", @@ -1266,7 +1437,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b79a3def", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{aligned}\n", @@ -1279,14 +1453,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9d99cb4e", + "metadata": { + "editable": true + }, "source": [ "Now, if $g_i = g(t_i)$ then" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "3b9dcc24", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -1305,7 +1485,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "65ce688e", + "metadata": { + "editable": true + }, "source": [ "for $i \\geq 1$ and $g_0 = g(t_0) = g(0) = g_0$.\n", "\n", @@ -1315,7 +1498,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 4, + "id": "d5497948", "metadata": { "collapsed": false, "editable": true @@ -1391,7 +1575,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "fa8f0bb4", + "metadata": { + "editable": true + }, "source": [ "## Solving the one dimensional Poisson equation\n", "\n", @@ -1400,7 +1587,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f765b0ba", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -1414,7 +1604,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "bd63b92e", + "metadata": { + "editable": true + }, "source": [ "where $f(x)$ is a given function for $x \\in (0,1)$.\n", "\n", @@ -1423,7 +1616,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "c0a7face", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{align*}\n", @@ -1435,19 +1631,24 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f71c3cb9", + "metadata": { + "editable": true + }, "source": [ "This equation can be solved numerically using programs where e.g Autograd and TensorFlow are used.\n", "The results from the networks can then be compared to the analytical solution.\n", "In addition, it could be interesting to see how a typical method for numerically solving second order ODEs compares to the neural networks.\n", "\n", - "\n", "Here, the function $g(x)$ to solve for follows the equation" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "35aa6a37", + "metadata": { + "editable": true + }, "source": [ "$$\n", "-g''(x) = f(x),\\qquad x \\in (0,1)\n", @@ -1456,14 +1657,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f8b35111", + "metadata": { + "editable": true + }, "source": [ "where $f(x)$ is a given function, along with the chosen conditions" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "33813514", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -1477,7 +1684,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "c70aadd4", + "metadata": { + "editable": true + }, "source": [ "In this example, we consider the case when $f(x) = (3x + x^2)\\exp(x)$.\n", "\n", @@ -1486,7 +1696,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d5719dee", + "metadata": { + "editable": true + }, "source": [ "$$\n", "g_t(x) = x \\cdot (1-x) \\cdot N(P,x)\n", @@ -1495,14 +1708,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a683041b", + "metadata": { + "editable": true + }, "source": [ "The analytical solution for this problem is" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "672cdaff", + "metadata": { + "editable": true + }, "source": [ "$$\n", "g(x) = x(1 - x)\\exp(x)\n", @@ -1511,7 +1730,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 5, + "id": "66e6bebb", "metadata": { "collapsed": false, "editable": true @@ -1672,7 +1892,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d425cba5", + "metadata": { + "editable": true + }, "source": [ "### Comparing with a numerical scheme\n", "\n", @@ -1691,7 +1914,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a2efa110", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -1705,14 +1931,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5011ee25", + "metadata": { + "editable": true + }, "source": [ "If $x_i = i \\Delta x = x_{i-1} + \\Delta x$ and $g_i = g(x_i)$ for $i = 1,\\dots N_x - 2$ with $N_x$ being the number of values for $x$, ([15](#approx)) becomes" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "705ee300", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{aligned}\n", @@ -1724,14 +1956,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b390796d", + "metadata": { + "editable": true + }, "source": [ "Since we know from our problem that" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "19c9ece4", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{aligned}\n", @@ -1743,7 +1981,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6ade1a7a", + "metadata": { + "editable": true + }, "source": [ "along with the conditions $g(0) = g(1) = 0$,\n", "the following scheme can be used to find an approximate solution for $g(x)$ numerically:" @@ -1751,7 +1992,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "78b16d02", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -1768,7 +2012,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f2bfcca4", + "metadata": { + "editable": true + }, "source": [ "for $i = 1, \\dots, N_x - 2$ where $g_0 = g_{N_x - 1} = 0$ and $f(x_i) = (3x_i + x_i^2)\\exp(x_i)$, which is given for our specific problem.\n", "\n", @@ -1777,7 +2024,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a6191528", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{aligned}\n", @@ -1811,17 +2061,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "aefef707", + "metadata": { + "editable": true + }, "source": [ "which makes it possible to solve for the vector $\\boldsymbol{g}$.\n", "\n", - "\n", "We can then compare the result from this numerical scheme with the output from our network using Autograd:" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 6, + "id": "471664cd", "metadata": { "collapsed": false, "editable": true @@ -2022,7 +2275,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6f02e86d", + "metadata": { + "editable": true + }, "source": [ "## Partial Differential Equations\n", "\n", @@ -2036,7 +2292,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ad5c63e8", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -2050,10 +2309,21 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "c7b98797", + "metadata": { + "editable": true + }, + "source": [ + "where $f$ is an expression involving all kinds of possible mixed derivatives of $g(x_1,\\dots,x_N)$ up to an order $n$. In order for the solution to be unique, some additional conditions must also be given." + ] + }, + { + "cell_type": "markdown", + "id": "52f9394e", + "metadata": { + "editable": true + }, "source": [ - "where $f$ is an expression involving all kinds of possible mixed derivatives of $g(x_1,\\dots,x_N)$ up to an order $n$. In order for the solution to be unique, some additional conditions must also be given.\n", - "\n", "### Type of problem\n", "\n", "The problem our network must solve for, is similar to the ODE case.\n", @@ -2064,7 +2334,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "4d489854", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{align*}\n", @@ -2075,15 +2348,24 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "82311538", + "metadata": { + "editable": true + }, "source": [ "where $h_1(x_1,\\dots,x_N)$ is a function that ensures $g_t(x_1,\\dots,x_N)$ satisfies some given conditions.\n", "The neural network $N(x_1,\\dots,x_N,P)$ has weights and biases described by $P$ and $h_2(x_1,\\dots,x_N,N(x_1,\\dots,x_N,P))$ is an expression using the output from the neural network in some way.\n", "\n", - "The role of the function $h_2(x_1,\\dots,x_N,N(x_1,\\dots,x_N,P))$, is to ensure that the output of $N(x_1,\\dots,x_N,P)$ is zero when $g_t(x_1,\\dots,x_N)$ is evaluated at the values of $x_1,\\dots,x_N$ where the given conditions must be satisfied. The function $h_1(x_1,\\dots,x_N)$ should alone make $g_t(x_1,\\dots,x_N)$ satisfy the conditions.\n", - "\n", - "\n", - "\n", + "The role of the function $h_2(x_1,\\dots,x_N,N(x_1,\\dots,x_N,P))$, is to ensure that the output of $N(x_1,\\dots,x_N,P)$ is zero when $g_t(x_1,\\dots,x_N)$ is evaluated at the values of $x_1,\\dots,x_N$ where the given conditions must be satisfied. The function $h_1(x_1,\\dots,x_N)$ should alone make $g_t(x_1,\\dots,x_N)$ satisfy the conditions." + ] + }, + { + "cell_type": "markdown", + "id": "9d134db4", + "metadata": { + "editable": true + }, + "source": [ "### Network requirements\n", "\n", "The network tries then the minimize the cost function following the\n", @@ -2099,7 +2381,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5a463498", + "metadata": { + "editable": true + }, "source": [ "$$\n", "C\\left(x_1, \\dots, x_N, P\\right) = \\left( f\\left(x_1, \\, \\dots \\, , x_N, \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_1}, \\dots , \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_N}, \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(x_1,\\dots,x_N) }{\\partial x_N^n} \\right) \\right)^2\n", @@ -2108,14 +2393,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "c40d8997", + "metadata": { + "editable": true + }, "source": [ "If we let $\\boldsymbol{x} = \\big( x_1, \\dots, x_N \\big)$ be an array containing the values for $x_1, \\dots, x_N$ respectively, the cost function can be reformulated into the following:" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "cc033de6", + "metadata": { + "editable": true + }, "source": [ "$$\n", "C\\left(\\boldsymbol{x}, P\\right) = f\\left( \\left( \\boldsymbol{x}, \\frac{\\partial g(\\boldsymbol{x}) }{\\partial x_1}, \\dots , \\frac{\\partial g(\\boldsymbol{x}) }{\\partial x_N}, \\frac{\\partial g(\\boldsymbol{x}) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(\\boldsymbol{x}) }{\\partial x_N^n} \\right) \\right)^2\n", @@ -2124,14 +2415,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ae18b4f4", + "metadata": { + "editable": true + }, "source": [ "If we also have $M$ different sets of values for $x_1, \\dots, x_N$, that is $\\boldsymbol{x}_i = \\big(x_1^{(i)}, \\dots, x_N^{(i)}\\big)$ for $i = 1,\\dots,M$ being the rows in matrix $X$, the cost function can be generalized into" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "44ca21bd", + "metadata": { + "editable": true + }, "source": [ "$$\n", "C\\left(X, P \\right) = \\sum_{i=1}^M f\\left( \\left( \\boldsymbol{x}_i, \\frac{\\partial g(\\boldsymbol{x}_i) }{\\partial x_1}, \\dots , \\frac{\\partial g(\\boldsymbol{x}_i) }{\\partial x_N}, \\frac{\\partial g(\\boldsymbol{x}_i) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(\\boldsymbol{x}_i) }{\\partial x_N^n} \\right) \\right)^2.\n", @@ -2140,7 +2437,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "fe94451e", + "metadata": { + "editable": true + }, "source": [ "## Example: The diffusion equation\n", "\n", @@ -2149,7 +2449,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "30a42273", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial g(x,t)}{\\partial t} = \\frac{\\partial^2 g(x,t)}{\\partial x^2}\n", @@ -2158,14 +2461,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "fd4077e7", + "metadata": { + "editable": true + }, "source": [ "where a possible choice of conditions are" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "dbbf2e4b", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{align*}\n", @@ -2178,18 +2487,22 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "83653db9", + "metadata": { + "editable": true + }, "source": [ "with $u(x)$ being some given function.\n", "\n", - "\n", - "\n", "For this case, we want to find $g(x,t)$ such that" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "d2c2ef4f", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -2203,14 +2516,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "8fb27bdb", + "metadata": { + "editable": true + }, "source": [ "and" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "dc9297ab", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{align*}\n", @@ -2223,7 +2542,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ab90784f", + "metadata": { + "editable": true + }, "source": [ "with $u(x) = \\sin(\\pi x)$.\n", "\n", @@ -2231,9 +2553,6 @@ "The deep neural network will follow the same structure as discussed in the examples solving the ODEs.\n", "First, we will look into how Autograd could be used in a network tailored to solve for bivariate functions.\n", "\n", - "\n", - "\n", - "\n", "The only change to do here, is to extend our network such that\n", "functions of multiple parameters are correctly handled. In this case\n", "we have two variables in our function to solve for, that is time $t$\n", @@ -2245,7 +2564,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 7, + "id": "b07e858c", "metadata": { "collapsed": false, "editable": true @@ -2300,7 +2620,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9a167ec1", + "metadata": { + "editable": true + }, "source": [ "The cost function must then iterate through the given arrays\n", "containing values for $x$ and $t$, defines a point $(x,t)$ the deep\n", @@ -2322,8 +2645,6 @@ "$$\n", "since $(0) = u(1) = 0$ and $u(x) = \\sin(\\pi x)$.\n", "\n", - "\n", - "\n", "The Jacobian is used because the program must find the derivative of\n", "the trial solution with respect to $x$ and $t$.\n", "\n", @@ -2345,7 +2666,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 8, + "id": "921e5969", "metadata": { "collapsed": false, "editable": true @@ -2392,7 +2714,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0bc00b69", + "metadata": { + "editable": true + }, "source": [ "### Setting up the network using Autograd; The full program\n", "\n", @@ -2408,14 +2733,14 @@ "Be aware, though, that it is fairly slow for the parameters used.\n", "A better result is possible, but requires more iterations, and thus longer time to complete.\n", "\n", - "\n", "Indeed, the program below is not optimal in its implementation, but rather serves as an example on how to implement and use a neural network to solve a PDE.\n", "Using TensorFlow results in a much better execution time. Try it!" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 9, + "id": "20734418", "metadata": { "collapsed": false, "editable": true @@ -2649,7 +2974,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "143f1c80", + "metadata": { + "editable": true + }, "source": [ "## Solving the wave equation with Neural Networks\n", "\n", @@ -2658,7 +2986,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "190bd4f2", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial^2 g(x,t)}{\\partial t^2} = c^2\\frac{\\partial^2 g(x,t)}{\\partial x^2}\n", @@ -2667,7 +2998,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9719cfc5", + "metadata": { + "editable": true + }, "source": [ "with $c$ being the specified wave speed.\n", "\n", @@ -2676,7 +3010,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f69ec7fd", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{align*}\n", @@ -2690,17 +3027,22 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6ce8c5a8", + "metadata": { + "editable": true + }, "source": [ "where $\\frac{\\partial g(x,t)}{\\partial t} \\Big |_{t = 0}$ means the derivative of $g(x,t)$ with respect to $t$ is evaluated at $t = 0$, and $u(x)$ and $v(x)$ being given functions.\n", "\n", - "\n", "The wave equation to solve for, is" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "4be700d7", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -2714,7 +3056,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "993f93ba", + "metadata": { + "editable": true + }, "source": [ "where $c$ is the given wave speed.\n", "The chosen conditions for this equation are" @@ -2722,7 +3067,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "2cb2a80f", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -2739,12 +3087,13 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "3f1dcfd4", + "metadata": { + "editable": true + }, "source": [ "In this example, let $c = 1$ and $u(x) = \\sin(\\pi x)$ and $v(x) = -\\pi\\sin(\\pi x)$.\n", "\n", - "\n", - "\n", "Setting up the network is done in similar matter as for the example of solving the diffusion equation.\n", "The only things we have to change, is the trial solution such that it satisfies the conditions from ([20](#condwave)) and the cost function.\n", "\n", @@ -2762,7 +3111,6 @@ "\n", "Note that this trial solution satisfies the conditions only if $u(0) = v(0) = u(1) = v(1) = 0$, which is the case in this example.\n", "\n", - "\n", "The analytical solution for our specific problem, is\n", "\n", "$$\n", @@ -2772,7 +3120,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 10, + "id": "230a9aef", "metadata": { "collapsed": false, "editable": true @@ -3003,7 +3352,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a23bd19a", + "metadata": { + "editable": true + }, "source": [ "## Resources on differential equations and deep learning\n", "\n", @@ -3019,5 +3371,5 @@ ], "metadata": {}, "nbformat": 4, - "nbformat_minor": 4 + "nbformat_minor": 5 } diff --git a/doc/LectureNotes/_build/html/_sources/chapter12.ipynb b/doc/LectureNotes/_build/html/_sources/chapter12.ipynb new file mode 100644 index 000000000..c0569826b --- /dev/null +++ b/doc/LectureNotes/_build/html/_sources/chapter12.ipynb @@ -0,0 +1,1577 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "b1ad0b52", + "metadata": { + "editable": true + }, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "id": "9548423d", + "metadata": { + "editable": true + }, + "source": [ + "# Convolutional Neural Networks\n", + "\n", + "Convolutional neural networks (CNNs) were developed during the last\n", + "decade of the previous century, with a focus on character recognition\n", + "tasks. Nowadays, CNNs are a central element in the spectacular success\n", + "of deep learning methods. The success in for example image\n", + "classifications have made them a central tool for most machine\n", + "learning practitioners.\n", + "\n", + "CNNs are very similar to ordinary Neural Networks.\n", + "They are made up of neurons that have learnable weights and\n", + "biases. Each neuron receives some inputs, performs a dot product and\n", + "optionally follows it with a non-linearity. The whole network still\n", + "expresses a single differentiable score function: from the raw image\n", + "pixels on one end to class scores at the other. And they still have a\n", + "loss function (for example Softmax) on the last (fully-connected) layer\n", + "and all the tips/tricks we developed for learning regular Neural\n", + "Networks still apply (back propagation, gradient descent etc etc).\n", + "\n", + "**CNN architectures make the explicit assumption that\n", + "the inputs are images, which allows us to encode certain properties\n", + "into the architecture. These then make the forward function more\n", + "efficient to implement and vastly reduce the amount of parameters in\n", + "the network.**\n", + "\n", + "Here we provide only a superficial overview, for the more interested, we recommend highly the course\n", + "[IN5400 – Machine Learning for Image Analysis](https://www.uio.no/studier/emner/matnat/ifi/IN5400/index-eng.html)\n", + "and the slides of [CS231](http://cs231n.github.io/convolutional-networks/).\n", + "\n", + "Another good read is the article here ." + ] + }, + { + "cell_type": "markdown", + "id": "315b8308", + "metadata": { + "editable": true + }, + "source": [ + "## Neural Networks vs CNNs\n", + "\n", + "Neural networks are defined as **affine transformations**, that is \n", + "a vector is received as input and is multiplied with a matrix of so-called weights (our unknown paramters) to produce an\n", + "output (to which a bias vector is usually added before passing the result\n", + "through a nonlinear activation function). This is applicable to any type of input, be it an\n", + "image, a sound clip or an unordered collection of features: whatever their\n", + "dimensionality, their representation can always be flattened into a vector\n", + "before the transformation.\n", + "\n", + "However, when we consider images, sound clips and many other similar kinds of data, these data have an intrinsic\n", + "structure. More formally, they share these important properties:\n", + "* They are stored as multi-dimensional arrays (think of the pixels of a figure) .\n", + "\n", + "* They feature one or more axes for which ordering matters (e.g., width and height axes for an image, time axis for a sound clip).\n", + "\n", + "* One axis, called the channel axis, is used to access different views of the data (e.g., the red, green and blue channels of a color image, or the left and right channels of a stereo audio track).\n", + "\n", + "These properties are not exploited when an affine transformation is applied; in\n", + "fact, all the axes are treated in the same way and the topological information\n", + "is not taken into account. Still, taking advantage of the implicit structure of\n", + "the data may prove very handy in solving some tasks, like computer vision and\n", + "speech recognition, and in these cases it would be best to preserve it. This is\n", + "where discrete convolutions come into play.\n", + "\n", + "A discrete convolution is a linear transformation that preserves this notion of\n", + "ordering. It is sparse (only a few input units contribute to a given output\n", + "unit) and reuses parameters (the same weights are applied to multiple locations\n", + "in the input).\n", + "\n", + "As an example, consider\n", + "an image of size $32\\times 32\\times 3$ (32 wide, 32 high, 3 color channels), so a\n", + "single fully-connected neuron in a first hidden layer of a regular\n", + "Neural Network would have $32\\times 32\\times 3 = 3072$ weights. This amount still\n", + "seems manageable, but clearly this fully-connected structure does not\n", + "scale to larger images. For example, an image of more respectable\n", + "size, say $200\\times 200\\times 3$, would lead to neurons that have \n", + "$200\\times 200\\times 3 = 120,000$ weights. \n", + "\n", + "We could have\n", + "several such neurons, and the parameters would add up quickly! Clearly,\n", + "this full connectivity is wasteful and the huge number of parameters\n", + "would quickly lead to possible overfitting.\n", + "\n", + "\n", + "\n", + "\n", + "

Figure 1: A regular 3-layer Neural Network.

\n", + "\n", + "\n", + "Convolutional Neural Networks take advantage of the fact that the\n", + "input consists of images and they constrain the architecture in a more\n", + "sensible way. \n", + "\n", + "In particular, unlike a regular Neural Network, the\n", + "layers of a CNN have neurons arranged in 3 dimensions: width,\n", + "height, depth. (Note that the word depth here refers to the third\n", + "dimension of an activation volume, not to the depth of a full Neural\n", + "Network, which can refer to the total number of layers in a network.)\n", + "\n", + "To understand it better, the above example of an image \n", + "with an input volume of\n", + "activations has dimensions $32\\times 32\\times 3$ (width, height,\n", + "depth respectively). \n", + "\n", + "The neurons in a layer will\n", + "only be connected to a small region of the layer before it, instead of\n", + "all of the neurons in a fully-connected manner. Moreover, the final\n", + "output layer could for this specific image have dimensions $1\\times 1 \\times 10$, \n", + "because by the\n", + "end of the CNN architecture we will reduce the full image into a\n", + "single vector of class scores, arranged along the depth\n", + "dimension. \n", + "\n", + "\n", + "\n", + "\n", + "

Figure 1: A CNN arranges its neurons in three dimensions (width, height, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D output volume of neuron activations. In this example, the red input layer holds the image, so its width and height would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels).

\n", + "" + ] + }, + { + "cell_type": "markdown", + "id": "a142cd5d", + "metadata": { + "editable": true + }, + "source": [ + "## Layers used to build CNNs\n", + "\n", + "A simple CNN is a sequence of layers, and every layer of a CNN\n", + "transforms one volume of activations to another through a\n", + "differentiable function. We use three main types of layers to build\n", + "CNN architectures: Convolutional Layer, Pooling Layer, and\n", + "Fully-Connected Layer (exactly as seen in regular Neural Networks). We\n", + "will stack these layers to form a full CNN architecture.\n", + "\n", + "A simple CNN for image classification could have the architecture:\n", + "\n", + "* **INPUT** ($32\\times 32 \\times 3$) will hold the raw pixel values of the image, in this case an image of width 32, height 32, and with three color channels R,G,B.\n", + "\n", + "* **CONV** (convolutional )layer will compute the output of neurons that are connected to local regions in the input, each computing a dot product between their weights and a small region they are connected to in the input volume. This may result in volume such as $[32\\times 32\\times 12]$ if we decided to use 12 filters.\n", + "\n", + "* **RELU** layer will apply an elementwise activation function, such as the $max(0,x)$ thresholding at zero. This leaves the size of the volume unchanged ($[32\\times 32\\times 12]$).\n", + "\n", + "* **POOL** (pooling) layer will perform a downsampling operation along the spatial dimensions (width, height), resulting in volume such as $[16\\times 16\\times 12]$.\n", + "\n", + "* **FC** (i.e. fully-connected) layer will compute the class scores, resulting in volume of size $[1\\times 1\\times 10]$, where each of the 10 numbers correspond to a class score, such as among the 10 categories of the MNIST images we considered above . As with ordinary Neural Networks and as the name implies, each neuron in this layer will be connected to all the numbers in the previous volume.\n", + "\n", + "CNNs transform the original image layer by layer from the original\n", + "pixel values to the final class scores. \n", + "\n", + "Observe that some layers contain\n", + "parameters and other don’t. In particular, the CNN layers perform\n", + "transformations that are a function of not only the activations in the\n", + "input volume, but also of the parameters (the weights and biases of\n", + "the neurons). On the other hand, the RELU/POOL layers will implement a\n", + "fixed function. The parameters in the CONV/FC layers will be trained\n", + "with gradient descent so that the class scores that the CNN computes\n", + "are consistent with the labels in the training set for each image.\n", + "\n", + "In summary:\n", + "\n", + "* A CNN architecture is in the simplest case a list of Layers that transform the image volume into an output volume (e.g. holding the class scores)\n", + "\n", + "* There are a few distinct types of Layers (e.g. CONV/FC/RELU/POOL are by far the most popular)\n", + "\n", + "* Each Layer accepts an input 3D volume and transforms it to an output 3D volume through a differentiable function\n", + "\n", + "* Each Layer may or may not have parameters (e.g. CONV/FC do, RELU/POOL don’t)\n", + "\n", + "* Each Layer may or may not have additional hyperparameters (e.g. CONV/FC/POOL do, RELU doesn’t)\n", + "\n", + "A dense neural network is representd by an affine operation (like matrix-matrix multiplication) where all parameters are included.\n", + "\n", + "The key idea in CNNs for say imaging is that in images neighbor pixels tend to be related! So we connect\n", + "only neighboring neurons in the input instead of connecting all with the first hidden layer.\n", + "\n", + "We say we perform a filtering (convolution is the mathematical operation)." + ] + }, + { + "cell_type": "markdown", + "id": "bb1e3617", + "metadata": { + "editable": true + }, + "source": [ + "## Mathematics of CNNs\n", + "\n", + "The mathematics of CNNs is based on the mathematical operation of\n", + "**convolution**. In mathematics (in particular in functional analysis),\n", + "convolution is represented by matheematical operation (integration,\n", + "summation etc) on two function in order to produce a third function\n", + "that expresses how the shape of one gets modified by the other.\n", + "Convolution has a plethora of applications in a variety of disciplines, spanning from statistics to signal processing, computer vision, solutions of differential equations,linear algebra, engineering, and yes, machine learning.\n", + "\n", + "Mathematically, convolution is defined as follows (one-dimensional example):\n", + "Let us define a continuous function $y(t)$ given by" + ] + }, + { + "cell_type": "markdown", + "id": "051485e5", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "y(t) = \\int x(a) w(t-a) da,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "799b8e5f", + "metadata": { + "editable": true + }, + "source": [ + "where $x(a)$ represents a so-called input and $w(t-a)$ is normally called the weight function or kernel.\n", + "\n", + "The above integral is written in a more compact form as" + ] + }, + { + "cell_type": "markdown", + "id": "3353345f", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "y(t) = \\left(x * w\\right)(t).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "ef341ee6", + "metadata": { + "editable": true + }, + "source": [ + "The discretized version reads" + ] + }, + { + "cell_type": "markdown", + "id": "ca8f7582", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "y(t) = \\sum_{a=-\\infty}^{a=\\infty}x(a)w(t-a).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "216e011f", + "metadata": { + "editable": true + }, + "source": [ + "Computing the inverse of the above convolution operations is known as deconvolution.\n", + "\n", + "How can we use this? And what does it mean? Let us study some familiar examples first." + ] + }, + { + "cell_type": "markdown", + "id": "5545738c", + "metadata": { + "editable": true + }, + "source": [ + "### Convolution Examples: Polynomial multiplication\n", + "\n", + "We have already met such an example in project 1 when we tried to set\n", + "up the design matrix for a two-dimensional function. This was an\n", + "example of polynomial multiplication. Let us recast such a problem in terms of the convolution operation.\n", + "Let us look a the following polynomials to second and third order, respectively:" + ] + }, + { + "cell_type": "markdown", + "id": "9f317908", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(t) = \\alpha_0+\\alpha_1 t+\\alpha_2 t^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "eb6f1070", + "metadata": { + "editable": true + }, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "9fe61b3c", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "s(t) = \\beta_0+\\beta_1 t+\\beta_2 t^2+\\beta_3 t^3.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "38c68ea7", + "metadata": { + "editable": true + }, + "source": [ + "The polynomial multiplication gives us a new polynomial of degree $5$" + ] + }, + { + "cell_type": "markdown", + "id": "3e91a9f1", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "z(t) = \\delta_0+\\delta_1 t+\\delta_2 t^2+\\delta_3 t^3+\\delta_4 t^4+\\delta_5 t^5.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "5e24ac44", + "metadata": { + "editable": true + }, + "source": [ + "Computing polynomial products can be implemented efficiently if we rewrite the more brute force multiplications using convolution.\n", + "We note first that the new coefficients are given as" + ] + }, + { + "cell_type": "markdown", + "id": "ac215eb9", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{split}\n", + "\\delta_0=&\\alpha_0\\beta_0\\\\\n", + "\\delta_1=&\\alpha_1\\beta_0+\\alpha_1\\beta_0\\\\\n", + "\\delta_2=&\\alpha_0\\beta_2+\\alpha_1\\beta_1+\\alpha_2\\beta_0\\\\\n", + "\\delta_3=&\\alpha_1\\beta_2+\\alpha_2\\beta_1+\\alpha_0\\beta_3\\\\\n", + "\\delta_4=&\\alpha_2\\beta_2+\\alpha_1\\beta_3\\\\\n", + "\\delta_5=&\\alpha_2\\beta_3.\\\\\n", + "\\end{split}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "eca51025", + "metadata": { + "editable": true + }, + "source": [ + "We note that $\\alpha_i=0$ except for $i\\in \\left\\{0,1,2\\right\\}$ and $\\beta_i=0$ except for $i\\in\\left\\{0,1,2,3\\right\\}$.\n", + "\n", + "We can then rewrite the coefficients $\\delta_j$ using a discrete convolution as" + ] + }, + { + "cell_type": "markdown", + "id": "0d517816", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\delta_j = \\sum_{i=-\\infty}^{i=\\infty}\\alpha_i\\beta_{j-i}=(\\alpha * \\beta)_j,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "8cf1037e", + "metadata": { + "editable": true + }, + "source": [ + "or as a double sum with restriction $l=i+j$" + ] + }, + { + "cell_type": "markdown", + "id": "efe18d99", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\delta_l = \\sum_{ij}\\alpha_i\\beta_{j}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "14ae41db", + "metadata": { + "editable": true + }, + "source": [ + "Do you see a potential drawback with these equations?\n", + "\n", + "Since we only have a finite number of $\\alpha$ and $\\beta$ values\n", + "which are non-zero, we can rewrite the above convolution expressions\n", + "as a matrix-vector multiplication" + ] + }, + { + "cell_type": "markdown", + "id": "046e108b", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{\\delta}=\\begin{bmatrix}\\alpha_0 & 0 & 0 & 0 \\\\\n", + " \\alpha_1 & \\alpha_0 & 0 & 0 \\\\\n", + "\t\t\t \\alpha_2 & \\alpha_1 & \\alpha_0 & 0 \\\\\n", + "\t\t\t 0 & \\alpha_2 & \\alpha_1 & \\alpha_0 \\\\\n", + "\t\t\t 0 & 0 & \\alpha_2 & \\alpha_1 \\\\\n", + "\t\t\t 0 & 0 & 0 & \\alpha_2\n", + "\t\t\t \\end{bmatrix}\\begin{bmatrix} \\beta_0 \\\\ \\beta_1 \\\\ \\beta_2 \\\\ \\beta_3\\end{bmatrix}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "0c73a726", + "metadata": { + "editable": true + }, + "source": [ + "The process is commutative and we can easily see that we can rewrite the multiplication in terms of a matrix holding $\\beta$ and a vector holding $\\alpha$.\n", + "In this case we have" + ] + }, + { + "cell_type": "markdown", + "id": "81bc84d0", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{\\delta}=\\begin{bmatrix}\\beta_0 & 0 & 0 \\\\\n", + " \\beta_1 & \\beta_0 & 0 \\\\\n", + "\t\t\t \\beta_2 & \\beta_1 & \\beta_0 \\\\\n", + "\t\t\t \\beta_3 & \\beta_2 & \\beta_1 \\\\\n", + "\t\t\t 0 & \\beta_3 & \\beta_2 \\\\\n", + "\t\t\t 0 & 0 & \\beta_3\n", + "\t\t\t \\end{bmatrix}\\begin{bmatrix} \\alpha_0 \\\\ \\alpha_1 \\\\ \\alpha_2\\end{bmatrix}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "6f314582", + "metadata": { + "editable": true + }, + "source": [ + "Note that the use of these matrices is for mathematical purposes only and not implementation purposes.\n", + "When implementing the above equation we do not encode (and allocate memory) the matrices explicitely.\n", + "We rather code the convolutions in the minimal memory footprint that they require.\n", + "\n", + "Does the number of floating point operations change here when we use the commutative property?" + ] + }, + { + "cell_type": "markdown", + "id": "e01eb276", + "metadata": { + "editable": true + }, + "source": [ + "### Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)\n", + "\n", + "For problems with so-called harmonic oscillations, given by for example the following differential equation" + ] + }, + { + "cell_type": "markdown", + "id": "1016b422", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "m\\frac{d^2x}{dt^2}+\\eta\\frac{dx}{dt}+x(t)=F(t),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "c237cb74", + "metadata": { + "editable": true + }, + "source": [ + "where $F(t)$ is an applied external force acting on the system (often called a driving force), one can use the theory of Fourier transformations to find the solutions of this type of equations.\n", + "\n", + "If one has several driving forces, $F(t)=\\sum_n F_n(t)$, one can find\n", + "the particular solution to each $F_n$, $x_{pn}(t)$, and the particular\n", + "solution for the entire driving force is then given by a series like" + ] + }, + { + "cell_type": "markdown", + "id": "4f2f2089", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + "x_p(t)=\\sum_nx_{pn}(t).\n", + "\\label{_auto1} \\tag{1}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "a32fa928", + "metadata": { + "editable": true + }, + "source": [ + "This is known as the principle of superposition. It only applies when\n", + "the homogenous equation is linear. If there were an anharmonic term\n", + "such as $x^3$ in the homogenous equation, then when one summed various\n", + "solutions, $x=(\\sum_n x_n)^2$, one would get cross\n", + "terms. Superposition is especially useful when $F(t)$ can be written\n", + "as a sum of sinusoidal terms, because the solutions for each\n", + "sinusoidal (sine or cosine) term is analytic. \n", + "\n", + "Driving forces are often periodic, even when they are not\n", + "sinusoidal. Periodicity implies that for some time $\\tau$" + ] + }, + { + "cell_type": "markdown", + "id": "0b6ca2c5", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{eqnarray}\n", + "F(t+\\tau)=F(t). \n", + "\\end{eqnarray}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "789cd636", + "metadata": { + "editable": true + }, + "source": [ + "One example of a non-sinusoidal periodic force is a square wave. Many\n", + "components in electric circuits are non-linear, e.g. diodes, which\n", + "makes many wave forms non-sinusoidal even when the circuits are being\n", + "driven by purely sinusoidal sources.\n", + "\n", + "The code here shows a typical example of such a square wave generated using the functionality included in the **scipy** Python package. We have used a period of $\\tau=0.2$." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "2c90eb89", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "%matplotlib inline\n", + "\n", + "import numpy as np\n", + "import math\n", + "from scipy import signal\n", + "import matplotlib.pyplot as plt\n", + "\n", + "# number of points \n", + "n = 500\n", + "# start and final times \n", + "t0 = 0.0\n", + "tn = 1.0\n", + "# Period \n", + "t = np.linspace(t0, tn, n, endpoint=False)\n", + "SqrSignal = np.zeros(n)\n", + "SqrSignal = 1.0+signal.square(2*np.pi*5*t)\n", + "plt.plot(t, SqrSignal)\n", + "plt.ylim(-0.5, 2.5)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "8e7b8f5c", + "metadata": { + "editable": true + }, + "source": [ + "For the sinusoidal example the\n", + "period is $\\tau=2\\pi/\\omega$. However, higher harmonics can also\n", + "satisfy the periodicity requirement. In general, any force that\n", + "satisfies the periodicity requirement can be expressed as a sum over\n", + "harmonics," + ] + }, + { + "cell_type": "markdown", + "id": "4453c025", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + "F(t)=\\frac{f_0}{2}+\\sum_{n>0} f_n\\cos(2n\\pi t/\\tau)+g_n\\sin(2n\\pi t/\\tau).\n", + "\\label{_auto2} \\tag{2}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "32d5f410", + "metadata": { + "editable": true + }, + "source": [ + "We can write down the answer for\n", + "$x_{pn}(t)$, by substituting $f_n/m$ or $g_n/m$ for $F_0/m$. By\n", + "writing each factor $2n\\pi t/\\tau$ as $n\\omega t$, with $\\omega\\equiv\n", + "2\\pi/\\tau$," + ] + }, + { + "cell_type": "markdown", + "id": "4a58698d", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\\label{eq:fourierdef1} \\tag{3}\n", + "F(t)=\\frac{f_0}{2}+\\sum_{n>0}f_n\\cos(n\\omega t)+g_n\\sin(n\\omega t).\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "5664f63d", + "metadata": { + "editable": true + }, + "source": [ + "The solutions for $x(t)$ then come from replacing $\\omega$ with\n", + "$n\\omega$ for each term in the particular solution," + ] + }, + { + "cell_type": "markdown", + "id": "0d9a2811", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{eqnarray}\n", + "x_p(t)&=&\\frac{f_0}{2k}+\\sum_{n>0} \\alpha_n\\cos(n\\omega t-\\delta_n)+\\beta_n\\sin(n\\omega t-\\delta_n),\\\\\n", + "\\nonumber\n", + "\\alpha_n&=&\\frac{f_n/m}{\\sqrt{((n\\omega)^2-\\omega_0^2)+4\\beta^2n^2\\omega^2}},\\\\\n", + "\\nonumber\n", + "\\beta_n&=&\\frac{g_n/m}{\\sqrt{((n\\omega)^2-\\omega_0^2)+4\\beta^2n^2\\omega^2}},\\\\\n", + "\\nonumber\n", + "\\delta_n&=&\\tan^{-1}\\left(\\frac{2\\beta n\\omega}{\\omega_0^2-n^2\\omega^2}\\right).\n", + "\\end{eqnarray}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "ac8b5fff", + "metadata": { + "editable": true + }, + "source": [ + "Because the forces have been applied for a long time, any non-zero\n", + "damping eliminates the homogenous parts of the solution, so one need\n", + "only consider the particular solution for each $n$.\n", + "\n", + "The problem is considered solved if one can find expressions for the\n", + "coefficients $f_n$ and $g_n$, even though the solutions are expressed\n", + "as an infinite sum. The coefficients can be extracted from the\n", + "function $F(t)$ by" + ] + }, + { + "cell_type": "markdown", + "id": "5bb70ae9", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{eqnarray}\n", + "\\label{eq:fourierdef2} \\tag{4}\n", + "f_n&=&\\frac{2}{\\tau}\\int_{-\\tau/2}^{\\tau/2} dt~F(t)\\cos(2n\\pi t/\\tau),\\\\\n", + "\\nonumber\n", + "g_n&=&\\frac{2}{\\tau}\\int_{-\\tau/2}^{\\tau/2} dt~F(t)\\sin(2n\\pi t/\\tau).\n", + "\\end{eqnarray}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "4f64082e", + "metadata": { + "editable": true + }, + "source": [ + "To check the consistency of these expressions and to verify\n", + "Eq. ([4](#eq:fourierdef2)), one can insert the expansion of $F(t)$ in\n", + "Eq. ([3](#eq:fourierdef1)) into the expression for the coefficients in\n", + "Eq. ([4](#eq:fourierdef2)) and see whether" + ] + }, + { + "cell_type": "markdown", + "id": "bdd9e4e8", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{eqnarray}\n", + "f_n&=?&\\frac{2}{\\tau}\\int_{-\\tau/2}^{\\tau/2} dt~\\left\\{\n", + "\\frac{f_0}{2}+\\sum_{m>0}f_m\\cos(m\\omega t)+g_m\\sin(m\\omega t)\n", + "\\right\\}\\cos(n\\omega t).\n", + "\\end{eqnarray}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e64a4a4c", + "metadata": { + "editable": true + }, + "source": [ + "Immediately, one can throw away all the terms with $g_m$ because they\n", + "convolute an even and an odd function. The term with $f_0/2$\n", + "disappears because $\\cos(n\\omega t)$ is equally positive and negative\n", + "over the interval and will integrate to zero. For all the terms\n", + "$f_m\\cos(m\\omega t)$ appearing in the sum, one can use angle addition\n", + "formulas to see that $\\cos(m\\omega t)\\cos(n\\omega\n", + "t)=(1/2)(\\cos[(m+n)\\omega t]+\\cos[(m-n)\\omega t]$. This will integrate\n", + "to zero unless $m=n$. In that case the $m=n$ term gives" + ] + }, + { + "cell_type": "markdown", + "id": "d51a4f59", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\\int_{-\\tau/2}^{\\tau/2}dt~\\cos^2(m\\omega t)=\\frac{\\tau}{2},\n", + "\\label{_auto3} \\tag{5}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "44804e35", + "metadata": { + "editable": true + }, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "7de0dbe5", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{eqnarray}\n", + "f_n&=?&\\frac{2}{\\tau}\\int_{-\\tau/2}^{\\tau/2} dt~f_n/2\\\\\n", + "\\nonumber\n", + "&=&f_n~\\checkmark.\n", + "\\end{eqnarray}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "4d1a02e0", + "metadata": { + "editable": true + }, + "source": [ + "The same method can be used to check for the consistency of $g_n$.\n", + "\n", + "The code here uses the Fourier series applied to a \n", + "square wave signal. The code here\n", + "visualizes the various approximations given by Fourier series compared\n", + "with a square wave with period $T=0.2$ (dimensionless time), width $0.1$ and max value of the force $F=2$. We\n", + "see that when we increase the number of components in the Fourier\n", + "series, the Fourier series approximation gets closer and closer to the\n", + "square wave signal." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "4ae0ec42", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "import math\n", + "from scipy import signal\n", + "import matplotlib.pyplot as plt\n", + "\n", + "# number of points \n", + "n = 500\n", + "# start and final times \n", + "t0 = 0.0\n", + "tn = 1.0\n", + "# Period \n", + "T =0.2\n", + "# Max value of square signal \n", + "Fmax= 2.0\n", + "# Width of signal \n", + "Width = 0.1\n", + "t = np.linspace(t0, tn, n, endpoint=False)\n", + "SqrSignal = np.zeros(n)\n", + "FourierSeriesSignal = np.zeros(n)\n", + "SqrSignal = 1.0+signal.square(2*np.pi*5*t+np.pi*Width/T)\n", + "a0 = Fmax*Width/T\n", + "FourierSeriesSignal = a0\n", + "Factor = 2.0*Fmax/np.pi\n", + "for i in range(1,500):\n", + " FourierSeriesSignal += Factor/(i)*np.sin(np.pi*i*Width/T)*np.cos(i*t*2*np.pi/T)\n", + "plt.plot(t, SqrSignal)\n", + "plt.plot(t, FourierSeriesSignal)\n", + "plt.ylim(-0.5, 2.5)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "acd75ab0", + "metadata": { + "editable": true + }, + "source": [ + "## Two-dimensional Objects\n", + "\n", + "We often use convolutions over more than one dimension at a time. If\n", + "we have a two-dimensional image $I$ as input, we can have a **filter**\n", + "defined by a two-dimensional **kernel** $K$. This leads to an output $S$" + ] + }, + { + "cell_type": "markdown", + "id": "0cdf3af8", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "S_(i,j)=(I * K)(i,j) = \\sum_m\\sum_n I(m,n)K(i-m,j-n).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "1640aae2", + "metadata": { + "editable": true + }, + "source": [ + "Convolution is a commutatitave process, which means we can rewrite this equation as" + ] + }, + { + "cell_type": "markdown", + "id": "11491f4c", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "S_(i,j)=(I * K)(i,j) = \\sum_m\\sum_n I(i-m,j-n)K(m,n).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "a1e3ef9d", + "metadata": { + "editable": true + }, + "source": [ + "Normally the latter is more straightforward to implement in a machine elarning library since there is less variation in the range of values of $m$ and $n$.\n", + "\n", + "Many deep learning libraries implement cross-correlation instead of convolution" + ] + }, + { + "cell_type": "markdown", + "id": "919fb5e9", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "S_(i,j)=(I * K)(i,j) = \\sum_m\\sum_n I(i+m,j-+)K(m,n).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "c42d55ac", + "metadata": { + "editable": true + }, + "source": [ + "## More on Dimensionalities\n", + "\n", + "In fields like signal processing (and imaging as well), one designs\n", + "so-called filters. These filters are defined by the convolutions and\n", + "are often hand-crafted. One may specify filters for smoothing, edge\n", + "detection, frequency reshaping, and similar operations. However with\n", + "neural networks the idea is to automatically learn the filters and use\n", + "many of them in conjunction with non-linear operations (activation\n", + "functions).\n", + "\n", + "As an example consider a neural network operating on sound sequence\n", + "data. Assume that we an input vector $\\boldsymbol{x}$ of length $d=10^6$. We\n", + "construct then a neural network with onle hidden layer only with\n", + "$10^4$ nodes. This means that we will have a weight matrix with\n", + "$10^4\\times 10^6=10^{10}$ weights to be determined, together with $10^4$ biases.\n", + "\n", + "Assume furthermore that we have an output layer which is meant to train whether the sound sequence represents a human voice (true) or something else (false).\n", + "It means that we have only one output node. But since this output node connects to $10^4$ nodes in the hidden layer, there are in total $10^4$ weights to be determined for the output layer, plus one bias. In total we have" + ] + }, + { + "cell_type": "markdown", + "id": "4ebd8c85", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\mathrm{NumberParameters}=10^{10}+10^4+10^4+1 \\approx 10^{10},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "6d886fa7", + "metadata": { + "editable": true + }, + "source": [ + "that is ten billion parameters to determine." + ] + }, + { + "cell_type": "markdown", + "id": "cd3b731e", + "metadata": { + "editable": true + }, + "source": [ + "## Further Dimensionality Remarks\n", + "\n", + "In today’s architecture one can train such neural networks, however\n", + "this is a huge number of parameters for the task at hand. In general,\n", + "it is a very wasteful and inefficient use of dense matrices as\n", + "parameters. Just as importantly, such trained network parameters are\n", + "very specific for the type of input data on which they were trained\n", + "and the network is not likely to generalize easily to variations in\n", + "the input.\n", + "\n", + "The main principles that justify convolutions is locality of\n", + "information and repetion of patterns within the signal. Sound samples\n", + "of the input in adjacent spots are much more likely to affect each\n", + "other than those that are very far away. Similarly, sounds are\n", + "repeated in multiple times in the signal. While slightly simplistic,\n", + "reasoning about such a sound example demonstrates this. The same\n", + "principles then apply to images and other similar data." + ] + }, + { + "cell_type": "markdown", + "id": "fc38055c", + "metadata": { + "editable": true + }, + "source": [ + "## CNNs in more detail, building convolutional neural networks in Tensorflow and Keras\n", + "\n", + "As discussed above, CNNs are neural networks built from the assumption that the inputs\n", + "to the network are 2D images. This is important because the number of features or pixels in images\n", + "grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network. \n", + "\n", + "As before, we still have our input, a hidden layer and an output. What's novel about convolutional networks\n", + "are the **convolutional** and **pooling** layers stacked in pairs between the input and the hidden layer.\n", + "In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D\n", + "matrices, typically 1 for each color dimension (Red, Green, Blue). \n", + "\n", + "It means that to represent the entire\n", + "dataset of images, we require a 4D matrix or **tensor**. This tensor has the dimensions:" + ] + }, + { + "cell_type": "markdown", + "id": "c153f6eb", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "(n_{inputs},\\, n_{pixels, width},\\, n_{pixels, height},\\, depth) .\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "f45ab21d", + "metadata": { + "editable": true + }, + "source": [ + "### The MNIST dataset again\n", + "\n", + "The MNIST dataset consists of grayscale images with a pixel size of\n", + "$28\\times 28$, meaning we require $28 \\times 28 = 724$ weights to each\n", + "neuron in the first hidden layer.\n", + "\n", + "If we were to analyze images of size $128\\times 128$ we would require\n", + "$128 \\times 128 = 16384$ weights to each neuron. Even worse if we were\n", + "dealing with color images, as most images are, we have an image matrix\n", + "of size $128\\times 128$ for each color dimension (Red, Green, Blue),\n", + "meaning 3 times the number of weights $= 49152$ are required for every\n", + "single neuron in the first hidden layer.\n", + "\n", + "Images typically have strong local correlations, meaning that a small\n", + "part of the image varies little from its neighboring regions. If for\n", + "example we have an image of a blue car, we can roughly assume that a\n", + "small blue part of the image is surrounded by other blue regions.\n", + "\n", + "Therefore, instead of connecting every single pixel to a neuron in the\n", + "first hidden layer, as we have previously done with deep neural\n", + "networks, we can instead connect each neuron to a small part of the\n", + "image (in all 3 RGB depth dimensions). The size of each small area is\n", + "fixed, and known as a [receptive](https://en.wikipedia.org/wiki/Receptive_field).\n", + "\n", + "The layers of a convolutional neural network arrange neurons in 3D: width, height and depth. \n", + "The input image is typically a square matrix of depth 3. \n", + "\n", + "A **convolution** is performed on the image which outputs\n", + "a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as **filters**.\n", + "\n", + "Each filter slides along the input image, taking the dot product\n", + "between each small part of the image and the filter, in all depth\n", + "dimensions. This is then passed through a non-linear function,\n", + "typically the **Rectified Linear (ReLu)** function, which serves as the\n", + "activation of the neurons in the first convolutional layer. This is\n", + "further passed through a **pooling layer**, which reduces the size of the\n", + "convolutional layer, e.g. by taking the maximum or average across some\n", + "small regions, and this serves as input to the next convolutional\n", + "layer." + ] + }, + { + "cell_type": "markdown", + "id": "4af74ca7", + "metadata": { + "editable": true + }, + "source": [ + "### Systematic reduction\n", + "\n", + "By systematically reducing the size of the input volume, through\n", + "convolution and pooling, the network should create representations of\n", + "small parts of the input, and then from them assemble representations\n", + "of larger areas. The final pooling layer is flattened to serve as\n", + "input to a hidden layer, such that each neuron in the final pooling\n", + "layer is connected to every single neuron in the hidden layer. This\n", + "then serves as input to the output layer, e.g. a softmax output for\n", + "classification." + ] + }, + { + "cell_type": "markdown", + "id": "2b435dc0", + "metadata": { + "editable": true + }, + "source": [ + "### Prerequisites: Collect and pre-process data" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "ffb4904f", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "# import necessary packages\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from sklearn import datasets\n", + "\n", + "\n", + "# ensure the same random numbers appear every time\n", + "np.random.seed(0)\n", + "\n", + "# display images in notebook\n", + "%matplotlib inline\n", + "plt.rcParams['figure.figsize'] = (12,12)\n", + "\n", + "\n", + "# download MNIST dataset\n", + "digits = datasets.load_digits()\n", + "\n", + "# define inputs and labels\n", + "inputs = digits.images\n", + "labels = digits.target\n", + "\n", + "# RGB images have a depth of 3\n", + "# our images are grayscale so they should have a depth of 1\n", + "inputs = inputs[:,:,:,np.newaxis]\n", + "\n", + "print(\"inputs = (n_inputs, pixel_width, pixel_height, depth) = \" + str(inputs.shape))\n", + "print(\"labels = (n_inputs) = \" + str(labels.shape))\n", + "\n", + "\n", + "# choose some random images to display\n", + "n_inputs = len(inputs)\n", + "indices = np.arange(n_inputs)\n", + "random_indices = np.random.choice(indices, size=5)\n", + "\n", + "for i, image in enumerate(digits.images[random_indices]):\n", + " plt.subplot(1, 5, i+1)\n", + " plt.axis('off')\n", + " plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')\n", + " plt.title(\"Label: %d\" % digits.target[random_indices[i]])\n", + "plt.show()" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "id": "bb2db46c", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "from tensorflow.keras import datasets, layers, models\n", + "from tensorflow.keras.layers import Input\n", + "from tensorflow.keras.models import Sequential #This allows appending layers to existing models\n", + "from tensorflow.keras.layers import Dense #This allows defining the characteristics of a particular layer\n", + "from tensorflow.keras import optimizers #This allows using whichever optimiser we want (sgd,adam,RMSprop)\n", + "from tensorflow.keras import regularizers #This allows using whichever regularizer we want (l1,l2,l1_l2)\n", + "from tensorflow.keras.utils import to_categorical #This allows using categorical cross entropy as the cost function\n", + "#from tensorflow.keras import Conv2D\n", + "#from tensorflow.keras import MaxPooling2D\n", + "#from tensorflow.keras import Flatten\n", + "\n", + "from sklearn.model_selection import train_test_split\n", + "\n", + "# representation of labels\n", + "labels = to_categorical(labels)\n", + "\n", + "# split into train and test data\n", + "# one-liner from scikit-learn library\n", + "train_size = 0.8\n", + "test_size = 1 - train_size\n", + "X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,\n", + " test_size=test_size)" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "id": "100a8d6d", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def create_convolutional_neural_network_keras(input_shape, receptive_field,\n", + " n_filters, n_neurons_connected, n_categories,\n", + " eta, lmbd):\n", + " model = Sequential()\n", + " model.add(layers.Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same',\n", + " activation='relu', kernel_regularizer=regularizers.l2(lmbd)))\n", + " model.add(layers.MaxPooling2D(pool_size=(2, 2)))\n", + " model.add(layers.Flatten())\n", + " model.add(layers.Dense(n_neurons_connected, activation='relu', kernel_regularizer=regularizers.l2(lmbd)))\n", + " model.add(layers.Dense(n_categories, activation='softmax', kernel_regularizer=regularizers.l2(lmbd)))\n", + " \n", + " sgd = optimizers.SGD(lr=eta)\n", + " model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])\n", + " \n", + " return model\n", + "\n", + "epochs = 100\n", + "batch_size = 100\n", + "input_shape = X_train.shape[1:4]\n", + "receptive_field = 3\n", + "n_filters = 10\n", + "n_neurons_connected = 50\n", + "n_categories = 10\n", + "\n", + "eta_vals = np.logspace(-5, 1, 7)\n", + "lmbd_vals = np.logspace(-5, 1, 7)" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "id": "635da5a7", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)\n", + " \n", + "for i, eta in enumerate(eta_vals):\n", + " for j, lmbd in enumerate(lmbd_vals):\n", + " CNN = create_convolutional_neural_network_keras(input_shape, receptive_field,\n", + " n_filters, n_neurons_connected, n_categories,\n", + " eta, lmbd)\n", + " CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)\n", + " scores = CNN.evaluate(X_test, Y_test)\n", + " \n", + " CNN_keras[i][j] = CNN\n", + " \n", + " print(\"Learning rate = \", eta)\n", + " print(\"Lambda = \", lmbd)\n", + " print(\"Test accuracy: %.3f\" % scores[1])\n", + " print()" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "id": "df145244", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "# visual representation of grid search\n", + "# uses seaborn heatmap, could probably do this in matplotlib\n", + "import seaborn as sns\n", + "\n", + "sns.set()\n", + "\n", + "train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", + "test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", + "\n", + "for i in range(len(eta_vals)):\n", + " for j in range(len(lmbd_vals)):\n", + " CNN = CNN_keras[i][j]\n", + "\n", + " train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1]\n", + " test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1]\n", + "\n", + " \n", + "fig, ax = plt.subplots(figsize = (10, 10))\n", + "sns.heatmap(train_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", + "ax.set_title(\"Training Accuracy\")\n", + "ax.set_ylabel(\"$\\eta$\")\n", + "ax.set_xlabel(\"$\\lambda$\")\n", + "plt.show()\n", + "\n", + "fig, ax = plt.subplots(figsize = (10, 10))\n", + "sns.heatmap(test_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", + "ax.set_title(\"Test Accuracy\")\n", + "ax.set_ylabel(\"$\\eta$\")\n", + "ax.set_xlabel(\"$\\lambda$\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "a6724f1c", + "metadata": { + "editable": true + }, + "source": [ + "## The CIFAR01 data set\n", + "\n", + "The CIFAR10 dataset contains 60,000 color images in 10 classes, with\n", + "6,000 images in each class. The dataset is divided into 50,000\n", + "training images and 10,000 testing images. The classes are mutually\n", + "exclusive and there is no overlap between them." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "id": "90c28005", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "import tensorflow as tf\n", + "\n", + "from tensorflow.keras import datasets, layers, models\n", + "import matplotlib.pyplot as plt\n", + "\n", + "# We import the data set\n", + "(train_images, train_labels), (test_images, test_labels) = datasets.cifar10.load_data()\n", + "\n", + "# Normalize pixel values to be between 0 and 1 by dividing by 255. \n", + "train_images, test_images = train_images / 255.0, test_images / 255.0" + ] + }, + { + "cell_type": "markdown", + "id": "759ea77a", + "metadata": { + "editable": true + }, + "source": [ + "To verify that the dataset looks correct, let's plot the first 25 images from the training set and display the class name below each image." + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "id": "494edadd", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "class_names = ['airplane', 'automobile', 'bird', 'cat', 'deer',\n", + " 'dog', 'frog', 'horse', 'ship', 'truck']\n", + "​\n", + "plt.figure(figsize=(10,10))\n", + "for i in range(25):\n", + " plt.subplot(5,5,i+1)\n", + " plt.xticks([])\n", + " plt.yticks([])\n", + " plt.grid(False)\n", + " plt.imshow(train_images[i], cmap=plt.cm.binary)\n", + " # The CIFAR labels happen to be arrays, \n", + " # which is why you need the extra index\n", + " plt.xlabel(class_names[train_labels[i][0]])\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "46d2ea5d", + "metadata": { + "editable": true + }, + "source": [ + "The six lines of code below define the convolutional base using a common pattern: a stack of Conv2D and MaxPooling2D layers.\n", + "\n", + "As input, a CNN takes tensors of shape (image_height, image_width, color_channels), ignoring the batch size. If you are new to these dimensions, color_channels refers to (R,G,B). In this example, you will configure our CNN to process inputs of shape (32, 32, 3), which is the format of CIFAR images. You can do this by passing the argument input_shape to our first layer." + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "id": "044310ec", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "model = models.Sequential()\n", + "model.add(layers.Conv2D(32, (3, 3), activation='relu', input_shape=(32, 32, 3)))\n", + "model.add(layers.MaxPooling2D((2, 2)))\n", + "model.add(layers.Conv2D(64, (3, 3), activation='relu'))\n", + "model.add(layers.MaxPooling2D((2, 2)))\n", + "model.add(layers.Conv2D(64, (3, 3), activation='relu'))\n", + "\n", + "# Let's display the architecture of our model so far.\n", + "\n", + "model.summary()" + ] + }, + { + "cell_type": "markdown", + "id": "738f34a7", + "metadata": { + "editable": true + }, + "source": [ + "You can see that the output of every Conv2D and MaxPooling2D layer is a 3D tensor of shape (height, width, channels). The width and height dimensions tend to shrink as you go deeper in the network. The number of output channels for each Conv2D layer is controlled by the first argument (e.g., 32 or 64). Typically, as the width and height shrink, you can afford (computationally) to add more output channels in each Conv2D layer.\n", + "\n", + "To complete our model, you will feed the last output tensor from the\n", + "convolutional base (of shape (4, 4, 64)) into one or more Dense layers\n", + "to perform classification. Dense layers take vectors as input (which\n", + "are 1D), while the current output is a 3D tensor. First, you will\n", + "flatten (or unroll) the 3D output to 1D, then add one or more Dense\n", + "layers on top. CIFAR has 10 output classes, so you use a final Dense\n", + "layer with 10 outputs and a softmax activation." + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "id": "e0fea435", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "model.add(layers.Flatten())\n", + "model.add(layers.Dense(64, activation='relu'))\n", + "model.add(layers.Dense(10))\n", + "Here's the complete architecture of our model.\n", + "\n", + "model.summary()" + ] + }, + { + "cell_type": "markdown", + "id": "62f1e50d", + "metadata": { + "editable": true + }, + "source": [ + "As you can see, our (4, 4, 64) outputs were flattened into vectors of shape (1024) before going through two Dense layers." + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "id": "e2d8f4f2", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "model.compile(optimizer='adam',\n", + " loss=tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True),\n", + " metrics=['accuracy'])\n", + "​\n", + "history = model.fit(train_images, train_labels, epochs=10, \n", + " validation_data=(test_images, test_labels))" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "id": "d2965f40", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "plt.plot(history.history['accuracy'], label='accuracy')\n", + "plt.plot(history.history['val_accuracy'], label = 'val_accuracy')\n", + "plt.xlabel('Epoch')\n", + "plt.ylabel('Accuracy')\n", + "plt.ylim([0.5, 1])\n", + "plt.legend(loc='lower right')\n", + "\n", + "test_loss, test_acc = model.evaluate(test_images, test_labels, verbose=2)\n", + "\n", + "print(test_acc)" + ] + } + ], + "metadata": {}, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/doc/LectureNotes/_build/html/_sources/chapter13.ipynb b/doc/LectureNotes/_build/html/_sources/chapter13.ipynb new file mode 100644 index 000000000..7d78135c1 --- /dev/null +++ b/doc/LectureNotes/_build/html/_sources/chapter13.ipynb @@ -0,0 +1,1873 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "6d2db899", + "metadata": { + "editable": true + }, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "id": "f4908a97", + "metadata": { + "editable": true + }, + "source": [ + "# Recurrent neural networks: Overarching view\n", + "\n", + "Till now our focus has been, including convolutional neural networks\n", + "as well, on feedforward neural networks. The output or the activations\n", + "flow only in one direction, from the input layer to the output layer.\n", + "\n", + "A recurrent neural network (RNN) looks very much like a feedforward\n", + "neural network, except that it also has connections pointing\n", + "backward. \n", + "\n", + "RNNs are used to analyze time series data such as stock prices, and\n", + "tell you when to buy or sell. In autonomous driving systems, they can\n", + "anticipate car trajectories and help avoid accidents. More generally,\n", + "they can work on sequences of arbitrary lengths, rather than on\n", + "fixed-sized inputs like all the nets we have discussed so far. For\n", + "example, they can take sentences, documents, or audio samples as\n", + "input, making them extremely useful for natural language processing\n", + "systems such as automatic translation and speech-to-text.\n", + "\n", + "More to text to be added" + ] + }, + { + "cell_type": "markdown", + "id": "43cb4913", + "metadata": { + "editable": true + }, + "source": [ + "## A simple example" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "cf6b9dab", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "%matplotlib inline\n", + "\n", + "# Start importing packages\n", + "import pandas as pd\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "import tensorflow as tf\n", + "from tensorflow.keras import datasets, layers, models\n", + "from tensorflow.keras.layers import Input\n", + "from tensorflow.keras.models import Model, Sequential \n", + "from tensorflow.keras.layers import Dense, SimpleRNN, LSTM, GRU\n", + "from tensorflow.keras import optimizers \n", + "from tensorflow.keras import regularizers \n", + "from tensorflow.keras.utils import to_categorical \n", + "\n", + "\n", + "\n", + "# convert into dataset matrix\n", + "def convertToMatrix(data, step):\n", + " X, Y =[], []\n", + " for i in range(len(data)-step):\n", + " d=i+step \n", + " X.append(data[i:d,])\n", + " Y.append(data[d,])\n", + " return np.array(X), np.array(Y)\n", + "\n", + "step = 4\n", + "N = 1000 \n", + "Tp = 800 \n", + "\n", + "t=np.arange(0,N)\n", + "x=np.sin(0.02*t)+2*np.random.rand(N)\n", + "df = pd.DataFrame(x)\n", + "df.head()\n", + "\n", + "plt.plot(df)\n", + "plt.show()\n", + "\n", + "values=df.values\n", + "train,test = values[0:Tp,:], values[Tp:N,:]\n", + "\n", + "# add step elements into train and test\n", + "test = np.append(test,np.repeat(test[-1,],step))\n", + "train = np.append(train,np.repeat(train[-1,],step))\n", + " \n", + "trainX,trainY =convertToMatrix(train,step)\n", + "testX,testY =convertToMatrix(test,step)\n", + "trainX = np.reshape(trainX, (trainX.shape[0], 1, trainX.shape[1]))\n", + "testX = np.reshape(testX, (testX.shape[0], 1, testX.shape[1]))\n", + "\n", + "model = Sequential()\n", + "model.add(SimpleRNN(units=32, input_shape=(1,step), activation=\"relu\"))\n", + "model.add(Dense(8, activation=\"relu\")) \n", + "model.add(Dense(1))\n", + "model.compile(loss='mean_squared_error', optimizer='rmsprop')\n", + "model.summary()\n", + "\n", + "model.fit(trainX,trainY, epochs=100, batch_size=16, verbose=2)\n", + "trainPredict = model.predict(trainX)\n", + "testPredict= model.predict(testX)\n", + "predicted=np.concatenate((trainPredict,testPredict),axis=0)\n", + "\n", + "trainScore = model.evaluate(trainX, trainY, verbose=0)\n", + "print(trainScore)\n", + "\n", + "index = df.index.values\n", + "plt.plot(index,df)\n", + "plt.plot(index,predicted)\n", + "plt.axvline(df.index[Tp], c=\"r\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "897a47c7", + "metadata": { + "editable": true + }, + "source": [ + "## An extrapolation example\n", + "\n", + "The following code provides an example of how recurrent neural\n", + "networks can be used to extrapolate to unknown values of physics data\n", + "sets. Specifically, the data sets used in this program come from\n", + "a quantum mechanical many-body calculation of energies as functions of the number of particles." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "6776ae2a", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "\n", + "# For matrices and calculations\n", + "import numpy as np\n", + "# For machine learning (backend for keras)\n", + "import tensorflow as tf\n", + "# User-friendly machine learning library\n", + "# Front end for TensorFlow\n", + "import tensorflow.keras\n", + "# Different methods from Keras needed to create an RNN\n", + "# This is not necessary but it shortened function calls \n", + "# that need to be used in the code.\n", + "from tensorflow.keras import datasets, layers, models\n", + "from tensorflow.keras.layers import Input\n", + "from tensorflow.keras import regularizers\n", + "from tensorflow.keras.models import Model, Sequential\n", + "from tensorflow.keras.layers import Dense, SimpleRNN, LSTM, GRU\n", + "# For timing the code\n", + "from timeit import default_timer as timer\n", + "# For plotting\n", + "import matplotlib.pyplot as plt\n", + "\n", + "\n", + "# The data set\n", + "datatype='VaryDimension'\n", + "X_tot = np.arange(2, 42, 2)\n", + "y_tot = np.array([-0.03077640549, -0.08336233266, -0.1446729567, -0.2116753732, -0.2830637392, -0.3581341341, -0.436462435, -0.5177783846,\n", + "\t-0.6019067271, -0.6887363571, -0.7782028952, -0.8702784034, -0.9649652536, -1.062292565, -1.16231451, \n", + "\t-1.265109911, -1.370782966, -1.479465113, -1.591317992, -1.70653767])" + ] + }, + { + "cell_type": "markdown", + "id": "35227d36", + "metadata": { + "editable": true + }, + "source": [ + "The way the recurrent neural networks are trained in this program\n", + "differs from how machine learning algorithms are usually trained.\n", + "Typically a machine learning algorithm is trained by learning the\n", + "relationship between the x data and the y data. In this program, the\n", + "recurrent neural network will be trained to recognize the relationship\n", + "in a sequence of y values. This is type of data formatting is\n", + "typically used time series forcasting, but it can also be used in any\n", + "extrapolation (time series forecasting is just a specific type of\n", + "extrapolation along the time axis). This method of data formatting\n", + "does not use the x data and assumes that the y data are evenly spaced.\n", + "\n", + "For a standard machine learning algorithm, the training data has the\n", + "form of (x,y) so the machine learning algorithm learns to assiciate a\n", + "y value with a given x value. This is useful when the test data has x\n", + "values within the same range as the training data. However, for this\n", + "application, the x values of the test data are outside of the x values\n", + "of the training data and the traditional method of training a machine\n", + "learning algorithm does not work as well. For this reason, the\n", + "recurrent neural network is trained on sequences of y values of the\n", + "form ((y1, y2), y3), so that the network is concerned with learning\n", + "the pattern of the y data and not the relation between the x and y\n", + "data. As long as the pattern of y data outside of the training region\n", + "stays relatively stable compared to what was inside the training\n", + "region, this method of training can produce accurate extrapolations to\n", + "y values far removed from the training data set." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "7dc577d1", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "# FORMAT_DATA\n", + "def format_data(data, length_of_sequence = 2): \n", + " \"\"\"\n", + " Inputs:\n", + " data(a numpy array): the data that will be the inputs to the recurrent neural\n", + " network\n", + " length_of_sequence (an int): the number of elements in one iteration of the\n", + " sequence patter. For a function approximator use length_of_sequence = 2.\n", + " Returns:\n", + " rnn_input (a 3D numpy array): the input data for the recurrent neural network. Its\n", + " dimensions are length of data - length of sequence, length of sequence, \n", + " dimnsion of data\n", + " rnn_output (a numpy array): the training data for the neural network\n", + " Formats data to be used in a recurrent neural network.\n", + " \"\"\"\n", + "\n", + " X, Y = [], []\n", + " for i in range(len(data)-length_of_sequence):\n", + " # Get the next length_of_sequence elements\n", + " a = data[i:i+length_of_sequence]\n", + " # Get the element that immediately follows that\n", + " b = data[i+length_of_sequence]\n", + " # Reshape so that each data point is contained in its own array\n", + " a = np.reshape (a, (len(a), 1))\n", + " X.append(a)\n", + " Y.append(b)\n", + " rnn_input = np.array(X)\n", + " rnn_output = np.array(Y)\n", + "\n", + " return rnn_input, rnn_output\n", + "\n", + "\n", + "# ## Defining the Recurrent Neural Network Using Keras\n", + "# \n", + "# The following method defines a simple recurrent neural network in keras consisting of one input layer, one hidden layer, and one output layer.\n", + "\n", + "def rnn(length_of_sequences, batch_size = None, stateful = False):\n", + " \"\"\"\n", + " Inputs:\n", + " length_of_sequences (an int): the number of y values in \"x data\". This is determined\n", + " when the data is formatted\n", + " batch_size (an int): Default value is None. See Keras documentation of SimpleRNN.\n", + " stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN.\n", + " Returns:\n", + " model (a Keras model): The recurrent neural network that is built and compiled by this\n", + " method\n", + " Builds and compiles a recurrent neural network with one hidden layer and returns the model.\n", + " \"\"\"\n", + " # Number of neurons in the input and output layers\n", + " in_out_neurons = 1\n", + " # Number of neurons in the hidden layer\n", + " hidden_neurons = 200\n", + " # Define the input layer\n", + " inp = Input(batch_shape=(batch_size, \n", + " length_of_sequences, \n", + " in_out_neurons)) \n", + " # Define the hidden layer as a simple RNN layer with a set number of neurons and add it to \n", + " # the network immediately after the input layer\n", + " rnn = SimpleRNN(hidden_neurons, \n", + " return_sequences=False,\n", + " stateful = stateful,\n", + " name=\"RNN\")(inp)\n", + " # Define the output layer as a dense neural network layer (standard neural network layer)\n", + " #and add it to the network immediately after the hidden layer.\n", + " dens = Dense(in_out_neurons,name=\"dense\")(rnn)\n", + " # Create the machine learning model starting with the input layer and ending with the \n", + " # output layer\n", + " model = Model(inputs=[inp],outputs=[dens])\n", + " # Compile the machine learning model using the mean squared error function as the loss \n", + " # function and an Adams optimizer.\n", + " model.compile(loss=\"mean_squared_error\", optimizer=\"adam\") \n", + " return model" + ] + }, + { + "cell_type": "markdown", + "id": "1978b6a1", + "metadata": { + "editable": true + }, + "source": [ + "## Predicting New Points With A Trained Recurrent Neural Network" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "id": "d4ad417a", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def test_rnn (x1, y_test, plot_min, plot_max):\n", + " \"\"\"\n", + " Inputs:\n", + " x1 (a list or numpy array): The complete x component of the data set\n", + " y_test (a list or numpy array): The complete y component of the data set\n", + " plot_min (an int or float): the smallest x value used in the training data\n", + " plot_max (an int or float): the largest x valye used in the training data\n", + " Returns:\n", + " None.\n", + " Uses a trained recurrent neural network model to predict future points in the \n", + " series. Computes the MSE of the predicted data set from the true data set, saves\n", + " the predicted data set to a csv file, and plots the predicted and true data sets w\n", + " while also displaying the data range used for training.\n", + " \"\"\"\n", + " # Add the training data as the first dim points in the predicted data array as these\n", + " # are known values.\n", + " y_pred = y_test[:dim].tolist()\n", + " # Generate the first input to the trained recurrent neural network using the last two \n", + " # points of the training data. Based on how the network was trained this means that it\n", + " # will predict the first point in the data set after the training data. All of the \n", + " # brackets are necessary for Tensorflow.\n", + " next_input = np.array([[[y_test[dim-2]], [y_test[dim-1]]]])\n", + " # Save the very last point in the training data set. This will be used later.\n", + " last = [y_test[dim-1]]\n", + "\n", + " # Iterate until the complete data set is created.\n", + " for i in range (dim, len(y_test)):\n", + " # Predict the next point in the data set using the previous two points.\n", + " next = model.predict(next_input)\n", + " # Append just the number of the predicted data set\n", + " y_pred.append(next[0][0])\n", + " # Create the input that will be used to predict the next data point in the data set.\n", + " next_input = np.array([[last, next[0]]], dtype=np.float64)\n", + " last = next\n", + "\n", + " # Print the mean squared error between the known data set and the predicted data set.\n", + " print('MSE: ', np.square(np.subtract(y_test, y_pred)).mean())\n", + " # Save the predicted data set as a csv file for later use\n", + " name = datatype + 'Predicted'+str(dim)+'.csv'\n", + " np.savetxt(name, y_pred, delimiter=',')\n", + " # Plot the known data set and the predicted data set. The red box represents the region that was used\n", + " # for the training data.\n", + " fig, ax = plt.subplots()\n", + " ax.plot(x1, y_test, label=\"true\", linewidth=3)\n", + " ax.plot(x1, y_pred, 'g-.',label=\"predicted\", linewidth=4)\n", + " ax.legend()\n", + " # Created a red region to represent the points used in the training data.\n", + " ax.axvspan(plot_min, plot_max, alpha=0.25, color='red')\n", + " plt.show()\n", + "\n", + "# Check to make sure the data set is complete\n", + "assert len(X_tot) == len(y_tot)\n", + "\n", + "# This is the number of points that will be used in as the training data\n", + "dim=12\n", + "\n", + "# Separate the training data from the whole data set\n", + "X_train = X_tot[:dim]\n", + "y_train = y_tot[:dim]\n", + "\n", + "\n", + "# Generate the training data for the RNN, using a sequence of 2\n", + "rnn_input, rnn_training = format_data(y_train, 2)\n", + "\n", + "\n", + "# Create a recurrent neural network in Keras and produce a summary of the \n", + "# machine learning model\n", + "model = rnn(length_of_sequences = rnn_input.shape[1])\n", + "model.summary()\n", + "\n", + "# Start the timer. Want to time training+testing\n", + "start = timer()\n", + "# Fit the model using the training data genenerated above using 150 training iterations and a 5%\n", + "# validation split. Setting verbose to True prints information about each training iteration.\n", + "hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, \n", + " verbose=True,validation_split=0.05)\n", + "\n", + "for label in [\"loss\",\"val_loss\"]:\n", + " plt.plot(hist.history[label],label=label)\n", + "\n", + "plt.ylabel(\"loss\")\n", + "plt.xlabel(\"epoch\")\n", + "plt.title(\"The final validation loss: {}\".format(hist.history[\"val_loss\"][-1]))\n", + "plt.legend()\n", + "plt.show()\n", + "\n", + "# Use the trained neural network to predict more points of the data set\n", + "test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1])\n", + "# Stop the timer and calculate the total time needed.\n", + "end = timer()\n", + "print('Time: ', end-start)" + ] + }, + { + "cell_type": "markdown", + "id": "e2cad4fc", + "metadata": { + "editable": true + }, + "source": [ + "Changing the size of the recurrent neural network and its parameters\n", + "can drastically change the results you get from the model. The below\n", + "code takes the simple recurrent neural network from above and adds a\n", + "second hidden layer, changes the number of neurons in the hidden\n", + "layer, and explicitly declares the activation function of the hidden\n", + "layers to be a sigmoid function. The loss function and optimizer can\n", + "also be changed but are kept the same as the above network. These\n", + "parameters can be tuned to provide the optimal result from the\n", + "network. For some ideas on how to improve the performance of a\n", + "[recurrent neural network](https://danijar.com/tips-for-training-recurrent-neural-networks)." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "id": "c39f1516", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def rnn_2layers(length_of_sequences, batch_size = None, stateful = False):\n", + " \"\"\"\n", + " Inputs:\n", + " length_of_sequences (an int): the number of y values in \"x data\". This is determined\n", + " when the data is formatted\n", + " batch_size (an int): Default value is None. See Keras documentation of SimpleRNN.\n", + " stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN.\n", + " Returns:\n", + " model (a Keras model): The recurrent neural network that is built and compiled by this\n", + " method\n", + " Builds and compiles a recurrent neural network with two hidden layers and returns the model.\n", + " \"\"\"\n", + " # Number of neurons in the input and output layers\n", + " in_out_neurons = 1\n", + " # Number of neurons in the hidden layer, increased from the first network\n", + " hidden_neurons = 500\n", + " # Define the input layer\n", + " inp = Input(batch_shape=(batch_size, \n", + " length_of_sequences, \n", + " in_out_neurons)) \n", + " # Create two hidden layers instead of one hidden layer. Explicitly set the activation\n", + " # function to be the sigmoid function (the default value is hyperbolic tangent)\n", + " rnn1 = SimpleRNN(hidden_neurons, \n", + " return_sequences=True, # This needs to be True if another hidden layer is to follow\n", + " stateful = stateful, activation = 'sigmoid',\n", + " name=\"RNN1\")(inp)\n", + " rnn2 = SimpleRNN(hidden_neurons, \n", + " return_sequences=False, activation = 'sigmoid',\n", + " stateful = stateful,\n", + " name=\"RNN2\")(rnn1)\n", + " # Define the output layer as a dense neural network layer (standard neural network layer)\n", + " #and add it to the network immediately after the hidden layer.\n", + " dens = Dense(in_out_neurons,name=\"dense\")(rnn2)\n", + " # Create the machine learning model starting with the input layer and ending with the \n", + " # output layer\n", + " model = Model(inputs=[inp],outputs=[dens])\n", + " # Compile the machine learning model using the mean squared error function as the loss \n", + " # function and an Adams optimizer.\n", + " model.compile(loss=\"mean_squared_error\", optimizer=\"adam\") \n", + " return model\n", + "\n", + "# Check to make sure the data set is complete\n", + "assert len(X_tot) == len(y_tot)\n", + "\n", + "# This is the number of points that will be used in as the training data\n", + "dim=12\n", + "\n", + "# Separate the training data from the whole data set\n", + "X_train = X_tot[:dim]\n", + "y_train = y_tot[:dim]\n", + "\n", + "\n", + "# Generate the training data for the RNN, using a sequence of 2\n", + "rnn_input, rnn_training = format_data(y_train, 2)\n", + "\n", + "\n", + "# Create a recurrent neural network in Keras and produce a summary of the \n", + "# machine learning model\n", + "model = rnn_2layers(length_of_sequences = 2)\n", + "model.summary()\n", + "\n", + "# Start the timer. Want to time training+testing\n", + "start = timer()\n", + "# Fit the model using the training data genenerated above using 150 training iterations and a 5%\n", + "# validation split. Setting verbose to True prints information about each training iteration.\n", + "hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, \n", + " verbose=True,validation_split=0.05)\n", + "\n", + "\n", + "# This section plots the training loss and the validation loss as a function of training iteration.\n", + "# This is not required for analyzing the couple cluster data but can help determine if the network is\n", + "# being overtrained.\n", + "for label in [\"loss\",\"val_loss\"]:\n", + " plt.plot(hist.history[label],label=label)\n", + "\n", + "plt.ylabel(\"loss\")\n", + "plt.xlabel(\"epoch\")\n", + "plt.title(\"The final validation loss: {}\".format(hist.history[\"val_loss\"][-1]))\n", + "plt.legend()\n", + "plt.show()\n", + "\n", + "# Use the trained neural network to predict more points of the data set\n", + "test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1])\n", + "# Stop the timer and calculate the total time needed.\n", + "end = timer()\n", + "print('Time: ', end-start)" + ] + }, + { + "cell_type": "markdown", + "id": "842c7602", + "metadata": { + "editable": true + }, + "source": [ + "## Other Types of Recurrent Neural Networks\n", + "\n", + "Besides a simple recurrent neural network layer, there are two other\n", + "commonly used types of recurrent neural network layers: Long Short\n", + "Term Memory (LSTM) and Gated Recurrent Unit (GRU). For a short\n", + "introduction to these layers see \n", + "and .\n", + "\n", + "The first network created below is similar to the previous network,\n", + "but it replaces the SimpleRNN layers with LSTM layers. The second\n", + "network below has two hidden layers made up of GRUs, which are\n", + "preceeded by two dense (feeddorward) neural network layers. These\n", + "dense layers \"preprocess\" the data before it reaches the recurrent\n", + "layers. This architecture has been shown to improve the performance\n", + "of recurrent neural networks (see the link above and also\n", + "." + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "id": "6f0e9b62", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def lstm_2layers(length_of_sequences, batch_size = None, stateful = False):\n", + " \"\"\"\n", + " Inputs:\n", + " length_of_sequences (an int): the number of y values in \"x data\". This is determined\n", + " when the data is formatted\n", + " batch_size (an int): Default value is None. See Keras documentation of SimpleRNN.\n", + " stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN.\n", + " Returns:\n", + " model (a Keras model): The recurrent neural network that is built and compiled by this\n", + " method\n", + " Builds and compiles a recurrent neural network with two LSTM hidden layers and returns the model.\n", + " \"\"\"\n", + " # Number of neurons on the input/output layer and the number of neurons in the hidden layer\n", + " in_out_neurons = 1\n", + " hidden_neurons = 250\n", + " # Input Layer\n", + " inp = Input(batch_shape=(batch_size, \n", + " length_of_sequences, \n", + " in_out_neurons)) \n", + " # Hidden layers (in this case they are LSTM layers instead if SimpleRNN layers)\n", + " rnn= LSTM(hidden_neurons, \n", + " return_sequences=True,\n", + " stateful = stateful,\n", + " name=\"RNN\", use_bias=True, activation='tanh')(inp)\n", + " rnn1 = LSTM(hidden_neurons, \n", + " return_sequences=False,\n", + " stateful = stateful,\n", + " name=\"RNN1\", use_bias=True, activation='tanh')(rnn)\n", + " # Output layer\n", + " dens = Dense(in_out_neurons,name=\"dense\")(rnn1)\n", + " # Define the midel\n", + " model = Model(inputs=[inp],outputs=[dens])\n", + " # Compile the model\n", + " model.compile(loss='mean_squared_error', optimizer='adam') \n", + " # Return the model\n", + " return model\n", + "\n", + "def dnn2_gru2(length_of_sequences, batch_size = None, stateful = False):\n", + " \"\"\"\n", + " Inputs:\n", + " length_of_sequences (an int): the number of y values in \"x data\". This is determined\n", + " when the data is formatted\n", + " batch_size (an int): Default value is None. See Keras documentation of SimpleRNN.\n", + " stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN.\n", + " Returns:\n", + " model (a Keras model): The recurrent neural network that is built and compiled by this\n", + " method\n", + " Builds and compiles a recurrent neural network with four hidden layers (two dense followed by\n", + " two GRU layers) and returns the model.\n", + " \"\"\" \n", + " # Number of neurons on the input/output layers and hidden layers\n", + " in_out_neurons = 1\n", + " hidden_neurons = 250\n", + " # Input layer\n", + " inp = Input(batch_shape=(batch_size, \n", + " length_of_sequences, \n", + " in_out_neurons)) \n", + " # Hidden Dense (feedforward) layers\n", + " dnn = Dense(hidden_neurons/2, activation='relu', name='dnn')(inp)\n", + " dnn1 = Dense(hidden_neurons/2, activation='relu', name='dnn1')(dnn)\n", + " # Hidden GRU layers\n", + " rnn1 = GRU(hidden_neurons, \n", + " return_sequences=True,\n", + " stateful = stateful,\n", + " name=\"RNN1\", use_bias=True)(dnn1)\n", + " rnn = GRU(hidden_neurons, \n", + " return_sequences=False,\n", + " stateful = stateful,\n", + " name=\"RNN\", use_bias=True)(rnn1)\n", + " # Output layer\n", + " dens = Dense(in_out_neurons,name=\"dense\")(rnn)\n", + " # Define the model\n", + " model = Model(inputs=[inp],outputs=[dens])\n", + " # Compile the mdoel\n", + " model.compile(loss='mean_squared_error', optimizer='adam') \n", + " # Return the model\n", + " return model\n", + "\n", + "# Check to make sure the data set is complete\n", + "assert len(X_tot) == len(y_tot)\n", + "\n", + "# This is the number of points that will be used in as the training data\n", + "dim=12\n", + "\n", + "# Separate the training data from the whole data set\n", + "X_train = X_tot[:dim]\n", + "y_train = y_tot[:dim]\n", + "\n", + "\n", + "# Generate the training data for the RNN, using a sequence of 2\n", + "rnn_input, rnn_training = format_data(y_train, 2)\n", + "\n", + "\n", + "# Create a recurrent neural network in Keras and produce a summary of the \n", + "# machine learning model\n", + "# Change the method name to reflect which network you want to use\n", + "model = dnn2_gru2(length_of_sequences = 2)\n", + "model.summary()\n", + "\n", + "# Start the timer. Want to time training+testing\n", + "start = timer()\n", + "# Fit the model using the training data genenerated above using 150 training iterations and a 5%\n", + "# validation split. Setting verbose to True prints information about each training iteration.\n", + "hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, \n", + " verbose=True,validation_split=0.05)\n", + "\n", + "\n", + "# This section plots the training loss and the validation loss as a function of training iteration.\n", + "# This is not required for analyzing the couple cluster data but can help determine if the network is\n", + "# being overtrained.\n", + "for label in [\"loss\",\"val_loss\"]:\n", + " plt.plot(hist.history[label],label=label)\n", + "\n", + "plt.ylabel(\"loss\")\n", + "plt.xlabel(\"epoch\")\n", + "plt.title(\"The final validation loss: {}\".format(hist.history[\"val_loss\"][-1]))\n", + "plt.legend()\n", + "plt.show()\n", + "\n", + "# Use the trained neural network to predict more points of the data set\n", + "test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1])\n", + "# Stop the timer and calculate the total time needed.\n", + "end = timer()\n", + "print('Time: ', end-start)\n", + "\n", + "\n", + "# ### Training Recurrent Neural Networks in the Standard Way (i.e. learning the relationship between the X and Y data)\n", + "# \n", + "# Finally, comparing the performace of a recurrent neural network using the standard data formatting to the performance of the network with time sequence data formatting shows the benefit of this type of data formatting with extrapolation.\n", + "\n", + "# Check to make sure the data set is complete\n", + "assert len(X_tot) == len(y_tot)\n", + "\n", + "# This is the number of points that will be used in as the training data\n", + "dim=12\n", + "\n", + "# Separate the training data from the whole data set\n", + "X_train = X_tot[:dim]\n", + "y_train = y_tot[:dim]\n", + "\n", + "# Reshape the data for Keras specifications\n", + "X_train = X_train.reshape((dim, 1))\n", + "y_train = y_train.reshape((dim, 1))\n", + "\n", + "\n", + "# Create a recurrent neural network in Keras and produce a summary of the \n", + "# machine learning model\n", + "# Set the sequence length to 1 for regular data formatting \n", + "model = rnn(length_of_sequences = 1)\n", + "model.summary()\n", + "\n", + "# Start the timer. Want to time training+testing\n", + "start = timer()\n", + "# Fit the model using the training data genenerated above using 150 training iterations and a 5%\n", + "# validation split. Setting verbose to True prints information about each training iteration.\n", + "hist = model.fit(X_train, y_train, batch_size=None, epochs=150, \n", + " verbose=True,validation_split=0.05)\n", + "\n", + "\n", + "# This section plots the training loss and the validation loss as a function of training iteration.\n", + "# This is not required for analyzing the couple cluster data but can help determine if the network is\n", + "# being overtrained.\n", + "for label in [\"loss\",\"val_loss\"]:\n", + " plt.plot(hist.history[label],label=label)\n", + "\n", + "plt.ylabel(\"loss\")\n", + "plt.xlabel(\"epoch\")\n", + "plt.title(\"The final validation loss: {}\".format(hist.history[\"val_loss\"][-1]))\n", + "plt.legend()\n", + "plt.show()\n", + "\n", + "# Use the trained neural network to predict the remaining data points\n", + "X_pred = X_tot[dim:]\n", + "X_pred = X_pred.reshape((len(X_pred), 1))\n", + "y_model = model.predict(X_pred)\n", + "y_pred = np.concatenate((y_tot[:dim], y_model.flatten()))\n", + "\n", + "# Plot the known data set and the predicted data set. The red box represents the region that was used\n", + "# for the training data.\n", + "fig, ax = plt.subplots()\n", + "ax.plot(X_tot, y_tot, label=\"true\", linewidth=3)\n", + "ax.plot(X_tot, y_pred, 'g-.',label=\"predicted\", linewidth=4)\n", + "ax.legend()\n", + "# Created a red region to represent the points used in the training data.\n", + "ax.axvspan(X_tot[0], X_tot[dim], alpha=0.25, color='red')\n", + "plt.show()\n", + "\n", + "# Stop the timer and calculate the total time needed.\n", + "end = timer()\n", + "print('Time: ', end-start)" + ] + }, + { + "cell_type": "markdown", + "id": "0752ba7f", + "metadata": { + "editable": true + }, + "source": [ + "# Generative Models\n", + "\n", + "**Generative models** describe a class of statistical models that are a contrast\n", + "to **discriminative models**. Informally we say that generative models can\n", + "generate new data instances while discriminative models discriminate between\n", + "different kinds of data instances. A generative model could generate new photos\n", + "of animals that look like 'real' animals while a discriminative model could tell\n", + "a dog from a cat. More formally, given a data set $x$ and a set of labels /\n", + "targets $y$. Generative models capture the joint probability $p(x, y)$, or\n", + "just $p(x)$ if there are no labels, while discriminative models capture the\n", + "conditional probability $p(y | x)$. Discriminative models generally try to draw\n", + "boundaries in the data space (often high dimensional), while generative models\n", + "try to model how data is placed throughout the space.\n", + "\n", + "**Note**: this material is thanks to Linus Ekstrøm." + ] + }, + { + "cell_type": "markdown", + "id": "784138f8", + "metadata": { + "editable": true + }, + "source": [ + "## Generative Adversarial Networks\n", + "\n", + "**Generative Adversarial Networks** are a type of unsupervised machine learning\n", + "algorithm proposed by [Goodfellow et. al](https://arxiv.org/pdf/1406.2661.pdf)\n", + "in 2014 (short and good article).\n", + "\n", + "The simplest formulation of\n", + "the model is based on a game theoretic approach, *zero sum game*, where we pit\n", + "two neural networks against one another. We define two rival networks, one\n", + "generator $g$, and one discriminator $d$. The generator directly produces\n", + "samples" + ] + }, + { + "cell_type": "markdown", + "id": "a42f89ee", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " x = g(z; \\theta^{(g)})\n", + "\\label{_auto1} \\tag{1}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "abe7212f", + "metadata": { + "editable": true + }, + "source": [ + "The discriminator attempts to distinguish between samples drawn from the\n", + "training data and samples drawn from the generator. In other words, it tries to\n", + "tell the difference between the fake data produced by $g$ and the actual data\n", + "samples we want to do prediction on. The discriminator outputs a probability\n", + "value given by" + ] + }, + { + "cell_type": "markdown", + "id": "0821eaee", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " d(x; \\theta^{(d)})\n", + "\\label{_auto2} \\tag{2}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "55e0ccf6", + "metadata": { + "editable": true + }, + "source": [ + "indicating the probability that $x$ is a real training example rather than a\n", + "fake sample the generator has generated. The simplest way to formulate the\n", + "learning process in a generative adversarial network is a zero-sum game, in\n", + "which a function" + ] + }, + { + "cell_type": "markdown", + "id": "f37ece14", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " v(\\theta^{(g)}, \\theta^{(d)})\n", + "\\label{_auto3} \\tag{3}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "d6d6d5fa", + "metadata": { + "editable": true + }, + "source": [ + "determines the reward for the discriminator, while the generator gets the\n", + "conjugate reward" + ] + }, + { + "cell_type": "markdown", + "id": "3c74b45c", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " -v(\\theta^{(g)}, \\theta^{(d)})\n", + "\\label{_auto4} \\tag{4}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "605ac8c9", + "metadata": { + "editable": true + }, + "source": [ + "During learning both of the networks maximize their own reward function, so that\n", + "the generator gets better and better at tricking the discriminator, while the\n", + "discriminator gets better and better at telling the difference between the fake\n", + "and real data. The generator and discriminator alternate on which one trains at\n", + "one time (i.e. for one epoch). In other words, we keep the generator constant\n", + "and train the discriminator, then we keep the discriminator constant to train\n", + "the generator and repeat. It is this back and forth dynamic which lets GANs\n", + "tackle otherwise intractable generative problems. As the generator improves with\n", + " training, the discriminator's performance gets worse because it cannot easily\n", + " tell the difference between real and fake. If the generator ends up succeeding\n", + " perfectly, the the discriminator will do no better than random guessing i.e.\n", + " 50\\%. This progression in the training poses a problem for the convergence\n", + " criteria for GANs. The discriminator feedback gets less meaningful over time,\n", + " if we continue training after this point then the generator is effectively\n", + " training on junk data which can undo the learning up to that point. Therefore,\n", + " we stop training when the discriminator starts outputting $1/2$ everywhere.\n", + "\n", + "At convergence we have" + ] + }, + { + "cell_type": "markdown", + "id": "cfec7462", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " g^* = \\underset{g}{\\mathrm{argmin}}\\hspace{2pt}\n", + " \\underset{d}{\\mathrm{max}}v(\\theta^{(g)}, \\theta^{(d)})\n", + "\\label{_auto5} \\tag{5}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "37c65d4c", + "metadata": { + "editable": true + }, + "source": [ + "The default choice for $v$ is" + ] + }, + { + "cell_type": "markdown", + "id": "c868a092", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " v(\\theta^{(g)}, \\theta^{(d)}) = \\mathbb{E}_{x\\sim p_\\mathrm{data}}\\log d(x)\n", + " + \\mathbb{E}_{x\\sim p_\\mathrm{model}}\n", + " \\log (1 - d(x))\n", + "\\label{_auto6} \\tag{6}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "ad465af3", + "metadata": { + "editable": true + }, + "source": [ + "The main motivation for the design of GANs is that the learning process requires\n", + "neither approximate inference (variational autoencoders for example) nor\n", + "approximation of a partition function. In the case where" + ] + }, + { + "cell_type": "markdown", + "id": "27858a4e", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " \\underset{d}{\\mathrm{max}}v(\\theta^{(g)}, \\theta^{(d)})\n", + "\\label{_auto7} \\tag{7}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "86006023", + "metadata": { + "editable": true + }, + "source": [ + "is convex in $\\theta^{(g)} then the procedure is guaranteed to converge and is\n", + "asymptotically consistent\n", + "( [Seth Lloyd on QuGANs](https://arxiv.org/pdf/1804.09139.pdf) ).\n", + "\n", + "This is in\n", + "general not the case and it is possible to get situations where the training\n", + "process never converges because the generator and discriminator chase one\n", + "another around in the parameter space indefinitely. A much deeper discussion on\n", + "the currently open research problem of GAN convergence is available\n", + "[here](https://www.deeplearningbook.org/contents/generative_models.html). To\n", + "anyone interested in learning more about GANs it is a highly recommended read.\n", + "Direct quote: \"In this best-performing formulation, the generator aims to\n", + "increase the log probability that the discriminator makes a mistake, rather than\n", + "aiming to decrease the log probability that the discriminator makes the correct\n", + "prediction.\" [Another interesting read](https://arxiv.org/abs/1701.00160)" + ] + }, + { + "cell_type": "markdown", + "id": "2fee38bd", + "metadata": { + "editable": true + }, + "source": [ + "## Writing Our First Generative Adversarial Network\n", + "Let us now move on to actually implementing a GAN in tensorflow. We will study\n", + "the performance of our GAN on the MNIST dataset. This code is based on and\n", + "adapted from the\n", + "[google tutorial](https://www.tensorflow.org/tutorials/generative/dcgan)\n", + "\n", + "First we import our libraries" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "id": "004a0b53", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "import os\n", + "import time\n", + "import numpy as np\n", + "import tensorflow as tf\n", + "import matplotlib.pyplot as plt\n", + "from tensorflow.keras import layers\n", + "from tensorflow.keras.utils import plot_model" + ] + }, + { + "cell_type": "markdown", + "id": "353af161", + "metadata": { + "editable": true + }, + "source": [ + "Next we define our hyperparameters and import our data the usual way" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "id": "8cbaf16a", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "BUFFER_SIZE = 60000\n", + "BATCH_SIZE = 256\n", + "EPOCHS = 30\n", + "\n", + "data = tf.keras.datasets.mnist.load_data()\n", + "(train_images, train_labels), (test_images, test_labels) = data\n", + "train_images = np.reshape(train_images, (train_images.shape[0],\n", + " 28,\n", + " 28,\n", + " 1)).astype('float32')\n", + "\n", + "# we normalize between -1 and 1\n", + "train_images = (train_images - 127.5) / 127.5\n", + "training_dataset = tf.data.Dataset.from_tensor_slices(\n", + " train_images).shuffle(BUFFER_SIZE).batch(BATCH_SIZE)" + ] + }, + { + "cell_type": "markdown", + "id": "822b8cc7", + "metadata": { + "editable": true + }, + "source": [ + "### MNIST and GANs\n", + "\n", + "Let's have a quick look" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "id": "52b5965c", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "plt.imshow(train_images[0], cmap='Greys')\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "21c5199c", + "metadata": { + "editable": true + }, + "source": [ + "Now we define our two models. This is where the 'magic' happens. There are a\n", + "huge amount of possible formulations for both models. A lot of engineering and\n", + "trial and error can be done here to try to produce better performing models. For\n", + "more advanced GANs this is by far the step where you can 'make or break' a\n", + "model.\n", + "\n", + "We start with the generator. As stated in the introductory text the generator\n", + "$g$ upsamples from a random sample to the shape of what we want to predict. In\n", + "our case we are trying to predict MNIST images ($28\\times 28$ pixels)." + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "id": "356759c7", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def generator_model():\n", + " \"\"\"\n", + " The generator uses upsampling layers tf.keras.layers.Conv2DTranspose() to\n", + " produce an image from a random seed. We start with a Dense layer taking this\n", + " random sample as an input and subsequently upsample through multiple\n", + " convolutional layers.\n", + " \"\"\"\n", + "\n", + " # we define our model\n", + " model = tf.keras.Sequential()\n", + "\n", + "\n", + " # adding our input layer. Dense means that every neuron is connected and\n", + " # the input shape is the shape of our random noise. The units need to match\n", + " # in some sense the upsampling strides to reach our desired output shape.\n", + " # we are using 100 random numbers as our seed\n", + " model.add(layers.Dense(units=7*7*BATCH_SIZE,\n", + " use_bias=False,\n", + " input_shape=(100, )))\n", + " # we normalize the output form the Dense layer\n", + " model.add(layers.BatchNormalization())\n", + " # and add an activation function to our 'layer'. LeakyReLU avoids vanishing\n", + " # gradient problem\n", + " model.add(layers.LeakyReLU())\n", + " model.add(layers.Reshape((7, 7, BATCH_SIZE)))\n", + " assert model.output_shape == (None, 7, 7, BATCH_SIZE)\n", + " # even though we just added four keras layers we think of everything above\n", + " # as 'one' layer\n", + "\n", + " # next we add our upscaling convolutional layers\n", + " model.add(layers.Conv2DTranspose(filters=128,\n", + " kernel_size=(5, 5),\n", + " strides=(1, 1),\n", + " padding='same',\n", + " use_bias=False))\n", + " model.add(layers.BatchNormalization())\n", + " model.add(layers.LeakyReLU())\n", + " assert model.output_shape == (None, 7, 7, 128)\n", + "\n", + " model.add(layers.Conv2DTranspose(filters=64,\n", + " kernel_size=(5, 5),\n", + " strides=(2, 2),\n", + " padding='same',\n", + " use_bias=False))\n", + " model.add(layers.BatchNormalization())\n", + " model.add(layers.LeakyReLU())\n", + " assert model.output_shape == (None, 14, 14, 64)\n", + "\n", + " model.add(layers.Conv2DTranspose(filters=1,\n", + " kernel_size=(5, 5),\n", + " strides=(2, 2),\n", + " padding='same',\n", + " use_bias=False,\n", + " activation='tanh'))\n", + " assert model.output_shape == (None, 28, 28, 1)\n", + "\n", + " return model" + ] + }, + { + "cell_type": "markdown", + "id": "854bcd6b", + "metadata": { + "editable": true + }, + "source": [ + "And there we have our 'simple' generator model. Now we move on to defining our\n", + "discriminator model $d$, which is a convolutional neural network based image\n", + "classifier." + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "id": "41473304", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def discriminator_model():\n", + " \"\"\"\n", + " The discriminator is a convolutional neural network based image classifier\n", + " \"\"\"\n", + "\n", + " # we define our model\n", + " model = tf.keras.Sequential()\n", + " model.add(layers.Conv2D(filters=64,\n", + " kernel_size=(5, 5),\n", + " strides=(2, 2),\n", + " padding='same',\n", + " input_shape=[28, 28, 1]))\n", + " model.add(layers.LeakyReLU())\n", + " # adding a dropout layer as you do in conv-nets\n", + " model.add(layers.Dropout(0.3))\n", + "\n", + "\n", + " model.add(layers.Conv2D(filters=128,\n", + " kernel_size=(5, 5),\n", + " strides=(2, 2),\n", + " padding='same'))\n", + " model.add(layers.LeakyReLU())\n", + " # adding a dropout layer as you do in conv-nets\n", + " model.add(layers.Dropout(0.3))\n", + "\n", + " model.add(layers.Flatten())\n", + " model.add(layers.Dense(1))\n", + "\n", + " return model" + ] + }, + { + "cell_type": "markdown", + "id": "353af567", + "metadata": { + "editable": true + }, + "source": [ + "Let us take a look at our models." + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "id": "f899d4e3", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "generator = generator_model()\n", + "plot_model(generator, show_shapes=True, rankdir='LR')" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "id": "87ef384b", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "discriminator = discriminator_model()\n", + "plot_model(discriminator, show_shapes=True, rankdir='LR')" + ] + }, + { + "cell_type": "markdown", + "id": "b2bef82d", + "metadata": { + "editable": true + }, + "source": [ + "Next we need a few helper objects we will use in training" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "id": "e397847a", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "cross_entropy = tf.keras.losses.BinaryCrossentropy(from_logits=True)\n", + "generator_optimizer = tf.keras.optimizers.Adam(1e-4)\n", + "discriminator_optimizer = tf.keras.optimizers.Adam(1e-4)" + ] + }, + { + "cell_type": "markdown", + "id": "db3396cc", + "metadata": { + "editable": true + }, + "source": [ + "The first object, *cross_entropy* is our loss function and the two others are\n", + "our optimizers. Notice we use the same learning rate for both $g$ and $d$. This\n", + "is because they need to improve their accuracy at approximately equal speeds to\n", + "get convergence (not necessarily exactly equal). Now we define our loss\n", + "functions" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "id": "931eaced", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def generator_loss(fake_output):\n", + " loss = cross_entropy(tf.ones_like(fake_output), fake_output)\n", + "\n", + " return loss" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "id": "0c4a44bb", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def discriminator_loss(real_output, fake_output):\n", + " real_loss = cross_entropy(tf.ones_like(real_output), real_output)\n", + " fake_loss = cross_entropy(tf.zeros_liks(fake_output), fake_output)\n", + " total_loss = real_loss + fake_loss\n", + "\n", + " return total_loss" + ] + }, + { + "cell_type": "markdown", + "id": "fcf8f066", + "metadata": { + "editable": true + }, + "source": [ + "Next we define a kind of seed to help us compare the learning process over\n", + "multiple training epochs." + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "id": "eea2bbee", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "noise_dimension = 100\n", + "n_examples_to_generate = 16\n", + "seed_images = tf.random.normal([n_examples_to_generate, noise_dimension])" + ] + }, + { + "cell_type": "markdown", + "id": "94a6e341", + "metadata": { + "editable": true + }, + "source": [ + "Now we have everything we need to define our training step, which we will apply\n", + "for every step in our training loop. Notice the @tf.function flag signifying\n", + "that the function is tensorflow 'compiled'. Removing this flag doubles the\n", + "computation time." + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "id": "8d48470b", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "@tf.function\n", + "def train_step(images):\n", + " noise = tf.random.normal([BATCH_SIZE, noise_dimension])\n", + "\n", + " with tf.GradientTape() as gen_tape, tf.GradientTape() as disc_tape:\n", + " generated_images = generator(noise, training=True)\n", + "\n", + " real_output = discriminator(images, training=True)\n", + " fake_output = discriminator(generated_images, training=True)\n", + "\n", + " gen_loss = generator_loss(fake_output)\n", + " disc_loss = discriminator_loss(real_output, fake_output)\n", + "\n", + " gradients_of_generator = gen_tape.gradient(gen_loss,\n", + " generator.trainable_variables)\n", + " gradients_of_discriminator = disc_tape.gradient(disc_loss,\n", + " discriminator.trainable_variables)\n", + " generator_optimizer.apply_gradients(zip(gradients_of_generator,\n", + " generator.trainable_variables))\n", + " discriminator_optimizer.apply_gradients(zip(gradients_of_discriminator,\n", + " discriminator.trainable_variables))\n", + "\n", + " return gen_loss, disc_loss" + ] + }, + { + "cell_type": "markdown", + "id": "62015b88", + "metadata": { + "editable": true + }, + "source": [ + "Next we define a helper function to produce an output over our training epochs\n", + "to see the predictive progression of our generator model. **Note**: I am including\n", + "this code here, but comment it out in the training loop." + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "id": "b189ed96", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def generate_and_save_images(model, epoch, test_input):\n", + " # we're making inferences here\n", + " predictions = model(test_input, training=False)\n", + "\n", + " fig = plt.figure(figsize=(4, 4))\n", + "\n", + " for i in range(predictions.shape[0]):\n", + " plt.subplot(4, 4, i+1)\n", + " plt.imshow(predictions[i, :, :, 0] * 127.5 + 127.5, cmap='gray')\n", + " plt.axis('off')\n", + "\n", + " plt.savefig(f'./images_from_seed_images/image_at_epoch_{str(epoch).zfill(3)}.png')\n", + " plt.close()\n", + " #plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "ba2c82d5", + "metadata": { + "editable": true + }, + "source": [ + "Setting up checkpoints to periodically save our model during training so that\n", + "everything is not lost even if the program were to somehow terminate while\n", + "training." + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "id": "a0e2fc8a", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "# Setting up checkpoints to save model during training\n", + "checkpoint_dir = './training_checkpoints'\n", + "checkpoint_prefix = os.path.join(checkpoint_dir, 'ckpt')\n", + "checkpoint = tf.train.Checkpoint(generator_optimizer=generator_optimizer,\n", + " discriminator_optimizer=discriminator_optimizer,\n", + " generator=generator,\n", + " discriminator=discriminator)" + ] + }, + { + "cell_type": "markdown", + "id": "4d93a0f7", + "metadata": { + "editable": true + }, + "source": [ + "Now we define our training loop" + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "id": "a1275556", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def train(dataset, epochs):\n", + " generator_loss_list = []\n", + " discriminator_loss_list = []\n", + "\n", + " for epoch in range(epochs):\n", + " start = time.time()\n", + "\n", + " for image_batch in dataset:\n", + " gen_loss, disc_loss = train_step(image_batch)\n", + " generator_loss_list.append(gen_loss.numpy())\n", + " discriminator_loss_list.append(disc_loss.numpy())\n", + "\n", + " #generate_and_save_images(generator, epoch + 1, seed_images)\n", + "\n", + " if (epoch + 1) % 15 == 0:\n", + " checkpoint.save(file_prefix=checkpoint_prefix)\n", + "\n", + " print(f'Time for epoch {epoch} is {time.time() - start}')\n", + "\n", + " #generate_and_save_images(generator, epochs, seed_images)\n", + "\n", + " loss_file = './data/lossfile.txt'\n", + " with open(loss_file, 'w') as outfile:\n", + " outfile.write(str(generator_loss_list))\n", + " outfile.write('\\n')\n", + " outfile.write('\\n')\n", + " outfile.write(str(discriminator_loss_list))\n", + " outfile.write('\\n')\n", + " outfile.write('\\n')" + ] + }, + { + "cell_type": "markdown", + "id": "6ff3a75a", + "metadata": { + "editable": true + }, + "source": [ + "To train simply call this function. **Warning**: this might take a long time so\n", + "there is a folder of a pretrained network already included in the repository." + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "id": "371ed41a", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "train(train_dataset, EPOCHS)" + ] + }, + { + "cell_type": "markdown", + "id": "654399f1", + "metadata": { + "editable": true + }, + "source": [ + "Now to avoid having to train and everything, which will take a while depending\n", + "on your computer setup we now load in the model which produced the above gif." + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "id": "dec4b560", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "checkpoint.restore(tf.train.latest_checkpoint(checkpoint_dir))\n", + "restored_generator = checkpoint.generator\n", + "restored_discriminator = checkpoint.discriminator\n", + "\n", + "print(restored_generator)\n", + "print(restored_discriminator)" + ] + }, + { + "cell_type": "markdown", + "id": "296bfa5c", + "metadata": { + "editable": true + }, + "source": [ + "We have successfully loaded in our latest model. Let us now play around a bit\n", + "and see what kind of things we can learn about this model. Our generator takes\n", + "an array of 100 numbers. One idea can be to try to systematically change our\n", + "input. Let us try and see what we get" + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "id": "eecfbb1f", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def generate_latent_points(number=100, scale_means=1, scale_stds=1):\n", + " latent_dim = 100\n", + " means = scale_means * tf.linspace(-1, 1, num=latent_dim)\n", + " stds = scale_stds * tf.linspace(-1, 1, num=latent_dim)\n", + " latent_space_value_range = tf.random.normal([number, latent_dim],\n", + " means,\n", + " stds,\n", + " dtype=tf.float64)\n", + "\n", + " return latent_space_value_range\n", + "\n", + "def generate_images(latent_points):\n", + " # notice we set training to false because we are making inferences\n", + " generated_images = restored_generator.predict(latent_points)\n", + "\n", + " return generated_images" + ] + }, + { + "cell_type": "code", + "execution_count": 25, + "id": "333a593d", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def plot_result(generated_images, number=100):\n", + " # obviously this assumes sqrt number is an int\n", + " fig, axs = plt.subplots(int(np.sqrt(number)), int(np.sqrt(number)),\n", + " figsize=(10, 10))\n", + "\n", + " for i in range(int(np.sqrt(number))):\n", + " for j in range(int(np.sqrt(number))):\n", + " axs[i, j].imshow(generated_images[i*j], cmap='Greys')\n", + " axs[i, j].axis('off')\n", + "\n", + " plt.show()" + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "id": "2f5f0154", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "generated_images = generate_images(generate_latent_points())\n", + "plot_result(generated_images)" + ] + }, + { + "cell_type": "markdown", + "id": "ff581bf2", + "metadata": { + "editable": true + }, + "source": [ + "We see that the generator generates images that look like MNIST\n", + "numbers: $1, 4, 7, 9$. Let's try to tweak it a bit more to see if we are able\n", + "to generate a similar plot where we generate every MNIST number. Let us now try\n", + "to 'move' a bit around in the latent space. **Note**: decrease the plot number if\n", + "these following cells take too long to run on your computer." + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "id": "d3617ad8", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "plot_number = 225\n", + "\n", + "generated_images = generate_images(generate_latent_points(number=plot_number,\n", + " scale_means=5,\n", + " scale_stds=1))\n", + "plot_result(generated_images, number=plot_number)\n", + "\n", + "generated_images = generate_images(generate_latent_points(number=plot_number,\n", + " scale_means=-5,\n", + " scale_stds=1))\n", + "plot_result(generated_images, number=plot_number)\n", + "\n", + "generated_images = generate_images(generate_latent_points(number=plot_number,\n", + " scale_means=1,\n", + " scale_stds=5))\n", + "plot_result(generated_images, number=plot_number)" + ] + }, + { + "cell_type": "markdown", + "id": "1a074f93", + "metadata": { + "editable": true + }, + "source": [ + "Again, we have found something interesting. *Moving* around using our means\n", + "takes us from digit to digit, while *moving* around using our standard\n", + "deviations seem to increase the number of different digits! In the last image\n", + "above, we can barely make out every MNIST digit. Let us make on last plot using\n", + "this information by upping the standard deviation of our Gaussian noises." + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "id": "4ef8937d", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "plot_number = 400\n", + "generated_images = generate_images(generate_latent_points(number=plot_number,\n", + " scale_means=1,\n", + " scale_stds=10))\n", + "plot_result(generated_images, number=plot_number)" + ] + }, + { + "cell_type": "markdown", + "id": "385a2d0a", + "metadata": { + "editable": true + }, + "source": [ + "A pretty cool result! We see that our generator indeed has learned a\n", + "distribution which qualitatively looks a whole lot like the MNIST dataset.\n", + "\n", + "Another interesting way to explore the latent space of our generator model is by\n", + "interpolating between the MNIST digits. This section is largely based on\n", + "[this excellent blogpost](https://machinelearningmastery.com/how-to-interpolate-and-perform-vector-arithmetic-with-faces-using-a-generative-adversarial-network/)\n", + "by Jason Brownlee.\n", + "\n", + "So let us start by defining a function to interpolate between two points in the\n", + "latent space." + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "id": "57de87b8", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def interpolation(point_1, point_2, n_steps=10):\n", + " ratios = np.linspace(0, 1, num=n_steps)\n", + " vectors = []\n", + " for i, ratio in enumerate(ratios):\n", + " vectors.append(((1.0 - ratio) * point_1 + ratio * point_2))\n", + "\n", + " return tf.stack(vectors)" + ] + }, + { + "cell_type": "markdown", + "id": "cfb76bb6", + "metadata": { + "editable": true + }, + "source": [ + "Now we have all we need to do our interpolation analysis." + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "id": "e25decef", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "plot_number = 100\n", + "latent_points = generate_latent_points(number=plot_number)\n", + "results = None\n", + "for i in range(0, 2*np.sqrt(plot_number), 2):\n", + " interpolated = interpolation(latent_points[i], latent_points[i+1])\n", + " generated_images = generate_images(interpolated)\n", + "\n", + " if results is None:\n", + " results = generated_images\n", + " else:\n", + " results = tf.stack((results, generated_images))\n", + "\n", + "plot_results(results, plot_number)" + ] + } + ], + "metadata": {}, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/doc/LectureNotes/_build/html/_sources/chapteroptimization.ipynb b/doc/LectureNotes/_build/html/_sources/chapteroptimization.ipynb index e8dc9fc17..5d83c5b50 100644 --- a/doc/LectureNotes/_build/html/_sources/chapteroptimization.ipynb +++ b/doc/LectureNotes/_build/html/_sources/chapteroptimization.ipynb @@ -2,7 +2,21 @@ "cells": [ { "cell_type": "markdown", - "metadata": {}, + "id": "4d72e1df", + "metadata": { + "editable": true + }, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "id": "fb6e8fcd", + "metadata": { + "editable": true + }, "source": [ "# Optimization, the central part of any Machine Learning algortithm\n", "\n", @@ -15,9 +29,6 @@ "analytically, however this is not possible in general and we must use\n", "some approximative/numerical method to compute the minimum.\n", "\n", - "\n", - "\n", - "\n", "In our discussion on Logistic Regression we studied the \n", "case of\n", "two classes, with $y_i$ either\n", @@ -28,7 +39,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a507b82a", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{align*}\n", @@ -40,12 +54,13 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "4071429e", + "metadata": { + "editable": true + }, "source": [ "where $\\boldsymbol{\\beta}$ are the weights we wish to extract from data, in our case $\\beta_0$ and $\\beta_1$. \n", "\n", - "\n", - "\n", "Our compact equations used a definition of a vector $\\boldsymbol{y}$ with $n$\n", "elements $y_i$, an $n\\times p$ matrix $\\boldsymbol{X}$ which contains the\n", "$x_i$ values and a vector $\\boldsymbol{p}$ of fitted probabilities\n", @@ -55,7 +70,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a3cca0b2", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}} = -\\boldsymbol{X}^T\\left(\\boldsymbol{y}-\\boldsymbol{p}\\right).\n", @@ -64,7 +82,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "12a4ad28", + "metadata": { + "editable": true + }, "source": [ "If we in addition define a diagonal matrix $\\boldsymbol{W}$ with elements \n", "$p(y_i\\vert x_i,\\boldsymbol{\\beta})(1-p(y_i\\vert x_i,\\boldsymbol{\\beta})$, we can obtain a compact expression of the second derivative as" @@ -72,7 +93,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5d6e7796", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial^2 \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}\\partial \\boldsymbol{\\beta}^T} = \\boldsymbol{X}^T\\boldsymbol{W}\\boldsymbol{X}.\n", @@ -81,12 +105,13 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b324ae78", + "metadata": { + "editable": true + }, "source": [ "This defines what is called the Hessian matrix.\n", "\n", - "\n", - "\n", "If we can set up these equations, Newton-Raphson's iterative method is normally the method of choice. It requires however that we can compute in an efficient way the matrices that define the first and second derivatives. \n", "\n", "Our iterative scheme is then given by" @@ -94,7 +119,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "7c39cc3f", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{\\beta}^{\\mathrm{new}} = \\boldsymbol{\\beta}^{\\mathrm{old}}-\\left(\\frac{\\partial^2 \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}\\partial \\boldsymbol{\\beta}^T}\\right)^{-1}_{\\boldsymbol{\\beta}^{\\mathrm{old}}}\\times \\left(\\frac{\\partial \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}}\\right)_{\\boldsymbol{\\beta}^{\\mathrm{old}}},\n", @@ -103,14 +131,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "10749f5d", + "metadata": { + "editable": true + }, "source": [ "or in matrix form as" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "3ec50136", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{\\beta}^{\\mathrm{new}} = \\boldsymbol{\\beta}^{\\mathrm{old}}-\\left(\\boldsymbol{X}^T\\boldsymbol{W}\\boldsymbol{X} \\right)^{-1}\\times \\left(-\\boldsymbol{X}^T(\\boldsymbol{y}-\\boldsymbol{p}) \\right)_{\\boldsymbol{\\beta}^{\\mathrm{old}}}.\n", @@ -119,13 +153,15 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0145d9eb", + "metadata": { + "editable": true + }, "source": [ "The right-hand side is computed with the old values of $\\beta$. \n", "\n", "If we can compute these matrices, in particular the Hessian, the above is often the easiest method to implement. \n", "\n", - "\n", "Let us quickly remind ourselves how we derive the above method.\n", "\n", "Perhaps the most celebrated of all one-dimensional root-finding\n", @@ -136,8 +172,6 @@ "numerically and/or your function is not of the smooth type, we\n", "normally discourage the use of this method.\n", "\n", - "\n", - "\n", "The Newton-Raphson formula consists geometrically of extending the\n", "tangent line at a current point until it crosses zero, then setting\n", "the next guess to the abscissa of that zero-crossing. The mathematics\n", @@ -147,7 +181,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "43c17534", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -160,7 +197,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "32ef04f0", + "metadata": { + "editable": true + }, "source": [ "For small enough values of the function and for well-behaved\n", "functions, the terms beyond linear are unimportant, hence we obtain" @@ -168,7 +208,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6700017c", + "metadata": { + "editable": true + }, "source": [ "$$\n", "f(x)+(s-x)f'(x)\\approx 0,\n", @@ -177,14 +220,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "8791dec4", + "metadata": { + "editable": true + }, "source": [ "yielding" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "69008872", + "metadata": { + "editable": true + }, "source": [ "$$\n", "s\\approx x-\\frac{f(x)}{f'(x)}.\n", @@ -193,14 +242,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "fce4ef1f", + "metadata": { + "editable": true + }, "source": [ "Having in mind an iterative procedure, it is natural to start iterating with" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "e64aef7e", + "metadata": { + "editable": true + }, "source": [ "$$\n", "x_{n+1}=x_n-\\frac{f(x_n)}{f'(x_n)}.\n", @@ -209,7 +264,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9acecc44", + "metadata": { + "editable": true + }, "source": [ "The above is Newton-Raphson's method. It has a simple geometric\n", "interpretation, namely $x_{n+1}$ is the point where the tangent from\n", @@ -223,16 +281,16 @@ "guess near such a local extremum, so that the first derivative nearly\n", "vanishes, then Newton-Raphson may fail totally\n", "\n", - "\n", - "\n", - "\n", "Newton's method can be generalized to systems of several non-linear equations\n", "and variables. Consider the case with two equations" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "18bc9fd6", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{array}{cc} f_1(x_1,x_2) &=0\\\\\n", @@ -242,14 +300,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "7dcad370", + "metadata": { + "editable": true + }, "source": [ "which we Taylor expand to obtain" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "f1724121", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{array}{cc} 0=f_1(x_1+h_1,x_2+h_2)=&f_1(x_1,x_2)+h_1\n", @@ -264,14 +328,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d04bfa9f", + "metadata": { + "editable": true + }, "source": [ "Defining the Jacobian matrix $\\boldsymbol{J}$ we have" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "a4d3a9e3", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{J}=\\left( \\begin{array}{cc}\n", @@ -283,14 +353,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d9d6ff69", + "metadata": { + "editable": true + }, "source": [ "we can rephrase Newton's method as" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "362a86a3", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\left(\\begin{array}{c} x_1^{n+1} \\\\ x_2^{n+1} \\end{array} \\right)=\n", @@ -301,14 +377,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "4b8c2937", + "metadata": { + "editable": true + }, "source": [ "where we have defined" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "7580aa9a", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\left(\\begin{array}{c} h_1^{n} \\\\ h_2^{n} \\end{array} \\right)=\n", @@ -319,18 +401,26 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a2e3adc3", + "metadata": { + "editable": true + }, "source": [ "We need thus to compute the inverse of the Jacobian matrix and it\n", "is to understand that difficulties may\n", "arise in case $\\boldsymbol{J}$ is nearly singular.\n", "\n", "It is rather straightforward to extend the above scheme to systems of\n", - "more than two non-linear equations. In our case, the Jacobian matrix is given by the Hessian that represents the second derivative of cost function. \n", - "\n", - "\n", - "\n", - "\n", + "more than two non-linear equations. In our case, the Jacobian matrix is given by the Hessian that represents the second derivative of cost function." + ] + }, + { + "cell_type": "markdown", + "id": "83a585c9", + "metadata": { + "editable": true + }, + "source": [ "## Steepest descent\n", "\n", "The basic idea of gradient descent is\n", @@ -343,7 +433,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e127ea11", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathbf{x}_{k+1} = \\mathbf{x}_k - \\gamma_k \\nabla F(\\mathbf{x}_k),\n", @@ -352,7 +445,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "bce435bc", + "metadata": { + "editable": true + }, "source": [ "with $\\gamma_k > 0$.\n", "\n", @@ -360,7 +456,6 @@ "F(\\mathbf{x}_k)$. This means that for a sufficiently small $\\gamma_k$\n", "we are always moving towards smaller function values, i.e a minimum.\n", "\n", - "\n", "The previous observation is the basis of the method of steepest\n", "descent, which is also referred to as just gradient descent (GD). One\n", "starts with an initial guess $\\mathbf{x}_0$ for a minimum of $F$ and\n", @@ -369,7 +464,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "2691da5f", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathbf{x}_{k+1} = \\mathbf{x}_k - \\gamma_k \\nabla F(\\mathbf{x}_k), \\ \\ k \\geq 0.\n", @@ -378,12 +476,14 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e7ae7322", + "metadata": { + "editable": true + }, "source": [ "The parameter $\\gamma_k$ is often referred to as the step length or\n", "the learning rate within the context of Machine Learning.\n", "\n", - "\n", "Ideally the sequence $\\{\\mathbf{x}_k \\}_{k=0}$ converges to a global\n", "minimum of the function $F$. In general we do not know if we are in a\n", "global or local minimum. In the special case when $F$ is a convex\n", @@ -402,9 +502,6 @@ "Note that the gradient is a function of $\\mathbf{x} =\n", "(x_1,\\cdots,x_n)$ which makes it expensive to compute numerically.\n", "\n", - "\n", - "\n", - "\n", "The gradient descent method \n", "is sensitive to the choice of learning rate $\\gamma_k$. This is due\n", "to the fact that we are only guaranteed that $F(\\mathbf{x}_{k+1}) \\leq\n", @@ -415,10 +512,16 @@ "\n", "Many of these shortcomings can be alleviated by introducing\n", "randomness. One such method is that of Stochastic Gradient Descent\n", - "(SGD), see below.\n", - "\n", - "\n", - "\n", + "(SGD), see below." + ] + }, + { + "cell_type": "markdown", + "id": "a6a44ea7", + "metadata": { + "editable": true + }, + "source": [ "## Convex functions\n", "\n", "Ideally we want our cost/loss function to be convex(concave).\n", @@ -433,11 +536,8 @@ "$\\mathbb{R}$. Examples of convex sets of $\\mathbb{R}^2$ are the\n", "regular polygons (triangles, rectangles, pentagons, etc...).\n", "\n", - "\n", - "\n", "**Convex function**: Let $X \\subset \\mathbb{R}^n$ be a convex set. Assume that the function $f: X \\rightarrow \\mathbb{R}$ is continuous, then $f$ is said to be convex if $$f(tx_1 + (1-t)x_2) \\leq tf(x_1) + (1-t)f(x_2) $$ for all $x_1, x_2 \\in X$ and for all $t \\in [0,1]$. If $\\leq$ is replaced with a strict inequaltiy in the definition, we demand $x_1 \\neq x_2$ and $t\\in(0,1)$ then $f$ is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting $f(x_1)$ and $f(x_2)$, the value of the function on the interval $[x_1,x_2]$ is always below the line as illustrated below.\n", "\n", - "\n", "In the following we state first and second-order conditions which\n", "ensures convexity of a function $f$. We write $D_f$ to denote the\n", "domain of $f$, i.e the subset of $R^n$ where $f$ is defined. For more\n", @@ -454,8 +554,6 @@ "make a drawing of $f(x) = x^2+1$ and draw the tangent line to $f(x)$ and\n", "note that it is always below the graph.\n", "\n", - "\n", - "\n", "**Second order condition.**\n", "\n", "Assume that $f$ is twice\n", @@ -465,12 +563,8 @@ "single-variable function this reduces to $f''(x) \\geq 0$. Geometrically this means that $f$ has nonnegative curvature\n", "everywhere.\n", "\n", - "\n", - "\n", "This condition is particularly useful since it gives us an procedure for determining if the function under consideration is convex, apart from using the definition.\n", "\n", - "\n", - "\n", "The next result is of great importance to us and the reason why we are\n", "going on about convex functions. In machine learning we frequently\n", "have to minimize a loss/cost function in order to find the best\n", @@ -487,11 +581,16 @@ "is minimal, where $f$ is convex and differentiable. Then, any point\n", "$x^*$ that satisfies $\\nabla f(x^*) = 0$ is a global minimum.\n", "\n", - "\n", - "\n", - "This result means that if we know that the cost/loss function is convex and we are able to find a minimum, we are guaranteed that it is a global minimum.\n", - "\n", - "\n", + "This result means that if we know that the cost/loss function is convex and we are able to find a minimum, we are guaranteed that it is a global minimum." + ] + }, + { + "cell_type": "markdown", + "id": "809f8f01", + "metadata": { + "editable": true + }, + "source": [ "### Some simple problems\n", "\n", "1. Show that $f(x)=x^2$ is convex for $x \\in \\mathbb{R}$ using the definition of convexity. Hint: If you re-write the definition, $f$ is convex if the following holds for all $x,y \\in D_f$ and any $\\lambda \\in [0,1]$ $\\lambda f(x)+(1-\\lambda)f(y)-f(\\lambda x + (1-\\lambda) y ) \\geq 0$.\n", @@ -502,7 +601,6 @@ "\n", " * $g(x) = -\\ln(x)$ is convex for $x \\in (0,\\infty)$.\n", "\n", - "\n", "3. Let $f(x) = x^2$ and $g(x) = e^x$. Show that $f(g(x))$ and $g(f(x))$ is convex for $x \\in \\mathbb{R}$. Also show that if $f(x)$ is any convex function than $h(x) = e^{f(x)}$ is convex.\n", "\n", "4. A norm is any function that satisfy the following properties\n", @@ -513,13 +611,18 @@ "\n", " * $f(x) \\leq 0$ for all $x \\in \\mathbb{R}^n$ with equality if and only if $x = 0$\n", "\n", - "\n", - "Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this).\n", - "\n", - "\n", + "Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this)." + ] + }, + { + "cell_type": "markdown", + "id": "f3b91277", + "metadata": { + "editable": true + }, + "source": [ "## Standard steepest descent\n", "\n", - "\n", "Before we proceed, we would like to discuss the approach called the\n", "**standard Steepest descent** (different from the above steepest descent discussion), which again leads to us having to be able\n", "to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).\n", @@ -533,7 +636,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ec752109", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{A}\\boldsymbol{x} = \\boldsymbol{b}.\n", @@ -542,14 +648,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "eb64c8e7", + "metadata": { + "editable": true + }, "source": [ "In the iterative process we end up with a problem like" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "7e99eb7f", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{r}= \\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x},\n", @@ -558,20 +670,24 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "3a3f6414", + "metadata": { + "editable": true + }, "source": [ "where $\\boldsymbol{r}$ is the so-called residual or error in the iterative process.\n", "\n", "When we have found the exact solution, $\\boldsymbol{r}=0$.\n", "\n", - "\n", - "\n", "The residual is zero when we reach the minimum of the quadratic equation" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "a88175a4", + "metadata": { + "editable": true + }, "source": [ "$$\n", "P(\\boldsymbol{x})=\\frac{1}{2}\\boldsymbol{x}^T\\boldsymbol{A}\\boldsymbol{x} - \\boldsymbol{x}^T\\boldsymbol{b},\n", @@ -580,19 +696,24 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6992cc4f", + "metadata": { + "editable": true + }, "source": [ "with the constraint that the matrix $\\boldsymbol{A}$ is positive definite and\n", "symmetric. This defines also the Hessian and we want it to be positive definite. \n", "\n", - "\n", "We denote the initial guess for $\\boldsymbol{x}$ as $\\boldsymbol{x}_0$. \n", "We can assume without loss of generality that" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "00e22e2b", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{x}_0=0,\n", @@ -601,14 +722,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "faab896d", + "metadata": { + "editable": true + }, "source": [ "or consider the system" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "02cb8061", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{A}\\boldsymbol{z} = \\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_0,\n", @@ -617,17 +744,22 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "fe52c448", + "metadata": { + "editable": true + }, "source": [ "instead.\n", "\n", - "\n", "One can show that the solution $\\boldsymbol{x}$ is also the unique minimizer of the quadratic form" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "f698b7fe", + "metadata": { + "editable": true + }, "source": [ "$$\n", "f(\\boldsymbol{x}) = \\frac{1}{2}\\boldsymbol{x}^T\\boldsymbol{A}\\boldsymbol{x} - \\boldsymbol{x}^T \\boldsymbol{x} , \\quad \\boldsymbol{x}\\in\\mathbf{R}^n.\n", @@ -636,7 +768,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "022bda7f", + "metadata": { + "editable": true + }, "source": [ "This suggests taking the first basis vector $\\boldsymbol{r}_1$ (see below for definition) \n", "to be the gradient of $f$ at $\\boldsymbol{x}=\\boldsymbol{x}_0$, \n", @@ -645,7 +780,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b64077bc", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{A}\\boldsymbol{x}_0-\\boldsymbol{b},\n", @@ -654,18 +792,23 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "dec06904", + "metadata": { + "editable": true + }, "source": [ "and \n", "$\\boldsymbol{x}_0=0$ it is equal $-\\boldsymbol{b}$.\n", "\n", - "\n", "We can compute the residual iteratively as" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "b566de75", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{r}_{k+1}=\\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_{k+1},\n", @@ -674,14 +817,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "2c2d16e7", + "metadata": { + "editable": true + }, "source": [ "which equals" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "6c97f03d", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{b}-\\boldsymbol{A}(\\boldsymbol{x}_k+\\alpha_k\\boldsymbol{r}_k),\n", @@ -690,14 +839,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "2b40818b", + "metadata": { + "editable": true + }, "source": [ "or" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "6619d064", + "metadata": { + "editable": true + }, "source": [ "$$\n", "(\\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_k)-\\alpha_k\\boldsymbol{A}\\boldsymbol{r}_k,\n", @@ -706,14 +861,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "467c71be", + "metadata": { + "editable": true + }, "source": [ "which gives" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "d58fd1af", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\alpha_k = \\frac{\\boldsymbol{r}_k^T\\boldsymbol{r}_k}{\\boldsymbol{r}_k^T\\boldsymbol{A}\\boldsymbol{r}_k}\n", @@ -722,14 +883,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "38e32957", + "metadata": { + "editable": true + }, "source": [ "leading to the iterative scheme" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "98043fd6", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{x}_{k+1}=\\boldsymbol{x}_k-\\alpha_k\\boldsymbol{r}_{k},\n", @@ -738,7 +905,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 1, + "id": "8c7efe84", "metadata": { "collapsed": false, "editable": true @@ -771,14 +939,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "3bdeb3c1", + "metadata": { + "editable": true + }, "source": [ "And then as countor plot" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 2, + "id": "e6e460db", "metadata": { "collapsed": false, "editable": true @@ -792,14 +964,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d165add7", + "metadata": { + "editable": true + }, "source": [ "Find guesses" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 3, + "id": "236615ae", "metadata": { "collapsed": false, "editable": true @@ -812,14 +988,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "4b90442c", + "metadata": { + "editable": true + }, "source": [ "Run it!" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 4, + "id": "0346fd1d", "metadata": { "collapsed": false, "editable": true @@ -837,14 +1017,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ecff8b5b", + "metadata": { + "editable": true + }, "source": [ "What happened?" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 5, + "id": "0d9b7732", "metadata": { "collapsed": false, "editable": true @@ -859,7 +1043,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b0f8920b", + "metadata": { + "editable": true + }, "source": [ "## Conjugate gradient method\n", "In the CG method we define so-called conjugate directions and two vectors \n", @@ -870,7 +1057,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "67b215c6", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{s}^T\\boldsymbol{A}\\boldsymbol{t}= 0.\n", @@ -879,7 +1069,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "efc1c192", + "metadata": { + "editable": true + }, "source": [ "The philosophy of the CG method is to perform searches in various conjugate directions\n", "of our vectors $\\boldsymbol{x}_i$ obeying the above criterion, namely" @@ -887,7 +1080,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "eb4e1832", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{x}_i^T\\boldsymbol{A}\\boldsymbol{x}_j= 0.\n", @@ -896,7 +1092,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "775cf999", + "metadata": { + "editable": true + }, "source": [ "Two vectors are conjugate if they are orthogonal with respect to \n", "this inner product. Being conjugate is a symmetric relation: if $\\boldsymbol{s}$ is conjugate to $\\boldsymbol{t}$, then $\\boldsymbol{t}$ is conjugate to $\\boldsymbol{s}$.\n", @@ -906,7 +1105,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "95893950", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{v}_i^T\\boldsymbol{A}\\boldsymbol{v}_j= \\lambda\\boldsymbol{v}_i^T\\boldsymbol{v}_j,\n", @@ -915,7 +1117,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "16bfad5c", + "metadata": { + "editable": true + }, "source": [ "which is zero unless $i=j$. \n", "\n", @@ -925,7 +1130,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e2577f1c", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{x}_{i+1}=\\boldsymbol{x}_{i}+\\alpha_i\\boldsymbol{p}_{i}.\n", @@ -934,7 +1142,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e4defaaf", + "metadata": { + "editable": true + }, "source": [ "We assume that $\\boldsymbol{p}_{i}$ is a sequence of $n$ mutually conjugate directions. \n", "Then the $\\boldsymbol{p}_{i}$ form a basis of $R^n$ and we can expand the solution \n", @@ -943,7 +1154,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b4632612", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{x} = \\sum^{n}_{i=1} \\alpha_i \\boldsymbol{p}_i.\n", @@ -952,14 +1166,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a0a06bf2", + "metadata": { + "editable": true + }, "source": [ "The coefficients are given by" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "1db7cc6a", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathbf{A}\\mathbf{x} = \\sum^{n}_{i=1} \\alpha_i \\mathbf{A} \\mathbf{p}_i = \\mathbf{b}.\n", @@ -968,14 +1188,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e7aa5384", + "metadata": { + "editable": true + }, "source": [ "Multiplying with $\\boldsymbol{p}_k^T$ from the left gives" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "884580b0", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{p}_k^T \\boldsymbol{A}\\boldsymbol{x} = \\sum^{n}_{i=1} \\alpha_i\\boldsymbol{p}_k^T \\boldsymbol{A}\\boldsymbol{p}_i= \\boldsymbol{p}_k^T \\boldsymbol{b},\n", @@ -984,14 +1210,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "934990e4", + "metadata": { + "editable": true + }, "source": [ "and we can define the coefficients $\\alpha_k$ as" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "8be96384", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\alpha_k = \\frac{\\boldsymbol{p}_k^T \\boldsymbol{b}}{\\boldsymbol{p}_k^T \\boldsymbol{A} \\boldsymbol{p}_k}\n", @@ -1000,7 +1232,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "44f4d8d5", + "metadata": { + "editable": true + }, "source": [ "If we choose the conjugate vectors $\\boldsymbol{p}_k$ carefully, \n", "then we may not need all of them to obtain a good approximation to the solution \n", @@ -1015,7 +1250,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a8739d7d", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{x}_0=0,\n", @@ -1024,14 +1262,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d871e171", + "metadata": { + "editable": true + }, "source": [ "or consider the system" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "7ed84e84", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{A}\\boldsymbol{z} = \\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_0,\n", @@ -1040,7 +1284,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b690f75b", + "metadata": { + "editable": true + }, "source": [ "instead.\n", "\n", @@ -1049,7 +1296,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "86ce9e8a", + "metadata": { + "editable": true + }, "source": [ "$$\n", "f(\\boldsymbol{x}) = \\frac{1}{2}\\boldsymbol{x}^T\\boldsymbol{A}\\boldsymbol{x} - \\boldsymbol{x}^T \\boldsymbol{x} , \\quad \\boldsymbol{x}\\in\\mathbf{R}^n.\n", @@ -1058,7 +1308,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "7968787f", + "metadata": { + "editable": true + }, "source": [ "This suggests taking the first basis vector $\\boldsymbol{p}_1$ \n", "to be the gradient of $f$ at $\\boldsymbol{x}=\\boldsymbol{x}_0$, \n", @@ -1067,7 +1320,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0d510b11", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{A}\\boldsymbol{x}_0-\\boldsymbol{b},\n", @@ -1076,7 +1332,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "c7c47a8e", + "metadata": { + "editable": true + }, "source": [ "and \n", "$\\boldsymbol{x}_0=0$ it is equal $-\\boldsymbol{b}$.\n", @@ -1088,7 +1347,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5dc0e4ac", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{r}_k=\\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_k.\n", @@ -1097,7 +1359,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0aecb4b3", + "metadata": { + "editable": true + }, "source": [ "Note that $\\boldsymbol{r}_k$ is the negative gradient of $f$ at \n", "$\\boldsymbol{x}=\\boldsymbol{x}_k$, \n", @@ -1110,7 +1375,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f3dfc701", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{p}_{k+1}=\\boldsymbol{r}_k-\\frac{\\boldsymbol{p}_k^T \\boldsymbol{A}\\boldsymbol{r}_k}{\\boldsymbol{p}_k^T\\boldsymbol{A}\\boldsymbol{p}_k} \\boldsymbol{p}_k.\n", @@ -1119,14 +1387,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "12e92892", + "metadata": { + "editable": true + }, "source": [ "We can also compute the residual iteratively as" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "1514b03f", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{r}_{k+1}=\\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_{k+1},\n", @@ -1135,14 +1409,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "7b09db3d", + "metadata": { + "editable": true + }, "source": [ "which equals" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "4a0638f8", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{b}-\\boldsymbol{A}(\\boldsymbol{x}_k+\\alpha_k\\boldsymbol{p}_k),\n", @@ -1151,14 +1431,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "298f5e46", + "metadata": { + "editable": true + }, "source": [ "or" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "acd35abb", + "metadata": { + "editable": true + }, "source": [ "$$\n", "(\\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_k)-\\alpha_k\\boldsymbol{A}\\boldsymbol{p}_k,\n", @@ -1167,14 +1453,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "79625f01", + "metadata": { + "editable": true + }, "source": [ "which gives" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "dc5888e2", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{r}_{k+1}=\\boldsymbol{r}_k-\\boldsymbol{A}\\boldsymbol{p}_{k},\n", @@ -1183,7 +1475,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d11b96a9", + "metadata": { + "editable": true + }, "source": [ "## Revisiting our Linear Regression Solvers\n", "\n", @@ -1203,7 +1498,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 6, + "id": "cf7c349e", "metadata": { "collapsed": false, "editable": true @@ -1217,7 +1513,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "adf34219", + "metadata": { + "editable": true + }, "source": [ "with $x_i \\in [0,1] $ is chosen randomly using a uniform distribution. Additionally we have a stochastic noise chosen according to a normal distribution $\\cal {N}(0,1)$. \n", "The linear regression model is given by" @@ -1225,7 +1524,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d28eb4e0", + "metadata": { + "editable": true + }, "source": [ "$$\n", "h_\\beta(x) = \\boldsymbol{y} = \\beta_0 + \\beta_1 x,\n", @@ -1234,14 +1536,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "60060fcb", + "metadata": { + "editable": true + }, "source": [ "such that" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "82983f16", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{y}_i = \\beta_0 + \\beta_1 x_i.\n", @@ -1250,7 +1558,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "60fc4bc0", + "metadata": { + "editable": true + }, "source": [ "Let $\\mathbf{y} = (y_1,\\cdots,y_n)^T$, $\\mathbf{\\boldsymbol{y}} = (\\boldsymbol{y}_1,\\cdots,\\boldsymbol{y}_n)^T$ and $\\beta = (\\beta_0, \\beta_1)^T$\n", "\n", @@ -1259,7 +1570,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0c5dac72", + "metadata": { + "editable": true + }, "source": [ "$$\n", "X \\equiv \\begin{bmatrix}\n", @@ -1272,14 +1586,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "002d8c9b", + "metadata": { + "editable": true + }, "source": [ "The cost/loss/risk function is given by (" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "b91d8b5a", + "metadata": { + "editable": true + }, "source": [ "$$\n", "C(\\beta) = \\frac{1}{n}||X\\beta-\\mathbf{y}||_{2}^{2} = \\frac{1}{n}\\sum_{i=1}^{100}\\left[ (\\beta_0 + \\beta_1 x_i)^2 - 2 y_i (\\beta_0 + \\beta_1 x_i) + y_i^2\\right]\n", @@ -1288,17 +1608,22 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d0fc6c20", + "metadata": { + "editable": true + }, "source": [ "and we want to find $\\beta$ such that $C(\\beta)$ is minimized.\n", "\n", - "\n", "Computing $\\partial C(\\beta) / \\partial \\beta_0$ and $\\partial C(\\beta) / \\partial \\beta_1$ we can show that the gradient can be written as" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "3f99dab5", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\nabla_{\\beta} C(\\beta) = \\frac{2}{n}\\begin{bmatrix} \\sum_{i=1}^{100} \\left(\\beta_0+\\beta_1x_i-y_i\\right) \\\\\n", @@ -1309,17 +1634,22 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e1305a22", + "metadata": { + "editable": true + }, "source": [ "where $X$ is the design matrix defined above.\n", "\n", - "\n", "The Hessian matrix of $C(\\beta)$ is given by" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "ef62073f", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{H} \\equiv \\begin{bmatrix}\n", @@ -1331,18 +1661,22 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "dfb78235", + "metadata": { + "editable": true + }, "source": [ "This result implies that $C(\\beta)$ is a convex function since the matrix $X^T X$ always is positive semi-definite.\n", "\n", - "\n", - "\n", "We can now write a program that minimizes $C(\\beta)$ using the gradient descent method with a constant learning rate $\\gamma$ according to" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "56cee86c", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\beta_{k+1} = \\beta_k - \\gamma \\nabla_\\beta C(\\beta_k), \\ k=0,1,\\cdots\n", @@ -1351,7 +1685,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6cc71454", + "metadata": { + "editable": true + }, "source": [ "We can use the expression we computed for the gradient and let use a\n", "$\\beta_0$ be chosen randomly and let $\\gamma = 0.001$. Stop iterating\n", @@ -1360,14 +1697,13 @@ "And finally we can compare our solution for $\\beta$ with the analytic result given by \n", "$\\beta= (X^TX)^{-1} X^T \\mathbf{y}$.\n", "\n", - "\n", - "\n", "Here is our simple example" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 7, + "id": "90fab6b7", "metadata": { "collapsed": false, "editable": true @@ -1424,14 +1760,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "48ba87fe", + "metadata": { + "editable": true + }, "source": [ "Alternatively, we can use **Scikit-Learn** as done here" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 8, + "id": "7522775d", "metadata": { "collapsed": false, "editable": true @@ -1458,14 +1798,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "90552e2f", + "metadata": { + "editable": true + }, "source": [ "We have also discussed Ridge regression where the loss function contains a regularized term given by the $L_2$ norm of $\\beta$," ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "5d7c032c", + "metadata": { + "editable": true + }, "source": [ "$$\n", "C_{\\text{ridge}}(\\beta) = \\frac{1}{n}||X\\beta -\\mathbf{y}||^2 + \\lambda ||\\beta||^2, \\ \\lambda \\geq 0.\n", @@ -1474,14 +1820,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ed86f1ba", + "metadata": { + "editable": true + }, "source": [ "In order to minimize $C_{\\text{ridge}}(\\beta)$ using GD we only have adjust the gradient as follows" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "0386e23c", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\nabla_\\beta C_{\\text{ridge}}(\\beta) = \\frac{2}{n}\\begin{bmatrix} \\sum_{i=1}^{100} \\left(\\beta_0+\\beta_1x_i-y_i\\right) \\\\\n", @@ -1492,14 +1844,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b65523fc", + "metadata": { + "editable": true + }, "source": [ "We can easily extend our program to minimize $C_{\\text{ridge}}(\\beta)$ using gradient descent and compare with the analytical solution given by" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "c62584ee", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\beta_{\\text{ridge}} = \\left(X^T X + \\lambda I_{2 \\times 2} \\right)^{-1} X^T \\mathbf{y}.\n", @@ -1508,7 +1866,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 9, + "id": "e8d43667", "metadata": { "collapsed": false, "editable": true @@ -1562,7 +1921,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "1ca83847", + "metadata": { + "editable": true + }, "source": [ "## Using gradient descent methods, limitations\n", "\n", @@ -1576,9 +1938,39 @@ "\n", "* **GD treats all directions in parameter space uniformly.** Another major drawback of GD is that unlike Newton's method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive. \n", "\n", - "* GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points.\n", + "* GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points." + ] + }, + { + "cell_type": "markdown", + "id": "dcf3e808", + "metadata": { + "editable": true + }, + "source": [ + "## Stochastic Gradient Descent (SGD)\n", "\n", - "## Stochastic Gradient Descent\n", + "In stochastic gradient descent, the extreme case is the case where we\n", + "have only one batch, that is we include the whole data set.\n", + "\n", + "This process is called Stochastic Gradient\n", + "Descent (SGD) (or also sometimes on-line gradient descent). This is\n", + "relatively less common to see because in practice due to vectorized\n", + "code optimizations it can be computationally much more efficient to\n", + "evaluate the gradient for 100 examples, than the gradient for one\n", + "example 100 times. Even though SGD technically refers to using a\n", + "single example at a time to evaluate the gradient, you will hear\n", + "people use the term SGD even when referring to mini-batch gradient\n", + "descent (i.e. mentions of MGD for “Minibatch Gradient Descent”, or BGD\n", + "for “Batch gradient descent” are rare to see), where it is usually\n", + "assumed that mini-batches are used. The size of the mini-batch is a\n", + "hyperparameter but it is not very common to cross-validate or bootstrap it. It is\n", + "usually based on memory constraints (if any), or set to some value,\n", + "e.g. 32, 64 or 128. We use powers of 2 in practice because many\n", + "vectorized operation implementations work faster when their inputs are\n", + "sized in powers of 2.\n", + "\n", + "In our notes with SGD we mean stochastic gradient descent with mini-batches.\n", "\n", "Stochastic gradient descent (SGD) and variants thereof address some of\n", "the shortcomings of the Gradient descent method discussed above.\n", @@ -1590,7 +1982,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "473b1af6", + "metadata": { + "editable": true + }, "source": [ "$$\n", "C(\\mathbf{\\beta}) = \\sum_{i=1}^n c_i(\\mathbf{x}_i,\n", @@ -1600,7 +1995,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "3353fe2a", + "metadata": { + "editable": true + }, "source": [ "This in turn means that the gradient can be\n", "computed as a sum over $i$-gradients" @@ -1608,7 +2006,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6e8e47c3", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\nabla_\\beta C(\\mathbf{\\beta}) = \\sum_i^n \\nabla_\\beta c_i(\\mathbf{x}_i,\n", @@ -1618,7 +2019,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a2eeb6ad", + "metadata": { + "editable": true + }, "source": [ "Stochasticity/randomness is introduced by only taking the\n", "gradient on a subset of the data called minibatches. If there are $n$\n", @@ -1626,8 +2030,6 @@ "minibatches. We denote these minibatches by $B_k$ where\n", "$k=1,\\cdots,n/M$.\n", "\n", - "\n", - "\n", "As an example, suppose we have $10$ data points $(\\mathbf{x}_1,\\cdots, \\mathbf{x}_{10})$ \n", "and we choose to have $M=5$ minibathces,\n", "then each minibatch contains two data points. In particular we have\n", @@ -1644,7 +2046,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5a7a0f8b", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\nabla_{\\beta}\n", @@ -1656,14 +2061,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b7b5884f", + "metadata": { + "editable": true + }, "source": [ "Thus a gradient descent step now looks like" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "6492d660", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\beta_{j+1} = \\beta_j - \\gamma_j \\sum_{i \\in B_k}^n \\nabla_\\beta c_i(\\mathbf{x}_i,\n", @@ -1673,7 +2084,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "584164f4", + "metadata": { + "editable": true + }, "source": [ "where $k$ is picked at random with equal\n", "probability from $[1,n/M]$. An iteration over the number of\n", @@ -1684,7 +2098,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 10, + "id": "42d97cf8", "metadata": { "collapsed": false, "editable": true @@ -1694,7 +2109,7 @@ "import numpy as np \n", "\n", "n = 100 #100 datapoints \n", - "M = 5 #size of each minibatch\n", + "M = 5 #size of each mini-batche\n", "m = int(n/M) #number of minibatches\n", "n_epochs = 10 #number of epochs\n", "\n", @@ -1709,7 +2124,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ca9c6c58", + "metadata": { + "editable": true + }, "source": [ "Taking the gradient only on a subset of the data has two important\n", "benefits. First, it introduces randomness which decreases the chance\n", @@ -1719,8 +2137,6 @@ "cheaper since we sum over the datapoints in the $k-th$ minibatch and not\n", "all $n$ datapoints.\n", "\n", - "\n", - "\n", "A natural question is when do we stop the search for a new minimum?\n", "One possibility is to compute the full gradient after a given number\n", "of epochs and check if the norm of the gradient is smaller than some\n", @@ -1732,8 +2148,6 @@ "compare the values of the cost function and keep the $\\beta$ that\n", "gave the lowest value.\n", "\n", - "\n", - "\n", "Another approach is to let the step length $\\gamma_j$ depend on the\n", "number of epochs in such a way that it becomes very small after a\n", "reasonable time such that we do not move at all.\n", @@ -1749,7 +2163,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 11, + "id": "d2921658", "metadata": { "collapsed": false, "editable": true @@ -1784,48 +2199,62 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "84469eb8", + "metadata": { + "editable": true + }, + "source": [ + "We note that we have defined several hyperparameters. These are now the number of epochs, the number of mini-batches and the parameters $t_0$ and $t_1$." + ] + }, + { + "cell_type": "markdown", + "id": "b4b94e7a", + "metadata": { + "editable": true + }, "source": [ "### Program for stochastic gradient" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 12, + "id": "71dcdb35", "metadata": { "collapsed": false, "editable": true }, "outputs": [], "source": [ + "# Importing various packages\n", "# Importing various packages\n", "from math import exp, sqrt\n", "from random import random, seed\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", - "from sklearn.linear_model import SGDRegressor\n", "\n", - "m = 100\n", - "x = 2*np.random.rand(m,1)\n", - "y = 4+3*x+np.random.randn(m,1)\n", + "n = 100\n", + "x = 2*np.random.rand(n,1)\n", + "y = 4+3*x+np.random.randn(n,1)\n", "\n", - "X = np.c_[np.ones((m,1)), x]\n", + "X = np.c_[np.ones((n,1)), x]\n", + "XT_X = X.T @ X\n", "theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)\n", "print(\"Own inversion\")\n", "print(theta_linreg)\n", - "sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1)\n", - "sgdreg.fit(x,y.ravel())\n", - "print(\"sgdreg from scikit\")\n", - "print(sgdreg.intercept_, sgdreg.coef_)\n", - "\n", + "# Hessian matrix\n", + "H = (2.0/n)* XT_X\n", + "EigValues, EigVectors = np.linalg.eig(H)\n", + "print(f\"Eigenvalues of Hessian Matrix:{EigValues}\")\n", "\n", "theta = np.random.randn(2,1)\n", - "eta = 0.1\n", + "eta = 1.0/np.max(EigValues)\n", "Niterations = 1000\n", "\n", "\n", "for iter in range(Niterations):\n", - " gradients = 2.0/m*X.T @ ((X @ theta)-y)\n", + " gradients = 2.0/n*X.T @ ((X @ theta)-y)\n", " theta -= eta*gradients\n", "print(\"theta from own gd\")\n", "print(theta)\n", @@ -1835,8 +2264,9 @@ "ypredict = Xnew.dot(theta)\n", "ypredict2 = Xnew.dot(theta_linreg)\n", "\n", - "\n", "n_epochs = 50\n", + "M = 5 #size of each minibatch\n", + "m = int(n/M) #number of minibatches\n", "t0, t1 = 5, 50\n", "def learning_schedule(t):\n", " return t0/(t+t1)\n", @@ -1844,16 +2274,20 @@ "theta = np.random.randn(2,1)\n", "\n", "for epoch in range(n_epochs):\n", + "# Can you figure out a better way of setting up the contributions to each batch?\n", " for i in range(m):\n", - " random_index = np.random.randint(m)\n", - " xi = X[random_index:random_index+1]\n", - " yi = y[random_index:random_index+1]\n", - " gradients = 2 * xi.T @ ((xi @ theta)-yi)\n", + " random_index = M*np.random.randint(m)\n", + " xi = X[random_index:random_index+M]\n", + " yi = y[random_index:random_index+M]\n", + " gradients = (2.0/M)* xi.T @ ((xi @ theta)-yi)\n", " eta = learning_schedule(epoch*m+i)\n", " theta = theta - eta*gradients\n", "print(\"theta from own sdg\")\n", "print(theta)\n", "\n", + "\n", + "\n", + "\n", "plt.plot(xnew, ypredict, \"r-\")\n", "plt.plot(xnew, ypredict2, \"b-\")\n", "plt.plot(x, y ,'ro')\n", @@ -1866,7 +2300,23 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6231b86f", + "metadata": { + "editable": true + }, + "source": [ + "In the above code, we have use replacement in setting up the\n", + "mini-batches. The discussion\n", + "[here](https://sebastianraschka.com/faq/docs/sgd-methods.html) may be\n", + "useful. More material will be added later." + ] + }, + { + "cell_type": "markdown", + "id": "8d50214c", + "metadata": { + "editable": true + }, "source": [ "## Momentum based GD\n", "\n", @@ -1878,7 +2328,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "45d43ca3", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathbf{v}_{t}=\\gamma \\mathbf{v}_{t-1}+\\eta_{t}\\nabla_\\theta E(\\boldsymbol{\\theta}_t) \\nonumber\n", @@ -1887,7 +2340,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "dcdd91bf", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -1902,7 +2358,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "c7519d1e", + "metadata": { + "editable": true + }, "source": [ "where we have introduced a momentum parameter $\\gamma$, with\n", "$0\\le\\gamma\\le 1$, and for brevity we dropped the explicit notation to\n", @@ -1918,7 +2377,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f4d4340d", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\Delta \\boldsymbol{\\theta}_{t+1} = \\gamma \\Delta \\boldsymbol{\\theta}_t -\\ \\eta_{t}\\nabla_\\theta E(\\boldsymbol{\\theta}_t),\n", @@ -1927,12 +2389,13 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "41ad532a", + "metadata": { + "editable": true + }, "source": [ "where we have defined $\\Delta \\boldsymbol{\\theta}_{t}= \\boldsymbol{\\theta}_t-\\boldsymbol{\\theta}_{t-1}$.\n", "\n", - "\n", - "\n", "Let us try to get more intuition from these equations. It is helpful\n", "to consider a simple physical analogy with a particle of mass $m$\n", "moving in a viscous medium with drag coefficient $\\mu$ and potential\n", @@ -1942,7 +2405,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "eef8ae92", + "metadata": { + "editable": true + }, "source": [ "$$\n", "m {d^2 \\mathbf{w} \\over dt^2} + \\mu {d \\mathbf{w} \\over dt }= -\\nabla_w E(\\mathbf{w}).\n", @@ -1951,14 +2417,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "80481a6d", + "metadata": { + "editable": true + }, "source": [ "We can discretize this equation in the usual way to get" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "6aab8db7", + "metadata": { + "editable": true + }, "source": [ "$$\n", "m { \\mathbf{w}_{t+\\Delta t}-2 \\mathbf{w}_{t} +\\mathbf{w}_{t-\\Delta t} \\over (\\Delta t)^2}+\\mu {\\mathbf{w}_{t+\\Delta t}- \\mathbf{w}_{t} \\over \\Delta t} = -\\nabla_w E(\\mathbf{w}).\n", @@ -1967,14 +2439,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "dd6414d2", + "metadata": { + "editable": true + }, "source": [ "Rearranging this equation, we can rewrite this as" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "b76a8372", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\Delta \\mathbf{w}_{t +\\Delta t}= - { (\\Delta t)^2 \\over m +\\mu \\Delta t} \\nabla_w E(\\mathbf{w})+ {m \\over m +\\mu \\Delta t} \\Delta \\mathbf{w}_t.\n", @@ -1983,7 +2461,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "07bbe6db", + "metadata": { + "editable": true + }, "source": [ "Notice that this equation is identical to previous one if we identify\n", "the position of the particle, $\\mathbf{w}$, with the parameters\n", @@ -1994,7 +2475,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "7a9a2ecf", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\gamma= {m \\over m +\\mu \\Delta t }, \\qquad \\eta = {(\\Delta t)^2 \\over m +\\mu \\Delta t}.\n", @@ -2003,7 +2487,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "fe5a2bbc", + "metadata": { + "editable": true + }, "source": [ "Thus, as the name suggests, the momentum parameter is proportional to\n", "the mass of the particle and effectively provides inertia.\n", @@ -2033,7 +2520,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "04540023", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathbf{v}_{t}=\\gamma \\mathbf{v}_{t-1}+\\eta_{t}\\nabla_\\theta E(\\boldsymbol{\\theta}_t +\\gamma \\mathbf{v}_{t-1}) \\nonumber\n", @@ -2042,7 +2532,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "dfbf53a5", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -2057,12 +2550,13 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "32ac3869", + "metadata": { + "editable": true + }, "source": [ "One of the major advantages of NAG is that it allows for the use of a larger learning rate than GDM for the same choice of $\\gamma$.\n", "\n", - "\n", - "\n", "In stochastic gradient descent, with and without momentum, we still\n", "have to specify a schedule for tuning the learning rates $\\eta_t$\n", "as a function of time. As discussed in the context of Newton's\n", @@ -2081,10 +2575,17 @@ "\n", "Recently, a number of methods have been introduced that accomplish\n", "this by tracking not only the gradient, but also the second moment of\n", - "the gradient. These methods include AdaGrad, AdaDelta, RMS-Prop, and\n", - "ADAM.\n", - "\n", - "\n", + "the gradient. These methods include AdaGrad, AdaDelta, Root Mean Squared Propagation (RMS-Prop), and\n", + "ADAM." + ] + }, + { + "cell_type": "markdown", + "id": "6f575b16", + "metadata": { + "editable": true + }, + "source": [ "### RMS prop\n", "\n", "In RMS prop, in addition to keeping a running average of the first\n", @@ -2095,7 +2596,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "274604c4", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -2110,7 +2614,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "cd679323", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathbf{s}_t =\\beta \\mathbf{s}_{t-1} +(1-\\beta)\\mathbf{g}_t^2 \\nonumber\n", @@ -2119,7 +2626,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0a7f8e9d", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{\\theta}_{t+1}=\\boldsymbol{\\theta}_t - \\eta_t { \\mathbf{g}_t \\over \\sqrt{\\mathbf{s}_t +\\epsilon}}, \\nonumber\n", @@ -2128,7 +2638,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "48ba8fff", + "metadata": { + "editable": true + }, "source": [ "where $\\beta$ controls the averaging time of the second moment and is\n", "typically taken to be about $\\beta=0.9$, $\\eta_t$ is a learning rate\n", @@ -2138,8 +2651,16 @@ "is clear from this formula that the learning rate is reduced in\n", "directions where the norm of the gradient is consistently large. This\n", "greatly speeds up the convergence by allowing us to use a larger\n", - "learning rate for flat directions.\n", - "\n", + "learning rate for flat directions." + ] + }, + { + "cell_type": "markdown", + "id": "83140ab2", + "metadata": { + "editable": true + }, + "source": [ "### ADAM optimizer\n", "\n", "A related algorithm is the ADAM optimizer. In ADAM, we keep a running\n", @@ -2158,7 +2679,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "61874dc3", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -2173,7 +2697,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "24e19d86", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathbf{m}_t = \\beta_1 \\mathbf{m}_{t-1} + (1-\\beta_1) \\mathbf{g}_t \\nonumber\n", @@ -2182,7 +2709,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "506f78ea", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathbf{s}_t =\\beta_2 \\mathbf{s}_{t-1} +(1-\\beta_2)\\mathbf{g}_t^2 \\nonumber\n", @@ -2191,7 +2721,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "cb4b8585", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{\\mathbf{m}}_t={\\mathbf{m}_t \\over 1-\\beta_1^t} \\nonumber\n", @@ -2200,7 +2733,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9b5b11b1", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{\\mathbf{s}}_t ={\\mathbf{s}_t \\over1-\\beta_2^t} \\nonumber\n", @@ -2209,7 +2745,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "92292a73", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{\\theta}_{t+1}=\\boldsymbol{\\theta}_t - \\eta_t { \\boldsymbol{\\mathbf{m}}_t \\over \\sqrt{\\boldsymbol{\\mathbf{s}}_t} +\\epsilon}, \\nonumber\n", @@ -2218,7 +2757,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "1a264832", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -2232,7 +2774,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6a307202", + "metadata": { + "editable": true + }, "source": [ "where $\\beta_1$ and $\\beta_2$ set the memory lifetime of the first and\n", "second moment and are typically taken to be $0.9$ and $0.99$\n", @@ -2248,7 +2793,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "3285f010", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\Delta \\theta_{t+1}= -\\eta_t { \\boldsymbol{m}_t \\over \\sqrt{\\sigma_t^2 + m_t^2 }+\\epsilon}.\n", @@ -2257,7 +2805,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "657349da", + "metadata": { + "editable": true + }, "source": [ "## Practical tips\n", "\n", @@ -2267,8 +2818,16 @@ "\n", "* **Monitor the out-of-sample performance.** Always monitor the performance of your model on a validation set (a small portion of the training data that is held out of the training process to serve as a proxy for the test set. If the validation error starts increasing, then the model is beginning to overfit. Terminate the learning process. This *early stopping* significantly improves performance in many settings.\n", "\n", - "* **Adaptive optimization methods don't always have good generalization.** Recent studies have shown that adaptive methods such as ADAM, RMSPorp, and AdaGrad tend to have poor generalization compared to SGD or SGD with momentum, particularly in the high-dimensional limit (i.e. the number of parameters exceeds the number of data points). Although it is not clear at this stage why these methods perform so well in training deep neural networks, simpler procedures like properly-tuned SGD may work as well or better in these applications.\n", - "\n", + "* **Adaptive optimization methods don't always have good generalization.** Recent studies have shown that adaptive methods such as ADAM, RMSPorp, and AdaGrad tend to have poor generalization compared to SGD or SGD with momentum, particularly in the high-dimensional limit (i.e. the number of parameters exceeds the number of data points). Although it is not clear at this stage why these methods perform so well in training deep neural networks, simpler procedures like properly-tuned SGD may work as well or better in these applications." + ] + }, + { + "cell_type": "markdown", + "id": "65044ac7", + "metadata": { + "editable": true + }, + "source": [ "## Automatic differentiation\n", "\n", "[Automatic differentiation (AD)](https://en.wikipedia.org/wiki/Automatic_differentiation), \n", @@ -2296,15 +2855,16 @@ "while numerical differentiation can introduce round-off errors in the\n", "discretization process and cancellation\n", "\n", - "\n", - "\n", "Python has tools for so-called **automatic differentiation**.\n", "Consider the following example" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "dacf05cf", + "metadata": { + "editable": true + }, "source": [ "$$\n", "f(x) = \\sin\\left(2\\pi x + x^2\\right)\n", @@ -2313,14 +2873,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "da4ad36e", + "metadata": { + "editable": true + }, "source": [ "which has the following derivative" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "f4c3e6c4", + "metadata": { + "editable": true + }, "source": [ "$$\n", "f'(x) = \\cos\\left(2\\pi x + x^2\\right)\\left(2\\pi + 2x\\right)\n", @@ -2329,14 +2895,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "1c8bfd4a", + "metadata": { + "editable": true + }, "source": [ "Using **autograd** we have" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 13, + "id": "e1d91b8b", "metadata": { "collapsed": false, "editable": true @@ -2381,7 +2951,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "cd1158a5", + "metadata": { + "editable": true + }, "source": [ "Here we\n", "experiment with what kind of functions Autograd is capable\n", @@ -2392,7 +2965,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 14, + "id": "e2e9faff", "metadata": { "collapsed": false, "editable": true @@ -2420,7 +2994,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e4a83059", + "metadata": { + "editable": true + }, "source": [ "To differentiate with respect to two (or more) arguments of a Python\n", "function, Autograd need to know at which variable the function if\n", @@ -2429,7 +3006,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 15, + "id": "f65983d8", "metadata": { "collapsed": false, "editable": true @@ -2473,14 +3051,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "1d369f97", + "metadata": { + "editable": true + }, "source": [ "Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable." ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 16, + "id": "46ea0652", "metadata": { "collapsed": false, "editable": true @@ -2508,7 +3090,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9cc6674a", + "metadata": { + "editable": true + }, "source": [ "Note that in this case, when sending an array as input argument, the\n", "output from Autograd is another array. This is the true gradient of\n", @@ -2520,7 +3105,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 17, + "id": "17813055", "metadata": { "collapsed": false, "editable": true @@ -2548,7 +3134,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 18, + "id": "6da49540", "metadata": { "collapsed": false, "editable": true @@ -2572,39 +3159,45 @@ ] }, { - "cell_type": "markdown", - "metadata": {}, + "cell_type": "code", + "execution_count": 19, + "id": "a8ff2c17", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], "source": [ - "1\n", - "8\n", - " \n", - "<\n", - "<\n", - "<\n", - "!\n", - "!\n", - "C\n", - "O\n", - "D\n", - "E\n", - "_\n", - "B\n", - "L\n", - "O\n", - "C\n", - "K\n", - " \n", - " \n", - "p\n", - "y\n", - "c\n", - "o\n", - "d" + "import autograd.numpy as np\n", + "from autograd import grad\n", + "def f6_for(x):\n", + " val = 0\n", + " for i in range(10):\n", + " val = val + x**i\n", + " return val\n", + "\n", + "def f6_while(x):\n", + " val = 0\n", + " i = 0\n", + " while i < 10:\n", + " val = val + x**i\n", + " i = i + 1\n", + " return val\n", + "\n", + "f6_for_grad = grad(f6_for)\n", + "f6_while_grad = grad(f6_while)\n", + "\n", + "x = 0.5\n", + "\n", + "# Print the computed derivaties of f6_for and f6_while\n", + "print(\"The computed derivative of f6_for at x = %g is: %g\"%(x,f6_for_grad(x)))\n", + "print(\"The computed derivative of f6_while at x = %g is: %g\"%(x,f6_while_grad(x)))" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 20, + "id": "ec67d4a3", "metadata": { "collapsed": false, "editable": true @@ -2624,7 +3217,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 21, + "id": "742a2d68", "metadata": { "collapsed": false, "editable": true @@ -2662,48 +3256,55 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e7be6348", + "metadata": { + "editable": true + }, "source": [ "Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input.\n", "\n", + "Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.\n", "\n", - "Autograd supports many features. However, there are some functions that are not supported (yet) by Autograd.\n", - "\n", - "Assigning a value to the variable being differentiated with respect to is an example thereof." + "Assigning a value to the variable being differentiated with respect to" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 22, + "id": "c551058c", "metadata": { "collapsed": false, "editable": true }, "outputs": [], "source": [ - "#import autograd.numpy as np\n", - "#from autograd import grad\n", - "#def f8(x): # Assume x is an array\n", - "# x[2] = 3\n", - "# return x*2\n", + "import autograd.numpy as np\n", + "from autograd import grad\n", + "def f8(x): # Assume x is an array\n", + " x[2] = 3\n", + " return x*2\n", "\n", - "#f8_grad = grad(f8)\n", + "f8_grad = grad(f8)\n", "\n", - "#x = 8.4\n", + "x = 8.4\n", "\n", - "#print(\"The derivative of f8 is:\",f8_grad(x))" + "print(\"The derivative of f8 is:\",f8_grad(x))" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "7a13b21d", + "metadata": { + "editable": true + }, "source": [ "Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible." ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 23, + "id": "19c7502b", "metadata": { "collapsed": false, "editable": true @@ -2725,7 +3326,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "4450885d", + "metadata": { + "editable": true + }, "source": [ "Here we are told that the 'dot' function does not belong to Autograd's\n", "version of a Numpy array. To overcome this, an alternative syntax\n", @@ -2734,7 +3338,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 24, + "id": "013fc7f8", "metadata": { "collapsed": false, "editable": true @@ -2759,14 +3364,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "8d360f7d", + "metadata": { + "editable": true + }, "source": [ "The documentation recommends to avoid inplace operations such as" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 25, + "id": "e29a24eb", "metadata": { "collapsed": false, "editable": true @@ -2781,13 +3390,233 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ad8fbbb7", + "metadata": { + "editable": true + }, "source": [ - "More examples will be added, in particular how to compare autograd with own codes for the gradients." + "## Using Autograd with OLS\n", + "\n", + "We conclude the part on optmization by showing how we can make codes\n", + "for linear regression and logistic regression using **autograd**. The\n", + "first example shows results with ordinary leats squares." + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "id": "904f65dc", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "# Using Autograd to calculate gradients for OLS\n", + "from random import random, seed\n", + "import numpy as np\n", + "import autograd.numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from autograd import grad\n", + "\n", + "def CostOLS(beta):\n", + " return (1.0/n)*np.sum((y-X @ beta)**2)\n", + "\n", + "n = 100\n", + "x = 2*np.random.rand(n,1)\n", + "y = 4+3*x+np.random.randn(n,1)\n", + "\n", + "X = np.c_[np.ones((n,1)), x]\n", + "XT_X = X.T @ X\n", + "theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)\n", + "print(\"Own inversion\")\n", + "print(theta_linreg)\n", + "# Hessian matrix\n", + "H = (2.0/n)* XT_X\n", + "EigValues, EigVectors = np.linalg.eig(H)\n", + "print(f\"Eigenvalues of Hessian Matrix:{EigValues}\")\n", + "\n", + "theta = np.random.randn(2,1)\n", + "eta = 1.0/np.max(EigValues)\n", + "Niterations = 1000\n", + "# define the gradient\n", + "training_gradient = grad(CostOLS)\n", + "\n", + "for iter in range(Niterations):\n", + " gradients = training_gradient(theta)\n", + " theta -= eta*gradients\n", + "print(\"theta from own gd\")\n", + "print(theta)\n", + "\n", + "xnew = np.array([[0],[2]])\n", + "Xnew = np.c_[np.ones((2,1)), xnew]\n", + "ypredict = Xnew.dot(theta)\n", + "ypredict2 = Xnew.dot(theta_linreg)\n", + "\n", + "plt.plot(xnew, ypredict, \"r-\")\n", + "plt.plot(xnew, ypredict2, \"b-\")\n", + "plt.plot(x, y ,'ro')\n", + "plt.axis([0,2.0,0, 15.0])\n", + "plt.xlabel(r'$x$')\n", + "plt.ylabel(r'$y$')\n", + "plt.title(r'Random numbers ')\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "ce338980", + "metadata": { + "editable": true + }, + "source": [ + "### Including Stochastic Gradient Descent with Autograd\n", + "\n", + "In this code we include the stochastic gradient descent approach discussed above. Note here that we specify which argument we are taking the derivative with respect to when using **autograd**." + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "id": "de261f10", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "# Using Autograd to calculate gradients using SGD\n", + "# OLS example\n", + "from random import random, seed\n", + "import numpy as np\n", + "import autograd.numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from autograd import grad\n", + "\n", + "# Note change from previous example\n", + "def CostOLS(y,X,theta):\n", + " return np.sum((y-X @ theta)**2)\n", + "\n", + "n = 100\n", + "x = 2*np.random.rand(n,1)\n", + "y = 4+3*x+np.random.randn(n,1)\n", + "\n", + "X = np.c_[np.ones((n,1)), x]\n", + "XT_X = X.T @ X\n", + "theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)\n", + "print(\"Own inversion\")\n", + "print(theta_linreg)\n", + "# Hessian matrix\n", + "H = (2.0/n)* XT_X\n", + "EigValues, EigVectors = np.linalg.eig(H)\n", + "print(f\"Eigenvalues of Hessian Matrix:{EigValues}\")\n", + "\n", + "theta = np.random.randn(2,1)\n", + "eta = 1.0/np.max(EigValues)\n", + "Niterations = 1000\n", + "\n", + "# Note that we request the derivative wrt third argument (theta, 2 here)\n", + "training_gradient = grad(CostOLS,2)\n", + "\n", + "for iter in range(Niterations):\n", + " gradients = (1.0/n)*training_gradient(y, X, theta)\n", + " theta -= eta*gradients\n", + "print(\"theta from own gd\")\n", + "print(theta)\n", + "\n", + "xnew = np.array([[0],[2]])\n", + "Xnew = np.c_[np.ones((2,1)), xnew]\n", + "ypredict = Xnew.dot(theta)\n", + "ypredict2 = Xnew.dot(theta_linreg)\n", + "\n", + "plt.plot(xnew, ypredict, \"r-\")\n", + "plt.plot(xnew, ypredict2, \"b-\")\n", + "plt.plot(x, y ,'ro')\n", + "plt.axis([0,2.0,0, 15.0])\n", + "plt.xlabel(r'$x$')\n", + "plt.ylabel(r'$y$')\n", + "plt.title(r'Random numbers ')\n", + "plt.show()\n", + "\n", + "n_epochs = 50\n", + "M = 5 #size of each minibatch\n", + "m = int(n/M) #number of minibatches\n", + "t0, t1 = 5, 50\n", + "def learning_schedule(t):\n", + " return t0/(t+t1)\n", + "\n", + "theta = np.random.randn(2,1)\n", + "\n", + "for epoch in range(n_epochs):\n", + "# Can you figure out a better way of setting up the contributions to each batch?\n", + " for i in range(m):\n", + " random_index = M*np.random.randint(m)\n", + " xi = X[random_index:random_index+M]\n", + " yi = y[random_index:random_index+M]\n", + " gradients = (1.0/M)*training_gradient(yi, xi, theta)\n", + " eta = learning_schedule(epoch*m+i)\n", + " theta = theta - eta*gradients\n", + "print(\"theta from own sdg\")\n", + "print(theta)" + ] + }, + { + "cell_type": "markdown", + "id": "ccd8829e", + "metadata": { + "editable": true + }, + "source": [ + "### And Logistic Regression" + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "id": "cc5811d1", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "import autograd.numpy as np\n", + "from autograd import grad\n", + "\n", + "def sigmoid(x):\n", + " return 0.5 * (np.tanh(x / 2.) + 1)\n", + "\n", + "def logistic_predictions(weights, inputs):\n", + " # Outputs probability of a label being true according to logistic model.\n", + " return sigmoid(np.dot(inputs, weights))\n", + "\n", + "def training_loss(weights):\n", + " # Training loss is the negative log-likelihood of the training labels.\n", + " preds = logistic_predictions(weights, inputs)\n", + " label_probabilities = preds * targets + (1 - preds) * (1 - targets)\n", + " return -np.sum(np.log(label_probabilities))\n", + "\n", + "# Build a toy dataset.\n", + "inputs = np.array([[0.52, 1.12, 0.77],\n", + " [0.88, -1.08, 0.15],\n", + " [0.52, 0.06, -1.30],\n", + " [0.74, -2.49, 1.39]])\n", + "targets = np.array([True, True, False, True])\n", + "\n", + "# Define a function that returns gradients of training loss using Autograd.\n", + "training_gradient_fun = grad(training_loss)\n", + "\n", + "# Optimize weights using gradient descent.\n", + "weights = np.array([0.0, 0.0, 0.0])\n", + "print(\"Initial loss:\", training_loss(weights))\n", + "for i in range(100):\n", + " weights -= training_gradient_fun(weights) * 0.01\n", + "\n", + "print(\"Trained loss:\", training_loss(weights))" ] } ], "metadata": {}, "nbformat": 4, - "nbformat_minor": 4 + "nbformat_minor": 5 } diff --git a/doc/LectureNotes/_build/html/_sources/clustering.ipynb b/doc/LectureNotes/_build/html/_sources/clustering.ipynb new file mode 100644 index 000000000..84bc644b4 --- /dev/null +++ b/doc/LectureNotes/_build/html/_sources/clustering.ipynb @@ -0,0 +1,649 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "c3edcae5", + "metadata": { + "editable": true + }, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "id": "4102577e", + "metadata": { + "editable": true + }, + "source": [ + "# Clustering and Unsupervised Learning\n", + "\n", + "In general terms cluster analysis, or clustering, is the task of grouping a\n", + "data-set into different distinct categories based on some measure of equality of\n", + "the data. This measure is often referred to as a **metric** or **similarity\n", + "measure** in the literature (note: sometimes we deal with a **dissimilarity\n", + "measure** instead). Usually, these metrics are formulated as some kind of\n", + "distance function between points in a high-dimensional space.\n", + "\n", + "The simplest, and also the most\n", + "common is the **Euclidean distance**.\n", + "\n", + "The simplest of all clustering algorithms is the **k-means algorithm**\n", + ", sometimes also referred to as *Lloyds algorithm*. It is the simplest and also\n", + "the most common. From its simplicity it obtains both strengths and weaknesses.\n", + "These will be discussed in more detail later. The $k$-means algorithm is a\n", + "**centroid based** clustering algorithm.\n", + "\n", + "Assume, we are given $n$ data points and we wish to split the data into $K < n$\n", + "different categories, or clusters. We label each cluster by an integer" + ] + }, + { + "cell_type": "markdown", + "id": "0deb3255", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "k\\in\\{1, \\cdots, K \\}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "cfd8fe00", + "metadata": { + "editable": true + }, + "source": [ + "In the basic k-means algorithm each point is assigned to only\n", + "one cluster $k$, and these assignments are *non-injective* i.e. many-to-one. We\n", + "can think of these mappings as an encoder $k = C(i)$, which assigns the $i$-th\n", + "data-point $\\bf x_i$ to the $k$-th cluster.\n", + "\n", + "$k$-means algorithm in words:\n", + "1. We start with guesses / random initializations of our $k$ cluster centers/centroids\n", + "\n", + "2. For each centroid the points that are most similar are identified\n", + "\n", + "3. Then we move / replace each centroid with a coordinate average of all the points that were assigned to that centroid.\n", + "\n", + "4. Iterate 2-3 until the centroids no longer move (to some tolerance)\n", + "\n", + "We assume we have $n$ data-points" + ] + }, + { + "cell_type": "markdown", + "id": "a29b7459", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\\label{eq:kmeanspoints} \\tag{1}\n", + " \\boldsymbol{x_i} = \\{x_{i, 1}, \\cdots, x_{i, p}\\}\\in\\mathbb{R}^p.\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "98f9e37b", + "metadata": { + "editable": true + }, + "source": [ + "which we wish to group into $K < n$ clusters. For our dissimilarity measure we\n", + "use the *squared Euclidean distance*" + ] + }, + { + "cell_type": "markdown", + "id": "d3c32572", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\\label{eq:squaredeuclidean} \\tag{2}\n", + " d(\\boldsymbol{x_i}, \\boldsymbol{x_i'}) = \\sum_{j=1}^p(x_{ij} - x_{i'j})^2\n", + " = ||\\boldsymbol{x_i} - \\boldsymbol{x_{i'}}||^2\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "29d24648", + "metadata": { + "editable": true + }, + "source": [ + "We define the so called *within-cluster point scatter* which gives us a\n", + "measure of how close each data point assigned to the same cluster tends to be to\n", + "the all the others." + ] + }, + { + "cell_type": "markdown", + "id": "fce5c797", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\\label{eq:withincluster} \\tag{3}\n", + " W(C) = \\frac{1}{2}\\sum_{k=1}^K\\sum_{C(i)=k}\n", + " \\sum_{C(i')=k}d(\\boldsymbol{x_i}, \\boldsymbol{x_{i'}}) =\n", + " \\sum_{k=1}^KN_k\\sum_{C(i)=k}||\\boldsymbol{x_i} - \\boldsymbol{\\overline{x_k}}||^2\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "674a26b7", + "metadata": { + "editable": true + }, + "source": [ + "where $\\boldsymbol{\\overline{x_k}}$ is the mean vector associated with the $k$-th\n", + "cluster, and $N_k = \\sum_{i=1}^nI(C(i) = k)$, where the $I()$ notation is\n", + "similar to the Kronecker delta (*Commonly used in statistics, it just means that\n", + "when $i = k$ we have the encoder $C(i)$*). In other words, the within-cluster\n", + "scatter measures the compactness of each cluster with respect to the data points\n", + "assigned to each cluster. This is the quantity that the $k$-means algorithm aims\n", + "to minimize. We refer to this quantity $W(C)$ as the within cluster scatter\n", + "because of its relation to the *total scatter*.\n", + "\n", + "We have" + ] + }, + { + "cell_type": "markdown", + "id": "f200e7ff", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\\label{eq:totalscatter} \\tag{4}\n", + " T = W(C) + B(C) = \\frac{1}{2}\\sum_{i=1}^n\n", + " \\sum_{i'=1}^nd(\\boldsymbol{x_i}, \\boldsymbol{x_{i'}})\n", + " = \\frac{1}{2}\\sum_{k=1}^K\\sum_{C(i)=k}\n", + " \\Big(\\sum_{C(i') = k}d(\\boldsymbol{x_i}, \\boldsymbol{x_{i'}})\n", + " + \\sum_{C(i')\\neq k}d(\\boldsymbol{x_i}, \\boldsymbol{x_{i'}})\\Big).\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "5471a94d", + "metadata": { + "editable": true + }, + "source": [ + "This is a quantity that is conserved throughout the $k$-means algorithm. It can\n", + "be thought of as the total amount of information in the data, and it is composed\n", + "of the aforementioned within-cluster scatter and the *between-cluster scatter*\n", + "$B(C)$. In methods such as principle component analysis the total scatter is not\n", + "conserved.\n", + "\n", + "Given a cluster mean $\\boldsymbol{m_k}$ we define the **total cluster variance**" + ] + }, + { + "cell_type": "markdown", + "id": "299a99ce", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\\label{eq:totalclustervariance} \\tag{5}\n", + " \\min_{C, \\{\\boldsymbol{m_k}\\}_1^K}\\sum_{k=1}^KN_k\\sum||\\boldsymbol{x_i} - \\boldsymbol{m_k}||^2\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "bdfa54ee", + "metadata": { + "editable": true + }, + "source": [ + "Now we have all the pieces necessary to formally revisit the $k$-means algorithm.\n", + "\n", + "The $k$-means clustering algorithm goes as follows \n", + "\n", + "1. For a given cluster assignment $C$, and $k$ cluster means $\\left\\{m_1, \\cdots, m_k\\right\\}$. We minimize the total cluster variance with respect to the cluster means $\\{m_k\\}$ yielding the means of the currently assigned clusters.\n", + "\n", + "2. Given a current set of $k$ means $\\{m_k\\}$ the total cluster variance is minimized by assigning each observation to the closest (current) cluster mean. That is $$C(i) = \\underset{1\\leq k\\leq K}{\\mathrm{argmin}} ||\\boldsymbol{x_i} - \\boldsymbol{m_k}||^2$$\n", + "\n", + "3. Steps 1 and 2 are repeated until the assignments do not change." + ] + }, + { + "cell_type": "markdown", + "id": "f5def86c", + "metadata": { + "editable": true + }, + "source": [ + "## Codes and Approaches\n", + "\n", + "1. Before we start we specify a number $k$ which is the number of clusters we want to try to separate our data into.\n", + "\n", + "2. We initially choose $k$ random data points in our data as our initial centroids, *or means* (this is where the name comes from).\n", + "\n", + "3. Assign each data point to their closest centroid, based on the squared Euclidean distance.\n", + "\n", + "4. For each of the $k$ cluster we update the centroid by calculating new mean values for all the data points in the cluster.\n", + "\n", + "5. Iteratively minimize the within cluster scatter by performing steps (3, 4) until the new assignments stop changing (can be to some tolerance) or until a maximum number of iterations have passed.\n", + "\n", + "Let us now program the most basic version of the algorithm using nothing but\n", + "Python with numpy arrays. This code is kept intentionally simple to gradually\n", + "progress our understanding. There is no vectorization of any kind, and even most\n", + "helper functions are not utilized.\n", + "\n", + "We need first a dataset to do our cluster analysis on. In our case\n", + "this is a plain *vanilla* data set using random numbers using a\n", + "Gaussian distribution." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "b0260188", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "%matplotlib inline\n", + "\n", + "import time\n", + "import numpy as np\n", + "import tensorflow as tf\n", + "from matplotlib import image\n", + "import matplotlib.pyplot as plt\n", + "from sklearn.cluster import KMeans\n", + "from IPython.display import display\n", + "\n", + "np.random.seed(2021)" + ] + }, + { + "cell_type": "markdown", + "id": "fe680e35", + "metadata": { + "editable": true + }, + "source": [ + "Next we define functions, for ease of use later, to generate Gaussians and to\n", + "set up our toy data set." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "9db2bbce", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def gaussian_points(dim=2, n_points=1000, mean_vector=np.array([0, 0]),\n", + " sample_variance=1):\n", + " \"\"\"\n", + " Very simple custom function to generate gaussian distributed point clusters\n", + " with variable dimension, number of points, means in each direction\n", + " (must match dim) and sample variance.\n", + "\n", + " Inputs:\n", + " dim (int)\n", + " n_points (int)\n", + " mean_vector (np.array) (where index 0 is x, index 1 is y etc.)\n", + " sample_variance (float)\n", + "\n", + " Returns:\n", + " data (np.array): with dimensions (dim x n_points)\n", + " \"\"\"\n", + "\n", + " mean_matrix = np.zeros(dim) + mean_vector\n", + " covariance_matrix = np.eye(dim) * sample_variance\n", + " data = np.random.multivariate_normal(mean_matrix, covariance_matrix,\n", + " n_points)\n", + " return data\n", + "\n", + "\n", + "\n", + "def generate_simple_clustering_dataset(dim=2, n_points=1000, plotting=True,\n", + " return_data=True):\n", + " \"\"\"\n", + " Toy model to illustrate k-means clustering\n", + " \"\"\"\n", + "\n", + " data1 = gaussian_points(mean_vector=np.array([5, 5]))\n", + " data2 = gaussian_points()\n", + " data3 = gaussian_points(mean_vector=np.array([1, 4.5]))\n", + " data4 = gaussian_points(mean_vector=np.array([5, 1]))\n", + " data = np.concatenate((data1, data2, data3, data4), axis=0)\n", + "\n", + " if plotting:\n", + " fig, ax = plt.subplots()\n", + " ax.scatter(data[:, 0], data[:, 1], alpha=0.2)\n", + " ax.set_title('Toy Model Dataset')\n", + " plt.show()\n", + "\n", + "\n", + " if return_data:\n", + " return data\n", + "\n", + "\n", + "data = generate_simple_clustering_dataset()" + ] + }, + { + "cell_type": "markdown", + "id": "c0bb8c76", + "metadata": { + "editable": true + }, + "source": [ + "With the above dataset we start\n", + "implementing the $k$-means algorithm." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "29a75065", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "\n", + "n_samples, dimensions = data.shape\n", + "n_clusters = 4\n", + "\n", + "# we randomly initialize our centroids\n", + "np.random.seed(2021)\n", + "centroids = data[np.random.choice(n_samples, n_clusters, replace=False), :]\n", + "distances = np.zeros((n_samples, n_clusters))\n", + "\n", + "# first we need to calculate the distance to each centroid from our data\n", + "for k in range(n_clusters):\n", + " for n in range(n_samples):\n", + " dist = 0\n", + " for d in range(dimensions):\n", + " dist += np.abs(data[n, d] - centroids[k, d])**2\n", + " distances[n, k] = dist\n", + "\n", + "# we initialize an array to keep track of to which cluster each point belongs\n", + "# the way we set it up here the index tracks which point and the value which\n", + "# cluster the point belongs to\n", + "cluster_labels = np.zeros(n_samples, dtype='int')\n", + "\n", + "# next we loop through our samples and for every point assign it to the cluster\n", + "# to which it has the smallest distance to\n", + "for n in range(n_samples):\n", + " # tracking variables (all of this is basically just an argmin)\n", + " smallest = 1e10\n", + " smallest_row_index = 1e10\n", + " for k in range(n_clusters):\n", + " if distances[n, k] < smallest:\n", + " smallest = distances[n, k]\n", + " smallest_row_index = k\n", + "\n", + " cluster_labels[n] = smallest_row_index" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "id": "9fae7fc9", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "fig = plt.figure()\n", + "ax = fig.add_subplot()\n", + "unique_cluster_labels = np.unique(cluster_labels)\n", + "for i in unique_cluster_labels:\n", + " ax.scatter(data[cluster_labels == i, 0],\n", + " data[cluster_labels == i, 1],\n", + " label = i,\n", + " alpha = 0.2)\n", + " ax.scatter(centroids[:, 0], centroids[:, 1], c='black')\n", + "\n", + "ax.set_title(\"First Grouping of Points to Centroids\")\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "90d2a873", + "metadata": { + "editable": true + }, + "source": [ + "So what do we have so far? We have 'picked' $k$ centroids at random from our\n", + "data points. There are other ways of more intelligently choosing their\n", + "initializations, however for our purposes randomly is fine. Then we have\n", + "initialized an array 'distances' which holds the information of the distance,\n", + "*or dissimilarity*, of every point to of our centroids. Finally, we have\n", + "initialized an array 'cluster_labels' which according to our distances array\n", + "holds the information of to which centroid every point is assigned. This was the\n", + "first pass of our algorithm. Essentially, all we need to do now is repeat the\n", + "distance and assignment steps above until we have reached a desired convergence\n", + "or a maximum amount of iterations." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "id": "378c29fc", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "\n", + "max_iterations = 100\n", + "tolerance = 1e-8\n", + "\n", + "for iteration in range(max_iterations):\n", + " prev_centroids = centroids.copy()\n", + " for k in range(n_clusters):\n", + " # this array will be used to update our centroid positions\n", + " vector_mean = np.zeros(dimensions)\n", + " mean_divisor = 0\n", + " for n in range(n_samples):\n", + " if cluster_labels[n] == k:\n", + " vector_mean += data[n, :]\n", + " mean_divisor += 1\n", + "\n", + " # update according to the k means\n", + " centroids[k, :] = vector_mean / mean_divisor\n", + "\n", + " # we find the dissimilarity\n", + " for k in range(n_clusters):\n", + " for n in range(n_samples):\n", + " dist = 0\n", + " for d in range(dimensions):\n", + " dist += np.abs(data[n, d] - centroids[k, d])**2\n", + " distances[n, k] = dist\n", + "\n", + " # assign each point\n", + " for n in range(n_samples):\n", + " smallest = 1e10\n", + " smallest_row_index = 1e10\n", + " for k in range(n_clusters):\n", + " if distances[n, k] < smallest:\n", + " smallest = distances[n, k]\n", + " smallest_row_index = k\n", + "\n", + " cluster_labels[n] = smallest_row_index\n", + "\n", + " # convergence criteria\n", + " centroid_difference = np.sum(np.abs(centroids - prev_centroids))\n", + " if centroid_difference < tolerance:\n", + " print(f'Converged at iteration {iteration}')\n", + " break\n", + "\n", + " elif iteration == max_iterations:\n", + " print(f'Did not converge in {max_iterations} iterations')" + ] + }, + { + "cell_type": "markdown", + "id": "545a6742", + "metadata": { + "editable": true + }, + "source": [ + "We now have a simple , un-optimized $k$-means\n", + "clustering implementation. Lets plot the final result" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "id": "d9d3973b", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "fig = plt.figure()\n", + "ax = fig.add_subplot()\n", + "unique_cluster_labels = np.unique(cluster_labels)\n", + "for i in unique_cluster_labels:\n", + " ax.scatter(data[cluster_labels == i, 0],\n", + " data[cluster_labels == i, 1],\n", + " label = i,\n", + " alpha = 0.2)\n", + " ax.scatter(centroids[:, 0], centroids[:, 1], c='black')\n", + "\n", + "ax.set_title(\"Final Result of K-means Clustering\")\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "id": "ee6a145f", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def naive_kmeans(data, n_clusters=4, max_iterations=100, tolerance=1e-8):\n", + " start_time = time.time()\n", + "\n", + " n_samples, dimensions = data.shape\n", + " n_clusters = 4\n", + " #np.random.seed(2021)\n", + " centroids = data[np.random.choice(n_samples, n_clusters, replace=False), :]\n", + " distances = np.zeros((n_samples, n_clusters))\n", + "\n", + " for k in range(n_clusters):\n", + " for n in range(n_samples):\n", + " dist = 0\n", + " for d in range(dimensions):\n", + " dist += np.abs(data[n, d] - centroids[k, d])**2\n", + " distances[n, k] = dist\n", + "\n", + " cluster_labels = np.zeros(n_samples, dtype='int')\n", + "\n", + " for n in range(n_samples):\n", + " smallest = 1e10\n", + " smallest_row_index = 1e10\n", + " for k in range(n_clusters):\n", + " if distances[n, k] < smallest:\n", + " smallest = distances[n, k]\n", + " smallest_row_index = k\n", + "\n", + " cluster_labels[n] = smallest_row_index\n", + "\n", + " for iteration in range(max_iterations):\n", + " prev_centroids = centroids.copy()\n", + " for k in range(n_clusters):\n", + " vector_mean = np.zeros(dimensions)\n", + " mean_divisor = 0\n", + " for n in range(n_samples):\n", + " if cluster_labels[n] == k:\n", + " vector_mean += data[n, :]\n", + " mean_divisor += 1\n", + "\n", + " centroids[k, :] = vector_mean / mean_divisor\n", + "\n", + " for k in range(n_clusters):\n", + " for n in range(n_samples):\n", + " dist = 0\n", + " for d in range(dimensions):\n", + " dist += np.abs(data[n, d] - centroids[k, d])**2\n", + " distances[n, k] = dist\n", + "\n", + " for n in range(n_samples):\n", + " smallest = 1e10\n", + " smallest_row_index = 1e10\n", + " for k in range(n_clusters):\n", + " if distances[n, k] < smallest:\n", + " smallest = distances[n, k]\n", + " smallest_row_index = k\n", + "\n", + " cluster_labels[n] = smallest_row_index\n", + "\n", + " centroid_difference = np.sum(np.abs(centroids - prev_centroids))\n", + " if centroid_difference < tolerance:\n", + " print(f'Converged at iteration {iteration}')\n", + " print(f'Runtime: {time.time() - start_time} seconds')\n", + "\n", + " return cluster_labels, centroids\n", + "\n", + " print(f'Did not converge in {max_iterations} iterations')\n", + " print(f'Runtime: {time.time() - start_time} seconds')\n", + "\n", + " return cluster_labels, centroids" + ] + } + ], + "metadata": {}, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/doc/LectureNotes/_build/html/chapter10.html b/doc/LectureNotes/_build/html/chapter10.html index bca7acf4b..3bc899723 100644 --- a/doc/LectureNotes/_build/html/chapter10.html +++ b/doc/LectureNotes/_build/html/chapter10.html @@ -5,7 +5,7 @@ - 13. Building a Feed Forward Neural Network — Applied Data Analysis and Machine Learning + 14. Building a Feed Forward Neural Network — Applied Data Analysis and Machine Learning @@ -53,8 +53,8 @@ - - + + @@ -199,6 +199,11 @@ 11. Basic ideas of the Principal Component Analysis (PCA) +
  • + + 12. Clustering and Unsupervised Learning + +

    • @@ -208,17 +213,27 @@

    @@ -294,151 +309,151 @@ @@ -452,8 +467,9 @@
    -
    -

    13. Building a Feed Forward Neural Network

    +
    +

    14. Building a Feed Forward Neural Network

    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 @@ -481,7 +497,7 @@ P(y = 1 \mid \hat{x}, \hat{\theta}) = 1 - P(y = 0 \mid \hat{x}, \hat{\theta}) ,

    where \(y \in \{0, 1\}\) and \(\hat{\theta}\) represents the weights and biases of our network.

    -

    13.1. Defining the cost function

    +

    14.1. Defining the cost function

    Our cost function is given as (see the Logistic regression lectures)

    \[ @@ -519,7 +535,7 @@ P(\mathcal{D} \mid \hat{\theta}) = \prod_{i=1}^n \prod_{c=0}^{C-1} [P(y_{ic} = 1

    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!

    -

    13.1.1. Example: binary classification problem

    +

    14.1.1. Example: binary classification problem

    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

    \[ @@ -561,7 +577,7 @@ Our cost function at the final layer

    In case we use another activation function than the logistic one, we need to evaluate other derivatives.

    -

    13.1.2. The Softmax function

    +

    14.1.2. The Softmax function

    In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation \(z_i^l\), that is we need

    \[ @@ -582,7 +598,7 @@ f(z_i^l) = \frac{\exp{(z_i^l)}}{\sum_{m=1}^K\exp{(z_m^l)}}.
    -

    13.2. Developing a code for doing neural networks with back propagation

    +

    14.2. Developing a code for doing neural networks with back propagation

    One can identify a set of key steps when using neural networks to solve supervised learning problems:

    1. Collect and pre-process data

      2. @@ -593,7 +609,7 @@ f(z_i^l) = \frac{\exp{(z_i^l)}}{\sum_{m=1}^K\exp{(z_m^l)}}.

    2. Adjust hyperparameters (if necessary, network architecture)

    -

    13.2.1. Collect and pre-process data

    +

    14.2.1. Collect and pre-process data

    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 @@ -683,12 +699,12 @@ labels = (n_inputs) = (1797,) X = (n_inputs, n_features) = (1797, 64)

    -_images/chapter10_33_1.png +_images/chapter10_39_1.png
    -

    13.2.2. Train and test datasets

    +

    14.2.2. Train and test datasets

    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 @@ -737,7 +753,7 @@ Number of test images: 360

    -

    13.2.3. Define model and architecture

    +

    14.2.3. Define model and architecture

    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

    \[ z = \sum_{i=1}^n w_i a_i ,\]
    @@ -767,7 +783,7 @@ We will be using the sigmoid function 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.

    -

    13.2.4. Layers

    +

    14.2.4. Layers

    • Input

    @@ -824,7 +840,7 @@ of values. Without it, any input with the value 0 will be mapped to zero (before
    -

    13.2.5. Feed-forward pass

    +

    14.2.5. Feed-forward pass

    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 \(l\):

    @@ -917,7 +933,7 @@ correct label for image 0: 6
    -

    13.2.6. Choose cost function and optimizer

    +

    14.2.6. Choose cost function and optimizer

    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. @@ -936,7 +952,7 @@ probability of the correct category \ you got the correct label. The probability of category \(c\) is given by the softmax function. The vector \(\hat{\theta}\) represents the parameters of our network, i.e. all the weights and biases.

    -

    13.2.7. Optimizing the cost function

    +

    14.2.7. Optimizing the cost function

    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

    @@ -962,7 +978,7 @@ We denote each minibatch \(B_k\)The various optmization methods, with codes and algorithms, are discussed in our lectures on Gradient descent approaches.

    -

    13.2.8. Regularization

    +

    14.2.8. Regularization

    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. @@ -985,7 +1001,7 @@ above. This is a clever use of the chain rule that allows us to calculate the gradient efficently.

    -

    13.2.9. Matrix multiplication

    +

    14.2.9. Matrix multiplication

    To more efficently train our network these equations are implemented using matrix operations.
    The error in the output layer is calculated simply as, with \(\hat{t}\) being our targets,

    @@ -1086,7 +1102,7 @@ the Hadamard product, meaning element-wise multiplication.

    Old accuracy on training data: 0.1440501043841336
    -
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
    
 
    
@@ -1098,7 +1114,7 @@ the Hadamard product, meaning element-wise multiplication.
    -

    13.3. Improving performance

    +

    14.3. Improving performance

    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 \(\lambda = 10^{-6},...,10^{-0}\).

    @@ -1215,7 +1231,7 @@ being realizations of this object with different hyperparameters. An implementat
    -

    13.4. Evaluate model performance on test data

    +

    14.4. Evaluate model performance on test data

    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 \(1\).

    @@ -1251,7 +1267,7 @@ The accuracy is as you would expect just the number of images correctly labeled
    -

    13.5. Adjust hyperparameters

    +

    14.5. Adjust hyperparameters

    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).

    @@ -1420,7 +1436,7 @@ Lambda = 10.0 Accuracy score on test set: 0.21944444444444444
    -
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
    
 
    
@@ -1429,7 +1445,7 @@ Lambda =  1e-05
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
    
 
    
@@ -1438,7 +1454,7 @@ Lambda =  0.0001
 Accuracy score on test set:  0.10555555555555556
    -
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
    
 
    
@@ -1447,7 +1463,7 @@ Lambda =  0.001
 Accuracy score on test set:  0.08888888888888889
    -
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
    
 
    
@@ -1456,7 +1472,7 @@ Lambda =  0.01
 Accuracy score on test set:  0.11388888888888889
    -
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
    
 
    
@@ -1465,7 +1481,7 @@ Lambda =  0.1
 Accuracy score on test set:  0.08611111111111111
    -
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
    
 
    
@@ -1474,7 +1490,7 @@ Lambda =  1.0
 Accuracy score on test set:  0.125
    -
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
    
 
    
@@ -1483,11 +1499,11 @@ Lambda =  10.0
 Accuracy score on test set:  0.09166666666666666
    -
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1496,11 +1512,11 @@ Lambda =  1e-05
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1509,11 +1525,11 @@ Lambda =  0.0001
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1522,11 +1538,11 @@ Lambda =  0.001
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1535,11 +1551,11 @@ Lambda =  0.01
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1548,7 +1564,7 @@ Lambda =  0.1
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
    
 
    
@@ -1557,11 +1573,11 @@ Lambda =  1.0
 Accuracy score on test set:  0.125
    -
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1570,11 +1586,11 @@ Lambda =  10.0
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1583,11 +1599,11 @@ Lambda =  1e-05
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1596,11 +1612,11 @@ Lambda =  0.0001
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1609,11 +1625,11 @@ Lambda =  0.001
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1622,11 +1638,11 @@ Lambda =  0.01
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1635,11 +1651,11 @@ Lambda =  0.1
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1648,11 +1664,11 @@ Lambda =  1.0
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1665,7 +1681,7 @@ Accuracy score on test set:  0.07777777777777778
    -

    13.6. Visualization

    +

    14.6. Visualization

    # visual representation of grid search
@@ -1705,25 +1721,25 @@ Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
    
 
    
-_images/chapter10_49_1.png
-_images/chapter10_49_2.png
+_images/chapter10_59_1.png
+_images/chapter10_59_2.png
    -

    13.7. scikit-learn implementation

    +

    14.7. scikit-learn implementation

    scikit-learn focuses more on traditional machine learning methods, such as regression, clustering, decision trees, etc. As such, it has only two types of @@ -2058,13 +2074,13 @@ Accuracy score on test set: 0.2 Learning rate = 10.0 Lambda = 0.001 Accuracy score on test set: 0.10555555555555556 + +Learning rate = 10.0 +Lambda = 0.01 +Accuracy score on test set: 0.06388888888888888 

    Learning rate  =  10.0
-Lambda =  0.01
-Accuracy score on test set:  0.06388888888888888
-
-Learning rate  =  10.0
 Lambda =  0.1
 Accuracy score on test set:  0.08888888888888889
    @@ -2082,7 +2098,7 @@ Accuracy score on test set: 0.09166666666666666
    -

    13.8. Visualization

    +

    14.8. Visualization

    # optional
@@ -2123,13 +2139,13 @@ Accuracy score on test set:  0.09166666666666666
    -_images/chapter10_53_0.png -_images/chapter10_53_1.png +_images/chapter10_63_0.png +_images/chapter10_63_1.png
    -

    13.9. Building neural networks in Tensorflow and Keras

    +

    14.9. Building neural networks in Tensorflow and Keras

    Now we want to build on the experience gained from our neural network implementation in NumPy and scikit-learn and use it to construct a neural network in Tensorflow. Once we have constructed a neural network in NumPy and Tensorflow, building one in Keras is really quite trivial, though the performance may suffer.

    @@ -2154,23 +2170,8 @@ will give an introduction to the lower level Python Application Program Interfaces (APIs), and see how we use them to build our graph. Then we will build (effectively) the same graph in Keras, to see just how simple solving a machine learning problem can be.

    -

    To install tensorflow on Unix/Linux systems, use pip as

    -
    -
    -
    pip3 install tensorflow
-
    -
    -
    -
    -
      File "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/2357089093.py", line 1
-    pip3 install tensorflow
-         ^
-SyntaxError: invalid syntax
-
    -
    -
    -
    -

    and/or if you use anaconda, just write (or install from the graphical user interface) +

    To install tensorflow on Unix/Linux systems, use pip as pip3 install tensorflow +and/or if you use anaconda, just write (or install from the graphical user interface) (current release of CPU-only TensorFlow)

    @@ -2179,6 +2180,14 @@ how simple solving a machine learning problem can be.

    +
    +
      File "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/2259440937.py", line 1
+    conda create -n tf tensorflow
+          ^
+SyntaxError: invalid syntax
+
    +
    +

    To install the current release of GPU TensorFlow

    @@ -2357,7 +2366,7 @@ If you have Anaconda installed you may run the following command

    -

    13.10. The Breast Cancer Data, now with Keras

    +

    14.10. The Breast Cancer Data, now with Keras

    import tensorflow as tf
@@ -2531,7 +2540,7 @@ If you have Anaconda installed you may run the following command
    -

    13.11. Fine-tuning neural network hyperparameters

    +

    14.11. Fine-tuning neural network hyperparameters

    The flexibility of neural networks is also one of their main drawbacks: there are many hyperparameters to tweak. Not only can you use any imaginable network topology (how neurons/nodes are interconnected), @@ -2562,7 +2571,7 @@ of training data. However, you will rarely have to train such networks from scra common to reuse parts of a pretrained state-of-the-art network that performs a similar task.

    -

    13.12. Which activation function should I use?

    +

    14.12. Which activation function should I use?

    The Back propagation algorithm we derived above works by going from the output layer to the input layer, propagating the error gradient on the way. Once the algorithm has computed the gradient of the cost @@ -2627,7 +2636,7 @@ it does not saturate for positive values (and also because it is quite fast to compute).

    -

    13.13. The RELU function family

    +

    14.13. The RELU function family

    The ReLU activation function suffers from a problem known as the dying ReLUs: during training, some neurons effectively die, meaning they stop outputting anything other than 0.

    @@ -2664,7 +2673,7 @@ bootstrap to evaluate other activation functions.

    -

    13.14. Batch Normalization

    +

    14.14. Batch Normalization

    Batch Normalization aims to address the vanishing/exploding gradients problems, and more generally the problem that the distribution of each layer’s inputs changes during training, as the parameters of the previous layers change.

    @@ -2677,7 +2686,7 @@ standard deviation. It does so by evaluating the mean and standard deviation of mini-batch, from this the name batch normalization.

    -

    13.15. Dropout

    +

    14.15. Dropout

    It is a fairly simple algorithm: at every training step, every neuron (including the input neurons but excluding the output neurons) has a probability \(p\) of being temporarily dropped out, meaning it will be entirely ignored during this training step, but it may be active during the next step.

    @@ -2686,7 +2695,7 @@ hyperparameter \(p\) is called It is viewed as one of the most popular regularization techniques.

    -

    13.16. Gradient Clipping

    +

    14.16. Gradient Clipping

    A popular technique to lessen the exploding gradients problem is to simply clip the gradients during backpropagation so that they never exceed some threshold (this is mostly useful for recurrent neural networks).

    @@ -2695,7 +2704,7 @@ networks).

    Normalization is preferred.

    -

    13.17. A top-down perspective on Neural networks

    +

    14.17. A top-down perspective on Neural networks

    The first thing we would like to do is divide the data into two or three parts. A training set, a validation or dev (development) set, and a test set. The test set is the data on which we want to make @@ -2729,7 +2738,7 @@ can serve as another important diagnostic when using DNNs for supervised learning.

    -

    13.18. Limitations of supervised learning with deep networks

    +

    14.18. Limitations of supervised learning with deep networks

    Like all statistical methods, supervised learning using neural networks has important limitations. This is especially important when one seeks to apply these methods, especially to physics problems. Like @@ -2782,7 +2791,7 @@ features).

    previous

    -

    12. Neural networks

    +

    13. Neural networks

    @@ -2790,7 +2799,7 @@ features).

    next

    -

    14. Solving Differential Equations with Deep Learning

    +

    15. Solving Differential Equations with Deep Learning

     diff --git a/doc/LectureNotes/_build/html/chapter11.html b/doc/LectureNotes/_build/html/chapter11.html index 19cb926de..59a43e75a 100644 --- a/doc/LectureNotes/_build/html/chapter11.html +++ b/doc/LectureNotes/_build/html/chapter11.html @@ -5,7 +5,7 @@ - 14. Solving Differential Equations with Deep Learning — Applied Data Analysis and Machine Learning + 15. Solving Differential Equations with Deep Learning — Applied Data Analysis and Machine Learning @@ -53,7 +53,8 @@ - + + @@ -198,6 +199,11 @@ 11. Basic ideas of the Principal Component Analysis (PCA) +
  • + + 12. Clustering and Unsupervised Learning + +

    • @@ -207,17 +213,27 @@

    @@ -293,90 +309,90 @@ @@ -390,8 +406,9 @@
    -
    -

    14. Solving Differential Equations with Deep Learning

    +
    +

    15. Solving Differential Equations with Deep Learning

    The Universal Approximation Theorem states that a neural network can approximate any function at a single hidden layer along with one input and output layer to any given precision.

    @@ -457,7 +474,7 @@ It might happen so that finding an analytical expression of the gradient of Luckily, there exists libraries that makes the job for us through automatic differentiation. Automatic differentiation is a method of finding the derivatives numerically with very high precision.

    -

    14.1. Example: Exponential decay

    +

    15.1. Example: Exponential decay

    An exponential decay of a quantity \(g(x)\) is described by the equation

    @@ -512,7 +529,7 @@ g_t(x, P) = g_0 + x \cdot N(x, P) \]
    -

    14.2. Reformulating the problem

    +

    15.2. Reformulating the problem

    We wish that our neural network manages to minimize a given cost function.

    A reformulation of out equation, (6), must therefore be done, such that it describes the problem a neural network can solve for.

    @@ -657,7 +674,7 @@ C(\boldsymbol{x}, P) = \frac{1}{N} \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i

    Here, gradient descent with a constant step size has been chosen.

    -

    14.3. Gradient descent

    +

    15.3. Gradient descent

    The idea of the gradient descent algorithm is to update parameters in a direction where the cost function decreases goes to a minimum.

    In general, the update of some parameters \(\boldsymbol{\omega}\) given a cost @@ -686,7 +703,7 @@ P_{\text{output},\text{new}} &= P_{\text{output}} - \lambda \nabla_{P_{\text \end{split}\]

    -

    14.4. The code for solving the ODE

    +

    15.4. The code for solving the ODE

    %matplotlib inline
@@ -846,12 +863,12 @@ P_{\text{output},\text{new}} &= P_{\text{output}} - \lambda \nabla_{P_{\text
 Max absolute difference: 0.0437499
    -_images/chapter11_47_2.png +_images/chapter11_50_2.png
    -

    14.5. The network with one input layer, specified number of hidden layers, and one output layer

    +

    15.5. The network with one input layer, specified number of hidden layers, and one output layer

    It is also possible to extend the construction of our network into a more general one, allowing the network to contain more than one hidden layers.

    The number of neurons within each hidden layer are given as a list of integers in the program below.

    @@ -1025,106 +1042,14 @@ Max absolute difference: 0.0437499 return array(a, dtype, copy=False, order=order)
    -
    ---------------------------------------------------------------------------
-KeyboardInterrupt                         Traceback (most recent call last)
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42376/2971492148.py in <module>
-    144     lmb = 0.001
-    145 
---> 146     P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)
-    147 
-    148     res = g_trial_deep(x,P)
-
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42376/2971492148.py in solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb)
-    119         # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases
-    120         # in the hidden layers and output layers evaluated at x.
---> 121         cost_deep_grad =  cost_function_deep_grad(P, x)
-    122 
-    123         for l in range(N_hidden+1):
-
-~/anaconda3/lib/python3.8/site-packages/autograd/wrap_util.py in nary_f(*args, **kwargs)
-     18             else:
-     19                 x = tuple(args[i] for i in argnum)
----> 20             return unary_operator(unary_f, x, *nary_op_args, **nary_op_kwargs)
-     21         return nary_f
-     22     return nary_operator
-
-~/anaconda3/lib/python3.8/site-packages/autograd/differential_operators.py in grad(fun, x)
-     23     arguments as `fun`, but returns the gradient instead. The function `fun`
-     24     should be scalar-valued. The gradient has the same type as the argument."""
----> 25     vjp, ans = _make_vjp(fun, x)
-     26     if not vspace(ans).size == 1:
-     27         raise TypeError("Grad only applies to real scalar-output functions. "
-
-~/anaconda3/lib/python3.8/site-packages/autograd/core.py in make_vjp(fun, x)
-      8 def make_vjp(fun, x):
-      9     start_node = VJPNode.new_root()
----> 10     end_value, end_node =  trace(start_node, fun, x)
-     11     if end_node is None:
-     12         def vjp(g): return vspace(x).zeros()
-
-~/anaconda3/lib/python3.8/site-packages/autograd/tracer.py in trace(start_node, fun, x)
-      8     with trace_stack.new_trace() as t:
-      9         start_box = new_box(x, t, start_node)
----> 10         end_box = fun(start_box)
-     11         if isbox(end_box) and end_box._trace == start_box._trace:
-     12             return end_box._value, end_box._node
-
-~/anaconda3/lib/python3.8/site-packages/autograd/wrap_util.py in unary_f(x)
-     13                 else:
-     14                     subargs = subvals(args, zip(argnum, x))
----> 15                 return fun(*subargs, **kwargs)
-     16             if isinstance(argnum, int):
-     17                 x = args[argnum]
-
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42376/2971492148.py in cost_function_deep(P, x)
-     67 
-     68     # Evaluate the trial function with the current parameters P
----> 69     g_t = g_trial_deep(x,P)
-     70 
-     71     # Find the derivative w.r.t x of the neural network
-
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42376/2971492148.py in g_trial_deep(x, params, g0)
-     57 # The trial solution using the deep neural network:
-     58 def g_trial_deep(x,params, g0 = 10):
----> 59     return g0 + x*deep_neural_network(params, x)
-     60 
-     61 # The right side of the ODE:
-
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42376/2971492148.py in deep_neural_network(deep_params, x)
-     37 
-     38         z_hidden = np.matmul(w_hidden, x_prev)
----> 39         x_hidden = sigmoid(z_hidden)
-     40 
-     41         # Update x_prev such that next layer can use the output from this layer
-
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42376/2971492148.py in sigmoid(z)
-      5 
-      6 def sigmoid(z):
-----> 7     return 1/(1 + np.exp(-z))
-      8 
-      9 # The neural network with one input layer and one output layer,
-
-~/anaconda3/lib/python3.8/site-packages/autograd/tracer.py in f_wrapped(*args, **kwargs)
-     40             if f_wrapped in notrace_primitives[node_constructor]:
-     41                 return f_wrapped(*argvals, **kwargs)
----> 42             parents = tuple(box._node for _     , box in boxed_args)
-     43             argnums = tuple(argnum    for argnum, _   in boxed_args)
-     44             ans = f_wrapped(*argvals, **kwargs)
-
-~/anaconda3/lib/python3.8/site-packages/autograd/tracer.py in <genexpr>(.0)
-     40             if f_wrapped in notrace_primitives[node_constructor]:
-     41                 return f_wrapped(*argvals, **kwargs)
----> 42             parents = tuple(box._node for _     , box in boxed_args)
-     43             argnums = tuple(argnum    for argnum, _   in boxed_args)
-     44             ans = f_wrapped(*argvals, **kwargs)
-
-KeyboardInterrupt: 
+
    Final cost: 0.119936
 
    
 
    
+_images/chapter11_52_3.png
    -

    14.5.1. Example: Population growth

    +

    15.5.1. Example: Population growth

    A logistic model of population growth assumes that a population converges toward an equilibrium. The population growth can be modeled by

    @@ -1335,11 +1260,25 @@ g(t) = \frac{Ag_0}{g_0 + (A - g_0)\exp(-\alpha A t)}
    +
    +
    Initial cost: 0.221805
+
    +
    +
    /Users/MortenImac/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray
+  return array(a, dtype, copy=False, order=order)
+
    +
    +
    Final cost: 0.000417932
+The max absolute difference between the solutions is: 0.00424909
+
    +
    +_images/chapter11_58_3.png +
    -

    14.6. Using forward Euler to solve the ODE

    +

    15.6. Using forward Euler to solve the ODE

    A straightforward way of solving an ODE numerically, is to use Euler’s method.

    Euler’s method uses Taylor series to approximate the value at a function \(f\) at a step \(\Delta x\) from \(x\):

    @@ -1452,10 +1391,27 @@ extending the program that uses the network using Autograd:

    +
    +
    Initial cost: 0.221805
+
    +
    +
    /Users/MortenImac/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray
+  return array(a, dtype, copy=False, order=order)
+
    +
    +
    Final cost: 0.000417932
+The max absolute difference between the solutions is: 0.00424909
+Max absolute difference between Euler method and analytical: 0.011225
+Max absolute difference between deep neural network and analytical: 0.00424909
+
    +
    +_images/chapter11_66_3.png +_images/chapter11_66_4.png +
    -

    14.7. Solving the one dimensional Poisson equation

    +

    15.7. Solving the one dimensional Poisson equation

    The Poisson equation for \(g(x)\) in one dimension is

    @@ -1657,9 +1613,23 @@ g(x) = x(1 - x)\exp(x)
    +
    +
    /Users/MortenImac/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray
+  return array(a, dtype, copy=False, order=order)
+
    +
    +
    Initial cost: 457.256
+
    +
    +
    Final cost: 0.00310113
+The max absolute difference between the solutions is: 0.000464088
+
    +
    +_images/chapter11_79_3.png +
    -

    14.7.1. Comparing with a numerical scheme

    +

    15.7.1. Comparing with a numerical scheme

    The Poisson equation is possible to solve using Taylor series to approximate the second derivative.

    Using Taylor series, the second derivative can be expressed as

    @@ -1933,11 +1903,27 @@ f(x_{N_x - 2})
    +
    +
    Initial cost: 457.256
+
    +
    +
    /Users/MortenImac/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray
+  return array(a, dtype, copy=False, order=order)
+
    +
    +
    Final cost: 0.00310113
+The max absolute difference between the analytical solution and DNN Autograd: 0.000464088
+The max absolute difference between the analytical solution and numerical scheme: 0.00266858
+
    +
    +_images/chapter11_91_3.png +_images/chapter11_91_4.png +
    -

    14.8. Partial Differential Equations

    +

    15.8. Partial Differential Equations

    A partial differential equation (PDE) has a solution here the function is defined by multiple variables. The equation may involve all kinds of combinations of which variables the function is differentiated with @@ -1953,7 +1939,7 @@ respect to.

    \]

    where \(f\) is an expression involving all kinds of possible mixed derivatives of \(g(x_1,\dots,x_N)\) up to an order \(n\). In order for the solution to be unique, some additional conditions must also be given.

    -

    14.8.1. Type of problem

    +

    15.8.1. Type of problem

    The problem our network must solve for, is similar to the ODE case. We must have a trial solution \(g_t\) at hand.

    For instance, the trial solution could be expressed as

    @@ -1968,7 +1954,7 @@ The neural network \(N(x_1,\dots,x_N,

    The role of the function \(h_2(x_1,\dots,x_N,N(x_1,\dots,x_N,P))\), is to ensure that the output of \(N(x_1,\dots,x_N,P)\) is zero when \(g_t(x_1,\dots,x_N)\) is evaluated at the values of \(x_1,\dots,x_N\) where the given conditions must be satisfied. The function \(h_1(x_1,\dots,x_N)\) should alone make \(g_t(x_1,\dots,x_N)\) satisfy the conditions.

    -

    14.8.2. Network requirements

    +

    15.8.2. Network requirements

    The network tries then the minimize the cost function following the same ideas as described for the ODE case, but now with more than one variables to consider. The concept still remains the same; find a set @@ -1994,7 +1980,7 @@ C\left(X, P \right) = \sum_{i=1}^M f\left( \left( \boldsymbol{x}_i, \frac{\parti

    -

    14.9. Example: The diffusion equation

    +

    15.9. Example: The diffusion equation

    In one spatial dimension, the equation reads

    \[ @@ -2162,7 +2148,7 @@ mixed derivatives of \(g(x,t)\)
    -

    14.9.1. Setting up the network using Autograd; The full program

    +

    15.9.1. Setting up the network using Autograd; The full program

    Having set up the network, along with the trial solution and cost function, we can now see how the deep neural network performs by comparing the results to the analytical solution.

    The analytical solution of our problem is

    @@ -2402,11 +2388,200 @@ Using TensorFlow results in a much better execution time. Try it!

    +
    +
    /Users/MortenImac/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray
+  return array(a, dtype, copy=False, order=order)
+
    +
    +
    Initial cost:  41.05505310046362
+
    +
    +
    ---------------------------------------------------------------------------
+KeyboardInterrupt                         Traceback (most recent call last)
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47448/73752910.py in <module>
+    141     lmb = 0.01
+    142 
+--> 143     P = solve_pde_deep_neural_network(x,t, num_hidden_neurons, num_iter, lmb)
+    144 
+    145     ## Store the results
+
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47448/73752910.py in solve_pde_deep_neural_network(x, t, num_neurons, num_iter, lmb)
+    118     # Let the update be done num_iter times
+    119     for i in range(num_iter):
+--> 120         cost_grad =  cost_function_grad(P, x , t)
+    121 
+    122         for l in range(N_hidden+1):
+
+~/anaconda3/lib/python3.8/site-packages/autograd/wrap_util.py in nary_f(*args, **kwargs)
+     18             else:
+     19                 x = tuple(args[i] for i in argnum)
+---> 20             return unary_operator(unary_f, x, *nary_op_args, **nary_op_kwargs)
+     21         return nary_f
+     22     return nary_operator
+
+~/anaconda3/lib/python3.8/site-packages/autograd/differential_operators.py in grad(fun, x)
+     23     arguments as `fun`, but returns the gradient instead. The function `fun`
+     24     should be scalar-valued. The gradient has the same type as the argument."""
+---> 25     vjp, ans = _make_vjp(fun, x)
+     26     if not vspace(ans).size == 1:
+     27         raise TypeError("Grad only applies to real scalar-output functions. "
+
+~/anaconda3/lib/python3.8/site-packages/autograd/core.py in make_vjp(fun, x)
+      8 def make_vjp(fun, x):
+      9     start_node = VJPNode.new_root()
+---> 10     end_value, end_node =  trace(start_node, fun, x)
+     11     if end_node is None:
+     12         def vjp(g): return vspace(x).zeros()
+
+~/anaconda3/lib/python3.8/site-packages/autograd/tracer.py in trace(start_node, fun, x)
+      8     with trace_stack.new_trace() as t:
+      9         start_box = new_box(x, t, start_node)
+---> 10         end_box = fun(start_box)
+     11         if isbox(end_box) and end_box._trace == start_box._trace:
+     12             return end_box._value, end_box._node
+
+~/anaconda3/lib/python3.8/site-packages/autograd/wrap_util.py in unary_f(x)
+     13                 else:
+     14                     subargs = subvals(args, zip(argnum, x))
+---> 15                 return fun(*subargs, **kwargs)
+     16             if isinstance(argnum, int):
+     17                 x = args[argnum]
+
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47448/73752910.py in cost_function(P, x, t)
+     78             g_t = g_trial(point,P)
+     79             g_t_jacobian = g_t_jacobian_func(point,P)
+---> 80             g_t_hessian = g_t_hessian_func(point,P)
+     81 
+     82             g_t_dt = g_t_jacobian[1]
+
+~/anaconda3/lib/python3.8/site-packages/autograd/wrap_util.py in nary_f(*args, **kwargs)
+     18             else:
+     19                 x = tuple(args[i] for i in argnum)
+---> 20             return unary_operator(unary_f, x, *nary_op_args, **nary_op_kwargs)
+     21         return nary_f
+     22     return nary_operator
+
+~/anaconda3/lib/python3.8/site-packages/autograd/differential_operators.py in hessian(fun, x)
+     76 def hessian(fun, x):
+     77     "Returns a function that computes the exact Hessian."
+---> 78     return jacobian(jacobian(fun))(x)
+     79 
+     80 @unary_to_nary
+
+~/anaconda3/lib/python3.8/site-packages/autograd/wrap_util.py in nary_f(*args, **kwargs)
+     18             else:
+     19                 x = tuple(args[i] for i in argnum)
+---> 20             return unary_operator(unary_f, x, *nary_op_args, **nary_op_kwargs)
+     21         return nary_f
+     22     return nary_operator
+
+~/anaconda3/lib/python3.8/site-packages/autograd/differential_operators.py in jacobian(fun, x)
+     59     jacobian_shape = ans_vspace.shape + vspace(x).shape
+     60     grads = map(vjp, ans_vspace.standard_basis())
+---> 61     return np.reshape(np.stack(grads), jacobian_shape)
+     62 
+     63 @unary_to_nary
+
+~/anaconda3/lib/python3.8/site-packages/autograd/numpy/numpy_wrapper.py in stack(arrays, axis)
+     86     # primitives defined in this file
+     87 
+---> 88     arrays = [array(arr) for arr in arrays]
+     89     if not arrays:
+     90         raise ValueError('need at least one array to stack')
+
+~/anaconda3/lib/python3.8/site-packages/autograd/numpy/numpy_wrapper.py in <listcomp>(.0)
+     86     # primitives defined in this file
+     87 
+---> 88     arrays = [array(arr) for arr in arrays]
+     89     if not arrays:
+     90         raise ValueError('need at least one array to stack')
+
+~/anaconda3/lib/python3.8/site-packages/autograd/core.py in vjp(g)
+     12         def vjp(g): return vspace(x).zeros()
+     13     else:
+---> 14         def vjp(g): return backward_pass(g, end_node)
+     15     return vjp, end_value
+     16 
+
+~/anaconda3/lib/python3.8/site-packages/autograd/core.py in backward_pass(g, end_node)
+     19     for node in toposort(end_node):
+     20         outgrad = outgrads.pop(node)
+---> 21         ingrads = node.vjp(outgrad[0])
+     22         for parent, ingrad in zip(node.parents, ingrads):
+     23             outgrads[parent] = add_outgrads(outgrads.get(parent), ingrad)
+
+~/anaconda3/lib/python3.8/site-packages/autograd/core.py in <lambda>(g)
+     65                     "VJP of {} wrt argnum 0 not defined".format(fun.__name__))
+     66             vjp = vjpfun(ans, *args, **kwargs)
+---> 67             return lambda g: (vjp(g),)
+     68         elif L == 2:
+     69             argnum_0, argnum_1 = argnums
+
+~/anaconda3/lib/python3.8/site-packages/autograd/numpy/numpy_vjps.py in <lambda>(g)
+    421     A_ndim = anp.ndim(A)
+    422     B_meta = anp.metadata(B)
+--> 423     return lambda g: matmul_adjoint_1(A, g, A_ndim, B_meta)
+    424 
+    425 defvjp(anp.matmul, matmul_vjp_0, matmul_vjp_1)
+
+~/anaconda3/lib/python3.8/site-packages/autograd/numpy/numpy_vjps.py in matmul_adjoint_1(A, G, A_ndim, B_meta)
+    408     else:  # We need to swap the last two axes of A
+    409         A = anp.swapaxes(A, A_ndim - 2, A_ndim - 1)
+--> 410     result = anp.matmul(A, G)
+    411     if B_is_vec:
+    412         result = anp.squeeze(result, anp.ndim(G) - 1)
+
+~/anaconda3/lib/python3.8/site-packages/autograd/tracer.py in f_wrapped(*args, **kwargs)
+     43             argnums = tuple(argnum    for argnum, _   in boxed_args)
+     44             ans = f_wrapped(*argvals, **kwargs)
+---> 45             node = node_constructor(ans, f_wrapped, argvals, kwargs, argnums, parents)
+     46             return new_box(ans, trace, node)
+     47         else:
+
+~/anaconda3/lib/python3.8/site-packages/autograd/core.py in __init__(self, value, fun, args, kwargs, parent_argnums, parents)
+     34             raise NotImplementedError("VJP of {} wrt argnums {} not defined"
+     35                                       .format(fun_name, parent_argnums))
+---> 36         self.vjp = vjpmaker(parent_argnums, value, args, kwargs)
+     37 
+     38     def initialize_root(self):
+
+~/anaconda3/lib/python3.8/site-packages/autograd/core.py in vjp_argnums(argnums, ans, args, kwargs)
+     75                     "VJP of {} wrt argnums 0, 1 not defined".format(fun.__name__))
+     76             vjp_0 = vjp_0_fun(ans, *args, **kwargs)
+---> 77             vjp_1 = vjp_1_fun(ans, *args, **kwargs)
+     78             return lambda g: (vjp_0(g), vjp_1(g))
+     79         else:
+
+~/anaconda3/lib/python3.8/site-packages/autograd/numpy/numpy_vjps.py in matmul_vjp_1(ans, A, B)
+    420 def matmul_vjp_1(ans, A, B):
+    421     A_ndim = anp.ndim(A)
+--> 422     B_meta = anp.metadata(B)
+    423     return lambda g: matmul_adjoint_1(A, g, A_ndim, B_meta)
+    424 
+
+~/anaconda3/lib/python3.8/site-packages/autograd/tracer.py in f_wrapped(*args, **kwargs)
+     59     def f_wrapped(*args, **kwargs):
+     60         argvals = map(getval, args)
+---> 61         return f_raw(*argvals, **kwargs)
+     62     f_wrapped._is_primitive = True
+     63     return f_wrapped
+
+~/anaconda3/lib/python3.8/site-packages/autograd/numpy/numpy_wrapper.py in metadata(A)
+    146 @notrace_primitive
+    147 def metadata(A):
+--> 148     return _np.shape(A), _np.ndim(A), _np.result_type(A), _np.iscomplexobj(A)
+    149 
+    150 @notrace_primitive
+
+KeyboardInterrupt: 
+
    +
    +
    -

    14.10. Solving the wave equation with Neural Networks

    +

    15.10. Solving the wave equation with Neural Networks

    The wave equation is

    \[ @@ -2693,7 +2868,7 @@ g(x,t) = \sin(\pi x)\cos(\pi t) - \sin(\pi x)\sin(\pi t)
    -

    14.11. Resources on differential equations and deep learning

    +

    15.11. Resources on differential equations and deep learning

    1. Artificial neural networks for solving ordinary and partial differential equations by I.E. Lagaris et al

    2. Neural networks for solving differential equations by A. Honchar

      3. @@ -2737,10 +2912,19 @@ g(x,t) = \sin(\pi x)\cos(\pi t) - \sin(\pi x)\sin(\pi t)

      previous

      -

      13. Building a Feed Forward Neural Network

      +

      14. Building a Feed Forward Neural Network

    + diff --git a/doc/LectureNotes/_build/html/chapter12.html b/doc/LectureNotes/_build/html/chapter12.html new file mode 100644 index 000000000..c1b605ab8 --- /dev/null +++ b/doc/LectureNotes/_build/html/chapter12.html @@ -0,0 +1,1509 @@ + + + + + + + + 16. Convolutional Neural Networks — Applied Data Analysis and Machine Learning + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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    +
    + + + + + + + + + +
    + + +
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    +

    16. Convolutional Neural Networks

    +

    Convolutional neural networks (CNNs) were developed during the last +decade of the previous century, with a focus on character recognition +tasks. Nowadays, CNNs are a central element in the spectacular success +of deep learning methods. The success in for example image +classifications have made them a central tool for most machine +learning practitioners.

    +

    CNNs are very similar to ordinary Neural Networks. +They are made up of neurons that have learnable weights and +biases. Each neuron receives some inputs, performs a dot product and +optionally follows it with a non-linearity. The whole network still +expresses a single differentiable score function: from the raw image +pixels on one end to class scores at the other. And they still have a +loss function (for example Softmax) on the last (fully-connected) layer +and all the tips/tricks we developed for learning regular Neural +Networks still apply (back propagation, gradient descent etc etc).

    +

    CNN architectures make the explicit assumption that +the inputs are images, which allows us to encode certain properties +into the architecture. These then make the forward function more +efficient to implement and vastly reduce the amount of parameters in +the network.

    +

    Here we provide only a superficial overview, for the more interested, we recommend highly the course +IN5400 – Machine Learning for Image Analysis +and the slides of CS231.

    +

    Another good read is the article here https://arxiv.org/pdf/1603.07285.pdf.

    +
    +

    16.1. Neural Networks vs CNNs

    +

    Neural networks are defined as affine transformations, that is +a vector is received as input and is multiplied with a matrix of so-called weights (our unknown paramters) to produce an +output (to which a bias vector is usually added before passing the result +through a nonlinear activation function). This is applicable to any type of input, be it an +image, a sound clip or an unordered collection of features: whatever their +dimensionality, their representation can always be flattened into a vector +before the transformation.

    +

    However, when we consider images, sound clips and many other similar kinds of data, these data have an intrinsic +structure. More formally, they share these important properties:

    +
      +

    • They are stored as multi-dimensional arrays (think of the pixels of a figure) .

      • +

    • They feature one or more axes for which ordering matters (e.g., width and height axes for an image, time axis for a sound clip).

      • +

    • One axis, called the channel axis, is used to access different views of the data (e.g., the red, green and blue channels of a color image, or the left and right channels of a stereo audio track).

      • +
    +

    These properties are not exploited when an affine transformation is applied; in +fact, all the axes are treated in the same way and the topological information +is not taken into account. Still, taking advantage of the implicit structure of +the data may prove very handy in solving some tasks, like computer vision and +speech recognition, and in these cases it would be best to preserve it. This is +where discrete convolutions come into play.

    +

    A discrete convolution is a linear transformation that preserves this notion of +ordering. It is sparse (only a few input units contribute to a given output +unit) and reuses parameters (the same weights are applied to multiple locations +in the input).

    +

    As an example, consider +an image of size \(32\times 32\times 3\) (32 wide, 32 high, 3 color channels), so a +single fully-connected neuron in a first hidden layer of a regular +Neural Network would have \(32\times 32\times 3 = 3072\) weights. This amount still +seems manageable, but clearly this fully-connected structure does not +scale to larger images. For example, an image of more respectable +size, say \(200\times 200\times 3\), would lead to neurons that have +\(200\times 200\times 3 = 120,000\) weights.

    +

    We could have +several such neurons, and the parameters would add up quickly! Clearly, +this full connectivity is wasteful and the huge number of parameters +would quickly lead to possible overfitting.

    + + +

    Figure 1: A regular 3-layer Neural Network.

    + +

    Convolutional Neural Networks take advantage of the fact that the +input consists of images and they constrain the architecture in a more +sensible way.

    +

    In particular, unlike a regular Neural Network, the +layers of a CNN have neurons arranged in 3 dimensions: width, +height, depth. (Note that the word depth here refers to the third +dimension of an activation volume, not to the depth of a full Neural +Network, which can refer to the total number of layers in a network.)

    +

    To understand it better, the above example of an image +with an input volume of +activations has dimensions \(32\times 32\times 3\) (width, height, +depth respectively).

    +

    The neurons in a layer will +only be connected to a small region of the layer before it, instead of +all of the neurons in a fully-connected manner. Moreover, the final +output layer could for this specific image have dimensions \(1\times 1 \times 10\), +because by the +end of the CNN architecture we will reduce the full image into a +single vector of class scores, arranged along the depth +dimension.

    + + +

    Figure 1: A CNN arranges its neurons in three dimensions (width, height, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D output volume of neuron activations. In this example, the red input layer holds the image, so its width and height would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels).

    +
    +
    +

    16.2. Layers used to build CNNs

    +

    A simple CNN is a sequence of layers, and every layer of a CNN +transforms one volume of activations to another through a +differentiable function. We use three main types of layers to build +CNN architectures: Convolutional Layer, Pooling Layer, and +Fully-Connected Layer (exactly as seen in regular Neural Networks). We +will stack these layers to form a full CNN architecture.

    +

    A simple CNN for image classification could have the architecture:

    +
      +

    • INPUT (\(32\times 32 \times 3\)) will hold the raw pixel values of the image, in this case an image of width 32, height 32, and with three color channels R,G,B.

      • +

    • CONV (convolutional )layer will compute the output of neurons that are connected to local regions in the input, each computing a dot product between their weights and a small region they are connected to in the input volume. This may result in volume such as \([32\times 32\times 12]\) if we decided to use 12 filters.

      • +

    • RELU layer will apply an elementwise activation function, such as the \(max(0,x)\) thresholding at zero. This leaves the size of the volume unchanged (\([32\times 32\times 12]\)).

      • +

    • POOL (pooling) layer will perform a downsampling operation along the spatial dimensions (width, height), resulting in volume such as \([16\times 16\times 12]\).

      • +

    • FC (i.e. fully-connected) layer will compute the class scores, resulting in volume of size \([1\times 1\times 10]\), where each of the 10 numbers correspond to a class score, such as among the 10 categories of the MNIST images we considered above . As with ordinary Neural Networks and as the name implies, each neuron in this layer will be connected to all the numbers in the previous volume.

      • +
    +

    CNNs transform the original image layer by layer from the original +pixel values to the final class scores.

    +

    Observe that some layers contain +parameters and other don’t. In particular, the CNN layers perform +transformations that are a function of not only the activations in the +input volume, but also of the parameters (the weights and biases of +the neurons). On the other hand, the RELU/POOL layers will implement a +fixed function. The parameters in the CONV/FC layers will be trained +with gradient descent so that the class scores that the CNN computes +are consistent with the labels in the training set for each image.

    +

    In summary:

    +
      +

    • A CNN architecture is in the simplest case a list of Layers that transform the image volume into an output volume (e.g. holding the class scores)

      • +

    • There are a few distinct types of Layers (e.g. CONV/FC/RELU/POOL are by far the most popular)

      • +

    • Each Layer accepts an input 3D volume and transforms it to an output 3D volume through a differentiable function

      • +

    • Each Layer may or may not have parameters (e.g. CONV/FC do, RELU/POOL don’t)

      • +

    • Each Layer may or may not have additional hyperparameters (e.g. CONV/FC/POOL do, RELU doesn’t)

      • +
    +

    A dense neural network is representd by an affine operation (like matrix-matrix multiplication) where all parameters are included.

    +

    The key idea in CNNs for say imaging is that in images neighbor pixels tend to be related! So we connect +only neighboring neurons in the input instead of connecting all with the first hidden layer.

    +

    We say we perform a filtering (convolution is the mathematical operation).

    +
    +
    +

    16.3. Mathematics of CNNs

    +

    The mathematics of CNNs is based on the mathematical operation of +convolution. In mathematics (in particular in functional analysis), +convolution is represented by matheematical operation (integration, +summation etc) on two function in order to produce a third function +that expresses how the shape of one gets modified by the other. +Convolution has a plethora of applications in a variety of disciplines, spanning from statistics to signal processing, computer vision, solutions of differential equations,linear algebra, engineering, and yes, machine learning.

    +

    Mathematically, convolution is defined as follows (one-dimensional example): +Let us define a continuous function \(y(t)\) given by

    +
    +\[ +y(t) = \int x(a) w(t-a) da, +\]
    +

    where \(x(a)\) represents a so-called input and \(w(t-a)\) is normally called the weight function or kernel.

    +

    The above integral is written in a more compact form as

    +
    +\[ +y(t) = \left(x * w\right)(t). +\]
    +

    The discretized version reads

    +
    +\[ +y(t) = \sum_{a=-\infty}^{a=\infty}x(a)w(t-a). +\]
    +

    Computing the inverse of the above convolution operations is known as deconvolution.

    +

    How can we use this? And what does it mean? Let us study some familiar examples first.

    +
    +

    16.3.1. Convolution Examples: Polynomial multiplication

    +

    We have already met such an example in project 1 when we tried to set +up the design matrix for a two-dimensional function. This was an +example of polynomial multiplication. Let us recast such a problem in terms of the convolution operation. +Let us look a the following polynomials to second and third order, respectively:

    +
    +\[ +p(t) = \alpha_0+\alpha_1 t+\alpha_2 t^2, +\]
    +

    and

    +
    +\[ +s(t) = \beta_0+\beta_1 t+\beta_2 t^2+\beta_3 t^3. +\]
    +

    The polynomial multiplication gives us a new polynomial of degree \(5\)

    +
    +\[ +z(t) = \delta_0+\delta_1 t+\delta_2 t^2+\delta_3 t^3+\delta_4 t^4+\delta_5 t^5. +\]
    +

    Computing polynomial products can be implemented efficiently if we rewrite the more brute force multiplications using convolution. +We note first that the new coefficients are given as

    +
    +\[\begin{split} +\begin{split} +\delta_0=&\alpha_0\beta_0\\ +\delta_1=&\alpha_1\beta_0+\alpha_1\beta_0\\ +\delta_2=&\alpha_0\beta_2+\alpha_1\beta_1+\alpha_2\beta_0\\ +\delta_3=&\alpha_1\beta_2+\alpha_2\beta_1+\alpha_0\beta_3\\ +\delta_4=&\alpha_2\beta_2+\alpha_1\beta_3\\ +\delta_5=&\alpha_2\beta_3.\\ +\end{split} +\end{split}\]
    +

    We note that \(\alpha_i=0\) except for \(i\in \left\{0,1,2\right\}\) and \(\beta_i=0\) except for \(i\in\left\{0,1,2,3\right\}\).

    +

    We can then rewrite the coefficients \(\delta_j\) using a discrete convolution as

    +
    +\[ +\delta_j = \sum_{i=-\infty}^{i=\infty}\alpha_i\beta_{j-i}=(\alpha * \beta)_j, +\]
    +

    or as a double sum with restriction \(l=i+j\)

    +
    +\[ +\delta_l = \sum_{ij}\alpha_i\beta_{j}. +\]
    +

    Do you see a potential drawback with these equations?

    +

    Since we only have a finite number of \(\alpha\) and \(\beta\) values +which are non-zero, we can rewrite the above convolution expressions +as a matrix-vector multiplication

    +
    +\[\begin{split} +\boldsymbol{\delta}=\begin{bmatrix}\alpha_0 & 0 & 0 & 0 \\ + \alpha_1 & \alpha_0 & 0 & 0 \\ + \alpha_2 & \alpha_1 & \alpha_0 & 0 \\ + 0 & \alpha_2 & \alpha_1 & \alpha_0 \\ + 0 & 0 & \alpha_2 & \alpha_1 \\ + 0 & 0 & 0 & \alpha_2 + \end{bmatrix}\begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3\end{bmatrix}. +\end{split}\]
    +

    The process is commutative and we can easily see that we can rewrite the multiplication in terms of a matrix holding \(\beta\) and a vector holding \(\alpha\). +In this case we have

    +
    +\[\begin{split} +\boldsymbol{\delta}=\begin{bmatrix}\beta_0 & 0 & 0 \\ + \beta_1 & \beta_0 & 0 \\ + \beta_2 & \beta_1 & \beta_0 \\ + \beta_3 & \beta_2 & \beta_1 \\ + 0 & \beta_3 & \beta_2 \\ + 0 & 0 & \beta_3 + \end{bmatrix}\begin{bmatrix} \alpha_0 \\ \alpha_1 \\ \alpha_2\end{bmatrix}. +\end{split}\]
    +

    Note that the use of these matrices is for mathematical purposes only and not implementation purposes. +When implementing the above equation we do not encode (and allocate memory) the matrices explicitely. +We rather code the convolutions in the minimal memory footprint that they require.

    +

    Does the number of floating point operations change here when we use the commutative property?

    +
    +
    +

    16.3.2. Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)

    +

    For problems with so-called harmonic oscillations, given by for example the following differential equation

    +
    +\[ +m\frac{d^2x}{dt^2}+\eta\frac{dx}{dt}+x(t)=F(t), +\]
    +

    where \(F(t)\) is an applied external force acting on the system (often called a driving force), one can use the theory of Fourier transformations to find the solutions of this type of equations.

    +

    If one has several driving forces, \(F(t)=\sum_n F_n(t)\), one can find +the particular solution to each \(F_n\), \(x_{pn}(t)\), and the particular +solution for the entire driving force is then given by a series like

    + +
    +
    +\[ +\begin{equation} +x_p(t)=\sum_nx_{pn}(t). +\label{_auto1} \tag{1} +\end{equation} +\]
    +

    This is known as the principle of superposition. It only applies when +the homogenous equation is linear. If there were an anharmonic term +such as \(x^3\) in the homogenous equation, then when one summed various +solutions, \(x=(\sum_n x_n)^2\), one would get cross +terms. Superposition is especially useful when \(F(t)\) can be written +as a sum of sinusoidal terms, because the solutions for each +sinusoidal (sine or cosine) term is analytic.

    +

    Driving forces are often periodic, even when they are not +sinusoidal. Periodicity implies that for some time \(\tau\)

    +
    +\[ +\begin{eqnarray} +F(t+\tau)=F(t). +\end{eqnarray} +\]
    +

    One example of a non-sinusoidal periodic force is a square wave. Many +components in electric circuits are non-linear, e.g. diodes, which +makes many wave forms non-sinusoidal even when the circuits are being +driven by purely sinusoidal sources.

    +

    The code here shows a typical example of such a square wave generated using the functionality included in the scipy Python package. We have used a period of \(\tau=0.2\).

    +
    +
    +
    %matplotlib inline
+
+import numpy as np
+import math
+from scipy import signal
+import matplotlib.pyplot as plt
+
+# number of points                                                                                       
+n = 500
+# start and final times                                                                                  
+t0 = 0.0
+tn = 1.0
+# Period                                                                                                 
+t = np.linspace(t0, tn, n, endpoint=False)
+SqrSignal = np.zeros(n)
+SqrSignal = 1.0+signal.square(2*np.pi*5*t)
+plt.plot(t, SqrSignal)
+plt.ylim(-0.5, 2.5)
+plt.show()
+
    +
    +
    +
    +_images/chapter12_35_0.png +
    +
    +

    For the sinusoidal example the +period is \(\tau=2\pi/\omega\). However, higher harmonics can also +satisfy the periodicity requirement. In general, any force that +satisfies the periodicity requirement can be expressed as a sum over +harmonics,

    + +
    +
    +\[ +\begin{equation} +F(t)=\frac{f_0}{2}+\sum_{n>0} f_n\cos(2n\pi t/\tau)+g_n\sin(2n\pi t/\tau). +\label{_auto2} \tag{2} +\end{equation} +\]
    +

    We can write down the answer for +\(x_{pn}(t)\), by substituting \(f_n/m\) or \(g_n/m\) for \(F_0/m\). By +writing each factor \(2n\pi t/\tau\) as \(n\omega t\), with \(\omega\equiv +2\pi/\tau\),

    + +
    +
    +\[ +\begin{equation} +\label{eq:fourierdef1} \tag{3} +F(t)=\frac{f_0}{2}+\sum_{n>0}f_n\cos(n\omega t)+g_n\sin(n\omega t). +\end{equation} +\]
    +

    The solutions for \(x(t)\) then come from replacing \(\omega\) with +\(n\omega\) for each term in the particular solution,

    +
    +\[\begin{split} +\begin{eqnarray} +x_p(t)&=&\frac{f_0}{2k}+\sum_{n>0} \alpha_n\cos(n\omega t-\delta_n)+\beta_n\sin(n\omega t-\delta_n),\\ +\nonumber +\alpha_n&=&\frac{f_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ +\nonumber +\beta_n&=&\frac{g_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ +\nonumber +\delta_n&=&\tan^{-1}\left(\frac{2\beta n\omega}{\omega_0^2-n^2\omega^2}\right). +\end{eqnarray} +\end{split}\]
    +

    Because the forces have been applied for a long time, any non-zero +damping eliminates the homogenous parts of the solution, so one need +only consider the particular solution for each \(n\).

    +

    The problem is considered solved if one can find expressions for the +coefficients \(f_n\) and \(g_n\), even though the solutions are expressed +as an infinite sum. The coefficients can be extracted from the +function \(F(t)\) by

    + +
    +
    +\[\begin{split} +\begin{eqnarray} +\label{eq:fourierdef2} \tag{4} +f_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\cos(2n\pi t/\tau),\\ +\nonumber +g_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\sin(2n\pi t/\tau). +\end{eqnarray} +\end{split}\]
    +

    To check the consistency of these expressions and to verify +Eq. (4), one can insert the expansion of \(F(t)\) in +Eq. (3) into the expression for the coefficients in +Eq. (4) and see whether

    +
    +\[ +\begin{eqnarray} +f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~\left\{ +\frac{f_0}{2}+\sum_{m>0}f_m\cos(m\omega t)+g_m\sin(m\omega t) +\right\}\cos(n\omega t). +\end{eqnarray} +\]
    +

    Immediately, one can throw away all the terms with \(g_m\) because they +convolute an even and an odd function. The term with \(f_0/2\) +disappears because \(\cos(n\omega t)\) is equally positive and negative +over the interval and will integrate to zero. For all the terms +\(f_m\cos(m\omega t)\) appearing in the sum, one can use angle addition +formulas to see that \(\cos(m\omega t)\cos(n\omega +t)=(1/2)(\cos[(m+n)\omega t]+\cos[(m-n)\omega t]\). This will integrate +to zero unless \(m=n\). In that case the \(m=n\) term gives

    + +
    +
    +\[ +\begin{equation} +\int_{-\tau/2}^{\tau/2}dt~\cos^2(m\omega t)=\frac{\tau}{2}, +\label{_auto3} \tag{5} +\end{equation} +\]
    +

    and

    +
    +\[\begin{split} +\begin{eqnarray} +f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~f_n/2\\ +\nonumber +&=&f_n~\checkmark. +\end{eqnarray} +\end{split}\]
    +

    The same method can be used to check for the consistency of \(g_n\).

    +

    The code here uses the Fourier series applied to a +square wave signal. The code here +visualizes the various approximations given by Fourier series compared +with a square wave with period \(T=0.2\) (dimensionless time), width \(0.1\) and max value of the force \(F=2\). We +see that when we increase the number of components in the Fourier +series, the Fourier series approximation gets closer and closer to the +square wave signal.

    +
    +
    +
    import numpy as np
+import math
+from scipy import signal
+import matplotlib.pyplot as plt
+
+# number of points                                                                                       
+n = 500
+# start and final times                                                                                  
+t0 = 0.0
+tn = 1.0
+# Period                                                                                                 
+T =0.2
+# Max value of square signal                                                                             
+Fmax= 2.0
+# Width of signal   
+Width = 0.1
+t = np.linspace(t0, tn, n, endpoint=False)
+SqrSignal = np.zeros(n)
+FourierSeriesSignal = np.zeros(n)
+SqrSignal = 1.0+signal.square(2*np.pi*5*t+np.pi*Width/T)
+a0 = Fmax*Width/T
+FourierSeriesSignal = a0
+Factor = 2.0*Fmax/np.pi
+for i in range(1,500):
+    FourierSeriesSignal += Factor/(i)*np.sin(np.pi*i*Width/T)*np.cos(i*t*2*np.pi/T)
+plt.plot(t, SqrSignal)
+plt.plot(t, FourierSeriesSignal)
+plt.ylim(-0.5, 2.5)
+plt.show()
+
    +
    +
    +
    +_images/chapter12_51_0.png +
    +
    +
    +
    +
    +

    16.4. Two-dimensional Objects

    +

    We often use convolutions over more than one dimension at a time. If +we have a two-dimensional image \(I\) as input, we can have a filter +defined by a two-dimensional kernel \(K\). This leads to an output \(S\)

    +
    +\[ +S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(m,n)K(i-m,j-n). +\]
    +

    Convolution is a commutatitave process, which means we can rewrite this equation as

    +
    +\[ +S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(i-m,j-n)K(m,n). +\]
    +

    Normally the latter is more straightforward to implement in a machine elarning library since there is less variation in the range of values of \(m\) and \(n\).

    +

    Many deep learning libraries implement cross-correlation instead of convolution

    +
    +\[ +S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(i+m,j-+)K(m,n). +\]
    +
    +
    +

    16.5. More on Dimensionalities

    +

    In fields like signal processing (and imaging as well), one designs +so-called filters. These filters are defined by the convolutions and +are often hand-crafted. One may specify filters for smoothing, edge +detection, frequency reshaping, and similar operations. However with +neural networks the idea is to automatically learn the filters and use +many of them in conjunction with non-linear operations (activation +functions).

    +

    As an example consider a neural network operating on sound sequence +data. Assume that we an input vector \(\boldsymbol{x}\) of length \(d=10^6\). We +construct then a neural network with onle hidden layer only with +\(10^4\) nodes. This means that we will have a weight matrix with +\(10^4\times 10^6=10^{10}\) weights to be determined, together with \(10^4\) biases.

    +

    Assume furthermore that we have an output layer which is meant to train whether the sound sequence represents a human voice (true) or something else (false). +It means that we have only one output node. But since this output node connects to \(10^4\) nodes in the hidden layer, there are in total \(10^4\) weights to be determined for the output layer, plus one bias. In total we have

    +
    +\[ +\mathrm{NumberParameters}=10^{10}+10^4+10^4+1 \approx 10^{10}, +\]
    +

    that is ten billion parameters to determine.

    +
    +
    +

    16.6. Further Dimensionality Remarks

    +

    In today’s architecture one can train such neural networks, however +this is a huge number of parameters for the task at hand. In general, +it is a very wasteful and inefficient use of dense matrices as +parameters. Just as importantly, such trained network parameters are +very specific for the type of input data on which they were trained +and the network is not likely to generalize easily to variations in +the input.

    +

    The main principles that justify convolutions is locality of +information and repetion of patterns within the signal. Sound samples +of the input in adjacent spots are much more likely to affect each +other than those that are very far away. Similarly, sounds are +repeated in multiple times in the signal. While slightly simplistic, +reasoning about such a sound example demonstrates this. The same +principles then apply to images and other similar data.

    +
    +
    +

    16.7. CNNs in more detail, building convolutional neural networks in Tensorflow and Keras

    +

    As discussed above, CNNs are neural networks built from the assumption that the inputs +to the network are 2D images. This is important because the number of features or pixels in images +grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network.

    +

    As before, we still have our input, a hidden layer and an output. What’s novel about convolutional networks +are the convolutional and pooling layers stacked in pairs between the input and the hidden layer. +In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D +matrices, typically 1 for each color dimension (Red, Green, Blue).

    +

    It means that to represent the entire +dataset of images, we require a 4D matrix or tensor. This tensor has the dimensions:

    +
    +\[ +(n_{inputs},\, n_{pixels, width},\, n_{pixels, height},\, depth) . +\]
    +
    +

    16.7.1. The MNIST dataset again

    +

    The MNIST dataset consists of grayscale images with a pixel size of +\(28\times 28\), meaning we require \(28 \times 28 = 724\) weights to each +neuron in the first hidden layer.

    +

    If we were to analyze images of size \(128\times 128\) we would require +\(128 \times 128 = 16384\) weights to each neuron. Even worse if we were +dealing with color images, as most images are, we have an image matrix +of size \(128\times 128\) for each color dimension (Red, Green, Blue), +meaning 3 times the number of weights \(= 49152\) are required for every +single neuron in the first hidden layer.

    +

    Images typically have strong local correlations, meaning that a small +part of the image varies little from its neighboring regions. If for +example we have an image of a blue car, we can roughly assume that a +small blue part of the image is surrounded by other blue regions.

    +

    Therefore, instead of connecting every single pixel to a neuron in the +first hidden layer, as we have previously done with deep neural +networks, we can instead connect each neuron to a small part of the +image (in all 3 RGB depth dimensions). The size of each small area is +fixed, and known as a receptive.

    +

    The layers of a convolutional neural network arrange neurons in 3D: width, height and depth.
    +The input image is typically a square matrix of depth 3.

    +

    A convolution is performed on the image which outputs +a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as filters.

    +

    Each filter slides along the input image, taking the dot product +between each small part of the image and the filter, in all depth +dimensions. This is then passed through a non-linear function, +typically the Rectified Linear (ReLu) function, which serves as the +activation of the neurons in the first convolutional layer. This is +further passed through a pooling layer, which reduces the size of the +convolutional layer, e.g. by taking the maximum or average across some +small regions, and this serves as input to the next convolutional +layer.

    +
    +
    +

    16.7.2. Systematic reduction

    +

    By systematically reducing the size of the input volume, through +convolution and pooling, the network should create representations of +small parts of the input, and then from them assemble representations +of larger areas. The final pooling layer is flattened to serve as +input to a hidden layer, such that each neuron in the final pooling +layer is connected to every single neuron in the hidden layer. This +then serves as input to the output layer, e.g. a softmax output for +classification.

    +
    +
    +

    16.7.3. Prerequisites: Collect and pre-process data

    +
    +
    +
    # 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
+
+# RGB images have a depth of 3
+# our images are grayscale so they should have a depth of 1
+inputs = inputs[:,:,:,np.newaxis]
+
+print("inputs = (n_inputs, pixel_width, pixel_height, depth) = " + str(inputs.shape))
+print("labels = (n_inputs) = " + str(labels.shape))
+
+
+# choose some random images to display
+n_inputs = len(inputs)
+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()
+
    +
    +
    +
    +
    inputs = (n_inputs, pixel_width, pixel_height, depth) = (1797, 8, 8, 1)
+labels = (n_inputs) = (1797,)
+
    +
    +_images/chapter12_67_1.png +
    +
    +
    +
    +
    from tensorflow.keras import datasets, layers, models
+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
+#from tensorflow.keras import Conv2D
+#from tensorflow.keras import MaxPooling2D
+#from tensorflow.keras import Flatten
+
+from sklearn.model_selection import train_test_split
+
+# representation of labels
+labels = to_categorical(labels)
+
+# split into train and test data
+# 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)
+
    +
    +
    +
    +
    +
    +
    def create_convolutional_neural_network_keras(input_shape, receptive_field,
+                                              n_filters, n_neurons_connected, n_categories,
+                                              eta, lmbd):
+    model = Sequential()
+    model.add(layers.Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same',
+              activation='relu', kernel_regularizer=regularizers.l2(lmbd)))
+    model.add(layers.MaxPooling2D(pool_size=(2, 2)))
+    model.add(layers.Flatten())
+    model.add(layers.Dense(n_neurons_connected, activation='relu', kernel_regularizer=regularizers.l2(lmbd)))
+    model.add(layers.Dense(n_categories, activation='softmax', kernel_regularizer=regularizers.l2(lmbd)))
+    
+    sgd = optimizers.SGD(lr=eta)
+    model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])
+    
+    return model
+
+epochs = 100
+batch_size = 100
+input_shape = X_train.shape[1:4]
+receptive_field = 3
+n_filters = 10
+n_neurons_connected = 50
+n_categories = 10
+
+eta_vals = np.logspace(-5, 1, 7)
+lmbd_vals = np.logspace(-5, 1, 7)
+
    +
    +
    +
    +
    +
    +
    CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+        
+for i, eta in enumerate(eta_vals):
+    for j, lmbd in enumerate(lmbd_vals):
+        CNN = create_convolutional_neural_network_keras(input_shape, receptive_field,
+                                              n_filters, n_neurons_connected, n_categories,
+                                              eta, lmbd)
+        CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
+        scores = CNN.evaluate(X_test, Y_test)
+        
+        CNN_keras[i][j] = CNN
+        
+        print("Learning rate = ", eta)
+        print("Lambda = ", lmbd)
+        print("Test accuracy: %.3f" % scores[1])
+        print()
+
    +
    +
    +
    +
    2021-12-08 06:58:03.630224: I tensorflow/core/platform/cpu_feature_guard.cc:142] This TensorFlow binary is optimized with oneAPI Deep Neural Network Library (oneDNN) to use the following CPU instructions in performance-critical operations:  SSE4.1 SSE4.2 AVX AVX2 FMA
+To enable them in other operations, rebuild TensorFlow with the appropriate compiler flags.
+/Users/MortenImac/anaconda3/lib/python3.8/site-packages/keras/optimizer_v2/optimizer_v2.py:355: UserWarning: The `lr` argument is deprecated, use `learning_rate` instead.
+  warnings.warn(
+
    +
    +
    2021-12-08 06:58:04.114437: I tensorflow/compiler/mlir/mlir_graph_optimization_pass.cc:185] None of the MLIR Optimization Passes are enabled (registered 2)
+
    +
    +
     1/12 [=>............................] - ETA: 1s - loss: 2.4411 - accuracy: 0.2500
+
    +
    +
    
+12/12 [==============================] - 0s 853us/step - loss: 2.4410 - accuracy: 0.1944
+
    +
    +
    Learning rate =  1e-05
+Lambda =  1e-05
+Test accuracy: 0.194
+
    +
    +
     1/12 [=>............................] - ETA: 0s - loss: 4.2597 - accuracy: 0.0938
+
    +
    +
    
+12/12 [==============================] - 0s 842us/step - loss: 3.4744 - accuracy: 0.1361
+
    +
    +
    Learning rate =  1e-05
+Lambda =  0.0001
+Test accuracy: 0.136
+
    +
    +
     1/12 [=>............................] - ETA: 1s - loss: 3.0466 - accuracy: 0.1562
+
    +
    +
    
+12/12 [==============================] - 0s 743us/step - loss: 2.9400 - accuracy: 0.1028
+
    +
    +
    Learning rate =  1e-05
+Lambda =  0.001
+Test accuracy: 0.103
+
    +
    +
     1/12 [=>............................] - ETA: 1s - loss: 3.8064 - accuracy: 0.1250
+
    +
    +
    
+12/12 [==============================] - 0s 912us/step - loss: 3.8175 - accuracy: 0.1056
+
    +
    +
    Learning rate =  1e-05
+Lambda =  0.01
+Test accuracy: 0.106
+
    +
    +
     1/12 [=>............................] - ETA: 1s - loss: 12.1612 - accuracy: 0.0938
+
    +
    +
    
+12/12 [==============================] - 0s 763us/step - loss: 12.3063 - accuracy: 0.1361
+
    +
    +
    Learning rate =  1e-05
+Lambda =  0.1
+Test accuracy: 0.136
+
    +
    +
     1/12 [=>............................] - ETA: 0s - loss: 91.8768 - accuracy: 0.2812
+
    +
    +
    
+12/12 [==============================] - 0s 922us/step - loss: 92.0923 - accuracy: 0.2972
+
    +
    +
    Learning rate =  1e-05
+Lambda =  1.0
+Test accuracy: 0.297
+
    +
    +
     1/12 [=>............................] - ETA: 0s - loss: 529.5700 - accuracy: 0.2188
+
    +
    +
    
+12/12 [==============================] - 0s 935us/step - loss: 529.7050 - accuracy: 0.1861
+
    +
    +
    Learning rate =  1e-05
+Lambda =  10.0
+Test accuracy: 0.186
+
    +
    +
     1/12 [=>............................] - ETA: 1s - loss: 1.2495 - accuracy: 0.5312
+
    +
    +
    
+12/12 [==============================] - 0s 750us/step - loss: 1.5138 - accuracy: 0.4694
+
    +
    +
    Learning rate =  0.0001
+Lambda =  1e-05
+Test accuracy: 0.469
+
    +
    +
     1/12 [=>............................] - ETA: 0s - loss: 1.4077 - accuracy: 0.6562
+
    +
    +
    
+12/12 [==============================] - 0s 954us/step - loss: 1.4837 - accuracy: 0.5611
+
    +
    +
    Learning rate =  0.0001
+Lambda =  0.0001
+Test accuracy: 0.561
+
    +
    +
     1/12 [=>............................] - ETA: 0s - loss: 1.5539 - accuracy: 0.5625
+
    +
    +
    
+12/12 [==============================] - 0s 932us/step - loss: 1.5615 - accuracy: 0.5639
+
    +
    +
    Learning rate =  0.0001
+Lambda =  0.001
+Test accuracy: 0.564
+
    +
    +
    ---------------------------------------------------------------------------
+KeyboardInterrupt                         Traceback (most recent call last)
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47647/2018906331.py in <module>
+      6                                               n_filters, n_neurons_connected, n_categories,
+      7                                               eta, lmbd)
+----> 8         CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
+      9         scores = CNN.evaluate(X_test, Y_test)
+     10 
+
+~/anaconda3/lib/python3.8/site-packages/keras/engine/training.py in fit(self, x, y, batch_size, epochs, verbose, callbacks, validation_split, validation_data, shuffle, class_weight, sample_weight, initial_epoch, steps_per_epoch, validation_steps, validation_batch_size, validation_freq, max_queue_size, workers, use_multiprocessing)
+   1182                 _r=1):
+   1183               callbacks.on_train_batch_begin(step)
+-> 1184               tmp_logs = self.train_function(iterator)
+   1185               if data_handler.should_sync:
+   1186                 context.async_wait()
+
+~/anaconda3/lib/python3.8/site-packages/tensorflow/python/eager/def_function.py in __call__(self, *args, **kwds)
+    883 
+    884       with OptionalXlaContext(self._jit_compile):
+--> 885         result = self._call(*args, **kwds)
+    886 
+    887       new_tracing_count = self.experimental_get_tracing_count()
+
+~/anaconda3/lib/python3.8/site-packages/tensorflow/python/eager/def_function.py in _call(self, *args, **kwds)
+    915       # In this case we have created variables on the first call, so we run the
+    916       # defunned version which is guaranteed to never create variables.
+--> 917       return self._stateless_fn(*args, **kwds)  # pylint: disable=not-callable
+    918     elif self._stateful_fn is not None:
+    919       # Release the lock early so that multiple threads can perform the call
+
+~/anaconda3/lib/python3.8/site-packages/tensorflow/python/eager/function.py in __call__(self, *args, **kwargs)
+   3037       (graph_function,
+   3038        filtered_flat_args) = self._maybe_define_function(args, kwargs)
+-> 3039     return graph_function._call_flat(
+   3040         filtered_flat_args, captured_inputs=graph_function.captured_inputs)  # pylint: disable=protected-access
+   3041 
+
+~/anaconda3/lib/python3.8/site-packages/tensorflow/python/eager/function.py in _call_flat(self, args, captured_inputs, cancellation_manager)
+   1961         and executing_eagerly):
+   1962       # No tape is watching; skip to running the function.
+-> 1963       return self._build_call_outputs(self._inference_function.call(
+   1964           ctx, args, cancellation_manager=cancellation_manager))
+   1965     forward_backward = self._select_forward_and_backward_functions(
+
+~/anaconda3/lib/python3.8/site-packages/tensorflow/python/eager/function.py in call(self, ctx, args, cancellation_manager)
+    589       with _InterpolateFunctionError(self):
+    590         if cancellation_manager is None:
+--> 591           outputs = execute.execute(
+    592               str(self.signature.name),
+    593               num_outputs=self._num_outputs,
+
+~/anaconda3/lib/python3.8/site-packages/tensorflow/python/eager/execute.py in quick_execute(op_name, num_outputs, inputs, attrs, ctx, name)
+     57   try:
+     58     ctx.ensure_initialized()
+---> 59     tensors = pywrap_tfe.TFE_Py_Execute(ctx._handle, device_name, op_name,
+     60                                         inputs, attrs, num_outputs)
+     61   except core._NotOkStatusException as e:
+
+KeyboardInterrupt: 
+
    +
    +
    +
    +
    +
    +
    # 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)):
+        CNN = CNN_keras[i][j]
+
+        train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1]
+        test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1]
+
+        
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Training Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Test Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
    +
    +
    +
    +
    +
    +
    +

    16.8. The CIFAR01 data set

    +

    The CIFAR10 dataset contains 60,000 color images in 10 classes, with +6,000 images in each class. The dataset is divided into 50,000 +training images and 10,000 testing images. The classes are mutually +exclusive and there is no overlap between them.

    +
    +
    +
    import tensorflow as tf
+
+from tensorflow.keras import datasets, layers, models
+import matplotlib.pyplot as plt
+
+# We import the data set
+(train_images, train_labels), (test_images, test_labels) = datasets.cifar10.load_data()
+
+# Normalize pixel values to be between 0 and 1 by dividing by 255. 
+train_images, test_images = train_images / 255.0, test_images / 255.0
+
    +
    +
    +
    +

    To verify that the dataset looks correct, let’s plot the first 25 images from the training set and display the class name below each image.

    +
    +
    +
    class_names = ['airplane', 'automobile', 'bird', 'cat', 'deer',
+               'dog', 'frog', 'horse', 'ship', 'truck']
+​
+plt.figure(figsize=(10,10))
+for i in range(25):
+    plt.subplot(5,5,i+1)
+    plt.xticks([])
+    plt.yticks([])
+    plt.grid(False)
+    plt.imshow(train_images[i], cmap=plt.cm.binary)
+    # The CIFAR labels happen to be arrays, 
+    # which is why you need the extra index
+    plt.xlabel(class_names[train_labels[i][0]])
+plt.show()
+
    +
    +
    +
    +

    The six lines of code below define the convolutional base using a common pattern: a stack of Conv2D and MaxPooling2D layers.

    +

    As input, a CNN takes tensors of shape (image_height, image_width, color_channels), ignoring the batch size. If you are new to these dimensions, color_channels refers to (R,G,B). In this example, you will configure our CNN to process inputs of shape (32, 32, 3), which is the format of CIFAR images. You can do this by passing the argument input_shape to our first layer.

    +
    +
    +
    model = models.Sequential()
+model.add(layers.Conv2D(32, (3, 3), activation='relu', input_shape=(32, 32, 3)))
+model.add(layers.MaxPooling2D((2, 2)))
+model.add(layers.Conv2D(64, (3, 3), activation='relu'))
+model.add(layers.MaxPooling2D((2, 2)))
+model.add(layers.Conv2D(64, (3, 3), activation='relu'))
+
+# Let's display the architecture of our model so far.
+
+model.summary()
+
    +
    +
    +
    +

    You can see that the output of every Conv2D and MaxPooling2D layer is a 3D tensor of shape (height, width, channels). The width and height dimensions tend to shrink as you go deeper in the network. The number of output channels for each Conv2D layer is controlled by the first argument (e.g., 32 or 64). Typically, as the width and height shrink, you can afford (computationally) to add more output channels in each Conv2D layer.

    +

    To complete our model, you will feed the last output tensor from the +convolutional base (of shape (4, 4, 64)) into one or more Dense layers +to perform classification. Dense layers take vectors as input (which +are 1D), while the current output is a 3D tensor. First, you will +flatten (or unroll) the 3D output to 1D, then add one or more Dense +layers on top. CIFAR has 10 output classes, so you use a final Dense +layer with 10 outputs and a softmax activation.

    +
    +
    +
    model.add(layers.Flatten())
+model.add(layers.Dense(64, activation='relu'))
+model.add(layers.Dense(10))
+Here's the complete architecture of our model.
+
+model.summary()
+
    +
    +
    +
    +

    As you can see, our (4, 4, 64) outputs were flattened into vectors of shape (1024) before going through two Dense layers.

    +
    +
    +
    model.compile(optimizer='adam',
+              loss=tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True),
+              metrics=['accuracy'])
+​
+history = model.fit(train_images, train_labels, epochs=10, 
+                    validation_data=(test_images, test_labels))
+
    +
    +
    +
    +
    +
    +
    plt.plot(history.history['accuracy'], label='accuracy')
+plt.plot(history.history['val_accuracy'], label = 'val_accuracy')
+plt.xlabel('Epoch')
+plt.ylabel('Accuracy')
+plt.ylim([0.5, 1])
+plt.legend(loc='lower right')
+
+test_loss, test_acc = model.evaluate(test_images,  test_labels, verbose=2)
+
+print(test_acc)
+
    +
    +
    +
    +
    +
    + + + + +
    + + + + + + +
    + + + + +
    + +
    +
    +
    +
    +

    + + By Morten Hjorth-Jensen
    + + © Copyright 2021.
    +

    +
    +
    +
    + + +
    +
    + + + + + + \ No newline at end of file diff --git a/doc/LectureNotes/_build/html/chapter13.html b/doc/LectureNotes/_build/html/chapter13.html new file mode 100644 index 000000000..f7f0db19c --- /dev/null +++ b/doc/LectureNotes/_build/html/chapter13.html @@ -0,0 +1,2386 @@ + + + + + + + + 17. Recurrent neural networks: Overarching view — Applied Data Analysis and Machine Learning + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    +
    + + + + + + + + +
    + +
    +
    +
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    + + +
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    +

    17. Recurrent neural networks: Overarching view

    +

    Till now our focus has been, including convolutional neural networks +as well, on feedforward neural networks. The output or the activations +flow only in one direction, from the input layer to the output layer.

    +

    A recurrent neural network (RNN) looks very much like a feedforward +neural network, except that it also has connections pointing +backward.

    +

    RNNs are used to analyze time series data such as stock prices, and +tell you when to buy or sell. In autonomous driving systems, they can +anticipate car trajectories and help avoid accidents. More generally, +they can work on sequences of arbitrary lengths, rather than on +fixed-sized inputs like all the nets we have discussed so far. For +example, they can take sentences, documents, or audio samples as +input, making them extremely useful for natural language processing +systems such as automatic translation and speech-to-text.

    +

    More to text to be added

    +
    +

    17.1. A simple example

    +
    +
    +
    %matplotlib inline
+
+# Start importing packages
+import pandas as pd
+import numpy as np
+import matplotlib.pyplot as plt
+import tensorflow as tf
+from tensorflow.keras import datasets, layers, models
+from tensorflow.keras.layers import Input
+from tensorflow.keras.models import Model, Sequential 
+from tensorflow.keras.layers import Dense, SimpleRNN, LSTM, GRU
+from tensorflow.keras import optimizers     
+from tensorflow.keras import regularizers           
+from tensorflow.keras.utils import to_categorical 
+
+
+
+# convert into dataset matrix
+def convertToMatrix(data, step):
+ X, Y =[], []
+ for i in range(len(data)-step):
+  d=i+step  
+  X.append(data[i:d,])
+  Y.append(data[d,])
+ return np.array(X), np.array(Y)
+
+step = 4
+N = 1000    
+Tp = 800    
+
+t=np.arange(0,N)
+x=np.sin(0.02*t)+2*np.random.rand(N)
+df = pd.DataFrame(x)
+df.head()
+
+plt.plot(df)
+plt.show()
+
+values=df.values
+train,test = values[0:Tp,:], values[Tp:N,:]
+
+# add step elements into train and test
+test = np.append(test,np.repeat(test[-1,],step))
+train = np.append(train,np.repeat(train[-1,],step))
+ 
+trainX,trainY =convertToMatrix(train,step)
+testX,testY =convertToMatrix(test,step)
+trainX = np.reshape(trainX, (trainX.shape[0], 1, trainX.shape[1]))
+testX = np.reshape(testX, (testX.shape[0], 1, testX.shape[1]))
+
+model = Sequential()
+model.add(SimpleRNN(units=32, input_shape=(1,step), activation="relu"))
+model.add(Dense(8, activation="relu")) 
+model.add(Dense(1))
+model.compile(loss='mean_squared_error', optimizer='rmsprop')
+model.summary()
+
+model.fit(trainX,trainY, epochs=100, batch_size=16, verbose=2)
+trainPredict = model.predict(trainX)
+testPredict= model.predict(testX)
+predicted=np.concatenate((trainPredict,testPredict),axis=0)
+
+trainScore = model.evaluate(trainX, trainY, verbose=0)
+print(trainScore)
+
+index = df.index.values
+plt.plot(index,df)
+plt.plot(index,predicted)
+plt.axvline(df.index[Tp], c="r")
+plt.show()
+
    +
    +
    +
    +_images/chapter13_3_0.png +
    2021-12-08 06:58:41.160539: I tensorflow/core/platform/cpu_feature_guard.cc:142] This TensorFlow binary is optimized with oneAPI Deep Neural Network Library (oneDNN) to use the following CPU instructions in performance-critical operations:  SSE4.1 SSE4.2 AVX AVX2 FMA
+To enable them in other operations, rebuild TensorFlow with the appropriate compiler flags.
+2021-12-08 06:58:41.346810: I tensorflow/compiler/mlir/mlir_graph_optimization_pass.cc:185] None of the MLIR Optimization Passes are enabled (registered 2)
+
    +
    +
    Model: "sequential"
+_________________________________________________________________
+Layer (type)                 Output Shape              Param #   
+=================================================================
+simple_rnn (SimpleRNN)       (None, 32)                1184      
+_________________________________________________________________
+dense (Dense)                (None, 8)                 264       
+_________________________________________________________________
+dense_1 (Dense)              (None, 1)                 9         
+=================================================================
+Total params: 1,457
+Trainable params: 1,457
+Non-trainable params: 0
+_________________________________________________________________
+Epoch 1/100
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    0.3469819128513336
+
    +
    +
    ---------------------------------------------------------------------------
+KeyError                                  Traceback (most recent call last)
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47724/2517560119.py in <module>
+     65 
+     66 index = df.index.values
+---> 67 plt.plot(index,df)
+     68 plt.plot(index,predicted)
+     69 plt.axvline(df.index[Tp], c="r")
+
+~/anaconda3/lib/python3.8/site-packages/matplotlib/pyplot.py in plot(scalex, scaley, data, *args, **kwargs)
+   3017 @_copy_docstring_and_deprecators(Axes.plot)
+   3018 def plot(*args, scalex=True, scaley=True, data=None, **kwargs):
+-> 3019     return gca().plot(
+   3020         *args, scalex=scalex, scaley=scaley,
+   3021         **({"data": data} if data is not None else {}), **kwargs)
+
+~/anaconda3/lib/python3.8/site-packages/matplotlib/axes/_axes.py in plot(self, scalex, scaley, data, *args, **kwargs)
+   1603         """
+   1604         kwargs = cbook.normalize_kwargs(kwargs, mlines.Line2D)
+-> 1605         lines = [*self._get_lines(*args, data=data, **kwargs)]
+   1606         for line in lines:
+   1607             self.add_line(line)
+
+~/anaconda3/lib/python3.8/site-packages/matplotlib/axes/_base.py in __call__(self, data, *args, **kwargs)
+    313                 this += args[0],
+    314                 args = args[1:]
+--> 315             yield from self._plot_args(this, kwargs)
+    316 
+    317     def get_next_color(self):
+
+~/anaconda3/lib/python3.8/site-packages/matplotlib/axes/_base.py in _plot_args(self, tup, kwargs, return_kwargs)
+    489         if len(xy) == 2:
+    490             x = _check_1d(xy[0])
+--> 491             y = _check_1d(xy[1])
+    492         else:
+    493             x, y = index_of(xy[-1])
+
+~/anaconda3/lib/python3.8/site-packages/matplotlib/cbook/__init__.py in _check_1d(x)
+   1360                     message='Support for multi-dimensional indexing')
+   1361 
+-> 1362                 ndim = x[:, None].ndim
+   1363                 # we have definitely hit a pandas index or series object
+   1364                 # cast to a numpy array.
+
+~/anaconda3/lib/python3.8/site-packages/pandas/core/frame.py in __getitem__(self, key)
+   3456             if self.columns.nlevels > 1:
+   3457                 return self._getitem_multilevel(key)
+-> 3458             indexer = self.columns.get_loc(key)
+   3459             if is_integer(indexer):
+   3460                 indexer = [indexer]
+
+~/anaconda3/lib/python3.8/site-packages/pandas/core/indexes/range.py in get_loc(self, key, method, tolerance)
+    386                 except ValueError as err:
+    387                     raise KeyError(key) from err
+--> 388             raise KeyError(key)
+    389         return super().get_loc(key, method=method, tolerance=tolerance)
+    390 
+
+KeyError: (slice(None, None, None), None)
+
    +
    +_images/chapter13_3_192.png +
    +
    +
    +
    +

    17.2. An extrapolation example

    +

    The following code provides an example of how recurrent neural +networks can be used to extrapolate to unknown values of physics data +sets. Specifically, the data sets used in this program come from +a quantum mechanical many-body calculation of energies as functions of the number of particles.

    +
    +
    +
    # For matrices and calculations
+import numpy as np
+# For machine learning (backend for keras)
+import tensorflow as tf
+# User-friendly machine learning library
+# Front end for TensorFlow
+import tensorflow.keras
+# Different methods from Keras needed to create an RNN
+# This is not necessary but it shortened function calls 
+# that need to be used in the code.
+from tensorflow.keras import datasets, layers, models
+from tensorflow.keras.layers import Input
+from tensorflow.keras import regularizers
+from tensorflow.keras.models import Model, Sequential
+from tensorflow.keras.layers import Dense, SimpleRNN, LSTM, GRU
+# For timing the code
+from timeit import default_timer as timer
+# For plotting
+import matplotlib.pyplot as plt
+
+
+# The data set
+datatype='VaryDimension'
+X_tot = np.arange(2, 42, 2)
+y_tot = np.array([-0.03077640549, -0.08336233266, -0.1446729567, -0.2116753732, -0.2830637392, -0.3581341341, -0.436462435, -0.5177783846,
+	-0.6019067271, -0.6887363571, -0.7782028952, -0.8702784034, -0.9649652536, -1.062292565, -1.16231451, 
+	-1.265109911, -1.370782966, -1.479465113, -1.591317992, -1.70653767])
+
    +
    +
    +
    +

    The way the recurrent neural networks are trained in this program +differs from how machine learning algorithms are usually trained. +Typically a machine learning algorithm is trained by learning the +relationship between the x data and the y data. In this program, the +recurrent neural network will be trained to recognize the relationship +in a sequence of y values. This is type of data formatting is +typically used time series forcasting, but it can also be used in any +extrapolation (time series forecasting is just a specific type of +extrapolation along the time axis). This method of data formatting +does not use the x data and assumes that the y data are evenly spaced.

    +

    For a standard machine learning algorithm, the training data has the +form of (x,y) so the machine learning algorithm learns to assiciate a +y value with a given x value. This is useful when the test data has x +values within the same range as the training data. However, for this +application, the x values of the test data are outside of the x values +of the training data and the traditional method of training a machine +learning algorithm does not work as well. For this reason, the +recurrent neural network is trained on sequences of y values of the +form ((y1, y2), y3), so that the network is concerned with learning +the pattern of the y data and not the relation between the x and y +data. As long as the pattern of y data outside of the training region +stays relatively stable compared to what was inside the training +region, this method of training can produce accurate extrapolations to +y values far removed from the training data set.

    +
    +
    +
    # FORMAT_DATA
+def format_data(data, length_of_sequence = 2):  
+    """
+        Inputs:
+            data(a numpy array): the data that will be the inputs to the recurrent neural
+                network
+            length_of_sequence (an int): the number of elements in one iteration of the
+                sequence patter.  For a function approximator use length_of_sequence = 2.
+        Returns:
+            rnn_input (a 3D numpy array): the input data for the recurrent neural network.  Its
+                dimensions are length of data - length of sequence, length of sequence, 
+                dimnsion of data
+            rnn_output (a numpy array): the training data for the neural network
+        Formats data to be used in a recurrent neural network.
+    """
+
+    X, Y = [], []
+    for i in range(len(data)-length_of_sequence):
+        # Get the next length_of_sequence elements
+        a = data[i:i+length_of_sequence]
+        # Get the element that immediately follows that
+        b = data[i+length_of_sequence]
+        # Reshape so that each data point is contained in its own array
+        a = np.reshape (a, (len(a), 1))
+        X.append(a)
+        Y.append(b)
+    rnn_input = np.array(X)
+    rnn_output = np.array(Y)
+
+    return rnn_input, rnn_output
+
+
+# ## Defining the Recurrent Neural Network Using Keras
+# 
+# The following method defines a simple recurrent neural network in keras consisting of one input layer, one hidden layer, and one output layer.
+
+def rnn(length_of_sequences, batch_size = None, stateful = False):
+    """
+        Inputs:
+            length_of_sequences (an int): the number of y values in "x data".  This is determined
+                when the data is formatted
+            batch_size (an int): Default value is None.  See Keras documentation of SimpleRNN.
+            stateful (a boolean): Default value is False.  See Keras documentation of SimpleRNN.
+        Returns:
+            model (a Keras model): The recurrent neural network that is built and compiled by this
+                method
+        Builds and compiles a recurrent neural network with one hidden layer and returns the model.
+    """
+    # Number of neurons in the input and output layers
+    in_out_neurons = 1
+    # Number of neurons in the hidden layer
+    hidden_neurons = 200
+    # Define the input layer
+    inp = Input(batch_shape=(batch_size, 
+                length_of_sequences, 
+                in_out_neurons))  
+    # Define the hidden layer as a simple RNN layer with a set number of neurons and add it to 
+    # the network immediately after the input layer
+    rnn = SimpleRNN(hidden_neurons, 
+                    return_sequences=False,
+                    stateful = stateful,
+                    name="RNN")(inp)
+    # Define the output layer as a dense neural network layer (standard neural network layer)
+    #and add it to the network immediately after the hidden layer.
+    dens = Dense(in_out_neurons,name="dense")(rnn)
+    # Create the machine learning model starting with the input layer and ending with the 
+    # output layer
+    model = Model(inputs=[inp],outputs=[dens])
+    # Compile the machine learning model using the mean squared error function as the loss 
+    # function and an Adams optimizer.
+    model.compile(loss="mean_squared_error", optimizer="adam")  
+    return model
+
    +
    +
    +
    +
    +
    +

    17.3. Predicting New Points With A Trained Recurrent Neural Network

    +
    +
    +
    def test_rnn (x1, y_test, plot_min, plot_max):
+    """
+        Inputs:
+            x1 (a list or numpy array): The complete x component of the data set
+            y_test (a list or numpy array): The complete y component of the data set
+            plot_min (an int or float): the smallest x value used in the training data
+            plot_max (an int or float): the largest x valye used in the training data
+        Returns:
+            None.
+        Uses a trained recurrent neural network model to predict future points in the 
+        series.  Computes the MSE of the predicted data set from the true data set, saves
+        the predicted data set to a csv file, and plots the predicted and true data sets w
+        while also displaying the data range used for training.
+    """
+    # Add the training data as the first dim points in the predicted data array as these
+    # are known values.
+    y_pred = y_test[:dim].tolist()
+    # Generate the first input to the trained recurrent neural network using the last two 
+    # points of the training data.  Based on how the network was trained this means that it
+    # will predict the first point in the data set after the training data.  All of the 
+    # brackets are necessary for Tensorflow.
+    next_input = np.array([[[y_test[dim-2]], [y_test[dim-1]]]])
+    # Save the very last point in the training data set.  This will be used later.
+    last = [y_test[dim-1]]
+
+    # Iterate until the complete data set is created.
+    for i in range (dim, len(y_test)):
+        # Predict the next point in the data set using the previous two points.
+        next = model.predict(next_input)
+        # Append just the number of the predicted data set
+        y_pred.append(next[0][0])
+        # Create the input that will be used to predict the next data point in the data set.
+        next_input = np.array([[last, next[0]]], dtype=np.float64)
+        last = next
+
+    # Print the mean squared error between the known data set and the predicted data set.
+    print('MSE: ', np.square(np.subtract(y_test, y_pred)).mean())
+    # Save the predicted data set as a csv file for later use
+    name = datatype + 'Predicted'+str(dim)+'.csv'
+    np.savetxt(name, y_pred, delimiter=',')
+    # Plot the known data set and the predicted data set.  The red box represents the region that was used
+    # for the training data.
+    fig, ax = plt.subplots()
+    ax.plot(x1, y_test, label="true", linewidth=3)
+    ax.plot(x1, y_pred, 'g-.',label="predicted", linewidth=4)
+    ax.legend()
+    # Created a red region to represent the points used in the training data.
+    ax.axvspan(plot_min, plot_max, alpha=0.25, color='red')
+    plt.show()
+
+# Check to make sure the data set is complete
+assert len(X_tot) == len(y_tot)
+
+# This is the number of points that will be used in as the training data
+dim=12
+
+# Separate the training data from the whole data set
+X_train = X_tot[:dim]
+y_train = y_tot[:dim]
+
+
+# Generate the training data for the RNN, using a sequence of 2
+rnn_input, rnn_training = format_data(y_train, 2)
+
+
+# Create a recurrent neural network in Keras and produce a summary of the 
+# machine learning model
+model = rnn(length_of_sequences = rnn_input.shape[1])
+model.summary()
+
+# Start the timer.  Want to time training+testing
+start = timer()
+# Fit the model using the training data genenerated above using 150 training iterations and a 5%
+# validation split.  Setting verbose to True prints information about each training iteration.
+hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, 
+                 verbose=True,validation_split=0.05)
+
+for label in ["loss","val_loss"]:
+    plt.plot(hist.history[label],label=label)
+
+plt.ylabel("loss")
+plt.xlabel("epoch")
+plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1]))
+plt.legend()
+plt.show()
+
+# Use the trained neural network to predict more points of the data set
+test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1])
+# Stop the timer and calculate the total time needed.
+end = timer()
+print('Time: ', end-start)
+
    +
    +
    +
    +

    Changing the size of the recurrent neural network and its parameters +can drastically change the results you get from the model. The below +code takes the simple recurrent neural network from above and adds a +second hidden layer, changes the number of neurons in the hidden +layer, and explicitly declares the activation function of the hidden +layers to be a sigmoid function. The loss function and optimizer can +also be changed but are kept the same as the above network. These +parameters can be tuned to provide the optimal result from the +network. For some ideas on how to improve the performance of a +recurrent neural network.

    +
    +
    +
    def rnn_2layers(length_of_sequences, batch_size = None, stateful = False):
+    """
+        Inputs:
+            length_of_sequences (an int): the number of y values in "x data".  This is determined
+                when the data is formatted
+            batch_size (an int): Default value is None.  See Keras documentation of SimpleRNN.
+            stateful (a boolean): Default value is False.  See Keras documentation of SimpleRNN.
+        Returns:
+            model (a Keras model): The recurrent neural network that is built and compiled by this
+                method
+        Builds and compiles a recurrent neural network with two hidden layers and returns the model.
+    """
+    # Number of neurons in the input and output layers
+    in_out_neurons = 1
+    # Number of neurons in the hidden layer, increased from the first network
+    hidden_neurons = 500
+    # Define the input layer
+    inp = Input(batch_shape=(batch_size, 
+                length_of_sequences, 
+                in_out_neurons))  
+    # Create two hidden layers instead of one hidden layer.  Explicitly set the activation
+    # function to be the sigmoid function (the default value is hyperbolic tangent)
+    rnn1 = SimpleRNN(hidden_neurons, 
+                    return_sequences=True,  # This needs to be True if another hidden layer is to follow
+                    stateful = stateful, activation = 'sigmoid',
+                    name="RNN1")(inp)
+    rnn2 = SimpleRNN(hidden_neurons, 
+                    return_sequences=False, activation = 'sigmoid',
+                    stateful = stateful,
+                    name="RNN2")(rnn1)
+    # Define the output layer as a dense neural network layer (standard neural network layer)
+    #and add it to the network immediately after the hidden layer.
+    dens = Dense(in_out_neurons,name="dense")(rnn2)
+    # Create the machine learning model starting with the input layer and ending with the 
+    # output layer
+    model = Model(inputs=[inp],outputs=[dens])
+    # Compile the machine learning model using the mean squared error function as the loss 
+    # function and an Adams optimizer.
+    model.compile(loss="mean_squared_error", optimizer="adam")  
+    return model
+
+# Check to make sure the data set is complete
+assert len(X_tot) == len(y_tot)
+
+# This is the number of points that will be used in as the training data
+dim=12
+
+# Separate the training data from the whole data set
+X_train = X_tot[:dim]
+y_train = y_tot[:dim]
+
+
+# Generate the training data for the RNN, using a sequence of 2
+rnn_input, rnn_training = format_data(y_train, 2)
+
+
+# Create a recurrent neural network in Keras and produce a summary of the 
+# machine learning model
+model = rnn_2layers(length_of_sequences = 2)
+model.summary()
+
+# Start the timer.  Want to time training+testing
+start = timer()
+# Fit the model using the training data genenerated above using 150 training iterations and a 5%
+# validation split.  Setting verbose to True prints information about each training iteration.
+hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, 
+                 verbose=True,validation_split=0.05)
+
+
+# This section plots the training loss and the validation loss as a function of training iteration.
+# This is not required for analyzing the couple cluster data but can help determine if the network is
+# being overtrained.
+for label in ["loss","val_loss"]:
+    plt.plot(hist.history[label],label=label)
+
+plt.ylabel("loss")
+plt.xlabel("epoch")
+plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1]))
+plt.legend()
+plt.show()
+
+# Use the trained neural network to predict more points of the data set
+test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1])
+# Stop the timer and calculate the total time needed.
+end = timer()
+print('Time: ', end-start)
+
    +
    +
    +
    +
    +
    +

    17.4. Other Types of Recurrent Neural Networks

    +

    Besides a simple recurrent neural network layer, there are two other +commonly used types of recurrent neural network layers: Long Short +Term Memory (LSTM) and Gated Recurrent Unit (GRU). For a short +introduction to these layers see https://medium.com/mindboard/lstm-vs-gru-experimental-comparison-955820c21e8b +and https://medium.com/mindboard/lstm-vs-gru-experimental-comparison-955820c21e8b.

    +

    The first network created below is similar to the previous network, +but it replaces the SimpleRNN layers with LSTM layers. The second +network below has two hidden layers made up of GRUs, which are +preceeded by two dense (feeddorward) neural network layers. These +dense layers “preprocess” the data before it reaches the recurrent +layers. This architecture has been shown to improve the performance +of recurrent neural networks (see the link above and also +https://arxiv.org/pdf/1807.02857.pdf.

    +
    +
    +
    def lstm_2layers(length_of_sequences, batch_size = None, stateful = False):
+    """
+        Inputs:
+            length_of_sequences (an int): the number of y values in "x data".  This is determined
+                when the data is formatted
+            batch_size (an int): Default value is None.  See Keras documentation of SimpleRNN.
+            stateful (a boolean): Default value is False.  See Keras documentation of SimpleRNN.
+        Returns:
+            model (a Keras model): The recurrent neural network that is built and compiled by this
+                method
+        Builds and compiles a recurrent neural network with two LSTM hidden layers and returns the model.
+    """
+    # Number of neurons on the input/output layer and the number of neurons in the hidden layer
+    in_out_neurons = 1
+    hidden_neurons = 250
+    # Input Layer
+    inp = Input(batch_shape=(batch_size, 
+                length_of_sequences, 
+                in_out_neurons)) 
+    # Hidden layers (in this case they are LSTM layers instead if SimpleRNN layers)
+    rnn= LSTM(hidden_neurons, 
+                    return_sequences=True,
+                    stateful = stateful,
+                    name="RNN", use_bias=True, activation='tanh')(inp)
+    rnn1 = LSTM(hidden_neurons, 
+                    return_sequences=False,
+                    stateful = stateful,
+                    name="RNN1", use_bias=True, activation='tanh')(rnn)
+    # Output layer
+    dens = Dense(in_out_neurons,name="dense")(rnn1)
+    # Define the midel
+    model = Model(inputs=[inp],outputs=[dens])
+    # Compile the model
+    model.compile(loss='mean_squared_error', optimizer='adam')  
+    # Return the model
+    return model
+
+def dnn2_gru2(length_of_sequences, batch_size = None, stateful = False):
+    """
+        Inputs:
+            length_of_sequences (an int): the number of y values in "x data".  This is determined
+                when the data is formatted
+            batch_size (an int): Default value is None.  See Keras documentation of SimpleRNN.
+            stateful (a boolean): Default value is False.  See Keras documentation of SimpleRNN.
+        Returns:
+            model (a Keras model): The recurrent neural network that is built and compiled by this
+                method
+        Builds and compiles a recurrent neural network with four hidden layers (two dense followed by
+        two GRU layers) and returns the model.
+    """    
+    # Number of neurons on the input/output layers and hidden layers
+    in_out_neurons = 1
+    hidden_neurons = 250
+    # Input layer
+    inp = Input(batch_shape=(batch_size, 
+                length_of_sequences, 
+                in_out_neurons)) 
+    # Hidden Dense (feedforward) layers
+    dnn = Dense(hidden_neurons/2, activation='relu', name='dnn')(inp)
+    dnn1 = Dense(hidden_neurons/2, activation='relu', name='dnn1')(dnn)
+    # Hidden GRU layers
+    rnn1 = GRU(hidden_neurons, 
+                    return_sequences=True,
+                    stateful = stateful,
+                    name="RNN1", use_bias=True)(dnn1)
+    rnn = GRU(hidden_neurons, 
+                    return_sequences=False,
+                    stateful = stateful,
+                    name="RNN", use_bias=True)(rnn1)
+    # Output layer
+    dens = Dense(in_out_neurons,name="dense")(rnn)
+    # Define the model
+    model = Model(inputs=[inp],outputs=[dens])
+    # Compile the mdoel
+    model.compile(loss='mean_squared_error', optimizer='adam')  
+    # Return the model
+    return model
+
+# Check to make sure the data set is complete
+assert len(X_tot) == len(y_tot)
+
+# This is the number of points that will be used in as the training data
+dim=12
+
+# Separate the training data from the whole data set
+X_train = X_tot[:dim]
+y_train = y_tot[:dim]
+
+
+# Generate the training data for the RNN, using a sequence of 2
+rnn_input, rnn_training = format_data(y_train, 2)
+
+
+# Create a recurrent neural network in Keras and produce a summary of the 
+# machine learning model
+# Change the method name to reflect which network you want to use
+model = dnn2_gru2(length_of_sequences = 2)
+model.summary()
+
+# Start the timer.  Want to time training+testing
+start = timer()
+# Fit the model using the training data genenerated above using 150 training iterations and a 5%
+# validation split.  Setting verbose to True prints information about each training iteration.
+hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, 
+                 verbose=True,validation_split=0.05)
+
+
+# This section plots the training loss and the validation loss as a function of training iteration.
+# This is not required for analyzing the couple cluster data but can help determine if the network is
+# being overtrained.
+for label in ["loss","val_loss"]:
+    plt.plot(hist.history[label],label=label)
+
+plt.ylabel("loss")
+plt.xlabel("epoch")
+plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1]))
+plt.legend()
+plt.show()
+
+# Use the trained neural network to predict more points of the data set
+test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1])
+# Stop the timer and calculate the total time needed.
+end = timer()
+print('Time: ', end-start)
+
+
+# ### Training Recurrent Neural Networks in the Standard Way (i.e. learning the relationship between the X and Y data)
+# 
+# Finally, comparing the performace of a recurrent neural network using the standard data formatting to the performance of the network with time sequence data formatting shows the benefit of this type of data formatting with extrapolation.
+
+# Check to make sure the data set is complete
+assert len(X_tot) == len(y_tot)
+
+# This is the number of points that will be used in as the training data
+dim=12
+
+# Separate the training data from the whole data set
+X_train = X_tot[:dim]
+y_train = y_tot[:dim]
+
+# Reshape the data for Keras specifications
+X_train = X_train.reshape((dim, 1))
+y_train = y_train.reshape((dim, 1))
+
+
+# Create a recurrent neural network in Keras and produce a summary of the 
+# machine learning model
+# Set the sequence length to 1 for regular data formatting 
+model = rnn(length_of_sequences = 1)
+model.summary()
+
+# Start the timer.  Want to time training+testing
+start = timer()
+# Fit the model using the training data genenerated above using 150 training iterations and a 5%
+# validation split.  Setting verbose to True prints information about each training iteration.
+hist = model.fit(X_train, y_train, batch_size=None, epochs=150, 
+                 verbose=True,validation_split=0.05)
+
+
+# This section plots the training loss and the validation loss as a function of training iteration.
+# This is not required for analyzing the couple cluster data but can help determine if the network is
+# being overtrained.
+for label in ["loss","val_loss"]:
+    plt.plot(hist.history[label],label=label)
+
+plt.ylabel("loss")
+plt.xlabel("epoch")
+plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1]))
+plt.legend()
+plt.show()
+
+# Use the trained neural network to predict the remaining data points
+X_pred = X_tot[dim:]
+X_pred = X_pred.reshape((len(X_pred), 1))
+y_model = model.predict(X_pred)
+y_pred = np.concatenate((y_tot[:dim], y_model.flatten()))
+
+# Plot the known data set and the predicted data set.  The red box represents the region that was used
+# for the training data.
+fig, ax = plt.subplots()
+ax.plot(X_tot, y_tot, label="true", linewidth=3)
+ax.plot(X_tot, y_pred, 'g-.',label="predicted", linewidth=4)
+ax.legend()
+# Created a red region to represent the points used in the training data.
+ax.axvspan(X_tot[0], X_tot[dim], alpha=0.25, color='red')
+plt.show()
+
+# Stop the timer and calculate the total time needed.
+end = timer()
+print('Time: ', end-start)
+
    +
    +
    +
    +
    +
    +
    +

    18. Generative Models

    +

    Generative models describe a class of statistical models that are a contrast +to discriminative models. Informally we say that generative models can +generate new data instances while discriminative models discriminate between +different kinds of data instances. A generative model could generate new photos +of animals that look like ‘real’ animals while a discriminative model could tell +a dog from a cat. More formally, given a data set \(x\) and a set of labels / +targets \(y\). Generative models capture the joint probability \(p(x, y)\), or +just \(p(x)\) if there are no labels, while discriminative models capture the +conditional probability \(p(y | x)\). Discriminative models generally try to draw +boundaries in the data space (often high dimensional), while generative models +try to model how data is placed throughout the space.

    +

    Note: this material is thanks to Linus Ekstrøm.

    +
    +

    18.1. Generative Adversarial Networks

    +

    Generative Adversarial Networks are a type of unsupervised machine learning +algorithm proposed by Goodfellow et. al +in 2014 (short and good article).

    +

    The simplest formulation of +the model is based on a game theoretic approach, zero sum game, where we pit +two neural networks against one another. We define two rival networks, one +generator \(g\), and one discriminator \(d\). The generator directly produces +samples

    + +
    +
    +\[ +\begin{equation} + x = g(z; \theta^{(g)}) +\label{_auto1} \tag{1} +\end{equation} +\]
    +

    The discriminator attempts to distinguish between samples drawn from the +training data and samples drawn from the generator. In other words, it tries to +tell the difference between the fake data produced by \(g\) and the actual data +samples we want to do prediction on. The discriminator outputs a probability +value given by

    + +
    +
    +\[ +\begin{equation} + d(x; \theta^{(d)}) +\label{_auto2} \tag{2} +\end{equation} +\]
    +

    indicating the probability that \(x\) is a real training example rather than a +fake sample the generator has generated. The simplest way to formulate the +learning process in a generative adversarial network is a zero-sum game, in +which a function

    + +
    +
    +\[ +\begin{equation} + v(\theta^{(g)}, \theta^{(d)}) +\label{_auto3} \tag{3} +\end{equation} +\]
    +

    determines the reward for the discriminator, while the generator gets the +conjugate reward

    + +
    +
    +\[ +\begin{equation} + -v(\theta^{(g)}, \theta^{(d)}) +\label{_auto4} \tag{4} +\end{equation} +\]
    +

    During learning both of the networks maximize their own reward function, so that +the generator gets better and better at tricking the discriminator, while the +discriminator gets better and better at telling the difference between the fake +and real data. The generator and discriminator alternate on which one trains at +one time (i.e. for one epoch). In other words, we keep the generator constant +and train the discriminator, then we keep the discriminator constant to train +the generator and repeat. It is this back and forth dynamic which lets GANs +tackle otherwise intractable generative problems. As the generator improves with +training, the discriminator’s performance gets worse because it cannot easily +tell the difference between real and fake. If the generator ends up succeeding +perfectly, the the discriminator will do no better than random guessing i.e. +50%. This progression in the training poses a problem for the convergence +criteria for GANs. The discriminator feedback gets less meaningful over time, +if we continue training after this point then the generator is effectively +training on junk data which can undo the learning up to that point. Therefore, +we stop training when the discriminator starts outputting \(1/2\) everywhere.

    +

    At convergence we have

    + +
    +
    +\[ +\begin{equation} + g^* = \underset{g}{\mathrm{argmin}}\hspace{2pt} + \underset{d}{\mathrm{max}}v(\theta^{(g)}, \theta^{(d)}) +\label{_auto5} \tag{5} +\end{equation} +\]
    +

    The default choice for \(v\) is

    + +
    +
    +\[ +\begin{equation} + v(\theta^{(g)}, \theta^{(d)}) = \mathbb{E}_{x\sim p_\mathrm{data}}\log d(x) + + \mathbb{E}_{x\sim p_\mathrm{model}} + \log (1 - d(x)) +\label{_auto6} \tag{6} +\end{equation} +\]
    +

    The main motivation for the design of GANs is that the learning process requires +neither approximate inference (variational autoencoders for example) nor +approximation of a partition function. In the case where

    + +
    +
    +\[ +\begin{equation} + \underset{d}{\mathrm{max}}v(\theta^{(g)}, \theta^{(d)}) +\label{_auto7} \tag{7} +\end{equation} +\]
    +

    is convex in $\theta^{(g)} then the procedure is guaranteed to converge and is +asymptotically consistent +( Seth Lloyd on QuGANs ).

    +

    This is in +general not the case and it is possible to get situations where the training +process never converges because the generator and discriminator chase one +another around in the parameter space indefinitely. A much deeper discussion on +the currently open research problem of GAN convergence is available +here. To +anyone interested in learning more about GANs it is a highly recommended read. +Direct quote: “In this best-performing formulation, the generator aims to +increase the log probability that the discriminator makes a mistake, rather than +aiming to decrease the log probability that the discriminator makes the correct +prediction.” Another interesting read

    +
    +
    +

    18.2. Writing Our First Generative Adversarial Network

    +

    Let us now move on to actually implementing a GAN in tensorflow. We will study +the performance of our GAN on the MNIST dataset. This code is based on and +adapted from the +google tutorial

    +

    First we import our libraries

    +
    +
    +
    import os
+import time
+import numpy as np
+import tensorflow as tf
+import matplotlib.pyplot as plt
+from tensorflow.keras import layers
+from tensorflow.keras.utils import plot_model
+
    +
    +
    +
    +

    Next we define our hyperparameters and import our data the usual way

    +
    +
    +
    BUFFER_SIZE = 60000
+BATCH_SIZE = 256
+EPOCHS = 30
+
+data = tf.keras.datasets.mnist.load_data()
+(train_images, train_labels), (test_images, test_labels) = data
+train_images = np.reshape(train_images, (train_images.shape[0],
+                                         28,
+                                         28,
+                                         1)).astype('float32')
+
+# we normalize between -1 and 1
+train_images = (train_images - 127.5) / 127.5
+training_dataset = tf.data.Dataset.from_tensor_slices(
+                      train_images).shuffle(BUFFER_SIZE).batch(BATCH_SIZE)
+
    +
    +
    +
    +
    +

    18.2.1. MNIST and GANs

    +

    Let’s have a quick look

    +
    +
    +
    plt.imshow(train_images[0], cmap='Greys')
+plt.show()
+
    +
    +
    +
    +

    Now we define our two models. This is where the ‘magic’ happens. There are a +huge amount of possible formulations for both models. A lot of engineering and +trial and error can be done here to try to produce better performing models. For +more advanced GANs this is by far the step where you can ‘make or break’ a +model.

    +

    We start with the generator. As stated in the introductory text the generator +\(g\) upsamples from a random sample to the shape of what we want to predict. In +our case we are trying to predict MNIST images (\(28\times 28\) pixels).

    +
    +
    +
    def generator_model():
+    """
+    The generator uses upsampling layers tf.keras.layers.Conv2DTranspose() to
+    produce an image from a random seed. We start with a Dense layer taking this
+    random sample as an input and subsequently upsample through multiple
+    convolutional layers.
+    """
+
+    # we define our model
+    model = tf.keras.Sequential()
+
+
+    # adding our input layer. Dense means that every neuron is connected and
+    # the input shape is the shape of our random noise. The units need to match
+    # in some sense the upsampling strides to reach our desired output shape.
+    # we are using 100 random numbers as our seed
+    model.add(layers.Dense(units=7*7*BATCH_SIZE,
+                           use_bias=False,
+                           input_shape=(100, )))
+    # we normalize the output form the Dense layer
+    model.add(layers.BatchNormalization())
+    # and add an activation function to our 'layer'. LeakyReLU avoids vanishing
+    # gradient problem
+    model.add(layers.LeakyReLU())
+    model.add(layers.Reshape((7, 7, BATCH_SIZE)))
+    assert model.output_shape == (None, 7, 7, BATCH_SIZE)
+    # even though we just added four keras layers we think of everything above
+    # as 'one' layer
+
+    # next we add our upscaling convolutional layers
+    model.add(layers.Conv2DTranspose(filters=128,
+                                     kernel_size=(5, 5),
+                                     strides=(1, 1),
+                                     padding='same',
+                                     use_bias=False))
+    model.add(layers.BatchNormalization())
+    model.add(layers.LeakyReLU())
+    assert model.output_shape == (None, 7, 7, 128)
+
+    model.add(layers.Conv2DTranspose(filters=64,
+                                     kernel_size=(5, 5),
+                                     strides=(2, 2),
+                                     padding='same',
+                                     use_bias=False))
+    model.add(layers.BatchNormalization())
+    model.add(layers.LeakyReLU())
+    assert model.output_shape == (None, 14, 14, 64)
+
+    model.add(layers.Conv2DTranspose(filters=1,
+                                     kernel_size=(5, 5),
+                                     strides=(2, 2),
+                                     padding='same',
+                                     use_bias=False,
+                                     activation='tanh'))
+    assert model.output_shape == (None, 28, 28, 1)
+
+    return model
+
    +
    +
    +
    +

    And there we have our ‘simple’ generator model. Now we move on to defining our +discriminator model \(d\), which is a convolutional neural network based image +classifier.

    +
    +
    +
    def discriminator_model():
+    """
+    The discriminator is a convolutional neural network based image classifier
+    """
+
+    # we define our model
+    model = tf.keras.Sequential()
+    model.add(layers.Conv2D(filters=64,
+                            kernel_size=(5, 5),
+                            strides=(2, 2),
+                            padding='same',
+                            input_shape=[28, 28, 1]))
+    model.add(layers.LeakyReLU())
+    # adding a dropout layer as you do in conv-nets
+    model.add(layers.Dropout(0.3))
+
+
+    model.add(layers.Conv2D(filters=128,
+                            kernel_size=(5, 5),
+                            strides=(2, 2),
+                            padding='same'))
+    model.add(layers.LeakyReLU())
+    # adding a dropout layer as you do in conv-nets
+    model.add(layers.Dropout(0.3))
+
+    model.add(layers.Flatten())
+    model.add(layers.Dense(1))
+
+    return model
+
    +
    +
    +
    +

    Let us take a look at our models.

    +
    +
    +
    generator = generator_model()
+plot_model(generator, show_shapes=True, rankdir='LR')
+
    +
    +
    +
    +
    +
    +
    discriminator = discriminator_model()
+plot_model(discriminator, show_shapes=True, rankdir='LR')
+
    +
    +
    +
    +

    Next we need a few helper objects we will use in training

    +
    +
    +
    cross_entropy = tf.keras.losses.BinaryCrossentropy(from_logits=True)
+generator_optimizer = tf.keras.optimizers.Adam(1e-4)
+discriminator_optimizer = tf.keras.optimizers.Adam(1e-4)
+
    +
    +
    +
    +

    The first object, cross_entropy is our loss function and the two others are +our optimizers. Notice we use the same learning rate for both \(g\) and \(d\). This +is because they need to improve their accuracy at approximately equal speeds to +get convergence (not necessarily exactly equal). Now we define our loss +functions

    +
    +
    +
    def generator_loss(fake_output):
+    loss = cross_entropy(tf.ones_like(fake_output), fake_output)
+
+    return loss
+
    +
    +
    +
    +
    +
    +
    def discriminator_loss(real_output, fake_output):
+    real_loss = cross_entropy(tf.ones_like(real_output), real_output)
+    fake_loss = cross_entropy(tf.zeros_liks(fake_output), fake_output)
+    total_loss = real_loss + fake_loss
+
+    return total_loss
+
    +
    +
    +
    +

    Next we define a kind of seed to help us compare the learning process over +multiple training epochs.

    +
    +
    +
    noise_dimension = 100
+n_examples_to_generate = 16
+seed_images = tf.random.normal([n_examples_to_generate, noise_dimension])
+
    +
    +
    +
    +

    Now we have everything we need to define our training step, which we will apply +for every step in our training loop. Notice the @tf.function flag signifying +that the function is tensorflow ‘compiled’. Removing this flag doubles the +computation time.

    +
    +
    +
    @tf.function
+def train_step(images):
+    noise = tf.random.normal([BATCH_SIZE, noise_dimension])
+
+    with tf.GradientTape() as gen_tape, tf.GradientTape() as disc_tape:
+        generated_images = generator(noise, training=True)
+
+        real_output = discriminator(images, training=True)
+        fake_output = discriminator(generated_images, training=True)
+
+        gen_loss = generator_loss(fake_output)
+        disc_loss = discriminator_loss(real_output, fake_output)
+
+    gradients_of_generator = gen_tape.gradient(gen_loss,
+                                            generator.trainable_variables)
+    gradients_of_discriminator = disc_tape.gradient(disc_loss,
+                                            discriminator.trainable_variables)
+    generator_optimizer.apply_gradients(zip(gradients_of_generator,
+                                            generator.trainable_variables))
+    discriminator_optimizer.apply_gradients(zip(gradients_of_discriminator,
+                                            discriminator.trainable_variables))
+
+    return gen_loss, disc_loss
+
    +
    +
    +
    +

    Next we define a helper function to produce an output over our training epochs +to see the predictive progression of our generator model. Note: I am including +this code here, but comment it out in the training loop.

    +
    +
    +
    def generate_and_save_images(model, epoch, test_input):
+    # we're making inferences here
+    predictions = model(test_input, training=False)
+
+    fig = plt.figure(figsize=(4, 4))
+
+    for i in range(predictions.shape[0]):
+        plt.subplot(4, 4, i+1)
+        plt.imshow(predictions[i, :, :, 0] * 127.5 + 127.5, cmap='gray')
+        plt.axis('off')
+
+    plt.savefig(f'./images_from_seed_images/image_at_epoch_{str(epoch).zfill(3)}.png')
+    plt.close()
+    #plt.show()
+
    +
    +
    +
    +

    Setting up checkpoints to periodically save our model during training so that +everything is not lost even if the program were to somehow terminate while +training.

    +
    +
    +
    # Setting up checkpoints to save model during training
+checkpoint_dir = './training_checkpoints'
+checkpoint_prefix = os.path.join(checkpoint_dir, 'ckpt')
+checkpoint = tf.train.Checkpoint(generator_optimizer=generator_optimizer,
+                            discriminator_optimizer=discriminator_optimizer,
+                            generator=generator,
+                            discriminator=discriminator)
+
    +
    +
    +
    +

    Now we define our training loop

    +
    +
    +
    def train(dataset, epochs):
+    generator_loss_list = []
+    discriminator_loss_list = []
+
+    for epoch in range(epochs):
+        start = time.time()
+
+        for image_batch in dataset:
+            gen_loss, disc_loss = train_step(image_batch)
+            generator_loss_list.append(gen_loss.numpy())
+            discriminator_loss_list.append(disc_loss.numpy())
+
+        #generate_and_save_images(generator, epoch + 1, seed_images)
+
+        if (epoch + 1) % 15 == 0:
+            checkpoint.save(file_prefix=checkpoint_prefix)
+
+        print(f'Time for epoch {epoch} is {time.time() - start}')
+
+    #generate_and_save_images(generator, epochs, seed_images)
+
+    loss_file = './data/lossfile.txt'
+    with open(loss_file, 'w') as outfile:
+        outfile.write(str(generator_loss_list))
+        outfile.write('\n')
+        outfile.write('\n')
+        outfile.write(str(discriminator_loss_list))
+        outfile.write('\n')
+        outfile.write('\n')
+
    +
    +
    +
    +

    To train simply call this function. Warning: this might take a long time so +there is a folder of a pretrained network already included in the repository.

    +
    +
    +
    train(train_dataset, EPOCHS)
+
    +
    +
    +
    +

    Now to avoid having to train and everything, which will take a while depending +on your computer setup we now load in the model which produced the above gif.

    +
    +
    +
    checkpoint.restore(tf.train.latest_checkpoint(checkpoint_dir))
+restored_generator = checkpoint.generator
+restored_discriminator = checkpoint.discriminator
+
+print(restored_generator)
+print(restored_discriminator)
+
    +
    +
    +
    +

    We have successfully loaded in our latest model. Let us now play around a bit +and see what kind of things we can learn about this model. Our generator takes +an array of 100 numbers. One idea can be to try to systematically change our +input. Let us try and see what we get

    +
    +
    +
    def generate_latent_points(number=100, scale_means=1, scale_stds=1):
+    latent_dim = 100
+    means = scale_means * tf.linspace(-1, 1, num=latent_dim)
+    stds = scale_stds * tf.linspace(-1, 1, num=latent_dim)
+    latent_space_value_range = tf.random.normal([number, latent_dim],
+                                                means,
+                                                stds,
+                                                dtype=tf.float64)
+
+    return latent_space_value_range
+
+def generate_images(latent_points):
+    # notice we set training to false because we are making inferences
+    generated_images = restored_generator.predict(latent_points)
+
+    return generated_images
+
    +
    +
    +
    +
    +
    +
    def plot_result(generated_images, number=100):
+    # obviously this assumes sqrt number is an int
+    fig, axs = plt.subplots(int(np.sqrt(number)), int(np.sqrt(number)),
+                            figsize=(10, 10))
+
+    for i in range(int(np.sqrt(number))):
+        for j in range(int(np.sqrt(number))):
+            axs[i, j].imshow(generated_images[i*j], cmap='Greys')
+            axs[i, j].axis('off')
+
+    plt.show()
+
    +
    +
    +
    +
    +
    +
    generated_images = generate_images(generate_latent_points())
+plot_result(generated_images)
+
    +
    +
    +
    +

    We see that the generator generates images that look like MNIST +numbers: \(1, 4, 7, 9\). Let’s try to tweak it a bit more to see if we are able +to generate a similar plot where we generate every MNIST number. Let us now try +to ‘move’ a bit around in the latent space. Note: decrease the plot number if +these following cells take too long to run on your computer.

    +
    +
    +
    plot_number = 225
+
+generated_images = generate_images(generate_latent_points(number=plot_number,
+                                                          scale_means=5,
+                                                          scale_stds=1))
+plot_result(generated_images, number=plot_number)
+
+generated_images = generate_images(generate_latent_points(number=plot_number,
+                                                          scale_means=-5,
+                                                          scale_stds=1))
+plot_result(generated_images, number=plot_number)
+
+generated_images = generate_images(generate_latent_points(number=plot_number,
+                                                          scale_means=1,
+                                                          scale_stds=5))
+plot_result(generated_images, number=plot_number)
+
    +
    +
    +
    +

    Again, we have found something interesting. Moving around using our means +takes us from digit to digit, while moving around using our standard +deviations seem to increase the number of different digits! In the last image +above, we can barely make out every MNIST digit. Let us make on last plot using +this information by upping the standard deviation of our Gaussian noises.

    +
    +
    +
    plot_number = 400
+generated_images = generate_images(generate_latent_points(number=plot_number,
+                                                          scale_means=1,
+                                                          scale_stds=10))
+plot_result(generated_images, number=plot_number)
+
    +
    +
    +
    +

    A pretty cool result! We see that our generator indeed has learned a +distribution which qualitatively looks a whole lot like the MNIST dataset.

    +

    Another interesting way to explore the latent space of our generator model is by +interpolating between the MNIST digits. This section is largely based on +this excellent blogpost +by Jason Brownlee.

    +

    So let us start by defining a function to interpolate between two points in the +latent space.

    +
    +
    +
    def interpolation(point_1, point_2, n_steps=10):
+    ratios = np.linspace(0, 1, num=n_steps)
+    vectors = []
+    for i, ratio in enumerate(ratios):
+        vectors.append(((1.0 - ratio) * point_1 + ratio * point_2))
+
+    return tf.stack(vectors)
+
    +
    +
    +
    +

    Now we have all we need to do our interpolation analysis.

    +
    +
    +
    plot_number = 100
+latent_points = generate_latent_points(number=plot_number)
+results = None
+for i in range(0, 2*np.sqrt(plot_number), 2):
+    interpolated = interpolation(latent_points[i], latent_points[i+1])
+    generated_images = generate_images(interpolated)
+
+    if results is None:
+        results = generated_images
+    else:
+        results = tf.stack((results, generated_images))
+
+plot_results(results, plot_number)
+
    +
    +
    +
    +
    +
    +
    + + + + +
    + + + + + + +
    + + + +
    + +
    +
    +
    +
    +

    + + By Morten Hjorth-Jensen
    + + © Copyright 2021.
    +

    +
    +
    +
    + + +
    +
    + + + + + + \ No newline at end of file diff --git a/doc/LectureNotes/_build/html/chapter9.html b/doc/LectureNotes/_build/html/chapter9.html index 3aac078fa..aca38d4f7 100644 --- a/doc/LectureNotes/_build/html/chapter9.html +++ b/doc/LectureNotes/_build/html/chapter9.html @@ -5,7 +5,7 @@ - 12. Neural networks — Applied Data Analysis and Machine Learning + 13. Neural networks — Applied Data Analysis and Machine Learning @@ -53,8 +53,8 @@ - - + + @@ -199,6 +199,11 @@ 11. Basic ideas of the Principal Component Analysis (PCA) +
  • + + 12. Clustering and Unsupervised Learning + +

    • @@ -208,17 +213,27 @@

    @@ -294,76 +309,76 @@

    @@ -208,17 +213,27 @@

    @@ -330,8 +345,8 @@
  • - - 7.7. Stochastic Gradient Descent + + 7.7. Stochastic Gradient Descent (SGD) @@ -379,7 +411,8 @@
    -
    +

    7. Optimization, the central part of any Machine Learning algortithm

    Almost every problem in machine learning and data science starts with a dataset \(X\), a model \(g(\beta)\), which is a function of the @@ -750,14 +783,14 @@ which equals

    -
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42573/483257001.py:18: MatplotlibDeprecationWarning: Calling gca() with keyword arguments was deprecated in Matplotlib 3.4. Starting two minor releases later, gca() will take no keyword arguments. The gca() function should only be used to get the current axes, or if no axes exist, create new axes with default keyword arguments. To create a new axes with non-default arguments, use plt.axes() or plt.subplot().
+
    /var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47735/483257001.py:18: MatplotlibDeprecationWarning: Calling gca() with keyword arguments was deprecated in Matplotlib 3.4. Starting two minor releases later, gca() will take no keyword arguments. The gca() function should only be used to get the current axes, or if no axes exist, create new axes with default keyword arguments. To create a new axes with non-default arguments, use plt.axes() or plt.subplot().
   ax = fig.gca(projection="3d")
 
    
 
    
-
    <mpl_toolkits.mplot3d.art3d.Poly3DCollection at 0x7fad10f9a280>
+
    <mpl_toolkits.mplot3d.art3d.Poly3DCollection at 0x7fd098df3280>
 
    
 
    
-_images/chapteroptimization_56_2.png
+_images/chapteroptimization_61_2.png
 
    
 
    
 
    And then as countor plot
    
@@ -770,7 +803,7 @@ which equals
    -_images/chapteroptimization_58_0.png +_images/chapteroptimization_63_0.png

    Find guesses

    @@ -812,10 +845,10 @@ which equals

    -
    [<matplotlib.lines.Line2D at 0x7fad20f69be0>]
+
    [<matplotlib.lines.Line2D at 0x7fd0a9063be0>]
 
    
 
    
-_images/chapteroptimization_64_1.png
+_images/chapteroptimization_69_1.png
    @@ -1069,16 +1102,14 @@ when \(||\nabla_\beta C(\beta_k) || \
    -
    [0.32903042 4.1484256 ]
-[[4.04621521]
- [3.00415763]]
+
    [0.28001319 4.21265216]
+[[3.96987657]
+ [3.02493054]]
+[[3.96987657]
+ [3.02493054]]
 
    
 
    
-
    [[4.04621521]
- [3.00415763]]
-
    
-
    
-_images/chapteroptimization_118_2.png
+_images/chapteroptimization_123_1.png

    Alternatively, we can use Scikit-Learn as done here

    @@ -1104,9 +1135,9 @@ when \(||\nabla_\beta C(\beta_k) || \
    -
    [[3.97230501]
- [3.14741468]]
-[3.94735055] [3.17084902]
+
    [[4.00275135]
+ [2.99724883]]
+[3.97065296] [3.07656896]
    @@ -1177,13 +1208,13 @@ C_{\text{ridge}}(\beta) = \frac{1}{n}||X\beta -\mathbf{y}||^2 + \lambda ||\beta|
    -
    [[3.92343595]
- [3.15258907]]
-[[3.815563  ]
- [3.23522201]]
+
    [[4.1533795 ]
+ [2.92819235]]
+[[4.06858699]
+ [2.99829953]]
 
    
 
    
-_images/chapteroptimization_127_1.png
+_images/chapteroptimization_132_1.png
    @@ -1198,8 +1229,27 @@ C_{\text{ridge}}(\beta) = \frac{1}{n}||X\beta -\mathbf{y}||^2 + \lambda ||\beta|

  • GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points.

    • -
    -

    7.7. Stochastic Gradient Descent

    +
    +

    7.7. Stochastic Gradient Descent (SGD)

    +

    In stochastic gradient descent, the extreme case is the case where we +have only one batch, that is we include the whole data set.

    +

    This process is called Stochastic Gradient +Descent (SGD) (or also sometimes on-line gradient descent). This is +relatively less common to see because in practice due to vectorized +code optimizations it can be computationally much more efficient to +evaluate the gradient for 100 examples, than the gradient for one +example 100 times. Even though SGD technically refers to using a +single example at a time to evaluate the gradient, you will hear +people use the term SGD even when referring to mini-batch gradient +descent (i.e. mentions of MGD for “Minibatch Gradient Descent”, or BGD +for “Batch gradient descent” are rare to see), where it is usually +assumed that mini-batches are used. The size of the mini-batch is a +hyperparameter but it is not very common to cross-validate or bootstrap it. It is +usually based on memory constraints (if any), or set to some value, +e.g. 32, 64 or 128. We use powers of 2 in practice because many +vectorized operation implementations work faster when their inputs are +sized in powers of 2.

    +

    In our notes with SGD we mean stochastic gradient descent with mini-batches.

    Stochastic gradient descent (SGD) and variants thereof address some of the shortcomings of the Gradient descent method discussed above.

    The underlying idea of SGD comes from the observation that the cost @@ -1256,7 +1306,7 @@ the number of minibatches, as exemplified in the code below.

    import numpy as np 
 
 n = 100 #100 datapoints 
-M = 5   #size of each minibatch
+M = 5   #size of each mini-batche
 m = int(n/M) #number of minibatches
 n_epochs = 10 #number of epochs
 
@@ -1332,38 +1382,39 @@ function.
    +

    We note that we have defined several hyperparameters. These are now the number of epochs, the number of mini-batches and the parameters \(t_0\) and \(t_1\).

    7.7.1. Program for stochastic gradient

    # Importing various packages
+# Importing various packages
 from math import exp, sqrt
 from random import random, seed
 import numpy as np
 import matplotlib.pyplot as plt
-from sklearn.linear_model import SGDRegressor
 
-m = 100
-x = 2*np.random.rand(m,1)
-y = 4+3*x+np.random.randn(m,1)
+n = 100
+x = 2*np.random.rand(n,1)
+y = 4+3*x+np.random.randn(n,1)
 
-X = np.c_[np.ones((m,1)), x]
+X = np.c_[np.ones((n,1)), x]
+XT_X = X.T @ X
 theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)
 print("Own inversion")
 print(theta_linreg)
-sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1)
-sgdreg.fit(x,y.ravel())
-print("sgdreg from scikit")
-print(sgdreg.intercept_, sgdreg.coef_)
-
+# Hessian matrix
+H = (2.0/n)* XT_X
+EigValues, EigVectors = np.linalg.eig(H)
+print(f"Eigenvalues of Hessian Matrix:{EigValues}")
 
 theta = np.random.randn(2,1)
-eta = 0.1
+eta = 1.0/np.max(EigValues)
 Niterations = 1000
 
 
 for iter in range(Niterations):
-    gradients = 2.0/m*X.T @ ((X @ theta)-y)
+    gradients = 2.0/n*X.T @ ((X @ theta)-y)
     theta -= eta*gradients
 print("theta from own gd")
 print(theta)
@@ -1373,8 +1424,9 @@ function.
    
 ypredict = Xnew.dot(theta)
 ypredict2 = Xnew.dot(theta_linreg)
 
-
 n_epochs = 50
+M = 5   #size of each minibatch
+m = int(n/M) #number of minibatches
 t0, t1 = 5, 50
 def learning_schedule(t):
     return t0/(t+t1)
@@ -1382,16 +1434,20 @@ function.
    
 theta = np.random.randn(2,1)
 
 for epoch in range(n_epochs):
+# Can you figure out a better way of setting up the contributions to each batch?
     for i in range(m):
-        random_index = np.random.randint(m)
-        xi = X[random_index:random_index+1]
-        yi = y[random_index:random_index+1]
-        gradients = 2 * xi.T @ ((xi @ theta)-yi)
+        random_index = M*np.random.randint(m)
+        xi = X[random_index:random_index+M]
+        yi = y[random_index:random_index+M]
+        gradients = (2.0/M)* xi.T @ ((xi @ theta)-yi)
         eta = learning_schedule(epoch*m+i)
         theta = theta - eta*gradients
 print("theta from own sdg")
 print(theta)
 
+
+
+
 plt.plot(xnew, ypredict, "r-")
 plt.plot(xnew, ypredict2, "b-")
 plt.plot(x, y ,'ro')
@@ -1405,23 +1461,24 @@ function.
    
 
    Own inversion
-[[4.15629539]
- [2.7293182 ]]
-sgdreg from scikit
-[4.0728785] [2.66410989]
+[[3.99775949]
+ [2.94659383]]
+Eigenvalues of Hessian Matrix:[0.36102113 4.18276924]
 theta from own gd
-[[4.15629539]
- [2.7293182 ]]
+[[3.99775949]
+ [2.94659383]]
+theta from own sdg
+[[3.96489434]
+ [2.98399675]]
    -
    theta from own sdg
-[[4.15454301]
- [2.72522848]]
-
    -
    -_images/chapteroptimization_141_2.png +_images/chapteroptimization_148_1.png
    +

    In the above code, we have use replacement in setting up the +mini-batches. The discussion +here may be +useful. More material will be added later.

    @@ -1538,7 +1595,7 @@ the steep computational price of calculating or approximating Hessians.

    Recently, a number of methods have been introduced that accomplish this by tracking not only the gradient, but also the second moment of -the gradient. These methods include AdaGrad, AdaDelta, RMS-Prop, and +the gradient. These methods include AdaGrad, AdaDelta, Root Mean Squared Propagation (RMS-Prop), and ADAM.

    7.8.1. RMS prop

    @@ -1724,7 +1781,7 @@ f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right)
    -_images/chapteroptimization_177_0.png +_images/chapteroptimization_188_0.png 
    The max absolute difference is: 1.77636e-15
    @@ -1907,28 +1964,42 @@ The analytical gradient of f4 at x = 2.7 is: 13.8759
    -

    1 -8

    -

    < -< -< -! -! -C -O -D -E -_ -B -L -O -C -K

    -

    p -y -c -o -d

    +
    +
    +
    import autograd.numpy as np
+from autograd import grad
+def f6_for(x):
+    val = 0
+    for i in range(10):
+        val = val + x**i
+    return val
+
+def f6_while(x):
+    val = 0
+    i = 0
+    while i < 10:
+        val = val + x**i
+        i = i + 1
+    return val
+
+f6_for_grad = grad(f6_for)
+f6_while_grad = grad(f6_while)
+
+x = 0.5
+
+# Print the computed derivaties of f6_for and f6_while
+print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x)))
+print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x)))
+
    +
    +
    +
    +
    The computed derivative of f6_for at x = 0.5 is: 3.95703
+The computed derivative of f6_while at x = 0.5 is: 3.95703
+
    +
    +
    +
    import autograd.numpy as np
@@ -1944,7 +2015,7 @@ d
    -
    The analytical derivative of f6 at x = 2.7 is: 37732.5
+
    The analytical derivative of f6 at x = 0.5 is: 3.95703
    @@ -1989,49 +2060,31 @@ The analytical derivative of f7 at n = 2 is: 1

    Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input.

    -

    Autograd supports many features. However, there are some functions that are not supported (yet) by Autograd.

    -

    Assigning a value to the variable being differentiated with respect to is an example thereof.

    -
    -
    -
    #import autograd.numpy as np
-#from autograd import grad
-#def f8(x): # Assume x is an array
-#    x[2] = 3
-#    return x*2
-
-#f8_grad = grad(f8)
-
-#x = 8.4
-
-#print("The derivative of f8 is:",f8_grad(x))
-
    -
    -
    -
    -

    Here, Autograd tells us that an ‘ArrayBox’ does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible.

    +

    Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.

    +

    Assigning a value to the variable being differentiated with respect to

    import autograd.numpy as np
 from autograd import grad
-def f9(a): # Assume a is an array with 2 elements
-    b = np.array([1.0,2.0])
-    return a.dot(b)
+def f8(x): # Assume x is an array
+    x[2] = 3
+    return x*2
 
-f9_grad = grad(f9)
+f8_grad = grad(f8)
 
-x = np.array([1.0,0.0])
+x = 8.4
 
-print("The derivative of f9 is:",f9_grad(x))
+print("The derivative of f8 is:",f8_grad(x))
 
    ---------------------------------------------------------------------------
-AttributeError                            Traceback (most recent call last)
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42573/546166676.py in <module>
-      9 x = np.array([1.0,0.0])
+TypeError                                 Traceback (most recent call last)
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47735/1122558214.py in <module>
+      9 x = 8.4
      10 
----> 11 print("The derivative of f9 is:",f9_grad(x))
+---> 11 print("The derivative of f8 is:",f8_grad(x))
 
 ~/anaconda3/lib/python3.8/site-packages/autograd/wrap_util.py in nary_f(*args, **kwargs)
      18             else:
@@ -2068,14 +2121,32 @@ AttributeError                            Traceback (most recent call last)
      16             if isinstance(argnum, int):
      17                 x = args[argnum]
 
-/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42573/546166676.py in f9(a)
-      3 def f9(a): # Assume a is an array with 2 elements
-      4     b = np.array([1.0,2.0])
-----> 5     return a.dot(b)
+/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47735/1122558214.py in f8(x)
+      2 from autograd import grad
+      3 def f8(x): # Assume x is an array
+----> 4     x[2] = 3
+      5     return x*2
       6 
-      7 f9_grad = grad(f9)
 
-AttributeError: 'ArrayBox' object has no attribute 'dot'
+TypeError: 'ArrayBox' object does not support item assignment
+
    +
    +
    +
    +

    Here, Autograd tells us that an ‘ArrayBox’ does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible.

    +
    +
    +
    import autograd.numpy as np
+from autograd import grad
+def f9(a): # Assume a is an array with 2 elements
+    b = np.array([1.0,2.0])
+    return a.dot(b)
+
+f9_grad = grad(f9)
+
+x = np.array([1.0,0.0])
+
+print("The derivative of f9 is:",f9_grad(x))
    @@ -2114,7 +2185,192 @@ which also computed the dot product can be used:

    -

    More examples will be added, in particular how to compare autograd with own codes for the gradients.

    + +
    +

    7.11. Using Autograd with OLS

    +

    We conclude the part on optmization by showing how we can make codes +for linear regression and logistic regression using autograd. The +first example shows results with ordinary leats squares.

    +
    +
    +
    # Using Autograd to calculate gradients for OLS
+from random import random, seed
+import numpy as np
+import autograd.numpy as np
+import matplotlib.pyplot as plt
+from autograd import grad
+
+def CostOLS(beta):
+    return (1.0/n)*np.sum((y-X @ beta)**2)
+
+n = 100
+x = 2*np.random.rand(n,1)
+y = 4+3*x+np.random.randn(n,1)
+
+X = np.c_[np.ones((n,1)), x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+print("Own inversion")
+print(theta_linreg)
+# Hessian matrix
+H = (2.0/n)* XT_X
+EigValues, EigVectors = np.linalg.eig(H)
+print(f"Eigenvalues of Hessian Matrix:{EigValues}")
+
+theta = np.random.randn(2,1)
+eta = 1.0/np.max(EigValues)
+Niterations = 1000
+# define the gradient
+training_gradient = grad(CostOLS)
+
+for iter in range(Niterations):
+    gradients = training_gradient(theta)
+    theta -= eta*gradients
+print("theta from own gd")
+print(theta)
+
+xnew = np.array([[0],[2]])
+Xnew = np.c_[np.ones((2,1)), xnew]
+ypredict = Xnew.dot(theta)
+ypredict2 = Xnew.dot(theta_linreg)
+
+plt.plot(xnew, ypredict, "r-")
+plt.plot(xnew, ypredict2, "b-")
+plt.plot(x, y ,'ro')
+plt.axis([0,2.0,0, 15.0])
+plt.xlabel(r'$x$')
+plt.ylabel(r'$y$')
+plt.title(r'Random numbers ')
+plt.show()
+
    +
    +
    +
    +
    +

    7.11.1. Including Stochastic Gradient Descent with Autograd

    +

    In this code we include the stochastic gradient descent approach discussed above. Note here that we specify which argument we are taking the derivative with respect to when using autograd.

    +
    +
    +
    # Using Autograd to calculate gradients using SGD
+# OLS example
+from random import random, seed
+import numpy as np
+import autograd.numpy as np
+import matplotlib.pyplot as plt
+from autograd import grad
+
+# Note change from previous example
+def CostOLS(y,X,theta):
+    return np.sum((y-X @ theta)**2)
+
+n = 100
+x = 2*np.random.rand(n,1)
+y = 4+3*x+np.random.randn(n,1)
+
+X = np.c_[np.ones((n,1)), x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+print("Own inversion")
+print(theta_linreg)
+# Hessian matrix
+H = (2.0/n)* XT_X
+EigValues, EigVectors = np.linalg.eig(H)
+print(f"Eigenvalues of Hessian Matrix:{EigValues}")
+
+theta = np.random.randn(2,1)
+eta = 1.0/np.max(EigValues)
+Niterations = 1000
+
+# Note that we request the derivative wrt third argument (theta, 2 here)
+training_gradient = grad(CostOLS,2)
+
+for iter in range(Niterations):
+    gradients = (1.0/n)*training_gradient(y, X, theta)
+    theta -= eta*gradients
+print("theta from own gd")
+print(theta)
+
+xnew = np.array([[0],[2]])
+Xnew = np.c_[np.ones((2,1)), xnew]
+ypredict = Xnew.dot(theta)
+ypredict2 = Xnew.dot(theta_linreg)
+
+plt.plot(xnew, ypredict, "r-")
+plt.plot(xnew, ypredict2, "b-")
+plt.plot(x, y ,'ro')
+plt.axis([0,2.0,0, 15.0])
+plt.xlabel(r'$x$')
+plt.ylabel(r'$y$')
+plt.title(r'Random numbers ')
+plt.show()
+
+n_epochs = 50
+M = 5   #size of each minibatch
+m = int(n/M) #number of minibatches
+t0, t1 = 5, 50
+def learning_schedule(t):
+    return t0/(t+t1)
+
+theta = np.random.randn(2,1)
+
+for epoch in range(n_epochs):
+# Can you figure out a better way of setting up the contributions to each batch?
+    for i in range(m):
+        random_index = M*np.random.randint(m)
+        xi = X[random_index:random_index+M]
+        yi = y[random_index:random_index+M]
+        gradients = (1.0/M)*training_gradient(yi, xi, theta)
+        eta = learning_schedule(epoch*m+i)
+        theta = theta - eta*gradients
+print("theta from own sdg")
+print(theta)
+
    +
    +
    +
    +
    +
    +

    7.11.2. And Logistic Regression

    +
    +
    +
    import autograd.numpy as np
+from autograd import grad
+
+def sigmoid(x):
+    return 0.5 * (np.tanh(x / 2.) + 1)
+
+def logistic_predictions(weights, inputs):
+    # Outputs probability of a label being true according to logistic model.
+    return sigmoid(np.dot(inputs, weights))
+
+def training_loss(weights):
+    # Training loss is the negative log-likelihood of the training labels.
+    preds = logistic_predictions(weights, inputs)
+    label_probabilities = preds * targets + (1 - preds) * (1 - targets)
+    return -np.sum(np.log(label_probabilities))
+
+# Build a toy dataset.
+inputs = np.array([[0.52, 1.12,  0.77],
+                   [0.88, -1.08, 0.15],
+                   [0.52, 0.06, -1.30],
+                   [0.74, -2.49, 1.39]])
+targets = np.array([True, True, False, True])
+
+# Define a function that returns gradients of training loss using Autograd.
+training_gradient_fun = grad(training_loss)
+
+# Optimize weights using gradient descent.
+weights = np.array([0.0, 0.0, 0.0])
+print("Initial loss:", training_loss(weights))
+for i in range(100):
+    weights -= training_gradient_fun(weights) * 0.01
+
+print("Trained loss:", training_loss(weights))
+
    +
    +
    +
    +
    diff --git a/doc/LectureNotes/_build/html/clustering.html b/doc/LectureNotes/_build/html/clustering.html new file mode 100644 index 000000000..41cdce5da --- /dev/null +++ b/doc/LectureNotes/_build/html/clustering.html @@ -0,0 +1,828 @@ + + + + + + + + 12. Clustering and Unsupervised Learning — Applied Data Analysis and Machine Learning + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    +
    + + + + + + + + +
    + +
    +
    +
    +
    + + + + +
    + + +
    + + + + + + +
    +
    + + + + + + + + + +
    + + +
    + +
    + Contents +
    + +
    +
    +
    +
    +
    + +
    + +
    +

    12. Clustering and Unsupervised Learning

    +

    In general terms cluster analysis, or clustering, is the task of grouping a +data-set into different distinct categories based on some measure of equality of +the data. This measure is often referred to as a metric or similarity +measure in the literature (note: sometimes we deal with a dissimilarity +measure instead). Usually, these metrics are formulated as some kind of +distance function between points in a high-dimensional space.

    +

    The simplest, and also the most +common is the Euclidean distance.

    +

    The simplest of all clustering algorithms is the k-means algorithm +, sometimes also referred to as Lloyds algorithm. It is the simplest and also +the most common. From its simplicity it obtains both strengths and weaknesses. +These will be discussed in more detail later. The \(k\)-means algorithm is a +centroid based clustering algorithm.

    +

    Assume, we are given \(n\) data points and we wish to split the data into \(K < n\) +different categories, or clusters. We label each cluster by an integer

    +
    +\[ +k\in\{1, \cdots, K \}. +\]
    +

    In the basic k-means algorithm each point is assigned to only +one cluster \(k\), and these assignments are non-injective i.e. many-to-one. We +can think of these mappings as an encoder \(k = C(i)\), which assigns the \(i\)-th +data-point \(\bf x_i\) to the \(k\)-th cluster.

    +

    \(k\)-means algorithm in words:

    +
      +

    1. We start with guesses / random initializations of our \(k\) cluster centers/centroids

      2. +

    2. For each centroid the points that are most similar are identified

      3. +

    3. Then we move / replace each centroid with a coordinate average of all the points that were assigned to that centroid.

      4. +

    4. Iterate 2-3 until the centroids no longer move (to some tolerance)

      5. +
    +

    We assume we have \(n\) data-points

    + +
    +
    +\[ +\begin{equation}\label{eq:kmeanspoints} \tag{1} + \boldsymbol{x_i} = \{x_{i, 1}, \cdots, x_{i, p}\}\in\mathbb{R}^p. +\end{equation} +\]
    +

    which we wish to group into \(K < n\) clusters. For our dissimilarity measure we +use the squared Euclidean distance

    + +
    +
    +\[ +\begin{equation}\label{eq:squaredeuclidean} \tag{2} + d(\boldsymbol{x_i}, \boldsymbol{x_i'}) = \sum_{j=1}^p(x_{ij} - x_{i'j})^2 + = ||\boldsymbol{x_i} - \boldsymbol{x_{i'}}||^2 +\end{equation} +\]
    +

    We define the so called within-cluster point scatter which gives us a +measure of how close each data point assigned to the same cluster tends to be to +the all the others.

    + +
    +
    +\[ +\begin{equation}\label{eq:withincluster} \tag{3} + W(C) = \frac{1}{2}\sum_{k=1}^K\sum_{C(i)=k} + \sum_{C(i')=k}d(\boldsymbol{x_i}, \boldsymbol{x_{i'}}) = + \sum_{k=1}^KN_k\sum_{C(i)=k}||\boldsymbol{x_i} - \boldsymbol{\overline{x_k}}||^2 +\end{equation} +\]
    +

    where \(\boldsymbol{\overline{x_k}}\) is the mean vector associated with the \(k\)-th +cluster, and \(N_k = \sum_{i=1}^nI(C(i) = k)\), where the \(I()\) notation is +similar to the Kronecker delta (Commonly used in statistics, it just means that +when \(i = k\) we have the encoder \(C(i)\)). In other words, the within-cluster +scatter measures the compactness of each cluster with respect to the data points +assigned to each cluster. This is the quantity that the \(k\)-means algorithm aims +to minimize. We refer to this quantity \(W(C)\) as the within cluster scatter +because of its relation to the total scatter.

    +

    We have

    + +
    +
    +\[ +\begin{equation}\label{eq:totalscatter} \tag{4} + T = W(C) + B(C) = \frac{1}{2}\sum_{i=1}^n + \sum_{i'=1}^nd(\boldsymbol{x_i}, \boldsymbol{x_{i'}}) + = \frac{1}{2}\sum_{k=1}^K\sum_{C(i)=k} + \Big(\sum_{C(i') = k}d(\boldsymbol{x_i}, \boldsymbol{x_{i'}}) + + \sum_{C(i')\neq k}d(\boldsymbol{x_i}, \boldsymbol{x_{i'}})\Big). +\end{equation} +\]
    +

    This is a quantity that is conserved throughout the \(k\)-means algorithm. It can +be thought of as the total amount of information in the data, and it is composed +of the aforementioned within-cluster scatter and the between-cluster scatter +\(B(C)\). In methods such as principle component analysis the total scatter is not +conserved.

    +

    Given a cluster mean \(\boldsymbol{m_k}\) we define the total cluster variance

    + +
    +
    +\[ +\begin{equation}\label{eq:totalclustervariance} \tag{5} + \min_{C, \{\boldsymbol{m_k}\}_1^K}\sum_{k=1}^KN_k\sum||\boldsymbol{x_i} - \boldsymbol{m_k}||^2 +\end{equation} +\]
    +

    Now we have all the pieces necessary to formally revisit the \(k\)-means algorithm.

    +

    The \(k\)-means clustering algorithm goes as follows

    +
      +

    1. For a given cluster assignment \(C\), and \(k\) cluster means \(\left\{m_1, \cdots, m_k\right\}\). We minimize the total cluster variance with respect to the cluster means \(\{m_k\}\) yielding the means of the currently assigned clusters.

      2. +

    2. Given a current set of \(k\) means \(\{m_k\}\) the total cluster variance is minimized by assigning each observation to the closest (current) cluster mean. That is $\(C(i) = \underset{1\leq k\leq K}{\mathrm{argmin}} ||\boldsymbol{x_i} - \boldsymbol{m_k}||^2\)$

      3. +

    3. Steps 1 and 2 are repeated until the assignments do not change.

      4. +
    +
    +

    12.1. Codes and Approaches

    +
      +

    1. Before we start we specify a number \(k\) which is the number of clusters we want to try to separate our data into.

      2. +

    2. We initially choose \(k\) random data points in our data as our initial centroids, or means (this is where the name comes from).

      3. +

    3. Assign each data point to their closest centroid, based on the squared Euclidean distance.

      4. +

    4. For each of the \(k\) cluster we update the centroid by calculating new mean values for all the data points in the cluster.

      5. +

    5. Iteratively minimize the within cluster scatter by performing steps (3, 4) until the new assignments stop changing (can be to some tolerance) or until a maximum number of iterations have passed.

      6. +
    +

    Let us now program the most basic version of the algorithm using nothing but +Python with numpy arrays. This code is kept intentionally simple to gradually +progress our understanding. There is no vectorization of any kind, and even most +helper functions are not utilized.

    +

    We need first a dataset to do our cluster analysis on. In our case +this is a plain vanilla data set using random numbers using a +Gaussian distribution.

    +
    +
    +
    %matplotlib inline
+
+import time
+import numpy as np
+import tensorflow as tf
+from matplotlib import image
+import matplotlib.pyplot as plt
+from sklearn.cluster import KMeans
+from IPython.display import display
+
+np.random.seed(2021)
+
    +
    +
    +
    +

    Next we define functions, for ease of use later, to generate Gaussians and to +set up our toy data set.

    +
    +
    +
    def gaussian_points(dim=2, n_points=1000, mean_vector=np.array([0, 0]),
+                    sample_variance=1):
+    """
+    Very simple custom function to generate gaussian distributed point clusters
+    with variable dimension, number of points, means in each direction
+    (must match dim) and sample variance.
+
+    Inputs:
+        dim (int)
+        n_points (int)
+        mean_vector (np.array) (where index 0 is x, index 1 is y etc.)
+        sample_variance (float)
+
+    Returns:
+        data (np.array): with dimensions (dim x n_points)
+    """
+
+    mean_matrix = np.zeros(dim) + mean_vector
+    covariance_matrix = np.eye(dim) * sample_variance
+    data = np.random.multivariate_normal(mean_matrix, covariance_matrix,
+                                    n_points)
+    return data
+
+
+
+def generate_simple_clustering_dataset(dim=2, n_points=1000, plotting=True,
+                                    return_data=True):
+    """
+    Toy model to illustrate k-means clustering
+    """
+
+    data1 = gaussian_points(mean_vector=np.array([5, 5]))
+    data2 = gaussian_points()
+    data3 = gaussian_points(mean_vector=np.array([1, 4.5]))
+    data4 = gaussian_points(mean_vector=np.array([5, 1]))
+    data = np.concatenate((data1, data2, data3, data4), axis=0)
+
+    if plotting:
+        fig, ax = plt.subplots()
+        ax.scatter(data[:, 0], data[:, 1], alpha=0.2)
+        ax.set_title('Toy Model Dataset')
+        plt.show()
+
+
+    if return_data:
+        return data
+
+
+data = generate_simple_clustering_dataset()
+
    +
    +
    +
    +_images/clustering_17_0.png +
    +
    +

    With the above dataset we start +implementing the \(k\)-means algorithm.

    +
    +
    +
    n_samples, dimensions = data.shape
+n_clusters = 4
+
+# we randomly initialize our centroids
+np.random.seed(2021)
+centroids = data[np.random.choice(n_samples, n_clusters, replace=False), :]
+distances = np.zeros((n_samples, n_clusters))
+
+# first we need to calculate the distance to each centroid from our data
+for k in range(n_clusters):
+    for n in range(n_samples):
+        dist = 0
+        for d in range(dimensions):
+            dist += np.abs(data[n, d] - centroids[k, d])**2
+            distances[n, k] = dist
+
+# we initialize an array to keep track of to which cluster each point belongs
+# the way we set it up here the index tracks which point and the value which
+# cluster the point belongs to
+cluster_labels = np.zeros(n_samples, dtype='int')
+
+# next we loop through our samples and for every point assign it to the cluster
+# to which it has the smallest distance to
+for n in range(n_samples):
+    # tracking variables (all of this is basically just an argmin)
+    smallest = 1e10
+    smallest_row_index = 1e10
+    for k in range(n_clusters):
+        if distances[n, k] < smallest:
+            smallest = distances[n, k]
+            smallest_row_index = k
+
+    cluster_labels[n] = smallest_row_index
+
    +
    +
    +
    +
    +
    +
    fig = plt.figure()
+ax = fig.add_subplot()
+unique_cluster_labels = np.unique(cluster_labels)
+for i in unique_cluster_labels:
+    ax.scatter(data[cluster_labels == i, 0],
+               data[cluster_labels == i, 1],
+               label = i,
+               alpha = 0.2)
+    ax.scatter(centroids[:, 0], centroids[:, 1], c='black')
+
+ax.set_title("First Grouping of Points to Centroids")
+
+plt.show()
+
    +
    +
    +
    +_images/clustering_20_0.png +
    +
    +

    So what do we have so far? We have ‘picked’ \(k\) centroids at random from our +data points. There are other ways of more intelligently choosing their +initializations, however for our purposes randomly is fine. Then we have +initialized an array ‘distances’ which holds the information of the distance, +or dissimilarity, of every point to of our centroids. Finally, we have +initialized an array ‘cluster_labels’ which according to our distances array +holds the information of to which centroid every point is assigned. This was the +first pass of our algorithm. Essentially, all we need to do now is repeat the +distance and assignment steps above until we have reached a desired convergence +or a maximum amount of iterations.

    +
    +
    +
    max_iterations = 100
+tolerance = 1e-8
+
+for iteration in range(max_iterations):
+    prev_centroids = centroids.copy()
+    for k in range(n_clusters):
+        # this array will be used to update our centroid positions
+        vector_mean = np.zeros(dimensions)
+        mean_divisor = 0
+        for n in range(n_samples):
+            if cluster_labels[n] == k:
+                vector_mean += data[n, :]
+                mean_divisor += 1
+
+        # update according to the k means
+        centroids[k, :] = vector_mean / mean_divisor
+
+    # we find the dissimilarity
+    for k in range(n_clusters):
+        for n in range(n_samples):
+            dist = 0
+            for d in range(dimensions):
+                dist += np.abs(data[n, d] - centroids[k, d])**2
+                distances[n, k] = dist
+
+    # assign each point
+    for n in range(n_samples):
+        smallest = 1e10
+        smallest_row_index = 1e10
+        for k in range(n_clusters):
+            if distances[n, k] < smallest:
+                smallest = distances[n, k]
+                smallest_row_index = k
+
+        cluster_labels[n] = smallest_row_index
+
+    # convergence criteria
+    centroid_difference = np.sum(np.abs(centroids - prev_centroids))
+    if centroid_difference < tolerance:
+        print(f'Converged at iteration {iteration}')
+        break
+
+    elif iteration == max_iterations:
+        print(f'Did not converge in {max_iterations} iterations')
+
    +
    +
    +
    +
    Converged at iteration 5
+
    +
    +
    +
    +

    We now have a simple , un-optimized \(k\)-means +clustering implementation. Lets plot the final result

    +
    +
    +
    fig = plt.figure()
+ax = fig.add_subplot()
+unique_cluster_labels = np.unique(cluster_labels)
+for i in unique_cluster_labels:
+    ax.scatter(data[cluster_labels == i, 0],
+               data[cluster_labels == i, 1],
+               label = i,
+               alpha = 0.2)
+    ax.scatter(centroids[:, 0], centroids[:, 1], c='black')
+
+ax.set_title("Final Result of K-means Clustering")
+
+plt.show()
+
    +
    +
    +
    +_images/clustering_24_0.png +
    +
    +
    +
    +
    def naive_kmeans(data, n_clusters=4, max_iterations=100, tolerance=1e-8):
+    start_time = time.time()
+
+    n_samples, dimensions = data.shape
+    n_clusters = 4
+    #np.random.seed(2021)
+    centroids = data[np.random.choice(n_samples, n_clusters, replace=False), :]
+    distances = np.zeros((n_samples, n_clusters))
+
+    for k in range(n_clusters):
+        for n in range(n_samples):
+            dist = 0
+            for d in range(dimensions):
+                dist += np.abs(data[n, d] - centroids[k, d])**2
+                distances[n, k] = dist
+
+    cluster_labels = np.zeros(n_samples, dtype='int')
+
+    for n in range(n_samples):
+        smallest = 1e10
+        smallest_row_index = 1e10
+        for k in range(n_clusters):
+            if distances[n, k] < smallest:
+                smallest = distances[n, k]
+                smallest_row_index = k
+
+        cluster_labels[n] = smallest_row_index
+
+    for iteration in range(max_iterations):
+        prev_centroids = centroids.copy()
+        for k in range(n_clusters):
+            vector_mean = np.zeros(dimensions)
+            mean_divisor = 0
+            for n in range(n_samples):
+                if cluster_labels[n] == k:
+                    vector_mean += data[n, :]
+                    mean_divisor += 1
+
+            centroids[k, :] = vector_mean / mean_divisor
+
+        for k in range(n_clusters):
+            for n in range(n_samples):
+                dist = 0
+                for d in range(dimensions):
+                    dist += np.abs(data[n, d] - centroids[k, d])**2
+                    distances[n, k] = dist
+
+        for n in range(n_samples):
+            smallest = 1e10
+            smallest_row_index = 1e10
+            for k in range(n_clusters):
+                if distances[n, k] < smallest:
+                    smallest = distances[n, k]
+                    smallest_row_index = k
+
+            cluster_labels[n] = smallest_row_index
+
+        centroid_difference = np.sum(np.abs(centroids - prev_centroids))
+        if centroid_difference < tolerance:
+            print(f'Converged at iteration {iteration}')
+            print(f'Runtime: {time.time() - start_time} seconds')
+
+            return cluster_labels, centroids
+
+    print(f'Did not converge in {max_iterations} iterations')
+    print(f'Runtime: {time.time() - start_time} seconds')
+
+    return cluster_labels, centroids
+
    +
    +
    +
    +
    +
    + + + + +
    + + + + + + +
    + + + + +
    + +
    +
    +
    +
    +

    + + By Morten Hjorth-Jensen
    + + © Copyright 2021.
    +

    +
    +
    +
    + + +
    +
    + + + + + + \ No newline at end of file diff --git a/doc/LectureNotes/_build/html/genindex.html b/doc/LectureNotes/_build/html/genindex.html index 5ce5d1fab..6b34d9aaa 100644 --- a/doc/LectureNotes/_build/html/genindex.html +++ b/doc/LectureNotes/_build/html/genindex.html @@ -188,6 +188,11 @@ 11. Basic ideas of the Principal Component Analysis (PCA) +
  • + + 12. Clustering and Unsupervised Learning + +

    • @@ -197,17 +202,27 @@

    diff --git a/doc/LectureNotes/_build/html/intro.html b/doc/LectureNotes/_build/html/intro.html index 32cd03a9c..9b986280a 100644 --- a/doc/LectureNotes/_build/html/intro.html +++ b/doc/LectureNotes/_build/html/intro.html @@ -196,6 +196,11 @@ 11. Basic ideas of the Principal Component Analysis (PCA) +
  • + + 12. Clustering and Unsupervised Learning + +

    • @@ -205,17 +210,27 @@

    diff --git a/doc/LectureNotes/_build/html/objects.inv b/doc/LectureNotes/_build/html/objects.inv index 7b981a738..70bca6a16 100644 Binary files a/doc/LectureNotes/_build/html/objects.inv and b/doc/LectureNotes/_build/html/objects.inv differ diff --git a/doc/LectureNotes/_build/html/search.html b/doc/LectureNotes/_build/html/search.html index 315741fbb..2f6153900 100644 --- a/doc/LectureNotes/_build/html/search.html +++ b/doc/LectureNotes/_build/html/search.html @@ -194,6 +194,11 @@ 11. Basic ideas of the Principal Component Analysis (PCA) +
  • + + 12. Clustering and Unsupervised Learning + +

    • @@ -203,17 +208,27 @@

    diff --git a/doc/LectureNotes/_build/html/searchindex.js b/doc/LectureNotes/_build/html/searchindex.js index cf2f29151..2acf2f68e 100644 --- a/doc/LectureNotes/_build/html/searchindex.js +++ b/doc/LectureNotes/_build/html/searchindex.js @@ -1 +1 @@ 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\ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter10.ipynb b/doc/LectureNotes/_build/jupyter_execute/chapter10.ipynb index 0c8ec17f0..3b986144d 100644 --- a/doc/LectureNotes/_build/jupyter_execute/chapter10.ipynb +++ b/doc/LectureNotes/_build/jupyter_execute/chapter10.ipynb @@ -2,7 +2,21 @@ "cells": [ { "cell_type": "markdown", - "metadata": {}, + "id": "7e3af66b", + "metadata": { + "editable": true + }, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "id": "64b4469b", + "metadata": { + "editable": true + }, "source": [ "# Building a Feed Forward Neural Network\n", "\n", @@ -27,7 +41,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a27e39fb", + "metadata": { + "editable": true + }, "source": [ "$$\n", "P(y = 0 \\mid \\hat{x}, \\hat{\\theta}) = \\frac{1}{1 + \\exp{(- \\hat{x}})} ,\n", @@ -36,14 +53,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6332f4a5", + "metadata": { + "editable": true + }, "source": [ "and" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "5d2a5fe5", + "metadata": { + "editable": true + }, "source": [ "$$\n", "P(y = 1 \\mid \\hat{x}, \\hat{\\theta}) = 1 - P(y = 0 \\mid \\hat{x}, \\hat{\\theta}) ,\n", @@ -52,13 +75,22 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "62bf3ae6", + "metadata": { + "editable": true + }, "source": [ "where $y \\in \\{0, 1\\}$ and $\\hat{\\theta}$ represents the weights and biases\n", - "of our network.\n", - "\n", - "\n", - "\n", + "of our network." + ] + }, + { + "cell_type": "markdown", + "id": "5fdb1150", + "metadata": { + "editable": true + }, + "source": [ "## Defining the cost function\n", "\n", "Our cost function is given as (see the Logistic regression lectures)" @@ -66,7 +98,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "64cfe441", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathcal{C}(\\hat{\\theta}) = - \\ln P(\\mathcal{D} \\mid \\hat{\\theta}) = - \\sum_{i=1}^n\n", @@ -76,7 +111,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "dbd78c70", + "metadata": { + "editable": true + }, "source": [ "This last equality means that we can interpret our *cost* function as a sum over the *loss* function\n", "for each point in the dataset $\\mathcal{L}_i(\\hat{\\theta})$. \n", @@ -87,10 +125,8 @@ "\n", "$y = 5 \\quad \\rightarrow \\quad \\hat{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$ and\n", "\n", - "\n", "$y = 1 \\quad \\rightarrow \\quad \\hat{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,$ \n", "\n", - "\n", "i.e. a binary bit string of length $C$, where $C = 10$ is the number of classes in the MNIST dataset (numbers from $0$ to $9$).. \n", "\n", "If $\\hat{x}_i$ is the $i$-th input (image), $y_{ic}$ refers to the $c$-th component of the $i$-th\n", @@ -100,7 +136,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "67834a87", + "metadata": { + "editable": true + }, "source": [ "$$\n", "P(y_{ic} = 1 \\mid \\hat{x}_i, \\hat{\\theta}) = \\frac{\\exp{((\\hat{a}_i^{hidden})^T \\hat{w}_c)}}\n", @@ -110,7 +149,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0521db23", + "metadata": { + "editable": true + }, "source": [ "which reduces to the logistic function in the binary case. \n", "The likelihood of this $C$-class classifier\n", @@ -119,7 +161,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "12a92031", + "metadata": { + "editable": true + }, "source": [ "$$\n", "P(\\mathcal{D} \\mid \\hat{\\theta}) = \\prod_{i=1}^n \\prod_{c=0}^{C-1} [P(y_{ic} = 1)]^{y_{ic}} .\n", @@ -128,14 +173,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e03ffaa9", + "metadata": { + "editable": true + }, "source": [ "Again we take the negative log-likelihood to define our cost function:" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "cfab4fa3", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathcal{C}(\\hat{\\theta}) = - \\log{P(\\mathcal{D} \\mid \\hat{\\theta})}.\n", @@ -144,13 +195,23 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "864840d8", + "metadata": { + "editable": true + }, "source": [ "See the logistic regression lectures for a full definition of the cost function.\n", "\n", - "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!\n", - "\n", - "\n", + "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!" + ] + }, + { + "cell_type": "markdown", + "id": "496bdba3", + "metadata": { + "editable": true + }, + "source": [ "### Example: binary classification problem\n", "\n", "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" @@ -158,7 +219,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "8646c18f", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathcal{C}(\\hat{\\beta}) = - \\sum_{i=1}^n \\left(y_i\\log{p(y_i \\vert x_i,\\hat{\\beta})}+(1-y_i)\\log{1-p(y_i \\vert x_i,\\hat{\\beta})}\\right),\n", @@ -167,14 +231,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "2dbfda71", + "metadata": { + "editable": true + }, "source": [ "where we had defined the logistic (sigmoid) function" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "5cd0a375", + "metadata": { + "editable": true + }, "source": [ "$$\n", "p(y_i =1\\vert x_i,\\hat{\\beta})=\\frac{\\exp{(\\beta_0+\\beta_1 x_i)}}{1+\\exp{(\\beta_0+\\beta_1 x_i)}},\n", @@ -183,14 +253,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "020de545", + "metadata": { + "editable": true + }, "source": [ "and" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "2efde2f5", + "metadata": { + "editable": true + }, "source": [ "$$\n", "p(y_i =0\\vert x_i,\\hat{\\beta})=1-p(y_i =1\\vert x_i,\\hat{\\beta}).\n", @@ -199,7 +275,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "bf9a596d", + "metadata": { + "editable": true + }, "source": [ "The parameters $\\hat{\\beta}$ were defined using a minimization method like gradient descent or Newton-Raphson's method. \n", "\n", @@ -209,7 +288,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0b392cc5", + "metadata": { + "editable": true + }, "source": [ "$$\n", "a_i^l = y_i = \\frac{\\exp{(z_i^l)}}{1+\\exp{(z_i^l)}},\n", @@ -218,14 +300,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5169434f", + "metadata": { + "editable": true + }, "source": [ "with" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "dcc8ed0a", + "metadata": { + "editable": true + }, "source": [ "$$\n", "z_i^l = \\sum_{j}w_{ij}^l a_j^{l-1}+b_i^l,\n", @@ -234,7 +322,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "4010628a", + "metadata": { + "editable": true + }, "source": [ "where the superscript $l-1$ indicates that these are the outputs from layer $l-1$.\n", "Our cost function at the final layer $l=L$ is now" @@ -242,7 +333,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "7bf89431", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathcal{C}(\\hat{W}) = - \\sum_{i=1}^n \\left(t_i\\log{a_i^L}+(1-t_i)\\log{(1-a_i^L)}\\right),\n", @@ -251,14 +345,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "79bb56ee", + "metadata": { + "editable": true + }, "source": [ "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" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "a27a06b5", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial \\mathcal{C}(\\hat{W})}{\\partial a_i^L} = \\frac{a_i^L-t_i}{a_i^L(1-a_i^L)}.\n", @@ -267,12 +367,21 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "cd6acfaf", + "metadata": { + "editable": true + }, + "source": [ + "In case we use another activation function than the logistic one, we need to evaluate other derivatives." + ] + }, + { + "cell_type": "markdown", + "id": "b9e59acd", + "metadata": { + "editable": true + }, "source": [ - "In case we use another activation function than the logistic one, we need to evaluate other derivatives. \n", - "\n", - "\n", - "\n", "### The Softmax function\n", "\n", "In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation $z_i^l$, that is we need" @@ -280,7 +389,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "eceaf7eb", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial f(z_i^l)}{\\partial w_{jk}^l} =\n", @@ -290,14 +402,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "bbc77522", + "metadata": { + "editable": true + }, "source": [ "For the Softmax function we have" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "88d4c18b", + "metadata": { + "editable": true + }, "source": [ "$$\n", "f(z_i^l) = \\frac{\\exp{(z_i^l)}}{\\sum_{m=1}^K\\exp{(z_m^l)}}.\n", @@ -306,14 +424,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b47f4092", + "metadata": { + "editable": true + }, "source": [ "Its derivative with respect to $z_j^l$ gives" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "65ec4336", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial f(z_i^l)}{\\partial z_j^l}= f(z_i^l)\\left(\\delta_{ij}-f(z_j^l)\\right),\n", @@ -322,14 +446,23 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "38f54e19", + "metadata": { + "editable": true + }, + "source": [ + "which in case of the simply binary model reduces to having $i=j$." + ] + }, + { + "cell_type": "markdown", + "id": "c0d8bc5e", + "metadata": { + "editable": true + }, "source": [ - "which in case of the simply binary model reduces to having $i=j$. \n", - "\n", - "\n", "## Developing a code for doing neural networks with back propagation\n", "\n", - "\n", "One can identify a set of key steps when using neural networks to solve supervised learning problems: \n", "\n", "1. Collect and pre-process data \n", @@ -342,8 +475,16 @@ "\n", "5. Evaluate model performance on test data \n", "\n", - "6. Adjust hyperparameters (if necessary, network architecture)\n", - "\n", + "6. Adjust hyperparameters (if necessary, network architecture)" + ] + }, + { + "cell_type": "markdown", + "id": "e3810222", + "metadata": { + "editable": true + }, + "source": [ "### Collect and pre-process data\n", "\n", "Here we will be using the MNIST dataset, which is readily available through the **scikit-learn**\n", @@ -389,6 +530,7 @@ { "cell_type": "code", "execution_count": 1, + "id": "5e7205ac", "metadata": { "collapsed": false, "editable": true @@ -412,7 +554,7 @@ }, "metadata": { "filenames": { - "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter10_33_1.png" + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter10_39_1.png" }, "needs_background": "light" }, @@ -468,7 +610,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "78721970", + "metadata": { + "editable": true + }, "source": [ "### Train and test datasets\n", "\n", @@ -486,6 +631,7 @@ { "cell_type": "code", "execution_count": 2, + "id": "dcc066a0", "metadata": { "collapsed": false, "editable": true @@ -532,7 +678,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "95cf3f87", + "metadata": { + "editable": true + }, "source": [ "### Define model and architecture\n", "\n", @@ -568,8 +717,16 @@ "\n", "$$ f(x) = \\sigma(x) = \\frac{1}{1 + e^{-x}} ,$$\n", "\n", - "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.\n", - "\n", + "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." + ] + }, + { + "cell_type": "markdown", + "id": "25c0cbe6", + "metadata": { + "editable": true + }, + "source": [ "### Layers\n", "\n", "* Input \n", @@ -602,7 +759,6 @@ "Since each neuron in the output layer is connected to the 50 inputs from the hidden layer we have 50x10 = 500\n", "weights to the output layer.\n", "\n", - "\n", "Typically weights are initialized with small values distributed around zero, drawn from a uniform\n", "or normal distribution. Setting all weights to zero means all neurons give the same output, making the network useless. \n", "\n", @@ -617,6 +773,7 @@ { "cell_type": "code", "execution_count": 3, + "id": "9f415ea2", "metadata": { "collapsed": false, "editable": true @@ -642,7 +799,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "afbff46b", + "metadata": { + "editable": true + }, "source": [ "### Feed-forward pass\n", "\n", @@ -664,7 +824,6 @@ "$$ a_{j}^{L} = \\frac{\\exp{(z_j^{L})}}\n", "{\\sum_{c=0}^{C-1} \\exp{(z_c^{L})}} .$$ \n", "\n", - "\n", "Since our data has the dimensions $X = (n_{inputs}, n_{features})$ and our weights to the hidden\n", "layer have the dimensions \n", "$W_{hidden} = (n_{features}, n_{hidden})$,\n", @@ -695,6 +854,7 @@ { "cell_type": "code", "execution_count": 4, + "id": "e05b2147", "metadata": { "collapsed": false, "editable": true @@ -757,7 +917,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6c8100ba", + "metadata": { + "editable": true + }, "source": [ "### Choose cost function and optimizer\n", "\n", @@ -770,10 +933,8 @@ "\n", "$$ y = 5 \\quad \\rightarrow \\quad \\hat{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$$ \n", "\n", - "\n", "$$ y = 1 \\quad \\rightarrow \\quad \\hat{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,$$ \n", "\n", - "\n", "i.e. a binary bit string of length $C$, where $C = 10$ is the number of classes in the MNIST dataset. \n", "\n", "Let $y_{ic}$ denote the $c$-th component of the $i$-th one-hot vector. \n", @@ -782,10 +943,16 @@ "In the one-hot representation only one of the terms in the loss function is non-zero, namely the\n", "probability of the correct category $c'$ \n", "(i.e. the category $c'$ such that $y_{ic'} = 1$). This means that the cross entropy loss only punishes you for how wrong\n", - "you got the correct label. The probability of category $c$ is given by the softmax function. The vector $\\hat{\\theta}$ represents the parameters of our network, i.e. all the weights and biases. \n", - "\n", - "\n", - "\n", + "you got the correct label. The probability of category $c$ is given by the softmax function. The vector $\\hat{\\theta}$ represents the parameters of our network, i.e. all the weights and biases." + ] + }, + { + "cell_type": "markdown", + "id": "720d19fd", + "metadata": { + "editable": true + }, + "source": [ "### Optimizing the cost function\n", "\n", "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\n", @@ -814,9 +981,16 @@ "\n", "2. It significantly speeds up the calculation, since we do not have to use the entire dataset to calculate the gradient. \n", "\n", - "The various optmization methods, with codes and algorithms, are discussed in our lectures on [Gradient descent approaches](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html).\n", - "\n", - "\n", + "The various optmization methods, with codes and algorithms, are discussed in our lectures on [Gradient descent approaches](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html)." + ] + }, + { + "cell_type": "markdown", + "id": "d0b1f84a", + "metadata": { + "editable": true + }, + "source": [ "### Regularization\n", "\n", "It is common to add an extra term to the cost function, proportional\n", @@ -834,7 +1008,6 @@ "\n", "i.e. we sum up all the weights squared. The factor $\\lambda$ is known as a regularization parameter.\n", "\n", - "\n", "In order to train the model, we need to calculate the derivative of\n", "the cost function with respect to every bias and weight in the\n", "network. In total our network has $(64 + 1)\\times 50=3250$ weights in\n", @@ -842,9 +1015,16 @@ "layer ($+1$ for the bias), and the gradient must be calculated for\n", "every parameter. We use the *backpropagation* algorithm discussed\n", "above. This is a clever use of the chain rule that allows us to\n", - "calculate the gradient efficently. \n", - "\n", - "\n", + "calculate the gradient efficently." + ] + }, + { + "cell_type": "markdown", + "id": "96417f62", + "metadata": { + "editable": true + }, + "source": [ "### Matrix multiplication\n", "\n", "To more efficently train our network these equations are implemented using matrix operations. \n", @@ -881,6 +1061,7 @@ { "cell_type": "code", "execution_count": 5, + "id": "c692ab5f", "metadata": { "collapsed": false, "editable": true @@ -897,7 +1078,7 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n" ] }, @@ -982,7 +1163,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9d0ad486", + "metadata": { + "editable": true + }, "source": [ "## Improving performance\n", "\n", @@ -997,7 +1181,6 @@ "If this does not improve network performance, you may want to consider altering the network architecture, adding more neurons or hidden layers. \n", "Andrew Ng goes through some of these considerations in this [video](https://youtu.be/F1ka6a13S9I). You can find a summary of the video [here](https://kevinzakka.github.io/2016/09/26/applying-deep-learning/). \n", "\n", - "\n", "It is very natural to think of the network as an object, with specific instances of the network\n", "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." ] @@ -1005,6 +1188,7 @@ { "cell_type": "code", "execution_count": 6, + "id": "84088dcb", "metadata": { "collapsed": false, "editable": true @@ -1114,7 +1298,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "61dae78a", + "metadata": { + "editable": true + }, "source": [ "## Evaluate model performance on test data\n", "\n", @@ -1130,6 +1317,7 @@ { "cell_type": "code", "execution_count": 7, + "id": "999a01b6", "metadata": { "collapsed": false, "editable": true @@ -1164,7 +1352,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "12b8be76", + "metadata": { + "editable": true + }, "source": [ "## Adjust hyperparameters\n", "\n", @@ -1175,6 +1366,7 @@ { "cell_type": "code", "execution_count": 8, + "id": "f022ba3c", "metadata": { "collapsed": false, "editable": true @@ -1464,7 +1656,7 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n" ] }, @@ -1482,7 +1674,7 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n" ] }, @@ -1500,7 +1692,7 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n" ] }, @@ -1518,7 +1710,7 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n" ] }, @@ -1536,7 +1728,7 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n" ] }, @@ -1554,7 +1746,7 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n" ] }, @@ -1572,7 +1764,7 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n" ] }, @@ -1590,11 +1782,11 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", " exp_term = np.exp(self.z_o)\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", " self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n" ] }, @@ -1612,11 +1804,11 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", " exp_term = np.exp(self.z_o)\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", " self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n" ] }, @@ -1634,11 +1826,11 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", " exp_term = np.exp(self.z_o)\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", " self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n" ] }, @@ -1656,11 +1848,11 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", " exp_term = np.exp(self.z_o)\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", " self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n" ] }, @@ -1678,11 +1870,11 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", " exp_term = np.exp(self.z_o)\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", " self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n" ] }, @@ -1700,7 +1892,7 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n" ] }, @@ -1718,11 +1910,11 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", " exp_term = np.exp(self.z_o)\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", " self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n" ] }, @@ -1740,11 +1932,11 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", " exp_term = np.exp(self.z_o)\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", " self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n" ] }, @@ -1762,11 +1954,11 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", " exp_term = np.exp(self.z_o)\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", " self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n" ] }, @@ -1784,11 +1976,11 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", " exp_term = np.exp(self.z_o)\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", " self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n" ] }, @@ -1806,11 +1998,11 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", " exp_term = np.exp(self.z_o)\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", " self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n" ] }, @@ -1828,11 +2020,11 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", " exp_term = np.exp(self.z_o)\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", " self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n" ] }, @@ -1850,11 +2042,11 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", " exp_term = np.exp(self.z_o)\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", " self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n" ] }, @@ -1872,11 +2064,11 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n", " exp_term = np.exp(self.z_o)\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n", " self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n" ] }, @@ -1916,7 +2108,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "118ee392", + "metadata": { + "editable": true + }, "source": [ "## Visualization" ] @@ -1924,6 +2119,7 @@ { "cell_type": "code", "execution_count": 9, + "id": "01cd7236", "metadata": { "collapsed": false, "editable": true @@ -1933,15 +2129,15 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n", - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/953065564.py:4: RuntimeWarning: overflow encountered in exp\n", " return 1/(1 + np.exp(-x))\n" ] }, @@ -1954,7 +2150,7 @@ }, "metadata": { "filenames": { - "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter10_49_1.png" + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter10_59_1.png" } }, "output_type": "display_data" @@ -1968,7 +2164,7 @@ }, "metadata": { "filenames": { - "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter10_49_2.png" + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter10_59_2.png" } }, "output_type": "display_data" @@ -2012,7 +2208,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "1cef52ca", + "metadata": { + "editable": true + }, "source": [ "## scikit-learn implementation\n", "\n", @@ -2032,6 +2231,7 @@ { "cell_type": "code", "execution_count": 10, + "id": "08650740", "metadata": { "collapsed": false, "editable": true @@ -2606,6 +2806,10 @@ "Learning rate = 10.0\n", "Lambda = 0.001\n", "Accuracy score on test set: 0.10555555555555556\n", + "\n", + "Learning rate = 10.0\n", + "Lambda = 0.01\n", + "Accuracy score on test set: 0.06388888888888888\n", "\n" ] }, @@ -2613,10 +2817,6 @@ "name": "stdout", "output_type": "stream", "text": [ - "Learning rate = 10.0\n", - "Lambda = 0.01\n", - "Accuracy score on test set: 0.06388888888888888\n", - "\n", "Learning rate = 10.0\n", "Lambda = 0.1\n", "Accuracy score on test set: 0.08888888888888889\n", @@ -2659,7 +2859,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9a5819b0", + "metadata": { + "editable": true + }, "source": [ "## Visualization" ] @@ -2667,6 +2870,7 @@ { "cell_type": "code", "execution_count": 11, + "id": "c76390e1", "metadata": { "collapsed": false, "editable": true @@ -2681,7 +2885,7 @@ }, "metadata": { "filenames": { - "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter10_53_0.png" + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter10_63_0.png" } }, "output_type": "display_data" @@ -2695,7 +2899,7 @@ }, "metadata": { "filenames": { - "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter10_53_1.png" + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter10_63_1.png" } }, "output_type": "display_data" @@ -2740,7 +2944,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6b973f8c", + "metadata": { + "editable": true + }, "source": [ "## Building neural networks in Tensorflow and Keras\n", "\n", @@ -2752,7 +2959,6 @@ "clear how to build one using an arbitrary number of hidden layers, using data structures such as Python lists or\n", "NumPy arrays.\n", "\n", - "\n", "Tensorflow is an open source library machine learning library\n", "developed by the Google Brain team for internal use. It was released\n", "under the Apache 2.0 open source license in November 9, 2015.\n", @@ -2775,12 +2981,15 @@ "Then we will build (effectively) the same graph in Keras, to see just\n", "how simple solving a machine learning problem can be.\n", "\n", - "To install tensorflow on Unix/Linux systems, use pip as" + "To install tensorflow on Unix/Linux systems, use pip as **pip3 install tensorflow**\n", + "and/or if you use **anaconda**, just write (or install from the graphical user interface)\n", + "(current release of CPU-only TensorFlow)" ] }, { "cell_type": "code", "execution_count": 12, + "id": "cb71108d", "metadata": { "collapsed": false, "editable": true @@ -2788,33 +2997,13 @@ "outputs": [ { "ename": "SyntaxError", - "evalue": "invalid syntax (2357089093.py, line 1)", + "evalue": "invalid syntax (2259440937.py, line 1)", "output_type": "error", "traceback": [ - "\u001b[0;36m File \u001b[0;32m\"/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42331/2357089093.py\"\u001b[0;36m, line \u001b[0;32m1\u001b[0m\n\u001b[0;31m pip3 install tensorflow\u001b[0m\n\u001b[0m ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m invalid syntax\n" + "\u001b[0;36m File \u001b[0;32m\"/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47411/2259440937.py\"\u001b[0;36m, line \u001b[0;32m1\u001b[0m\n\u001b[0;31m conda create -n tf tensorflow\u001b[0m\n\u001b[0m ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m invalid syntax\n" ] } ], - "source": [ - "pip3 install tensorflow" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and/or if you use **anaconda**, just write (or install from the graphical user interface)\n", - "(current release of CPU-only TensorFlow)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], "source": [ "conda create -n tf tensorflow\n", "conda activate tf" @@ -2822,14 +3011,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "8cf89ee7", + "metadata": { + "editable": true + }, "source": [ "To install the current release of GPU TensorFlow" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 13, + "id": "ace3d31e", "metadata": { "collapsed": false, "editable": true @@ -2842,7 +3035,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f5d7454d", + "metadata": { + "editable": true + }, "source": [ "Keras is a high level [neural network](https://en.wikipedia.org/wiki/Application_programming_interface)\n", "that supports Tensorflow, CTNK and Theano as backends. \n", @@ -2851,7 +3047,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 14, + "id": "024068e3", "metadata": { "collapsed": false, "editable": true @@ -2863,19 +3060,22 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e36b74d0", + "metadata": { + "editable": true + }, "source": [ "You can look up the [instructions here](https://keras.io/) for more information.\n", "\n", "We will to a large extent use **keras** in this course. \n", "\n", - "\n", "Let us look again at the MINST data set." ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 15, + "id": "42dbd743", "metadata": { "collapsed": false, "editable": true @@ -2929,7 +3129,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 16, + "id": "eba6a34d", "metadata": { "collapsed": false, "editable": true @@ -2957,7 +3158,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 17, + "id": "da80e8b6", "metadata": { "collapsed": false, "editable": true @@ -2986,7 +3188,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 18, + "id": "87273539", "metadata": { "collapsed": false, "editable": true @@ -3012,7 +3215,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 19, + "id": "8cd653a0", "metadata": { "collapsed": false, "editable": true @@ -3054,14 +3258,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "31009a5d", + "metadata": { + "editable": true + }, "source": [ "## The Breast Cancer Data, now with Keras" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 20, + "id": "991bf288", "metadata": { "collapsed": false, "editable": true @@ -3238,7 +3446,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d197746e", + "metadata": { + "editable": true + }, "source": [ "## Fine-tuning neural network hyperparameters\n", "\n", @@ -3256,7 +3467,6 @@ "training a neural network on a large dataset takes a lot of time, you\n", "will only be able to explore a tiny part of the hyperparameter space.\n", "\n", - "\n", "* You can use randomized search.\n", "\n", "* Or use tools like [Oscar](http://oscar.calldesk.ai/), which implements more complex algorithms to help you find a good set of hyperparameters quickly. \n", @@ -3272,12 +3482,16 @@ "as large image classification or speech recognition, typically require networks with dozens of layers\n", "and they need a huge amount\n", "of training data. However, you will rarely have to train such networks from scratch: it is much more\n", - "common to reuse parts of a pretrained state-of-the-art network that performs a similar task.\n", - "\n", - "\n", - "\n", - "\n", - "\n", + "common to reuse parts of a pretrained state-of-the-art network that performs a similar task." + ] + }, + { + "cell_type": "markdown", + "id": "3614fe93", + "metadata": { + "editable": true + }, + "source": [ "## Which activation function should I use?\n", "\n", "The Back propagation algorithm we derived above works by going from\n", @@ -3286,7 +3500,6 @@ "function with regards to each parameter in the network, it uses these\n", "gradients to update each parameter with a Gradient Descent (GD) step.\n", "\n", - "\n", "Unfortunately for us, the gradients often get smaller and smaller as the\n", "algorithm progresses down to the first hidden layers. As a result, the\n", "GD update leaves the lower layer connection weights\n", @@ -3302,9 +3515,6 @@ "neural networks suffer from unstable gradients, different layers may\n", "learn at widely different speeds\n", "\n", - "\n", - "\n", - "\n", "Although this unfortunate behavior has been empirically observed for\n", "quite a while (it was one of the reasons why deep neural networks were\n", "mostly abandoned for a long time), it is only around 2010 that\n", @@ -3327,8 +3537,6 @@ "(the hyperbolic tangent function has a mean of 0 and behaves slightly\n", "better than the logistic function in deep networks).\n", "\n", - "\n", - "\n", "Looking at the logistic activation function, when inputs become large\n", "(negative or positive), the function saturates at 0 or 1, with a\n", "derivative extremely close to 0. Thus when backpropagation kicks in,\n", @@ -3347,8 +3555,6 @@ "its inputs, and we also need the gradients to have equal variance\n", "before and after flowing through a layer in the reverse direction.\n", "\n", - "\n", - "\n", "One of the insights in the 2010 paper by Glorot and Bengio was that\n", "the vanishing/exploding gradients problems were in part due to a poor\n", "choice of activation function. Until then most people had assumed that\n", @@ -3357,9 +3563,16 @@ "that other activation functions behave much better in deep neural\n", "networks, in particular the ReLU activation function, mostly because\n", "it does not saturate for positive values (and also because it is quite\n", - "fast to compute).\n", - "\n", - "\n", + "fast to compute)." + ] + }, + { + "cell_type": "markdown", + "id": "0816093d", + "metadata": { + "editable": true + }, + "source": [ "## The RELU function family\n", "\n", "The ReLU activation function suffers from a problem known as the dying\n", @@ -3380,7 +3593,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d979c7db", + "metadata": { + "editable": true + }, "source": [ "$$\n", "ELU(z) = \\left\\{\\begin{array}{cc} \\alpha\\left( \\exp{(z)}-1\\right) & z < 0,\\\\ z & z \\ge 0.\\end{array}\\right.\n", @@ -3389,7 +3605,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "eadc7692", + "metadata": { + "editable": true + }, "source": [ "In general it seems that the ELU activation function is better than\n", "the leaky ReLU function (and its variants), which is better than\n", @@ -3404,8 +3623,6 @@ "spare time and computing power, you can use cross-validation or\n", "bootstrap to evaluate other activation functions.\n", "\n", - "\n", - "\n", "In most cases you can use the ReLU activation function in the hidden layers (or one of its variants).\n", "\n", "It is a bit faster to compute than other activation functions, and the gradient descent optimization does in general not get stuck.\n", @@ -3414,8 +3631,16 @@ "\n", "* For classification the softmax activation function is generally a good choice for classification tasks (when the classes are mutually exclusive).\n", "\n", - "* For regression tasks, you can simply use no activation function at all.\n", - "\n", + "* For regression tasks, you can simply use no activation function at all." + ] + }, + { + "cell_type": "markdown", + "id": "a78e9091", + "metadata": { + "editable": true + }, + "source": [ "## Batch Normalization\n", "\n", "Batch Normalization\n", @@ -3428,8 +3653,16 @@ "learn the optimal scale and mean of the inputs for each layer.\n", "In order to zero-center and normalize the inputs, the algorithm needs to estimate the inputs’ mean and\n", "standard deviation. It does so by evaluating the mean and standard deviation of the inputs over the current\n", - "mini-batch, from this the name batch normalization.\n", - "\n", + "mini-batch, from this the name batch normalization." + ] + }, + { + "cell_type": "markdown", + "id": "04e309c6", + "metadata": { + "editable": true + }, + "source": [ "## Dropout\n", "\n", "It is a fairly simple algorithm: at every training step, every neuron (including the input neurons but\n", @@ -3438,8 +3671,16 @@ "\n", "The\n", "hyperparameter $p$ is called the dropout rate, and it is typically set to 50%. After training, the neurons are not dropped anymore.\n", - " It is viewed as one of the most popular regularization techniques.\n", - "\n", + " It is viewed as one of the most popular regularization techniques." + ] + }, + { + "cell_type": "markdown", + "id": "f9eb2036", + "metadata": { + "editable": true + }, + "source": [ "## Gradient Clipping\n", "\n", "A popular technique to lessen the exploding gradients problem is to simply clip the gradients during\n", @@ -3449,12 +3690,18 @@ "This technique is called Gradient Clipping.\n", "\n", "In general however, Batch\n", - "Normalization is preferred.\n", - "\n", - "\n", + "Normalization is preferred." + ] + }, + { + "cell_type": "markdown", + "id": "95301882", + "metadata": { + "editable": true + }, + "source": [ "## A top-down perspective on Neural networks\n", "\n", - "\n", "The first thing we would like to do is divide the data into two or three\n", "parts. A training set, a validation or dev (development) set, and a\n", "test set. The test set is the data on which we want to make\n", @@ -3465,7 +3712,6 @@ "do not use any of the test data to train the algorithm. This is a\n", "cardinal sin in ML. Then:\n", "\n", - "\n", "* Estimate optimal error rate\n", "\n", "* Minimize underfitting (bias) on training data set.\n", @@ -3489,9 +3735,16 @@ "the test data. The difference between the performance of the algorithm\n", "on these two validation sets quantifies the train-test mismatch. This\n", "can serve as another important diagnostic when using DNNs for\n", - "supervised learning.\n", - "\n", - "\n", + "supervised learning." + ] + }, + { + "cell_type": "markdown", + "id": "e1da8c6e", + "metadata": { + "editable": true + }, + "source": [ "## Limitations of supervised learning with deep networks\n", "\n", "Like all statistical methods, supervised learning using neural\n", @@ -3504,8 +3757,6 @@ "\n", "Here we list some of the important limitations of supervised neural network based models. \n", "\n", - "\n", - "\n", "* **Need labeled data**. All supervised learning methods, DNNs for supervised learning require labeled data. Often, labeled data is harder to acquire than unlabeled data (e.g. one must pay for human experts to label images).\n", "\n", "* **Supervised neural networks are extremely data intensive.** DNNs are data hungry. They perform best when data is plentiful. This is doubly so for supervised methods where the data must also be labeled. The utility of DNNs is extremely limited if data is hard to acquire or the datasets are small (hundreds to a few thousand samples). In this case, the performance of other methods that utilize hand-engineered features can exceed that of DNNs.\n", @@ -3533,5 +3784,5 @@ } }, "nbformat": 4, - "nbformat_minor": 4 + "nbformat_minor": 5 } \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter10.py b/doc/LectureNotes/_build/jupyter_execute/chapter10.py index fe19214e6..cc472478c 100644 --- a/doc/LectureNotes/_build/jupyter_execute/chapter10.py +++ b/doc/LectureNotes/_build/jupyter_execute/chapter10.py @@ -1,6 +1,9 @@ #!/usr/bin/env python # coding: utf-8 +# + # # Building a Feed Forward Neural Network # # We are now gong to develop an example based on the MNIST data @@ -33,9 +36,7 @@ # where $y \in \{0, 1\}$ and $\hat{\theta}$ represents the weights and biases # of our network. -# -# -# + # ## Defining the cost function # # Our cost function is given as (see the Logistic regression lectures) @@ -54,10 +55,8 @@ # # $y = 5 \quad \rightarrow \quad \hat{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$ and # -# # $y = 1 \quad \rightarrow \quad \hat{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 $\hat{x}_i$ is the $i$-th input (image), $y_{ic}$ refers to the $c$-th component of the $i$-th @@ -86,8 +85,7 @@ # 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! -# -# + # ### Example: binary classification problem # # 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 @@ -136,10 +134,8 @@ # \frac{\partial \mathcal{C}(\hat{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 use another activation function than the logistic one, we need to evaluate other derivatives. + # ### The Softmax function # # In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation $z_i^l$, that is we need @@ -161,12 +157,10 @@ # \frac{\partial f(z_i^l)}{\partial z_j^l}= f(z_i^l)\left(\delta_{ij}-f(z_j^l)\right), # $$ -# which in case of the simply binary model reduces to having $i=j$. -# -# +# which in case of the simply binary model reduces to having $i=j$. + # ## Developing a code for doing neural networks with back propagation # -# # One can identify a set of key steps when using neural networks to solve supervised learning problems: # # 1. Collect and pre-process data @@ -180,7 +174,7 @@ # 5. Evaluate model performance on test data # # 6. Adjust hyperparameters (if necessary, network architecture) -# + # ### Collect and pre-process data # # Here we will be using the MNIST dataset, which is readily available through the **scikit-learn** @@ -350,7 +344,7 @@ print("Number of test images: " + str(len(X_test))) # $$ f(x) = \sigma(x) = \frac{1}{1 + e^{-x}} ,$$ # # which is inspired by probability theory (see logistic regression) and was most commonly used until about 2011. See the discussion below concerning other activation functions. -# + # ### Layers # # * Input @@ -383,7 +377,6 @@ print("Number of test images: " + str(len(X_test))) # Since each neuron in the output layer is connected to the 50 inputs from the hidden layer we have 50x10 = 500 # weights to the output layer. # -# # Typically weights are initialized with small values distributed around zero, drawn from a uniform # or normal distribution. Setting all weights to zero means all neurons give the same output, making the network useless. # @@ -434,7 +427,6 @@ output_bias = np.zeros(n_categories) + 0.01 # $$ a_{j}^{L} = \frac{\exp{(z_j^{L})}} # {\sum_{c=0}^{C-1} \exp{(z_c^{L})}} .$$ # -# # Since our data has the dimensions $X = (n_{inputs}, n_{features})$ and our weights to the hidden # layer have the dimensions # $W_{hidden} = (n_{features}, n_{hidden})$, @@ -512,10 +504,8 @@ print("correct label for image 0: " + str(Y_train[0])) # # $$ y = 5 \quad \rightarrow \quad \hat{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$$ # -# # $$ y = 1 \quad \rightarrow \quad \hat{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,$$ # -# # 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. @@ -524,10 +514,8 @@ print("correct label for image 0: " + str(Y_train[0])) # In the one-hot representation only one of the terms in the loss function is non-zero, namely the # probability of the correct category $c'$ # (i.e. the category $c'$ such that $y_{ic'} = 1$). This means that the cross entropy loss only punishes you for how wrong -# you got the correct label. The probability of category $c$ is given by the softmax function. The vector $\hat{\theta}$ represents the parameters of our network, i.e. all the weights and biases. -# -# -# +# you got the correct label. The probability of category $c$ is given by the softmax function. The vector $\hat{\theta}$ represents the parameters of our network, i.e. all the weights and biases. + # ### Optimizing the cost function # # 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 @@ -557,8 +545,7 @@ print("correct label for image 0: " + str(Y_train[0])) # 2. It significantly speeds up the calculation, since we do not have to use the entire dataset to calculate the gradient. # # The various optmization methods, with codes and algorithms, are discussed in our lectures on [Gradient descent approaches](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html). -# -# + # ### Regularization # # It is common to add an extra term to the cost function, proportional @@ -576,7 +563,6 @@ print("correct label for image 0: " + str(Y_train[0])) # # 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 @@ -584,9 +570,8 @@ print("correct label for image 0: " + str(Y_train[0])) # 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. -# -# +# calculate the gradient efficently. + # ### Matrix multiplication # # To more efficently train our network these equations are implemented using matrix operations. @@ -705,7 +690,6 @@ print("New accuracy on training data: " + str(accuracy_score(predict(X_train), Y # 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](https://youtu.be/F1ka6a13S9I). You can find a summary of the video [here](https://kevinzakka.github.io/2016/09/26/applying-deep-learning/). # -# # 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. @@ -1000,7 +984,6 @@ plt.show() # clear how to build one using an arbitrary number of hidden layers, using data structures such as Python lists or # NumPy arrays. # -# # Tensorflow is an open source library machine learning library # developed by the Google Brain team for internal use. It was released # under the Apache 2.0 open source license in November 9, 2015. @@ -1023,18 +1006,11 @@ plt.show() # Then we will build (effectively) the same graph in Keras, to see just # how simple solving a machine learning problem can be. # -# To install tensorflow on Unix/Linux systems, use pip as - -# In[12]: - - -pip3 install tensorflow - - +# To install tensorflow on Unix/Linux systems, use pip as **pip3 install tensorflow** # and/or if you use **anaconda**, just write (or install from the graphical user interface) # (current release of CPU-only TensorFlow) -# In[ ]: +# In[12]: conda create -n tf tensorflow @@ -1043,7 +1019,7 @@ conda activate tf # To install the current release of GPU TensorFlow -# In[ ]: +# In[13]: conda create -n tf-gpu tensorflow-gpu @@ -1054,7 +1030,7 @@ conda activate tf-gpu # that supports Tensorflow, CTNK and Theano as backends. # If you have Anaconda installed you may run the following command -# In[ ]: +# In[14]: conda install keras @@ -1064,10 +1040,9 @@ conda install keras # # We will to a large extent use **keras** in this course. # -# # Let us look again at the MINST data set. -# In[ ]: +# In[15]: # import necessary packages @@ -1115,7 +1090,7 @@ for i, image in enumerate(digits.images[random_indices]): plt.show() -# In[ ]: +# In[16]: from tensorflow.keras.layers import Input @@ -1137,7 +1112,7 @@ X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=t test_size=test_size) -# In[ ]: +# In[17]: @@ -1160,7 +1135,7 @@ def create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories return model -# In[ ]: +# In[18]: DNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object) @@ -1180,7 +1155,7 @@ for i, eta in enumerate(eta_vals): print() -# In[ ]: +# In[19]: # optional @@ -1218,7 +1193,7 @@ plt.show() # ## The Breast Cancer Data, now with Keras -# In[ ]: +# In[20]: @@ -1405,7 +1380,6 @@ plot_data(eta,n_neuron,Test_accuracy, 'testing') # training a neural network on a large dataset takes a lot of time, you # will only be able to explore a tiny part of the hyperparameter space. # -# # * You can use randomized search. # # * Or use tools like [Oscar](http://oscar.calldesk.ai/), which implements more complex algorithms to help you find a good set of hyperparameters quickly. @@ -1422,11 +1396,7 @@ plot_data(eta,n_neuron,Test_accuracy, 'testing') # and they need a huge amount # of training data. However, you will rarely have to train such networks from scratch: it is much more # common to reuse parts of a pretrained state-of-the-art network that performs a similar task. -# -# -# -# -# + # ## Which activation function should I use? # # The Back propagation algorithm we derived above works by going from @@ -1435,7 +1405,6 @@ plot_data(eta,n_neuron,Test_accuracy, 'testing') # function with regards to each parameter in the network, it uses these # gradients to update each parameter with a Gradient Descent (GD) step. # -# # Unfortunately for us, the gradients often get smaller and smaller as the # algorithm progresses down to the first hidden layers. As a result, the # GD update leaves the lower layer connection weights @@ -1451,9 +1420,6 @@ plot_data(eta,n_neuron,Test_accuracy, 'testing') # neural networks suffer from unstable gradients, different layers may # learn at widely different speeds # -# -# -# # Although this unfortunate behavior has been empirically observed for # quite a while (it was one of the reasons why deep neural networks were # mostly abandoned for a long time), it is only around 2010 that @@ -1476,8 +1442,6 @@ plot_data(eta,n_neuron,Test_accuracy, 'testing') # (the hyperbolic tangent function has a mean of 0 and behaves slightly # better than the logistic function in deep networks). # -# -# # Looking at the logistic activation function, when inputs become large # (negative or positive), the function saturates at 0 or 1, with a # derivative extremely close to 0. Thus when backpropagation kicks in, @@ -1496,8 +1460,6 @@ plot_data(eta,n_neuron,Test_accuracy, 'testing') # its inputs, and we also need the gradients to have equal variance # before and after flowing through a layer in the reverse direction. # -# -# # One of the insights in the 2010 paper by Glorot and Bengio was that # the vanishing/exploding gradients problems were in part due to a poor # choice of activation function. Until then most people had assumed that @@ -1507,8 +1469,7 @@ plot_data(eta,n_neuron,Test_accuracy, 'testing') # networks, in particular the ReLU activation function, mostly because # it does not saturate for positive values (and also because it is quite # fast to compute). -# -# + # ## The RELU function family # # The ReLU activation function suffers from a problem known as the dying @@ -1543,8 +1504,6 @@ plot_data(eta,n_neuron,Test_accuracy, 'testing') # spare time and computing power, you can use cross-validation or # bootstrap to evaluate other activation functions. # -# -# # In most cases you can use the ReLU activation function in the hidden layers (or one of its variants). # # It is a bit faster to compute than other activation functions, and the gradient descent optimization does in general not get stuck. @@ -1554,7 +1513,7 @@ plot_data(eta,n_neuron,Test_accuracy, 'testing') # * For classification the softmax activation function is generally a good choice for classification tasks (when the classes are mutually exclusive). # # * For regression tasks, you can simply use no activation function at all. -# + # ## Batch Normalization # # Batch Normalization @@ -1568,7 +1527,7 @@ plot_data(eta,n_neuron,Test_accuracy, 'testing') # In order to zero-center and normalize the inputs, the algorithm needs to estimate the inputs’ mean and # standard deviation. It does so by evaluating the mean and standard deviation of the inputs over the current # mini-batch, from this the name batch normalization. -# + # ## Dropout # # It is a fairly simple algorithm: at every training step, every neuron (including the input neurons but @@ -1578,7 +1537,7 @@ plot_data(eta,n_neuron,Test_accuracy, 'testing') # The # hyperparameter $p$ is called the dropout rate, and it is typically set to 50%. After training, the neurons are not dropped anymore. # It is viewed as one of the most popular regularization techniques. -# + # ## Gradient Clipping # # A popular technique to lessen the exploding gradients problem is to simply clip the gradients during @@ -1589,11 +1548,9 @@ plot_data(eta,n_neuron,Test_accuracy, 'testing') # # In general however, Batch # Normalization is preferred. -# -# + # ## A top-down perspective on Neural networks # -# # The first thing we would like to do is divide the data into two or three # parts. A training set, a validation or dev (development) set, and a # test set. The test set is the data on which we want to make @@ -1604,7 +1561,6 @@ plot_data(eta,n_neuron,Test_accuracy, 'testing') # do not use any of the test data to train the algorithm. This is a # cardinal sin in ML. Then: # -# # * Estimate optimal error rate # # * Minimize underfitting (bias) on training data set. @@ -1629,8 +1585,7 @@ plot_data(eta,n_neuron,Test_accuracy, 'testing') # on these two validation sets quantifies the train-test mismatch. This # can serve as another important diagnostic when using DNNs for # supervised learning. -# -# + # ## Limitations of supervised learning with deep networks # # Like all statistical methods, supervised learning using neural @@ -1643,8 +1598,6 @@ plot_data(eta,n_neuron,Test_accuracy, 'testing') # # Here we list some of the important limitations of supervised neural network based models. # -# -# # * **Need labeled data**. All supervised learning methods, DNNs for supervised learning require labeled data. Often, labeled data is harder to acquire than unlabeled data (e.g. one must pay for human experts to label images). # # * **Supervised neural networks are extremely data intensive.** DNNs are data hungry. They perform best when data is plentiful. This is doubly so for supervised methods where the data must also be labeled. The utility of DNNs is extremely limited if data is hard to acquire or the datasets are small (hundreds to a few thousand samples). In this case, the performance of other methods that utilize hand-engineered features can exceed that of DNNs. diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter10_39_1.png b/doc/LectureNotes/_build/jupyter_execute/chapter10_39_1.png new file mode 100644 index 000000000..91a4496c5 Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapter10_39_1.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter10_59_1.png b/doc/LectureNotes/_build/jupyter_execute/chapter10_59_1.png new file mode 100644 index 000000000..6d1467029 Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapter10_59_1.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter10_59_2.png b/doc/LectureNotes/_build/jupyter_execute/chapter10_59_2.png new file mode 100644 index 000000000..b7c47b451 Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapter10_59_2.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter10_63_0.png b/doc/LectureNotes/_build/jupyter_execute/chapter10_63_0.png new file mode 100644 index 000000000..59f332230 Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapter10_63_0.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter10_63_1.png b/doc/LectureNotes/_build/jupyter_execute/chapter10_63_1.png new file mode 100644 index 000000000..10b90707a Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapter10_63_1.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter11.ipynb b/doc/LectureNotes/_build/jupyter_execute/chapter11.ipynb index 31bbda3dc..861b2acf6 100644 --- a/doc/LectureNotes/_build/jupyter_execute/chapter11.ipynb +++ b/doc/LectureNotes/_build/jupyter_execute/chapter11.ipynb @@ -2,7 +2,21 @@ "cells": [ { "cell_type": "markdown", - "metadata": {}, + "id": "7660a7d5", + "metadata": { + "editable": true + }, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "id": "7174820b", + "metadata": { + "editable": true + }, "source": [ "# Solving Differential Equations with Deep Learning\n", "\n", @@ -10,7 +24,6 @@ "approximate any function at a single hidden layer along with one input\n", "and output layer to any given precision. \n", "\n", - "\n", "An ordinary differential equation (ODE) is an equation involving functions having one variable.\n", "\n", "In general, an ordinary differential equation looks like" @@ -18,7 +31,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ac838a12", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -32,7 +48,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ccee4286", + "metadata": { + "editable": true + }, "source": [ "where $g(x)$ is the function to find, and $g^{(n)}(x)$ is the $n$-th derivative of $g(x)$.\n", "\n", @@ -42,14 +61,15 @@ "Along with ([1](#ode)), some additional conditions of the function $g(x)$ are typically given\n", "for the solution to be unique.\n", "\n", - "\n", - "\n", "Let the trial solution $g_t(x)$ be" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "39d7dabd", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -64,7 +84,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "74f33170", + "metadata": { + "editable": true + }, "source": [ "where $h_1(x)$ is a function that makes $g_t(x)$ satisfy a given set\n", "of conditions, $N(x,P)$ a neural network with weights and biases\n", @@ -77,11 +100,8 @@ "\n", "But what about the network $N(x,P)$?\n", "\n", - "\n", "As described previously, an optimization method could be used to minimize the parameters of a neural network, that being its weights and biases, through backward propagation.\n", "\n", - "\n", - "\n", "For the minimization to be defined, we need to have a cost function at hand to minimize.\n", "\n", "It is given that $f\\left(x, \\, g(x), \\, g'(x), \\, g''(x), \\, \\dots \\, , \\, g^{(n)}(x)\\right)$ should be equal to zero in ([1](#ode)).\n", @@ -92,7 +112,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "2565ab16", + "metadata": { + "editable": true + }, "source": [ "$$\n", "C\\left(x, P\\right) = \\big(f\\left(x, \\, g(x), \\, g'(x), \\, g''(x), \\, \\dots \\, , \\, g^{(n)}(x)\\right)\\big)^2\n", @@ -101,7 +124,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "83d4ee25", + "metadata": { + "editable": true + }, "source": [ "If $N$ inputs are given as a vector $\\boldsymbol{x}$ with elements $x_i$ for $i = 1,\\dots,N$,\n", "the cost function becomes" @@ -109,7 +135,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a7a3a709", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -123,20 +152,28 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d3ca8026", + "metadata": { + "editable": true + }, "source": [ "The neural net should then find the parameters $P$ that minimizes the cost function in\n", "([3](#cost)) for a set of $N$ training samples $x_i$.\n", "\n", - "\n", - "\n", "To perform the minimization using gradient descent, the gradient of $C\\left(\\boldsymbol{x}, P\\right)$ is needed.\n", "It might happen so that finding an analytical expression of the gradient of $C(\\boldsymbol{x}, P)$ from ([3](#cost)) gets too messy, depending on which cost function one desires to use.\n", "\n", "Luckily, there exists libraries that makes the job for us through automatic differentiation.\n", - "Automatic differentiation is a method of finding the derivatives numerically with very high precision.\n", - "\n", - "\n", + "Automatic differentiation is a method of finding the derivatives numerically with very high precision." + ] + }, + { + "cell_type": "markdown", + "id": "9bde4bd4", + "metadata": { + "editable": true + }, + "source": [ "### Example: Exponential decay\n", "\n", "An exponential decay of a quantity $g(x)$ is described by the equation" @@ -144,7 +181,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e74337c3", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -158,7 +198,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "273549a4", + "metadata": { + "editable": true + }, "source": [ "with $g(0) = g_0$ for some chosen initial value $g_0$.\n", "\n", @@ -167,7 +210,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "db3c6623", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -182,18 +228,22 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "8547e3c8", + "metadata": { + "editable": true + }, "source": [ "Having an analytical solution at hand, it is possible to use it to compare how well a neural network finds a solution of ([4](#solve_expdec)).\n", "\n", - "\n", - "\n", "The program will use a neural network to solve" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "48341fc6", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -207,19 +257,24 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d25b1e02", + "metadata": { + "editable": true + }, "source": [ "where $g(0) = g_0$ with $\\gamma$ and $g_0$ being some chosen values.\n", "\n", "In this example, $\\gamma = 2$ and $g_0 = 10$.\n", "\n", - "\n", "To begin with, a trial solution $g_t(t)$ must be chosen. A general trial solution for ordinary differential equations could be" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "7c5dd91e", + "metadata": { + "editable": true + }, "source": [ "$$\n", "g_t(x, P) = h_1(x) + h_2(x, N(x, P))\n", @@ -228,12 +283,13 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "24223062", + "metadata": { + "editable": true + }, "source": [ "with $h_1(x)$ ensuring that $g_t(x)$ satisfies some conditions and $h_2(x,N(x, P))$ an expression involving $x$ and the output from the neural network $N(x,P)$ with $P $ being the collection of the weights and biases for each layer. For now, it is assumed that the network consists of one input layer, one hidden layer, and one output layer.\n", "\n", - "\n", - "\n", "In this network, there are no weights and bias at the input layer, so $P = \\{ P_{\\text{hidden}}, P_{\\text{output}} \\}$.\n", "If there are $N_{\\text{hidden} }$ neurons in the hidden layer, then $P_{\\text{hidden}}$ is a $N_{\\text{hidden} } \\times (1 + N_{\\text{input}})$ matrix, given that there are $N_{\\text{input}}$ neurons in the input layer.\n", "\n", @@ -247,7 +303,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ebf04383", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -261,7 +320,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ea4a0013", + "metadata": { + "editable": true + }, "source": [ "### Reformulating the problem\n", "\n", @@ -277,7 +339,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "2351b84f", + "metadata": { + "editable": true + }, "source": [ "$$\n", "g_t(x, P) = g_0 + x \\cdot N(x, P)\n", @@ -286,14 +351,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f6cc00e4", + "metadata": { + "editable": true + }, "source": [ "has been chosen such that it already solves the condition $g(0) = g_0$. What remains, is to find $P$ such that" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "84e066a1", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -307,11 +378,13 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "feff9ab3", + "metadata": { + "editable": true + }, "source": [ "is fulfilled as *best as possible*.\n", "\n", - "\n", "The left hand side and right hand side of ([8](#nnmin)) must be computed separately, and then the neural network must choose weights and biases, contained in $P$, such that the sides are equal as best as possible.\n", "This means that the absolute or squared difference between the sides must be as close to zero, ideally equal to zero.\n", "In this case, the difference squared shows to be an appropriate measurement of how erroneous the trial solution is with respect to $P$ of the neural network.\n", @@ -321,7 +394,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0c6f0e79", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\min_{P}\\Big\\{ \\big(g_t'(x, P) - ( -\\gamma g_t(x, P) \\big)^2 \\Big\\}\n", @@ -330,7 +406,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9302a1dd", + "metadata": { + "editable": true + }, "source": [ "(the notation $\\min_{P}\\{ f(x, P) \\}$ means that we desire to find $P$ that yields the minimum of $f(x, P)$)\n", "\n", @@ -339,7 +418,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f7f204bb", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\min_{P_{\\text{hidden} }, \\ P_{\\text{output} }}\\Big\\{ \\big(g_t'(x, \\{ P_{\\text{hidden} }, P_{\\text{output} }\\}) - ( -\\gamma g_t(x, \\{ P_{\\text{hidden} }, P_{\\text{output} }\\}) \\big)^2 \\Big\\}\n", @@ -348,18 +430,22 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "8d61d75f", + "metadata": { + "editable": true + }, "source": [ "for an input value $x$.\n", "\n", - "\n", - "\n", "If the neural network evaluates $g_t(x, P)$ at more values for $x$, say $N$ values $x_i$ for $i = 1, \\dots, N$, then the *total* error to minimize becomes" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "10d3aec9", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -373,14 +459,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5e296388", + "metadata": { + "editable": true + }, "source": [ "Letting $\\boldsymbol{x}$ be a vector with elements $x_i$ and $C(\\boldsymbol{x}, P) = \\frac{1}{N} \\sum_i \\big(g_t'(x_i, P) - ( -\\gamma g_t(x_i, P) \\big)^2$ denote the cost function, the minimization problem that our network must solve, becomes" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "fe010d79", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\min_{P} C(\\boldsymbol{x}, P)\n", @@ -389,7 +481,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "add715f9", + "metadata": { + "editable": true + }, "source": [ "In terms of $P_{\\text{hidden} }$ and $P_{\\text{output} }$, this could also be expressed as\n", "\n", @@ -397,20 +492,21 @@ "\\min_{P_{\\text{hidden} }, \\ P_{\\text{output} }} C(\\boldsymbol{x}, \\{P_{\\text{hidden} }, P_{\\text{output} }\\})\n", "$$\n", "\n", - "\n", "For simplicity, it is assumed that the input is an array $\\boldsymbol{x} = (x_1, \\dots, x_N)$ with $N$ elements. It is at these points the neural network should find $P$ such that it fulfills ([9](#min)).\n", "\n", "First, the neural network must feed forward the inputs.\n", "This means that $\\boldsymbol{x}s$ must be passed through an input layer, a hidden layer and a output layer. The input layer in this case, does not need to process the data any further.\n", "The input layer will consist of $N_{\\text{input} }$ neurons, passing its element to each neuron in the hidden layer. The number of neurons in the hidden layer will be $N_{\\text{hidden} }$.\n", "\n", - "\n", "For the $i$-th in the hidden layer with weight $w_i^{\\text{hidden} }$ and bias $b_i^{\\text{hidden} }$, the weighting from the $j$-th neuron at the input layer is:" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "cb9b22eb", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{aligned}\n", @@ -429,14 +525,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "49f3c493", + "metadata": { + "editable": true + }, "source": [ "The result after weighting the inputs at the $i$-th hidden neuron can be written as a vector:" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "a352bd61", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{aligned}\n", @@ -456,7 +558,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e5636ad7", + "metadata": { + "editable": true + }, "source": [ "The vector $\\boldsymbol{p}_{i, \\text{hidden}}^T$ constitutes each row in $P_{\\text{hidden} }$, which contains the weights for the neural network to minimize according to ([9](#min)).\n", "\n", @@ -467,7 +572,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "68bb8b9c", + "metadata": { + "editable": true + }, "source": [ "$$\n", "f(z) = \\frac{1}{1 + \\exp{(-z)}}\n", @@ -476,7 +584,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ff6e6536", + "metadata": { + "editable": true + }, "source": [ "It is possible to use other activations functions for the hidden layer also.\n", "\n", @@ -494,14 +605,15 @@ "and biases $b_i^{\\text{output}}$. In this case,\n", "it is assumes that the number of neurons in the output layer is one.\n", "\n", - "\n", - "\n", "The procedure of weighting the output neuron $j$ in the hidden layer to the $i$-th neuron in the output layer is similar as for the hidden layer described previously." ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "6fb4c55b", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{aligned}\n", @@ -519,14 +631,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "09c31d8d", + "metadata": { + "editable": true + }, "source": [ "Expressing $z_{1,j}^{\\text{output}}$ as a vector gives the following way of weighting the inputs from the hidden layer:" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "f81fe361", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{z}_{1}^{\\text{output}} =\n", @@ -542,11 +660,13 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "c309a4ce", + "metadata": { + "editable": true + }, "source": [ "In this case we seek a continuous range of values since we are approximating a function. This means that after computing $\\boldsymbol{z}_{1}^{\\text{output}}$ the neural network has finished its feed forward step, and $\\boldsymbol{z}_{1}^{\\text{output}}$ is the final output of the network.\n", "\n", - "\n", "The next step is to decide how the parameters should be changed such that they minimize the cost function.\n", "\n", "The chosen cost function for this problem is" @@ -554,7 +674,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ea47ae29", + "metadata": { + "editable": true + }, "source": [ "$$\n", "C(\\boldsymbol{x}, P) = \\frac{1}{N} \\sum_i \\big(g_t'(x_i, P) - ( -\\gamma g_t(x_i, P) \\big)^2\n", @@ -563,12 +686,23 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9dd83767", + "metadata": { + "editable": true + }, "source": [ "In order to minimize the cost function, an optimization method must be chosen.\n", "\n", - "Here, gradient descent with a constant step size has been chosen.\n", - "\n", + "Here, gradient descent with a constant step size has been chosen." + ] + }, + { + "cell_type": "markdown", + "id": "531a7b4f", + "metadata": { + "editable": true + }, + "source": [ "### Gradient descent\n", "\n", "The idea of the gradient descent algorithm is to update parameters in\n", @@ -581,7 +715,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e10c204a", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{\\omega}_{\\text{new} } = \\boldsymbol{\\omega} - \\lambda \\nabla_{\\boldsymbol{\\omega}} C(\\boldsymbol{x}, \\boldsymbol{\\omega})\n", @@ -590,7 +727,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9f4fa48e", + "metadata": { + "editable": true + }, "source": [ "for a number of iterations or until $ \\big|\\big| \\boldsymbol{\\omega}_{\\text{new} } - \\boldsymbol{\\omega} \\big|\\big|$ becomes smaller than some given tolerance.\n", "\n", @@ -609,7 +749,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "4652fc79", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{aligned}\n", @@ -621,7 +764,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "41ae0626", + "metadata": { + "editable": true + }, "source": [ "### The code for solving the ODE" ] @@ -629,6 +775,7 @@ { "cell_type": "code", "execution_count": 1, + "id": "af7c165f", "metadata": { "collapsed": false, "editable": true @@ -658,7 +805,7 @@ }, "metadata": { "filenames": { - "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter11_47_2.png" + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter11_50_2.png" }, "needs_background": "light" }, @@ -816,7 +963,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "875a0c0b", + "metadata": { + "editable": true + }, "source": [ "## The network with one input layer, specified number of hidden layers, and one output layer\n", "\n", @@ -828,6 +978,7 @@ { "cell_type": "code", "execution_count": 2, + "id": "00684827", "metadata": { "collapsed": false, "editable": true @@ -849,27 +1000,26 @@ ] }, { - "ename": "KeyboardInterrupt", - "evalue": "", - "output_type": "error", - "traceback": [ - "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", - "\u001b[0;31mKeyboardInterrupt\u001b[0m Traceback (most recent call last)", - "\u001b[0;32m/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42376/2971492148.py\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 144\u001b[0m \u001b[0mlmb\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m0.001\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 145\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 146\u001b[0;31m \u001b[0mP\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msolve_ode_deep_neural_network\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnum_hidden_neurons\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnum_iter\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlmb\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 147\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 148\u001b[0m \u001b[0mres\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mg_trial_deep\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mP\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", - "\u001b[0;32m/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42376/2971492148.py\u001b[0m in \u001b[0;36msolve_ode_deep_neural_network\u001b[0;34m(x, num_neurons, num_iter, lmb)\u001b[0m\n\u001b[1;32m 119\u001b[0m \u001b[0;31m# The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 120\u001b[0m \u001b[0;31m# in the hidden layers and output layers evaluated at x.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 121\u001b[0;31m \u001b[0mcost_deep_grad\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcost_function_deep_grad\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mP\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 122\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 123\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ml\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mN_hidden\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", - "\u001b[0;32m~/anaconda3/lib/python3.8/site-packages/autograd/wrap_util.py\u001b[0m in \u001b[0;36mnary_f\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m 18\u001b[0m \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 19\u001b[0m \u001b[0mx\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mtuple\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[0;32min\u001b[0m \u001b[0margnum\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 20\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0munary_operator\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0munary_f\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0mnary_op_args\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mnary_op_kwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 21\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mnary_f\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 22\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mnary_operator\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", - "\u001b[0;32m~/anaconda3/lib/python3.8/site-packages/autograd/differential_operators.py\u001b[0m in \u001b[0;36mgrad\u001b[0;34m(fun, x)\u001b[0m\n\u001b[1;32m 23\u001b[0m \u001b[0marguments\u001b[0m \u001b[0;32mas\u001b[0m\u001b[0;31m \u001b[0m\u001b[0;31m`\u001b[0m\u001b[0mfun\u001b[0m\u001b[0;31m`\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mbut\u001b[0m \u001b[0mreturns\u001b[0m \u001b[0mthe\u001b[0m \u001b[0mgradient\u001b[0m \u001b[0minstead\u001b[0m\u001b[0;34m.\u001b[0m \u001b[0mThe\u001b[0m \u001b[0mfunction\u001b[0m\u001b[0;31m \u001b[0m\u001b[0;31m`\u001b[0m\u001b[0mfun\u001b[0m\u001b[0;31m`\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 24\u001b[0m should be scalar-valued. The gradient has the same type as the argument.\"\"\"\n\u001b[0;32m---> 25\u001b[0;31m \u001b[0mvjp\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mans\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_make_vjp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfun\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 26\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mvspace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mans\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msize\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 27\u001b[0m raise TypeError(\"Grad only applies to real scalar-output functions. \"\n", - "\u001b[0;32m~/anaconda3/lib/python3.8/site-packages/autograd/core.py\u001b[0m in \u001b[0;36mmake_vjp\u001b[0;34m(fun, x)\u001b[0m\n\u001b[1;32m 8\u001b[0m \u001b[0;32mdef\u001b[0m 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trial function with the current parameters P\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 69\u001b[0;31m \u001b[0mg_t\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mg_trial_deep\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mP\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 70\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 71\u001b[0m \u001b[0;31m# Find the derivative w.r.t x of the neural network\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", - "\u001b[0;32m/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42376/2971492148.py\u001b[0m in \u001b[0;36mg_trial_deep\u001b[0;34m(x, params, g0)\u001b[0m\n\u001b[1;32m 57\u001b[0m \u001b[0;31m# The trial solution using the deep neural network:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 58\u001b[0m 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\u001b[0mtuple\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mbox\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_node\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0m_\u001b[0m \u001b[0;34m,\u001b[0m \u001b[0mbox\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mboxed_args\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 43\u001b[0m \u001b[0margnums\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mtuple\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margnum\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0margnum\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0m_\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mboxed_args\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 44\u001b[0m \u001b[0mans\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mf_wrapped\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margvals\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", - "\u001b[0;31mKeyboardInterrupt\u001b[0m: " + "name": "stdout", + "output_type": "stream", + "text": [ + "Final cost: 0.119936\n" ] + }, + { + "data": { + "image/png": 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    " + ] + }, + "metadata": { + "filenames": { + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter11_52_3.png" + }, + "needs_background": "light" + }, + "output_type": "display_data" } ], "source": [ @@ -1035,7 +1185,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5420c490", + "metadata": { + "editable": true + }, "source": [ "### Example: Population growth\n", "\n", @@ -1045,7 +1198,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "1436ed3c", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -1059,7 +1215,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a590bdb6", + "metadata": { + "editable": true + }, "source": [ "where $g(t)$ is the population density at time $t$, $\\alpha > 0$ the growth rate and $A > 0$ is the maximum population number in the environment.\n", "Also, at $t = 0$ the population has the size $g(0) = g_0$, where $g_0$ is some chosen constant.\n", @@ -1069,15 +1228,16 @@ "using a library like TensorFlow is recommended.\n", "Here, we stay with a more simple approach and implement for comparison, the simple forward Euler method.\n", "\n", - "\n", - "\n", "Here, we will model a population $g(t)$ in an environment having carrying capacity $A$.\n", "The population follows the model" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "48d788d6", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -1091,13 +1251,15 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "169c6c25", + "metadata": { + "editable": true + }, "source": [ "where $g(0) = g_0$.\n", "\n", "In this example, we let $\\alpha = 2$, $A = 1$, and $g_0 = 1.2$.\n", "\n", - "\n", "We will get a slightly different trial solution, as the boundary conditions are different\n", "compared to the case for exponential decay.\n", "\n", @@ -1115,19 +1277,57 @@ "g(t) = \\frac{Ag_0}{g_0 + (A - g_0)\\exp(-\\alpha A t)}\n", "$$\n", "\n", - "\n", - "\n", "The network will be the similar as for the exponential decay example, but with some small modifications for our problem." ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 3, + "id": "6d2e33bf", "metadata": { "collapsed": false, "editable": true }, - "outputs": [], + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Initial cost: 0.221805\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/Users/MortenImac/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray\n", + " return array(a, dtype, copy=False, order=order)\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Final cost: 0.000417932\n", + "The max absolute difference between the solutions is: 0.00424909\n" + ] + }, + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
    " + ] + }, + "metadata": { + "filenames": { + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter11_58_3.png" + }, + "needs_background": "light" + }, + "output_type": "display_data" + } + ], "source": [ "import autograd.numpy as np\n", "from autograd import grad, elementwise_grad\n", @@ -1296,7 +1496,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e31ec549", + "metadata": { + "editable": true + }, "source": [ "## Using forward Euler to solve the ODE\n", "\n", @@ -1313,7 +1516,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "2e9ee105", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{aligned}\n", @@ -1325,7 +1531,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e876da16", + "metadata": { + "editable": true + }, "source": [ "along with the condition that $g(0) = g_0$.\n", "\n", @@ -1336,7 +1545,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b79a3def", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{aligned}\n", @@ -1349,14 +1561,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9d99cb4e", + "metadata": { + "editable": true + }, "source": [ "Now, if $g_i = g(t_i)$ then" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "3b9dcc24", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -1375,7 +1593,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "65ce688e", + "metadata": { + "editable": true + }, "source": [ "for $i \\geq 1$ and $g_0 = g(t_0) = g(0) = g_0$.\n", "\n", @@ -1385,12 +1606,69 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 4, + "id": "d5497948", "metadata": { "collapsed": false, "editable": true }, - "outputs": [], + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Initial cost: 0.221805\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/Users/MortenImac/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray\n", + " return array(a, dtype, copy=False, order=order)\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Final cost: 0.000417932\n", + "The max absolute difference between the solutions is: 0.00424909\n", + "Max absolute difference between Euler method and analytical: 0.011225\n", + "Max absolute difference between deep neural network and analytical: 0.00424909\n" + ] + }, + { + "data": { + "image/png": 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\n", 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+ "text/plain": [ + "
    " + ] + }, + "metadata": { + "filenames": { + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter11_66_4.png" + }, + "needs_background": "light" + }, + "output_type": "display_data" + } + ], "source": [ "# Assume that all function definitions from the example program using Autograd\n", "# are located here.\n", @@ -1461,7 +1739,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "fa8f0bb4", + "metadata": { + "editable": true + }, "source": [ "## Solving the one dimensional Poisson equation\n", "\n", @@ -1470,7 +1751,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f765b0ba", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -1484,7 +1768,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "bd63b92e", + "metadata": { + "editable": true + }, "source": [ "where $f(x)$ is a given function for $x \\in (0,1)$.\n", "\n", @@ -1493,7 +1780,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "c0a7face", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{align*}\n", @@ -1505,19 +1795,24 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f71c3cb9", + "metadata": { + "editable": true + }, "source": [ "This equation can be solved numerically using programs where e.g Autograd and TensorFlow are used.\n", "The results from the networks can then be compared to the analytical solution.\n", "In addition, it could be interesting to see how a typical method for numerically solving second order ODEs compares to the neural networks.\n", "\n", - "\n", "Here, the function $g(x)$ to solve for follows the equation" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "35aa6a37", + "metadata": { + "editable": true + }, "source": [ "$$\n", "-g''(x) = f(x),\\qquad x \\in (0,1)\n", @@ -1526,14 +1821,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f8b35111", + "metadata": { + "editable": true + }, "source": [ "where $f(x)$ is a given function, along with the chosen conditions" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "33813514", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -1547,7 +1848,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "c70aadd4", + "metadata": { + "editable": true + }, "source": [ "In this example, we consider the case when $f(x) = (3x + x^2)\\exp(x)$.\n", "\n", @@ -1556,7 +1860,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d5719dee", + "metadata": { + "editable": true + }, "source": [ "$$\n", "g_t(x) = x \\cdot (1-x) \\cdot N(P,x)\n", @@ -1565,14 +1872,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a683041b", + "metadata": { + "editable": true + }, "source": [ "The analytical solution for this problem is" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "672cdaff", + "metadata": { + "editable": true + }, "source": [ "$$\n", "g(x) = x(1 - x)\\exp(x)\n", @@ -1581,12 +1894,52 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 5, + "id": "66e6bebb", "metadata": { "collapsed": false, "editable": true }, - "outputs": [], + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/Users/MortenImac/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray\n", + " return array(a, dtype, copy=False, order=order)\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Initial cost: 457.256\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Final cost: 0.00310113\n", + "The max absolute difference between the solutions is: 0.000464088\n" + ] + }, + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
    " + ] + }, + "metadata": { + "filenames": { + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter11_79_3.png" + }, + "needs_background": "light" + }, + "output_type": "display_data" + } + ], "source": [ "import autograd.numpy as np\n", "from autograd import grad, elementwise_grad\n", @@ -1742,7 +2095,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d425cba5", + "metadata": { + "editable": true + }, "source": [ "### Comparing with a numerical scheme\n", "\n", @@ -1761,7 +2117,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a2efa110", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -1775,14 +2134,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5011ee25", + "metadata": { + "editable": true + }, "source": [ "If $x_i = i \\Delta x = x_{i-1} + \\Delta x$ and $g_i = g(x_i)$ for $i = 1,\\dots N_x - 2$ with $N_x$ being the number of values for $x$, ([15](#approx)) becomes" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "705ee300", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{aligned}\n", @@ -1794,14 +2159,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b390796d", + "metadata": { + "editable": true + }, "source": [ "Since we know from our problem that" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "19c9ece4", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{aligned}\n", @@ -1813,7 +2184,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6ade1a7a", + "metadata": { + "editable": true + }, "source": [ "along with the conditions $g(0) = g(1) = 0$,\n", "the following scheme can be used to find an approximate solution for $g(x)$ numerically:" @@ -1821,7 +2195,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "78b16d02", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -1838,7 +2215,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f2bfcca4", + "metadata": { + "editable": true + }, "source": [ "for $i = 1, \\dots, N_x - 2$ where $g_0 = g_{N_x - 1} = 0$ and $f(x_i) = (3x_i + x_i^2)\\exp(x_i)$, which is given for our specific problem.\n", "\n", @@ -1847,7 +2227,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a6191528", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{aligned}\n", @@ -1881,22 +2264,80 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "aefef707", + "metadata": { + "editable": true + }, "source": [ "which makes it possible to solve for the vector $\\boldsymbol{g}$.\n", "\n", - "\n", "We can then compare the result from this numerical scheme with the output from our network using Autograd:" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 6, + "id": "471664cd", "metadata": { "collapsed": false, "editable": true }, - "outputs": [], + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Initial cost: 457.256\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/Users/MortenImac/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray\n", + " return array(a, dtype, copy=False, order=order)\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Final cost: 0.00310113\n", + "The max absolute difference between the analytical solution and DNN Autograd: 0.000464088\n", + "The max absolute difference between the analytical solution and numerical scheme: 0.00266858\n" + ] + }, + { + "data": { + "image/png": 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MI2+bj/S4mrse+C2VsGPkXRmm4/kIrADPXx4/pnnP7e/xfNNkBZ5vOn7lva418h3Es6F7Gc+b6CV4/qptqju8mRbh+VjhbjzHdjX5cVtrq/B8Y2Yanr1rD+M5Jmhdyx7VN+67Fk+hG4bnywl78bw5tvfO8gyeb25uA/6L5yO9I93fGjzHy8zHsxEYjOcYptbwAJ7n/r/GmIN4DqYf6/29ZXi/Rev9eGKc9zZz8WwsPz3MZTiG9cda+wGebxG+ZYwZ2cgso4EvjTEl3uw/sNZutdYW4nmj/hGej8HuBM601jb60Zy19lAR74rnTaHZrLXvAn8HPsFzsO9876TKRuY91vW+4f29hOc4uNvxrGNr8BwTOtH7XBwy3vtcFeMpzO2A0dbalQ3ussh88zxhPzzMr74cz7Fe6/Acs3abN89HeI6/eRXP3qVewEUtfXx49rDux7OH4Tk8B6ofen1+H/itd539FZ7n9LCsta/j2U68aDwfc63C89rHu36cD/wZz3rThyO/vg4VwUJjzFfeny/Gc3xQPp6PtX/tXY8b8y8g2/uaeuNIub35FuP5YtODeJ6PTXiOz2rM+3jW5Q14PgqroIkfrbZwO/Nv73wNP4r8E54/EIqMMXc0crtG1yE8x/bF4lmfFwDvNSW711RgtXddfwDPca8V1tr1eD4u/Yf3fs/Cc2qnqkbu42jb5qM9ruasB37LeP7IExEJLMaYAXje4KOt43PMBTLvnulnrbXN2Xsvbcy7B+tZPF++qDva/BIYtCdMRAKGMeYc70dmyXj2trytAibBzvvR3g+Ax1XAgotKmIgEkuvxfPy9Gc8xKDe6jSPiW949vkV4von7N6dhpNXp40gRERERB7QnTERERMQBvztx2dF07NjRZmZmuo4hIiIiclRLlizZa61NbWxawJWwzMxMFi9e7DqGiIiIyFEZYxqOLvI1fRwpIiIi4oBKmIiIiIgDKmEiIiIiDgTcMWEiIiLSOqqrq8nLy6OiosJ1lIAXExNDRkYGkZGRTb6NSpiIiEiIysvLIzExkczMTIwxruMELGsthYWF5OXlkZWV1eTb6eNIERGREFVRUUGHDh1UwI6RMYYOHTo0e4+iSpiIiEgIUwFrHS15HlXCRERERBxQCRMREZGA9tRTT3HzzTcfdZ78/PyvL1977bWsWbOm2b9rzpw5nHnmmc2+XWNUwkRERCToNSxhjz/+ONnZ2Q4TqYSJiIiIQzNmzGDkyJEMHDiQmTNnApCQkMD//d//MXToUMaNG8fu3bsBePvttxk7dizDhw/n5JNP/vr6Qw4ePEhWVhbV1dUAFBcXk5mZyX/+8x8WL17MpZdeyrBhwygvL2fy5MlfD4P43nvvMWLECIYOHcpJJ50EwMKFC5kwYQLDhw9nwoQJrF+/vtUfu05RISIiIvzm7dWsyS9u1fvM7tqOX5818IjzPPHEE6SkpFBeXs7o0aM577zzKC0tZdy4cfzhD3/gzjvv5LHHHuMXv/gFkyZNYsGCBRhjePzxx7nnnnu49957v76vxMREJk+ezKxZs5gxYwYvvvgi5513Hueffz4PPfQQf/3rXxk1atQ3fn9BQQHf+973+PTTT8nKymLfvn0A9O/fn08//ZSIiAg+/PBDfv7zn/Pqq6+26vOjEiYiIiLO/P3vf+f1118HIDc3l40bNxIVFfX1cVcjR47kgw8+ADznNbvwwgvZuXMnVVVVjZ6T69prr+Wee+5hxowZPPnkkzz22GNH/P0LFizg+OOP//q+UlJSADhw4ABXXnklGzduxBjz9d611qQSJiIiIkfdY+ULc+bM4cMPP2T+/PnExcUxefJkKioqiIyM/PqUD+Hh4dTU1ABwyy238MMf/pCzzz6bOXPmcNddd33rPidOnMi2bduYO3cutbW1DBo06IgZrLWNnl7il7/8JSeeeCKvv/4627ZtY/Lkycf8eBvSMWEiIiLixIEDB0hOTiYuLo5169axYMGCo86fnp4OwL///e/DznfFFVdw8cUXc/XVV399XWJiIgcPHvzWvOPHj2fu3Lls3boV4OuPI+v/rqeeeqpZj6upVMJERETEialTp1JTU8OQIUP45S9/ybhx4444/1133cX555/PcccdR8eOHQ8736WXXsr+/fu5+OKLv77uqquu4oYbbvj6wPxDUlNTmTlzJueeey5Dhw7lwgsvBODOO+/kZz/7GRMnTqS2tvYYH2njjLXWJ3fsK6NGjbKHvs0gIiIiLbd27VoGDBjgOkare+WVV3jzzTd55pln2vT3NvZ8GmOWWGtHNTa/jgkTERGRoHHLLbfw7rvvMnv2bNdRjkolTERERILGP/7xD9cRmkzHhImIiIg4oBImIiIi4oBKmIiIiIgDOiZMREQ86uqgphxbVUZVRSnlZSVUlB2koqyEqvISqspLqa4oo7ayhNrKUuqqyqirKofqMkx1OaamnPDacsJrKwirq6Zi1A2Mn3qJ60cl4rdUwkREAkFtDVSXQXW59//6P5dTXVFKVXkJleWl1FSWUF1RSm1lGXXesnRoPlNTTlhNBeG15UTUVhBRV0FUXSVRtoIYqgAwQLT331FjWUO5iaGSaKrCoqk2MVSHx5BQu4/282/lq3YdGTHhVF8+MyIBSyVMRKStHdhB7cLHqSnbT01FGbVVZdiqMmxVqbdYlRNWb69SZF0F4bbmiHcZ6f0XX++6ShtBBVGUE025jaKcGCqIoioshpqwJGrCY6iNjKEuIhbr/Wei4iAqjvDoeMKj4oiIiScqJp7ImHii4xKIjk0gNi6BuPh2xCUkEhUVTYIxJDTIU7JvJ/senEyP969hY8os+vQf0spPokjgUwkTEWlDtmA9pY+fRUxFASXEU040FTaKcm9Z8vzcjnI6Um6jqTDR1IbHUBce6ylLkbGYyDhMVCxhUfGERcURHpNAVEwckbEJRMXGExObQExsPPGxMcRHR5AQHUH76Ai6RocTHRHeJo8zIaULZVe8SsRTpxH14oXsuvEj0jp3bZPfLYFj27ZtTJs2jUmTJjFv3jzS09N58803mTZtGmPHjuWTTz6hqKiIf/3rXxx33HGu47Y6lTARkTZSl7uEin+fQ3k13JfxTzIGjCU+Opz46Iivy1LHKM//h66PjghrdHDhQNApcxDbzn6Krm9exPqZ3yH+9v+SmNBwn5n4jXd/CrtWtu59pg2GaX8+4iwbN27khRde4LHHHuOCCy7g1VdfBaCmpoaFCxcye/ZsfvOb3/Dhhx+2bjY/oBImItIGajZ9Qs3zF1NYk8Drgx/iF+edSlhYYJar5sgccQpr9/6FwfNuZ97DlzH6h68QGaG3HvmfrKwshg0bBsDIkSPZtm0bAOeee+63rgs2eiWIiPhY5Yo3CHvtWrbVpTF//ExumTo+YPdutcSAU7/Lin3bmLDuAT569DamfP8fIfX4A8ZR9lj5SnT0/74CEh4e/vXg2oeuDw8Pp6bmyMdEBiqdJ0xExIfK5j9BxGtXs6Iuk5WnPM/V0yaEZAEZcuFvWNlpOicVPMNHz//VdRwRv6ASJiLiIwc//Atx79/OF3WD2XvOS5x/XAh/Q9AYBl33OOviRzN5wx/5/L2XXCcScc5Ya11naJZRo0bZxYsXu44hInJ41lL01k9JWvoIs+xEUi55nPH99M1AgKrSInbdP5mU6l1sOvNVho2e6DpSSFu7di0DBgxwHSNoNPZ8GmOWWGtHNTa/9oSJiLSm2hr2v3AdSUsf4SUzlW7XPqsCVk9UfBJJ33uTirBY0mZdzuYtG11HEnFGJUxEpLVUV7DvyYtI3vAy/wq/kNE3Ps6QbimuU/mddp17UHvxSyRSSu0z51Owd6/rSCJOqISJiLSGimIKZ55FSt4HPBRzHWfc+nd6dkp0ncpvde47hoKpj9KzbjvbHr2Q0vIK15FCVqAdluSvWvI8qoSJiByrkgL2PXwq7fYs4W/tfsylt/6BtPYxrlP5vcxxM9g0+jeMrl7Mlw9dQ01NretIIScmJobCwkIVsWNkraWwsJCYmOa97nWeMBGRY2D3b6do5lnElu3kH51/ww3X3khclDatTdX/zFtZVbiVKVuf4N1//YKp1/0xJE/h4UpGRgZ5eXkUFBS4jhLwYmJiyMjIaNZttKUQEWmhut1rKXn8LMKqSnm0x73ccuVlRIbrA4bmGnT5vax9MIdpOx/mv6/04NTzb3AdKWRERkaSlZXlOkbI0tZCRKQFanIWUj7zVCqqqnl+wCPcetXlKmAtFRZGvxueZUvMIE5Y9Qs+/3iW60QibUJbDBGRZqpc/yE1T57F3uoY3hvzFDdceHZIjAPpS2FRsXS98XX2RaSSPfd6li//ynUkEZ9TCRMRaYbSr14h7IUL2VrbicUnvcAVZ5yoY5haSUz7TsRd/RrhBpJev4StuTmuI4n4lEqYiEgTFX8+k9i3rmVFXS9yp7/CeSc0ehJsOQbtMwZQft6zpLGX4icvYG/RAdeRRHxGJUxE5GisZd97f6Ldhz/mMzuMqkte5dSR/VynClppgyeTP/l+htatZc0/L6e8stp1JBGfUAkTETmSujr2vnoHKQv+zCxzPMnX/Ifx/bu5ThX0siZfzobBd3B85Vw++ect1NbpPFYSfFTCREQOp7aaPc9eQ8dVj/Ny+BkMuPE5hnRPdZ0qZPQ99xesS/8Opxe9wKyn/uQ6jkirUwkTEWlMdTl7Hj+fTlte46noSzj+ln/Rs1M716lCizH0/+6jbGo3jtO3/4X33nzOdSKRVqUSJiLSUMUB9jx8Oh3z5/BowveZ8YMHSEuKdZ0qNIVHkHXjf9gZncXEr37EF59/4jqRSKtRCRMRqcce3E3Bg6eQvG85j3T8GZff+juS4qJcxwpp4bHt6Hj9m1SGx9Hrg++ycu1a15FEWoVKmIiIV92+bex/cArxB7fxeLc/8r3v/1jjQPqJ2A7diLj8FRJNOVEvXcT2/N2uI4kcM5UwERGgeucqDj58EmEV+3m+39+5/rvXaxgiP5OUNYLisx6nFzns+tdF7DtY5jqSyDHRFkZEQl7llvlUPjaViuoaZo38F9dcfKGGIfJTXUaeSd6E3zO29isWP/xdKqpqXEcSaTGVMBEJaaWr38c+PZ2CmjjmT36eS8+epmGI/FzmqTexqe91nFr+Lu8++jPqdA4xCVAqYSISsooWvkDUfy5hq01j81mvMuPEia4jSRP1vuhuNnU6jXMKZ/Lmcw+6jiPSIiphIhKS9n7yMO1m38gy24cDF77JyaMHu44kzREWRq/v/Ztt8UM4fdNveG/2664TiTSbSpiIhBZr2f32b+k492d8xkhirn6DcdlZrlNJC5jIWDJueJ39kZ0Y8+XNfPHll64jiTSLSpiIhI66Ona+dBudl9zLu2GTybjhFQZnprlOJccgIrEj7a99g/CwMDJmX8HqTVtcRxJpMpUwEQkNtdXkP3UFXdY9xStR0xl+6wv0Skt2nUpaQWxaX2ovep40s4+aZy8id88+15FEmsSnJcwYM9UYs94Ys8kY89MjzDfaGFNrjPmOL/OISIiqKiP/0XPpmvM2z8Zfyck/eIy0pDjXqaQVpfQ7jn2n/oPBbGDzzMs5UFrpOpLIUfmshBljwoGHgGlANnCxMSb7MPPdDbzvqywiErps+X7yH5pG2u7PeDL5B5z7g/tIio92HUt8oMuEi8gd+TMm13zOJw/fTGVNretIIkfkyz1hY4BN1tot1toq4EVgeiPz3QK8CuzxYRYRCUF1xbvY8/eT6Vi0kqfSf82lN92lYYiCXI8z72RL5sXMKH2ZNx/7PdbqHGLiv3xZwtKB3HqX87zXfc0Ykw6cAzxypDsyxlxnjFlsjFlcUFDQ6kFFJPhU793Cvn+cSEJZLi/2uZerrr2NqAgdBhv0jKHn5Q+yLWUS5+76G6+99JTrRCKH5cstUmOnnG74J8nfgJ9Ya4+4z9haO9NaO8paOyo1NbW18olIkKrIW0HpP08ivKqYWcMe5fJLr9IwRKEkPIIe17/I7thenLb2p7z34X9dJxJplC9LWB7Qrd7lDCC/wTyjgBeNMduA7wAPG2Nm+DCTiAS5kg2fU/OvaZTXwOfHPcMF55yjYYhCkIlOpNMNb1IZkciwz65j/rIVriOJfIsvS9gioI8xJssYEwVcBLxVfwZrbZa1NtNamwm8AnzfWvuGDzOJSBDbv+wdIp4/hz11iayd9h/OOnmK60jiUGRSOrFXvUaiqSTl9UtZuy3PdSSRb/BZCbPW1gA34/nW41rgZWvtamPMDcaYG3z1e0UkNO354hkS37iCzTadgu+8wZRxo1xHEj8Q120Ilec+RS+zgwP/voT8wmLXkUS+ZgLtmyOjRo2yixcvdh1DRPxI/vt/o+v8X7OYbGKueJlBPbsd/UYSUvI/mUnXuT/m3ahTmXj7c7SLjXIdSUKEMWaJtbbRvwr1VSERCVzWkvvaL+k6/9fMDRtLyvVvq4BJo7qeeB05g25iWtV/ee+RO6murXMdSUQlTEQCVF0dOc/eRLcVf+e9yJPoe8ur9OzS0XUq8WPdz/sD27uewQUHnuTlJ+/XOcTEOZUwEQk8NVVsf/wSum9+jjfizmPsD16gS3Ki61Ti74yhx3efJK/dcL6T+0deee0/rhNJiFMJE5GAYitL2P7wdHrkv8tLSddy6m0zSU7QMETSRBHRpN/wGkXRXTl5xe28P/cz14kkhKmEiUjAqCvdR97fTyOjcD4vdL6Dc27+i4YhkmYzcSkkf+9NwsLDGfDxd1m4ar3rSBKiVMJEJCBUF+1g99+n0KlkHa/0/D0XXv8LDUMkLRaV2pPwS1+msyki+pVL2bhDwxdL29MWTET8XsWujRQ9OIXEip3MHvx3LrjiJg1DJMcsodc4Ss78J4PZRN6/Lmf3gTLXkSTEqISJiF87uG0p5TNPIby6hM8mPME537lUwxBJq+kw6jvsHvdLTqxbwOf/vInSyhrXkSSEqISJiN/at2YO5qkzqKg1rDj1ZaaddobrSBKEupz2Q/L6XM55Fa/x6qN3UaNziEkbUQkTEb+0a9EbxL98Pntse/LOeZPJEye6jiTByhgyLn6AvE4ncGnhgzz37GM6h5i0CZUwEfE7OZ8+Q8dZV7OJbpRd+g6jhw1xHUmCXVg4Gde+wJ6Efnxnyy955Z1ZrhNJCFAJExG/cnDXJlI//hGrTD9ivzebQX17uY4koSIqns7XvUFlZBLHL76JD+ZrnGLxLZUwEfEf1rLr2eupsWFEXvAEPdPTXCeSEBPWvgsJ17xGYlgV3d+7isXrtrqOJEFMJUxE/MbG9/5Jn5LFfJF1CwMHZLuOIyEqqssg6i54hl4mn+oXL2fTzv2uI0mQUgkTEb9QtjeHLl/+jqXhg5h8yZ2u40iISxhwMgdOvo/xrGTd49dQUFzhOpIEIZUwEXHPWnKevoFwW0PE9AeJiYp0nUiEDpOuYtfw2ziz9iPef+QOyqp0DjFpXSphIuLcpo+fon/xF3yacT2Dhwx3HUfka2ln30V+j+lcVvYMzz32V2rrdOoKaT0qYSLiVPn+3XT87FesDuvLcZf/0nUckW8yhq6XP87O5NFcsecvPP3CczqHmLQalTARcWrLMzcRZ0upOuMB4mKiXccR+baIKLpc9x+KYzM4Z8OdvPL+x64TSZBQCRMRZzZ9+hID933Ap12uZvjICa7jiBxebDIdrnuTsIgoxs6/no8WrXKdSIKASpiIOFFxcB9Jn/yEjSaTcVf83nUckaMKS8kk+or/0NkU0+GdK/lqc77rSBLgVMJExIkNT99KUt0Bik/7Gwlxsa7jiDRJdI/RVM6YyRCzmYLnr6dOB+rLMVAJE5E2t2XB2wwpeJtPUy9m5LgTXccRaZZ2w2awuc81nFLzGUvXrHEdRwKYSpiItKmqsmLi3v8h2+jKyCvvdh1HpEW6nXQDYcay67NnXEeRAKYSJiJtavUzd9CproC9U+6lfWKi6zgiLRKT1odtsQPpu+sdKnQSV2khlTARaTNbv/qQofkv83nKOYw6/nTXcUSOSe3gC+ljclm4YK7rKBKgVMJEpE1UV5QS+c6t7DSpDL7iXtdxRI5Z5vGXUU0E5Yufdx1FApRKmIi0iRXP/YyMuh3kH/dnkpNTXMcROWbhCR3YkjyR4Qc+pLC41HUcCUAqYSLic9tXfcHQnGf4ot3pjD7pPNdxRFpNwpjL6GSKWPLJG66jSABSCRMRn6qpqqDu9ZvYZ5Lof8UDruOItKr00dM5aBKIWvOy6ygSgFTCRMSnlr5wF1m1W9k67nd06NjJdRyR1hURTV7XqYypmM+WHbtcp5EAoxImIj6Tu24JQ7fMZGH8iYw57VLXcUR8ovNxVxFnKln/8XOuo0iAUQkTEZ+oramh4pUbKTFxZF3+IMYY15FEfCKl3yR2R3Sl49Y3NIyRNItKmIj4xJKX/kCfmvVsHPErUtMyXMcR8R1j2N9rBiNrV7J89WrXaSSAqISJSKvbsXkVQzb8g69ixzPmzGtdxxHxuR4nXk2Ysez8QsMYSdOphIlIq6qrreXASzdSRSTpl/0TE6bNjAS/2LS+bI0dSJ9dszSMkTSZto4i0qoWvXof2VUrWDvkJ3ROz3IdR6TN1A66iD7ksmjBp66jSIBQCRORVrMzZyMDV9/LyujhjDnnVtdxRNpU1uTLqCKC8sX6lqQ0jUqYiLQKW1fHnudvJIw6Olz0iD6GlJATHp/C1uSJDNMwRtJE2kqKSKtY+ObDDK1YxKoBt9E1q7/rOCJOxI+5lE6miK/mvOE6igQAlTAROWZ78nPov/yPrIvMZtT5P3EdR8SZjNEzPMMYrdYwRnJ0KmEickysteQ8+31ibBUJFzxCWHi460gi7kREk9t1KmMq5rFVwxjJUaiEicgxWTj7SUaVfcby3jeQ0Weo6zgizqUddxWxpor1n+gAfTkylTARabGCPTvpteguNkf0ZuTFv3YdR8QvfD2M0ZY3sFbDGMnhqYSJSItYa9n09K20tyVEnvsw4RGRriOJ+AdjKOx1DiNqV7J8zRrXacSPqYSJSIss+uAlxpf8l+WZV9M9e6zrOCJ+JfPQMEafaRgjOTyVMBFptn2Fe+k+7+fkhHdn2CW/dx1HxO/EpfVha+wg+ux6R8MYyWGphIlIs61++nZS7T7s2Q8SER3rOo6IX6oZdAG9yWXJl3NdRxE/pRImIs2y8JM3Oe7AWyzPuIQeQ09wHUfEb/WcfDlVRFCmYYzkMFTCRKTJig4U0WXuneSHdWHw5fe4jiPi18LjU9iSNJFhRR+y/2CZ6zjih1TCRKTJlv37Trqxi8ppfyMyJsF1HBG/Fz/6UlLNAb765HXXUcQPqYSJSJMs+eK/HFf4Mss7n0PW6Kmu44gEhG5jZ1BsEonQMEbSCJUwETmq4pISkj/8IfvCOtD/ivtdxxEJHBHR5HbxDGO0LX+36zTiZ1TCROSoFj39c3raXIpP+SvR8cmu44gElLTjriTWVLFOwxhJAyphInJESxd+xvG7n2Vlh2n0mnCO6zgiAadD/0nsiuhKx82vaxgj+QaVMBE5rNLyCmLf/QEHwxLoc8U/XMcRCUzGsK+nZxijFRrGSOpRCRORw/rimbvobzez/4Q/EtM+1XUckYCVOcU7jNHnT7uOIn5EJUxEGrVs6SJO2PE4a5Im02vyZa7jiAS0uLQ+bIkdRO+ds6is1jBG4qESJiLfUl5ZTdjbt1Bposm84mHXcUSCwqFhjBYv+NR1FPETKmEi8i1zn/0jQ+rWsnvCXcSlpLuOIxIUep5wGVVEUL74WddRxE+ohInIN6xctYLjch5iQ+JY+pxyres4IkEjIqEDm5MmMbToQ4pKNIyRqISJSD0VVTVUvX4zxhjSL58JxriOJBJU4rzDGC3RMEaCSpiI1PPxC/cxsnY5O0b/jPhOma7jiASd7mOnU2wSidIwRoJKmIh4rV63nklb7mdL3FD6TLvVdRyRoGQiosnpMpVR5fPI2alhjEKdSpiIUFldQ/ErtxBlaki97DEI06ZBxFe+HsboYx2gH+q0pRURPvzPI4yv+ZLcYbeT2LWf6zgiQa1j/0nsCu9KBw1jFPJUwkRC3LrNWxi//s9sjxlAn7N/4jqOSPAzhsJeMxheu4qVa9e6TiMOqYSJhLDq2jp2vXgbiaac5EtmQli460giISFzyjWeYYw++7frKOKQSphICHv/tSeYXD2Xbdk30q77ENdxREJGfFpvNscOotfOWVRV17qOI46ohImEqE05eYxe9Xt2RPWkz7m/ch1HJOR8PYzRl3NdRxFHVMJEQlBNbR1bn7udjuYA8Rc8AhFRriOJhJxeJ1zuGcZokb4lGapUwkRC0Htvvcgplf9lS5/vktR7rOs4IiEpIiGFzUmTGFL0IQdKyl3HEQdUwkRCzJYduxm67Ffsisyg9/m/cx1HJKQdGsboqzmvuo4iDqiEiYSQ2jrL2md/TLrZS/S5D2Oi4lxHEglph4Yxilj1H9dRxAGVMJEQ8u7s15lW9hZbMi8mecAJruOIhDzPMEanMap8Hrk797iOI21MJUwkRGzfXUj2ov9jX0Qnel18j+s4IuLV+biriTVVrNUwRiFHJUwkBNTVWZY+81N6mnzM9Acw0YmuI4mIV2r/iewMT6fjFg1jFGpUwkRCwLsfvMeZB19hS8Y5dBgyzXUcEanPO4zRsJqVrF67xnUaaUMqYSJBLm/vAXrN+wkHw5PIuvR+13FEpBGZU672DmP0tOso0oZUwkSCmLWWBU//kv5mOzXT7sXEJruOJCKNSEjrw+aYQfTc+Y6GMQohKmEiQez9T+Zw1oHn2JZ2Gqmjz3UdR0SOoHrQBfQij68WahijUKESJhKkdu0vpcunP6YyLI7ulz7kOo6IHEWvyZ5hjEo1jFHIUAkTCULWWuY8/VuGspHKk/9IWGKq60gichSRCSlsSprEkP0fcqBUwxiFApUwkSD0wefzmL7vCbZ3PIHUCZe5jiMiTXRoGKOln7zmOoq0AZUwkSCzp7iM5I9+TF1YBBmX/ROMcR1JRJqox9jpHDCJRKx62XUUaQMqYSJBxFrLf5++m9GspvSE3xCelO46kog0g2cYo6mMKv+CvJ27XccRH1MJEwkiH335FdMLHiUveQydTvie6zgi0gKdJ11JjKlmnYYxCnoqYSJBovBgBTHv/YhIU0fapTP1MaRIgOo0YBL54el00DBGQU8lTCRIzHruASaxlAMTfkZExyzXcUSkpYyhsNd0htasYs26ta7TiA+phIkEgU+WrOKsnX8nv90QOp98q+s4InKMMqd8lzBjyf/0366jiA+phIkEuKKyKmrfuYN4U0nqpY9BWLjrSCJyjBLT+rApdjA9d75DdY2GMQpWKmEiAe715x/hZDuf/aNvJ7Jzf9dxRKSVVA883zOM0ZcaxihYqYSJBLBPl6/nzNy/sju+H52n3uk6joi0ot5fD2P0nOso4iMqYSIBqriimoNv3kmyKSH54pkQHuk6koi0ov8NY/SBhjEKUiphIgHqlRef5Iy6ORQOvZGojGGu44iID8SOupSO5gDL57zqOor4gEqYSACat2YLp239M3tjMul81q9cxxERH8kc5xnGKHylhjEKRiphIgGmvKqWXa/+jC5mH4kXPQoR0a4jiYiPmIhotqdNZWT5PHbs0jBGwUYlTCTAzPrvu5xb+x67B1xFdOY413FExMe+HsboEx2gH2xUwkQCSHFFNWmL/0JJWCJdpv/GdRwRaQOdsz3DGKVsek3DGAUZlTCRAPLu268wiWUcHH0rxLR3HUdE2oIx7O05g6E1q1i3XsMYBROVMJEAsfdgBb1X3U9ReEe6nHyL6zgi0oYODWO0Q8MYBRWVMJEA8cGbzzDSrKdq0h0QGes6joi0oXZderMpZjA98zWMUTBRCRMJAPn7Sxm28R/sjUqn0/HXuo4jIg5UDTyfnuSxbOEc11GklaiEiQSAua8/ygCzHXPiz3VmfJEQdWgYo5KF+pZksFAJE/FzW3cXMX77o+yO6UWHsZe4jiMijkQlprAxaRKD939IcZmGMQoGKmEifm7Ba/8g0+wiZupdEKaXrEgoix15CR3NAZZ98prrKNIKfLpFN8ZMNcasN8ZsMsb8tJHp040xK4wxy4wxi40xk3yZRyTQrM3ZzeRdT7AjYTDth57lOo6IOJY1fgYHSCR81Uuuo0gr8FkJM8aEAw8B04Bs4GJjTHaD2T4ChlprhwHfBR73VR6RQLTi9fvoYvbR/szfgzGu44iIYyYimm1dpjKybB75u/e4jiPHyJd7wsYAm6y1W6y1VcCLwPT6M1hrS+z/Tv8bD+hUwCJeSzdu55R9z5KTPI6E/pNdxxERP5F23FWeYYw+esZ1FDlGvixh6UBuvct53uu+wRhzjjFmHTALz96wbzHGXOf9uHJxQUGBT8KK+BNrLZvevJsUU0LqjD+4jiMifqTzgImeYYw2v65hjAKcL0tYY5+dfGttsda+bq3tD8wAftfYHVlrZ1prR1lrR6WmprZuShE/tGDVBqYdfJVtnU4mtsco13FExJ94hzEaVruS9RrGKKD5soTlAd3qXc4A8g83s7X2U6CXMaajDzOJ+D1rLbtn/YlYU0XXcxv9u0REQlzmiZ4PjjSMUWDzZQlbBPQxxmQZY6KAi4C36s9gjOltjOdoY2PMCCAKKPRhJhG/N2fRcqaVv0NOxtlEpTX8LouICLTr6hnGKCv/HWo0jFHA8lkJs9bWADcD7wNrgZettauNMTcYY27wznYesMoYswzPNykvtPqAW0JYbZ2l7IM/YIyl2zm/cR1HRPxYpXcYo6WL5rqOIi0U4cs7t9bOBmY3uO6Rej/fDdztywwigeTDz+dxWtWH5Pa+lKwOma7jiIgf6zP5ciqX/I7Shc/C+Cmu40gL6PTbIn6iqqaO8Ll/pNpE0WP6L13HERE/F5WYwsb2kxi070NKNIxRQFIJE/ET//34A06u/ZzdA68mrF1n13FEJADEjPIMY7T0k1ddR5EWUAkT8QPlVbUkzb+bEpNAjzN/4jqOiASIXuOnc4BEIla97DqKtIBKmIgfeP/dN5hkl7B/+E2Y2GTXcUQkQJiIaLZ2mcrwsnns2rPbdRxpJpUwEceKy6votvQvFIWl0G3qba7jiEiA6TzpSmJMNWs/etZ1FGkmlTARxz58+3lGspbScT+EqDjXcUQkwHTJnkR+eDrJmzSMUaBRCRNxqPBgOf1X38/eiC6kT7nedRwRCUTGsKfnOQyrXcmGDRrGKJCohIk4NOf1x8k226g94ScQEeU6jogEqJ5TrgY0jFGgUQkTcWTn/oMM3/wwu6Kz6DzxCtdxRCSAtevSm40xg8ncoWGMAolKmIgjX7z6ID1NPuEn/wrCwl3HEZEAV5l9AT3JY/nCOa6jSBOphIk4sH1XIRNyHyM3LpvUUee4jiMiQaDPlMuoJJKSRc+5jiJNpBIm4sCS1+6jqykk4YzfgjGu44hIEIhOSGFD+4kM2veBhjEKECphIm1s/fZ8jt/9NNvajSZ54Cmu44hIEIkddQkdTDHL5rzmOoo0gUqYSBtb+/rddDTFdJj+e9dRRCTI9Bo/gyISCV/5kuso0gQqYSJtaPmGLUzZ/xKbO0wmsdc413FEJMh4hjGapmGMAoRKmEgbynnrjySYCrqc8wfXUUQkSH09jNHHGsbI36mEibSRRStWc8rBN9jc5QziMga5jiMiQapr9kR2hKfTYaOOC/N3KmEibcBay97ZvyfcWLqf+zvXcUQkmBlDQc9zGFK7io3r17hOI0egEibSBj5fuIiTy99nW4/vEJ3a03UcEQlyWSd6hjHK+/Qpt0HkiFTCRHysts5S9eEfqDXhZJ3za9dxRCQEtO/amw3eYYxqa+tcx5HDUAkT8bG5n83hxKq55Pa9koikrq7jiEiIqMw+nyx2aBgjP6YSJuJD1bV1RH36R8rC4ug1/f9cxxGRENJ3yuWeYYwW6luS/kolTMSHPv7gHSbVLmT3oOsIi092HUdEQkh0Qgrr209i4L4PKNUwRn5JJUzERyqqauj45Z8pCkui55l3uI4jIiEoxjuM0fK5Ol2FP1IJE/GRj2a/xEi7mqKRP8BEJ7iOIyIhqM/4GRTRjrAVGsbIH6mEifjAwfIqMpfdS0F4ZzJPu8l1HBEJUSYiiq1ppzG8bB57Cva4jiMNqISJ+MCcN59gIJupmHgnRES7jiMiIazTcVcRbapZ+9EzrqNIAyphIq1s38Fystf+nZ2RPeg2+WrXcUQkxKV7hzFK1jBGfkclTKSVff7ag/QyO7BT/g/Cwl3HEZFQZwx7smZ4hjHaoGGM/IlKmEgr2llYxIgtj5AT05+u4y5wHUdEBICsKd8FYMen/3acROpTCRNpRYtfvZ8Ms5eYqXeBMa7jiIgAkNS1N+tjhtAj720NY+RHVMJEWsn2nQWM3/EEWxOG02noVNdxRES+4dAwRisWzXEdRbxUwkRaycpX/0xHU0z7M3+vvWAi4nf6nniZZxijL/UtSX+hEibSCjZsz+H4gufZmHQcKf0nuY4jIvItMYmeYYyy931IWbmGMfIHKmEirWDT638kgXI6n/N711FERA4rZqR3GKM5Ol2FP1AJEzlGK9et58T9r7Kh01Ta9RjmOo6IyGH1Hj+D/bQjbKWGMfIHKmEix2jn278j0tTS7bzfuY4iInJEYZFRbEs7jWGl8yjQMEbOqYSJHIPFy5YyuWQ2G9PPJT6tj+s4IiJHlTrJO4zRxzpA3zWVMJEWstZS/O7vsCacrPPuch1HRKRJMgZOJC88gyQNY+ScSphIC81f8AWTKz5mS89LiUnJcB1HRKRpjGFP5nSG1Kxis4YxckolTKQF6uosdR/9njITS+9zfuE6johIs2RNuRqAvE+fchskxKmEibTAZ3PfY1LNfPIGXENkYkfXcUREmiU5vQ/ro4fQI+8dDWPkkEqYSDNV19YR/9kfKTLt6Xv2na7jiIi0SEX2d8hkB6s0jJEzKmEizfTpe68wqm4Fe4bdTFhsO9dxRERapN+Uy6m0kRQvfNZ1lJClEibSDBVVNaQtvoeCsFT6nH6L6zgiIi329TBGhR9QXl7hOk5IUgkTaYZP3/k3A+1GDoz9ESYy1nUcEZFjEjXKO4zR3FddRwlJKmEiTXSwrIKeK+4nPyKD3id/z3UcEZFj1tc7jJFZoWGMXFAJE2mieW88Qm9yqTr+5xAe4TqOiMgxC4uMYkvaVO8wRrtdxwk5KmEiTbC/uITs9Q+RE9WbzEkXu44jItJqOk28kmhTzbqPNIxRW1MJE2mCha/eTzezh7BTfg1hetmISPDoNsg7jNGm111HCTl6NxE5it2F+xi+7XE2xw0lY9RZruOIiLQuY9idOZ3BNavYslHDGLUllTCRo1j2yt10MkXET/stGOM6johIqzs0jFHu3H87ThJaVMJEjiA3fydj859hXbsJpA2e7DqOiIhPpHiHMcrc8TZ1GsaozaiEiRzBulf/QJIpJfXs37mOIiLiU+XZ59PD7mDV4jmuo4QMlTCRw9i0ZQsT9r7Mmg6n0KH3KNdxRER8qv+Uy6mwkRR/qWGM2opKmMhhbH/jt0RTTfo52gsmIsEvJjHZO4zRfzWMURtRCRNpxOo1q5h04G3WdZlO+4wBruOIiLSJqJGXkGIOskLDGLUJlTCRRux9x/NNyKzzfuM6iohIm+k34dAwRi+6jhISVMJEGli6ZAGTSv/Lhu4XEp/aw3UcEZE2ExYZxea0qQwtnc/egj2u4wQ9lTCReqy1lP/3t1SYaPqc92vXcURE2lyniVd4hjH6WMMY+ZpKmEg9C7/4iAmVX7Cl99XEtO/kOo6ISJvrPmgSuWEZJG18zXWUoKcSJuJVV2eJmPN7DpBI/3N/6jqOiIgbxrA7awaDalaxfct612mCmkqYiNe8j95gZM1ScgfdSGRckus4IiLOdB49A4Cdyz9yGyTIqYSJANU1tSTN/xMFpgPZZ93uOo6IiFPpvYdRZqOp2/GV6yhBTSVMBJj37vMMqltPwYjbCIuOcx1HRMSpsIhIcqJ7k1y0ynWUoKYSJiGvoqqarl/9hfzwrgyYdoPrOCIifqE4ZTCZ1ZuprKp0HSVoqYRJyJv/5kz62O0cHH8nJiLKdRwREb8Q1W0ksaaK7Wv1kaSvqIRJSCspK6f36r+zPbIn/aZc6TqOiIjf6Nx/PACFGxc4ThK8VMIkpC187QG6sYvayb+AML0cREQOScvK5iBxkL/UdZSgpXcdCVn7iw4wcOMjbIoeRM8J57qOIyLiV0xYODkx/ehwYLXrKEFLJUxC1lev/YXOZj9Rp90FxriOIyLid0o7DCazZivlZWWuowQllTAJSXsK9jBy+5OsjR9D9xGnuI4jIuKXYnqMJsrUsnXNl66jBCWVMAlJq1/5I0mmhKQzf+s6ioiI3+qaPQGAoo0qYb6gEiYhJy8vh9G7XmBl+xPpMmC86zgiIn6rY3ov9tGO8F06ON8XVMIk5Gx67bfEUknajN+7jiIi4t+MYUfsAFIPrnWdJCiphElI2bJ5PeML32Bl6pmkZg1yHUdExO+Vpw6hR20OxcVFrqMEHZUwCSn5b9yFwZJ5no4FExFpiris0YQby/ZVOmlra1MJk5CxdtVXjCt+j9Xp3yGpS0/XcUREAkKG9+D84s06OL+1qYRJyDgw6y6qTCR9zvu16ygiIgEjqXM39pgORO1e5jpK0FEJk5CwfNGnjCufy7rMK0jo0NV1HBGRgJIfn03nUh2c39pUwiToWWup/eA3HCCB7PP+z3UcEZGAU9VpKN3tTvbt3eM6SlBRCZOg99VnsxhRtZjN/b5HTGKy6zgiIgEnsedoAHJWf+E4SXBpcgkzxiQbYwYaY3oaY1TeJCDU1dYR/ekf2GuSGXzOHa7jiIgEpG6DJgFQumWx4yTBJeJIE40x7YGbgIuBKKAAiAE6G2MWAA9baz/xeUqRFlr04UuMrVnDsiG/omNMgus4IiIBKSGpIztMGjEFy1xHCSpHLGHAK8DTwHHW2qL6E4wxI4HLjTE9rbX/8lE+kRarqakh5cu7yQ9LY/BZt7iOIyIS0HYnZtO1eAXWWowxruMEhSOWMGvtKUeYtgRY0uqJRFrJl+88wcS6rawY8xe6Rka5jiMiEtBq0obTpfhjdu/KpXOX7q7jBIUmHdtljLmmweVwY4xOtiR+q6Kykm7L72NbeCaDp15z9BuIiMgRJfUeA0De6nmOkwSPph5gf5IxZrYxposxZhCwAEj0YS6RY7LojQfpbndSPulnmLBw13FERAJe94HjqbWGim06OL+1HO2YMACstZcYYy4EVgJlwMXWWn1PVfxSSWkJfdY+xIaoAQyYfKHrOCIiQSEmvj3bwrsRt3eF6yhBo6kfR/YBfgC8CmzDc0B+nA9zibTY0tfuJ41CzEm/Ah08KiLSagrbZ9O9Yh22rs51lKDQ1I8j3wZ+aa29HjgB2Ags8lkqkRaqqCinz+anWBc9mD5jT3cdR0QkqNR1GUEHDrAjZ5PrKEGhqSVsjLX2IwDrcS8ww2epRFpoyazHSGMvduIPXUcREQk6KX3GAbBzjQ7Obw1HLGHGmEkA1trihtOstRuNMe28B+qLOFdTU0P6qkfZGt6T/pNmuI4jIhJ0umePptqGU5WrM1S1hqMdmH+eMeYe4D085wQ7dMb83sCJQA/gRz5NKNJEX33wPGNsHstH3YcJ08haIiKtLTI6js2RWSQWrnQdJSgc7WSttxtjkoHvAOcDaUA5sBZ4RN+QFH9h6+pot/gf5Js0Bp9yhes4IiJBa3/7gfTZ+yG1tXWEh+sP3mNx1GfPWrsfaAesAD4APgf2Av2NMcN8mk6kiZZ/Pov+tRvIH3QdYRGRruOIiASv9BG0N6Xkbl7lOknAa2qFHQncAHQBugLXAZOBx4wxd/ommkgzfHE/hSQx5IwbXScREQlqqf3GA7B73XzHSQJfU0tYB2CEtfYOa+2PgFFAKnA8cJWPsok0ydqlnzGscgmbe11BVIxOXyci4ksZfYdTYSOp1cH5x6ypJaw7UFXvcjXQw1pbDlS2eiqRZij58K8cJJaBZ9/uOoqISNALj4xie1Rv2hfp48hj1aRhi4DngQXGmDe9l88CXjDGxANrfJJMpAm2b1zBiJK5LMm4gjHtU1zHEREJCQeSBzNw95tUV1cTGanjcFuqSXvCrLW/A74HFAEHgBustb+11pZaay/1YT6RI8qffQ81RND77DtcRxERCRkR3UYQbyrZtm6p6ygBral7wrDWLsFzrjARv7BrxzZG7nuX5alnMbpzd9dxRERCRuf+42EJFG5cQJ/BY1zHCVg6wYcErM1v/YVwask4Q1/QFRFpS117DaaEWOyOr1xHCWgqYRKQigoLGLrrVZa3n0KXrGzXcUREQooJCycnui8pRatdRwloKmESkFa/dT8JppyUU3/iOoqISEgq6TCYrJotVFSUu44SsHxawowxU40x640xm4wxP21k+qXGmBXef/OMMUN9mUeCQ1npQfpvf5YVsWPIHDTWdRwRkZAU1W0kUaaGrWsWuY4SsHxWwowx4cBDwDQgG7jYGNPwc6OtwAnW2iHA74CZvsojwWPZ2w/TgQNEnfBD11FEREJW14ETACja+KXjJIHLl3vCxgCbrLVbrLVVwIvA9PozWGvnecemBFgAZPgwjwSB6uoqeqz/F+sjB9B/7FTXcUREQlZqRl+KSMTs1GkqWsqXJSwdyK13Oc973eFcA7zb2ARjzHXGmMXGmMUFBQWtGFECzdJ3nyDd7qZy3K1gjOs4IiIhy4SFkRvTj9RiHZzfUr4sYY29Q9pGZzTmRDwlrNGjrK21M621o6y1o1JTU1sxogSSuto6Oi57mO1h3Rh84oWu44iIhLyy1KH0qM2hpKTYdZSA5MsSlgd0q3c5A8hvOJMxZgjwODDdWlvowzwS4JbP+Q8967ZTMPRGTFi46zgiIiEvrscoIkwd21YtcB0lIPmyhC0C+hhjsowxUcBFwFv1ZzDGdAdeAy631m7wYRYJcNZaohY8wC5SGTrtWtdxREQESB80EYDizQsdJwlMTR62qLmstTXGmJuB94Fw4Alr7WpjzA3e6Y8AvwI6AA8bz/E9NdbaUb7KJIFr7cIPGFi9mi/7/4S0qGjXcUREBEhJ60EBKUTsWuY6SkDyWQkDsNbOBmY3uO6Rej9fC2i3hhxV5Zz72E87hpx1i+soIiJST358fzqXrHEdIyDpjPni97asXsjw8vms73EJsfGJruOIiEg9FanD6GF3ULRfh3U3l0qY+L3C9++hzEaTffaPXEcREZEGEnqOBmD7qi8cJwk8KmHi1/K3rWf4gY9YkXYu7Tp0ch1HREQa6OY9c37pFg1f1FwqYeLXct65mzoMWWf92HUUERFpRLsOaeSbzkTtWe46SsBRCRO/Vbg7j2EFb7Es+TQ6Z/RyHUdERA5jd8IAupaudR0j4KiEid/a8Pa9RFFD52mNDqQgIiJ+orrzMLqyh4LdO1xHCSgqYeKXSor3MzDvJZYlHEePfsNcxxERkSNo12ssAHmr5jlOElhUwsQvrXrrAdpRSvxJd7iOIiIiR9F90DjqrKFsuw7Obw6VMPE7lRVl9Nz0b1ZGDaPfiBNcxxERkaOIS0whLzyd2IIVrqMEFJUw8TsrZj1KJ/ZhJ93uOoqIiDRRQeJAupWvw1rrOkrAUAkTv1JbU0PaqplsDO/N4Elnu44jIiJNVNdlGKnsZ2feFtdRAoZKmPiVFR8+QzebT/GomzFhWj1FRAJFUm/Pwfn5a+Y7ThI49C4nfsPW1ZG4+EFyTFeGnXK56zgiItIM3QeOocaGUbl9sesoAUMlTPzGmi/epnfNJvIHXkd4RITrOCIi0gzRsYnkRPQgoVAH5zeVSpj4Dfv5/ewhhWFnXO86ioiItEBh+0H0qFhPXW2d6ygBQSVM/MLmZZ8yqHIpm3pdSUxsnOs4IiLSAqbrcJJMCXlbNYRRU6iEiV8o/vAeim08g86+1XUUERFpoQ59xwGwa50Ozm8KlTBxLm/jMoYe/JxVGRfQrn2K6zgiItJC3fqPpNJGUpO7xHWUgKASJs7tnH0PlUTS52wNUSQiEsgiomLIiexJ+30rXUcJCCph4tTeHVsYuu89lnU8i9TOGa7jiIjIMdqfPIjMqo3U1NS4juL3VMLEqc1v30MYlowzfuI6ioiItIKIjBHEmwpyNi53HcXvqYSJM8X79zB452ssaXcS3Xr2cx1HRERaQcd+4wEoWLfAcRL/pxImzqx9817iTCUpp97pOoqIiLSSjN5DKbPR1O34ynUUv6cSJk5UlBbTb9tzfBUzjj6Dx7iOIyIirSQsIoLt0X1ILlrlOorfUwkTJ1a+/SBJHCTyhB+6jiIiIq2sOGUwmdWbqayscB3Fr6mESZurqaqk+/p/sTpyIIPGneo6joiItLKo7qOIMdVsX6fzhR2JSpi0uZXv/4vOdi/lY36AMcZ1HBERaWVpAzwH5xdu0MH5R6ISJm3K1tWSsvSfbA7LZMSU813HERERH0jrMYBi4jH5S11H8WsqYdKmVs95mR51OewZ+n3CwrX6iYgEIxMWRk5MPzoWr3Ydxa/pXVDajrVEzf8bO+jMyGlXu04jIiI+VNphCD1qtlNWVuI6it9SCZM2s3Hhe/StXse2ft8lKirKdRwREfGhmB6jiDS1bF/9pesofkslTNpM5Zx7KaQ9Q8+6yXUUERHxsfSBEwDYv2mh4yT+SyVM2kTOmgUMKl/Emu6XkpCQ6DqOiIj4WMeuPSkkiYidOjj/cFTCpE0UvncPJTaWgWff7jqKiIi0BWPIi+tPp5I1rpP4LZUw8bnd29Yy5MDHLOt8LikdO7mOIyIibaQidQjda/M4ULTPdRS/pBImPpc7625qiCDrrB+7jiIiIm0oLnMMYcaSu1onbW2MSpj41IE9uQze8w5LkqeR3i3LdRwREWlD3Qd5Ds4v3qKD8xujEiY+tfGtvxJBDZ2n3eE6ioiItLH2qensMqlE7V7mOopfUgkTnykr3ke/vJdYEn8CvfoNdR1HREQc2Bk/gLTSta5j+CWVMPGZtW/dTyLlxJ+kvWAiIqGqqtNQMuwu9hXsch3F76iEiU9UV5SSuelplkaNZODI41zHERERRxJ7jQEgZ/U8x0n8j0qY+MSq2Y/QgSLqJt7mOoqIiDjUfdBEAMq2LnKcxP+ohEmrq6uppvOqmawN78fwSWe6jiMiIg4ltO9AXlhXYgqWu47id1TCpNWt+ehputbt4sDImwgL1yomIhLqdidkk162Dmut6yh+Re+Q0rqsJX7Rg2w1GYw89VLXaURExA/Upg2jM4Xs2ZnjOopfUQmTVrXhi9fJqtlCXvZ1REZEuI4jIiJ+oH3vsQDsWP2F4yT+RSVMWpX97H520pFRZ17nOoqIiPiJHgPHUWsNFdsXu47iV1TCpNVsX/Yx/SpXsKHnlcTGxrqOIyIifiImvh254d2J37vCdRS/ohImrab4w79SZBMYevYtrqOIiIif2dt+IN0q1mPr6lxH8RsqYdIqdm5cyuCSL1iefhFJScmu44iIiJ+p6zKCFIrZsX2j6yh+QyVMWsXud/9MmY2m39k/dB1FRET8UIe+noPzd63VmfMPUQmTY7Z/xyYGFn7A4g5nk5aW7jqOiIj4oe4DRlNlw6nKWeI6it9QCZNjtvWde7BAtzN/7DqKiIj4qcjoWLZH9iRx30rXUfyGSpgck5L9uxiw83UWtTuFrJ79XMcRERE/VpQ0kMzK9dTW1rqO4hdUwuSYbHjzr0TbapJPucN1FBER8XMmfSSJppzcTatcR/ELKmHSYpWlRfTa9gKLYyeQPWS06zgiIuLnUvt5Ds7fs26+4yT+QSVMWmztOw/SnhIijr/ddRQREQkAGX1HUG6jqM3TwfmgEiYtVFdVQca6f7E8YgjDx5/sOo6IiASA8IhItkf1pv1+fRwJKmHSQqvff5yOdh/lY27FGOM6joiIBIjilEFkVW+iurrKdRTnVMKk2WxtDcnL/smGsJ6MmnKu6zgiIhJAwruNItZUsW3dUtdRnFMJk2ZbP/dFMmrz2D3k+0REhLuOIyIiASSt/3gA9m1Y4DiJeyph0jzWEjX/AXJIY/S0K12nERGRANO150AOEovdoYPzVcKkWbYumk3P6g1s7nsNMdFRruOIiEiAMWHh5ET3I+XAGtdRnFMJk2apnHMve2wyI8/+vusoIiISoEo6DCazZgsV5WWuozilEiZNlr96Hv3LlrCq+6W0S0hwHUdERAJUdI9RRJlatq1d5DqKUyph0mSF/72bYhvH4LNvcx1FREQCWNfsCQAUbfzScRK3VMKkSQq3r2Zg0VyWdD6P1NRU13FERCSApab3Zj/tMDu/ch3FKZUwaZLcd+6migh6nflj11FERCTAmbAwcmP7k1oc2gfnq4TJUR0syGHgnnf4MukMunfv4TqOiIgEgfKOQ+lRm0PJwQOuozijEiZHtfnNezBY0qbe4TqKiIgEidiskYQby/ZV811HcUYlTI6oongvffL+w5fxk+k3YLDrOCIiEiQysicCULwldL8hqRImR7T+7fuJp4L4KdoLJiIirSclrTt76EDErtAdQ1IlTA6rtrKUHhufZlHkaIaOnOA6joiIBJn8+P50LlnrOoYzKmFyWGtnP0wSxdROvA1jjOs4IiISZCo7DaO7zefAvgLXUZxQCZNG2ZoqUlfOZGX4AMYcf4brOCIiEoQSskYDsH3VPMdJ3FAJk0at/+gpOtftoWj4TYSFaS+YiIi0vm6DPAfnl2xd6DiJGyph8m11dcQvepBNpjtjTrvYdRoREQlS7VI6scOkEb1nuesoTqiEybdsmf8a3Wq2kzPgeqIjI1zHERGRILY7YQBdS0Pz4HyVMPkma7Gf3ssOUhlz1rWu04iISJCrThtOF/ZSsCvXdZQ2pxIm35C3/GN6Va5hXdZVJMTGuI4jIiJBrn0vz8H5eatD7+B8lTD5hoMf3kOhbcews29yHUVEREJA94HjqbOG8m2LXUdpcyph8rWCTUsYULKAZV0vokNysus4IiISAuISk8kNzyBu7wrXUdqcSph8bffsP1NiY+h31u2uo4iISAgpSMwmo3wdtq7OdZQ2pRImABTnb2BA4Qcs7DCdjK5dXccREZEQUtd1BB0pYlfeFtdR2pRKmACw7e27qSGc7mdooG4REWlbyX3GApC/JrQOzlcJE8r37aTfzjdZkHgqvXv1dR1HRERCTPfsMVTbcCpzQuvgfJUwYePbfyHS1pB8qvaCiYhI24uOiScnIpOEwpWuo7QplbAQV126n55bX2BBzCSGDBnpOo6IiISofe2z6VGxnrra0Dk4XyUsxK2f9XcSKCP8OH0jUkRE3DHpI2lvSsnbusZ1lDajEhbCbHU5XdY+yZKIYYyeMMV1HBERCWEd+o0DYPfa+Y6TtB2VsBC2/v2ZdLD7KRt9C2FhxnUcEREJYd36jqDCRlKTGzoH56uEharaGpKWPsxq04dxJ53jOo2IiIS4iKhotkf1ot3+Va6jtBmVsBC1+dPnSavdxe4hNxIZEe46joiICEVJg8is2khNdbXrKG1CJSwUWUvk/L+xla6MP/0K12lEREQAiMgYSbypJGfjMtdR2oRKWAjKXfwO3as2s6nPtcRGR7qOIyIiAkCn/uMBKFj/peMkbUMlLARVfvJXdtkURp11vesoIiIiX0vvNZhSG4PNW+I6SptQCQsxBRu+pHfZMlZkXEJyuwTXcURERL4WFhHB9ui+JB8IjYPzVcJCzO737+WgjSX7zFtcRxEREfmW4g6DyazeSmVluesoPqcSFkLKCrbTr/BDFiafQUaXNNdxREREviW620iiTTU5a4P/fGEqYSFk86z7CLN1pJ3yA9dRREREGpWWPQGAwo0LHSfxPZWwEFFbcZDMbS/zZcwkBg4c4jqOiIhIo9K696OIBMyOr1xH8TmVsBCx/t1/kkgZZsJNrqOIiIgclgkLIzemHx2LV7uO4nMqYaGgrpaUVf9iZVh/xhw31XUaERGRIyrtOJQetdspKz3oOopPqYSFgC2fv0xa7S72DrqWcA3ULSIifi62xygiTB3bVwf3SVtVwkKAnfcgeXRizDQNUSQiIv4v3XtwftEmlbAWM8ZMNcasN8ZsMsb8tJHp/Y0x840xlcaYO3yZJVTtXvM5vSpWsbbHZcTHRruOIyIiclQd07MoIJnwXctcR/GpCF/dsTEmHHgIOAXIAxYZY96y1q6pN9s+4FZghq9yhLq9H9xHrI1j8Jnfdx1FRESkyfLj+tP54JqjzxjAfLknbAywyVq7xVpbBbwITK8/g7V2j7V2EVDtwxwhq3jXZvrv+5jFHc4mLTXVdRwREZEmq+g0lG51OzhQtM91FJ/xZQlLB3LrXc7zXtdsxpjrjDGLjTGLCwoKWiVcKNg26z7qCCP9tNtcRxEREWmWuMzRhBlLzqp5rqP4jC9LWGNfw7MtuSNr7Uxr7Shr7ahU7dFpkuqyInrlvsqXccfTr98A13FERESapfugiQCUbAneM+f7soTlAd3qXc4A8n34+6Se9bMfIp5yoibd7DqKiIhIs7Xv2IWdphORe5a7juIzvixhi4A+xpgsY0wUcBHwlg9/n3jZ2mo6rXmS5eEDGTX+JNdxREREWmRn/AC6lK51HcNnfFbCrLU1wM3A+8Ba4GVr7WpjzA3GmBsAjDFpxpg84IfAL4wxecaYdr7KFCo2zX2BTnUF7B96HWE6OauIiASo6s7DSLe72bcnOD9I89kpKgCstbOB2Q2ue6Tez7vwfEwprcVawr98iBzSGDf1UtdpREREWiyx1xjYDLmr55PS6TzXcVqdzpgfZHasnEPPynVszLqCmKhI13FERERarPtAz5nzS7cG58H5Pt0TJm2v6KP7ibfxDDnrRtdRREREjklC+xRywtKJLVjhOopPaE9YEDmQt54BRZ+ypNO5pKakuI4jIiJyzPYkZpNevg5rW3SWK7+mEhZEts2+lxrCyJz2A9dRREREWkVd2nA6sY89+dtcR2l1KmFBovJgIX3y32RhwhR69ezjOo6IiEiraN97LAB5q4PvzPkqYUFi/awHiaOCuBNudR1FRESk1fQYOJYaG0bl9kWuo7Q6lbAgYGuq6LL+3yyNGMrw0ce5jiMiItJqYuISyYnoTkLhStdRWp1KWBBY//HTpNpCSoZfjzE6OauIiASXwnYD6VaxHltX5zpKq1IJC3TWErPoEbaSzphTL3CdRkREpPV1HUEyB8nftt51klalEhbgcpd9QGb1Rrb2uYroSJ2cVUREgk9yH8/B+fnr5jtO0rpUwgLcwY//xj6byPAzrncdRURExCd6ZI+mykZQk7PYdZRWpRIWwPZtX0P2wS9YlvYdkpPau44jIiLiE5FRMWyL7EnivuA6OF8lLIDlvPtXKm0kvU7XyVlFRCS4FSUNIrNyI7W1ta6jtBqVsABVcaCAfrveZmHiyfTokeU6joiIiE+FZYwgwZSTuzF4xpFUCQtQ6995gFiqaDdFe8FERCT4pfYdB8CeIDo4XyUsANVVVZCx6VmWRI5gyPBxruOIiIj4XEbf4ZTZaOp2fOU6SqtRCQtA6z56kg52P5Wjv6+Ts4qISEgIj4hge1Rvkvavch2l1aiEBRprSVjyKJtNd0ZPOdd1GhERkTZTnDKYzOpNVFdXuY7SKlTCAszWRbPoXrOV3H5XExkR7jqOiIhIm4noNooYU832tUtcR2kVKmEBpnzu39lr2zP8jOtcRxEREWlTaQM8x0EXbljgOEnrUAkLIAVblpNd+iUr0i+gfWKC6zgiIiJtqmvWQIqJh/ylrqO0CpWwALLj3XupsJH0O0OnpRARkdBjwsLIie5LyoHgODhfJSxAlO7byYCC2SxqP5X09G6u44iIiDhR0mEwmTXbqCgvdR3lmKmEBYgN7/yNaKpJOeU211FEREScie4xmkhTy7Y1C11HOWYqYQGgtrKMzC0vsChqDAMHj3IdR0RExJmu2eMBKNoY+Afnq4QFgLX/fZxkDlA39vuuo4iIiDjVKb0XhbQnbOcy11GOmUqYv7OW9stmssH0ZNTk6a7TiIiIOGXCwsiL7U/qwTWuoxwzlTA/t2n+G3SrzWXnwGsID9fiEhERKe84hO61uZQcLHId5ZjoXd3PVX/+D3aTzMjTv+s6ioiIiF+IyxpNuLFsXzXfdZRjohLmx3ZtWMyAsiWs6XYxCXFxruOIiIj4hYyBEwAo3hzY35BUCfNju96/jzIbzYAzb3UdRURExG+kdO7GLjoSuXuZ6yjHRCXMTx3cm8vAve+xOPl00jp3cR1HRETEr+yMH0BayVrXMY6JSpif2vT23winjrRTb3cdRURExO9UdhpKht1JUeEe11FaTCXMD1VXlNBz+4ssjh1P3+yhruOIiIj4ncSeYwDIWfWF4yQtpxLmh9a8N5P2lGDG3+w6ioiIiF/KGOQ5OL906yLHSVpOJczP2LpaOqx8nHVhfRg5aZrrOCIiIn6pfXIqeaYL0XuWu47SYiphfmbj56+SUbuDgsHXEqaTs4qIiBzW7sRsupatcx2jxfQu72fq5j3ITjoyatrVrqOIiIj4terOw0hjLwU7c1xHaRGVMD+yY818+lcsZ32PS4mNiXYdR0RExK8l9fYcnL9j9TzHSVpGJcyP7P3gPkpsDAPP0gH5IiIiR9N94DhqraF8W2AenK8S5icO7N5G9r6P+KrDWaR27OQ6joiIiN+LS0giN7wbcYUrXEdpEZUwP7HpnfsIo470qT90HUVERCRgFLQbSLfyddi6OtdRmk0lzA9Ulh2gT+4rLI47jl59s13HERERCRh1XYeTQjE7cze5jtJsKmF+YM3sR2hHKVHH3eI6ioiISEBJ6T0WgF1rA+/gfJUwx2xtDZ3XPMHq8AEMG3+K6zgiIiIBpXv2GKpsOBXbl7iO0mwqYY6tm/sSXet2UTTsOowxruOIiIgElOiYOLZHZtFuX+AdnK8S5lj4lw+RRydGnXaZ6ygiIiIBaX/7gXSv2EBdba3rKM2iEuZQzoq59K1czZaelxMdFeU6joiISEAyXUfQzpSRt2W16yjNohLm0P6P/kaxjWOQTs4qIiLSYh37jQNgz7r5jpM0j0qYI4U7NjGwaA7LOs0gJTnFdRwREZGA1a3fCMptFDW5gXVwvkqYI9tm3YvF0H3aba6jiIiIBLSIyCi2R/Wm/f5VrqM0i0qYAxUl++mX/zpLEk4gs2c/13FEREQC3oHkQfSo2kRNdZXrKE2mEubAmlkPkUA5cSfc6jqKiIhIUAjPGEGcqSRnwzLXUZpMJayN1dVU03Xdv1kZMYjBoye7jiMiIhIUOvcfD8De9QscJ2k6lbA2tubj50izeygdcb1OzioiItJK0nsN5qCNxe4InIPzVcLakrXELHqYXLow8tRLXKcREREJGmHh4eRE9yX5wBrXUZpMJawNbVn6Eb2r17Ot75VERkS4jiMiIhJUDnYYQmb1Fiory11HaRKVsDZU8skDHLDxDDnjRtdRREREgk5U95FEmRq2r17kOkqTqIS1kYLt6xhU/BnL086jffsk13FERESCTpfsCQDs2/Sl4yRNoxLWRra/ey81hNHr9NtdRxEREQlKad36sJ9EwvK/ch2lSVTC2kBp0V6yd73JV+1OIr1HT9dxREREgpIJCyM3pj8diwPj4HyVsDawdtY/iKOSdlNucx1FREQkqJV1HEL32hzKSotdRzkqlTAfq62uovumZ1geOYzs4RNdxxEREQlqMT1GEWHq2L7K/0/aqhLmY6s+eIpOtpCq0fpGpIiIiK9lDPTs8DiweaHjJEenEuZL1pL41aNsM+kMn/Id12lERESCXseuPdhDChG7lrqOclQqYT60cdH79KzZRF7/7xKhk7OKiIi0ify4/nQ66P8H56uE+VD53AfYRyLDzrjedRQREZGQUdFpGN1tPgeKCl1HOSKVMB/ZtWUlg0rms6brBSQkJLqOIyIiEjLiM0cBkLvqC8dJjkwlzEfy3r2PaiLofcZtrqOIiIiElO6DPQfnH9zi38MXqYT5wMH9uxm45x2+SjqFtPTuruOIiIiElPYd0thhOhO1e5nrKEekEuYD6975O7Gmig4na4giERERF3bFD6Br6VrXMY5IJayV1VSWk7X5OZZFj6Tv4DGu44iIiISk6rThdKGAwj07XEc5LJWwVrby/SfpyH5qx97sOoqIiEjISuzp2RGSt2qe4ySHpxLWimxdHUnLZ7IlrAfDT5jhOo6IiEjI6jFoPHXWULbNfw/OVwlrRRsWvENW7VZ2ZV9DWLieWhEREVcS2iWTG55BbMEK11EOS02hFVV99g/20p7hp3/PdRQREZGQtydxABnla7F1da6jNEolrJXs2LCUweULWdftImLj4lzHERERCXl1XYbTkSL27NzmOkqjVMJayc7376PCRtLvzB+4jiIiIiJA+95jAdjhpwfnq4S1ggMFOxi8912WpkwjtXO66zgiIiICZGaPpdqGU5njnwfnq4S1gvXvPEC0qabzaT90HUVERES8YuISyInoQXzhStdRGqUSdoyqKsrovf0FlsaMpWf/4a7jiIiISD2F7QfSvWK9Xx6crxJ2jFbOnkkKxYRN1MlZRURE/E7X4SRRQv42/xvCSCXsGNi6OlJXPc6msCyGTDzTdRwRERFpIKXPOAB2rp3vOMm3qYQdgzWfv073ulz2Dr4OE6anUkRExN/0GDCKShtJTc5i11G+Rc3hGNh5D7KHFIaffrXrKCIiItKIyKhotkX2pN3+Va6jfItKWAvlrPmSQRVfsTHzEqKjY13HERERkcMoShpEj8qN1NbUuI7yDSphLbTng/sps9Fkn6WTs4qIiPizsIwRxJsK8jYudx3lG1TCWqBwVw5D9v2X5R3PJLlDJ9dxRERE5Ag69RsPwO71Cxwn+SaVsBbYNOt+Iqij69TbXUcRERGRo8joM5RSG4PNW+I6yjeohDVTRdlB+uW+zLL4CfToM9h1HBERETmK8IgItkf3IanIvw7OVwlrppWzHiWJEqKOu9V1FBEREWmi4uRBZFZvoaqy0nWUr6mENUNdbS1pa59gfXgfBo491XUcERERaaLI7iOJNtXkrPOf84WphDXDqrkv061uB8XDrtfJWUVERAJI2oAJABRu8J+D89UkmiHiy4fZSUeGTb3SdRQRERFphq6ZAzhAPGbnUtdRvqYS1kRbVnxBduUKtva6nMjIKNdxREREpBlMWBg50f3ocGC16yhfUwlrov0f3U+pjWHgmbe4jiIiIiItUNJxCN1rtlNRVuI6CqAS1iQFO7YwpOhjVnSeTvvkDq7jiIiISAvE9BhNpKll25ovXUcBVMKaZPOs+wmjju7TfuQ6ioiIiLRQ12zPmfOLNqqEBYSykiKy819lWeLxpGf1cx1HREREWqhT1yz2kkT4rmWuowAqYUe16p2HaUcp8SdooG4REZFAZsLC2BHbn04H17iOAqiEHVFtTQ3p659ibeQA+o8+yXUcEREROUZlqUPpVptHSfF+11FUwo5kxccvkG53UzbiBtdRREREpBXEZY0mzFi2r5rvOopK2JHELnqYfNOZoSdf6jqKiIiItIJuAz1nzj+4xf3B+Sphh7FxySf0r15DTp8riIiMdB1HREREWkFKp3R2kkrE7uWuo6iEHU7xnAcotnEMOvMm11FERESkFe1MGECXEvcH5/u0hBljphpj1htjNhljftrIdGOM+bt3+gpjzAhf5mmqXTkbGFo8l9VdziGhXbLrOCIiItKKKjsNI93upmjvLqc5fFbCjDHhwEPANCAbuNgYk91gtmlAH++/64B/+ipPc2ybdR8WQ9YZP3QdRURERFpZYs/RAOSunuc0hy/3hI0BNllrt1hrq4AXgekN5pkOPG09FgBJxpguPsx0VAcP7GPQrjdY3u5E0rr1dhlFREREfKD7IM/B+SVbFzrN4csSlg7k1ruc572uufNgjLnOGLPYGLO4oKCg1YPWt3vbWorDEmk/RSdnFRERCUbtkjqyLjIb6mq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\n", + "text/plain": [ + "
    " + ] + }, + "metadata": { + "filenames": { + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter11_91_4.png" + }, + "needs_background": "light" + }, + "output_type": "display_data" + } + ], "source": [ "import autograd.numpy as np\n", "from autograd import grad, elementwise_grad\n", @@ -2092,7 +2533,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6f02e86d", + "metadata": { + "editable": true + }, "source": [ "## Partial Differential Equations\n", "\n", @@ -2106,7 +2550,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ad5c63e8", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -2120,10 +2567,21 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "c7b98797", + "metadata": { + "editable": true + }, + "source": [ + "where $f$ is an expression involving all kinds of possible mixed derivatives of $g(x_1,\\dots,x_N)$ up to an order $n$. In order for the solution to be unique, some additional conditions must also be given." + ] + }, + { + "cell_type": "markdown", + "id": "52f9394e", + "metadata": { + "editable": true + }, "source": [ - "where $f$ is an expression involving all kinds of possible mixed derivatives of $g(x_1,\\dots,x_N)$ up to an order $n$. In order for the solution to be unique, some additional conditions must also be given.\n", - "\n", "### Type of problem\n", "\n", "The problem our network must solve for, is similar to the ODE case.\n", @@ -2134,7 +2592,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "4d489854", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{align*}\n", @@ -2145,15 +2606,24 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "82311538", + "metadata": { + "editable": true + }, "source": [ "where $h_1(x_1,\\dots,x_N)$ is a function that ensures $g_t(x_1,\\dots,x_N)$ satisfies some given conditions.\n", "The neural network $N(x_1,\\dots,x_N,P)$ has weights and biases described by $P$ and $h_2(x_1,\\dots,x_N,N(x_1,\\dots,x_N,P))$ is an expression using the output from the neural network in some way.\n", "\n", - "The role of the function $h_2(x_1,\\dots,x_N,N(x_1,\\dots,x_N,P))$, is to ensure that the output of $N(x_1,\\dots,x_N,P)$ is zero when $g_t(x_1,\\dots,x_N)$ is evaluated at the values of $x_1,\\dots,x_N$ where the given conditions must be satisfied. The function $h_1(x_1,\\dots,x_N)$ should alone make $g_t(x_1,\\dots,x_N)$ satisfy the conditions.\n", - "\n", - "\n", - "\n", + "The role of the function $h_2(x_1,\\dots,x_N,N(x_1,\\dots,x_N,P))$, is to ensure that the output of $N(x_1,\\dots,x_N,P)$ is zero when $g_t(x_1,\\dots,x_N)$ is evaluated at the values of $x_1,\\dots,x_N$ where the given conditions must be satisfied. The function $h_1(x_1,\\dots,x_N)$ should alone make $g_t(x_1,\\dots,x_N)$ satisfy the conditions." + ] + }, + { + "cell_type": "markdown", + "id": "9d134db4", + "metadata": { + "editable": true + }, + "source": [ "### Network requirements\n", "\n", "The network tries then the minimize the cost function following the\n", @@ -2169,7 +2639,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5a463498", + "metadata": { + "editable": true + }, "source": [ "$$\n", "C\\left(x_1, \\dots, x_N, P\\right) = \\left( f\\left(x_1, \\, \\dots \\, , x_N, \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_1}, \\dots , \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_N}, \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(x_1,\\dots,x_N) }{\\partial x_N^n} \\right) \\right)^2\n", @@ -2178,14 +2651,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "c40d8997", + "metadata": { + "editable": true + }, "source": [ "If we let $\\boldsymbol{x} = \\big( x_1, \\dots, x_N \\big)$ be an array containing the values for $x_1, \\dots, x_N$ respectively, the cost function can be reformulated into the following:" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "cc033de6", + "metadata": { + "editable": true + }, "source": [ "$$\n", "C\\left(\\boldsymbol{x}, P\\right) = f\\left( \\left( \\boldsymbol{x}, \\frac{\\partial g(\\boldsymbol{x}) }{\\partial x_1}, \\dots , \\frac{\\partial g(\\boldsymbol{x}) }{\\partial x_N}, \\frac{\\partial g(\\boldsymbol{x}) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(\\boldsymbol{x}) }{\\partial x_N^n} \\right) \\right)^2\n", @@ -2194,14 +2673,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ae18b4f4", + "metadata": { + "editable": true + }, "source": [ "If we also have $M$ different sets of values for $x_1, \\dots, x_N$, that is $\\boldsymbol{x}_i = \\big(x_1^{(i)}, \\dots, x_N^{(i)}\\big)$ for $i = 1,\\dots,M$ being the rows in matrix $X$, the cost function can be generalized into" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "44ca21bd", + "metadata": { + "editable": true + }, "source": [ "$$\n", "C\\left(X, P \\right) = \\sum_{i=1}^M f\\left( \\left( \\boldsymbol{x}_i, \\frac{\\partial g(\\boldsymbol{x}_i) }{\\partial x_1}, \\dots , \\frac{\\partial g(\\boldsymbol{x}_i) }{\\partial x_N}, \\frac{\\partial g(\\boldsymbol{x}_i) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(\\boldsymbol{x}_i) }{\\partial x_N^n} \\right) \\right)^2.\n", @@ -2210,7 +2695,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "fe94451e", + "metadata": { + "editable": true + }, "source": [ "## Example: The diffusion equation\n", "\n", @@ -2219,7 +2707,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "30a42273", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial g(x,t)}{\\partial t} = \\frac{\\partial^2 g(x,t)}{\\partial x^2}\n", @@ -2228,14 +2719,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "fd4077e7", + "metadata": { + "editable": true + }, "source": [ "where a possible choice of conditions are" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "dbbf2e4b", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{align*}\n", @@ -2248,18 +2745,22 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "83653db9", + "metadata": { + "editable": true + }, "source": [ "with $u(x)$ being some given function.\n", "\n", - "\n", - "\n", "For this case, we want to find $g(x,t)$ such that" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "d2c2ef4f", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -2273,14 +2774,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "8fb27bdb", + "metadata": { + "editable": true + }, "source": [ "and" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "dc9297ab", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{align*}\n", @@ -2293,7 +2800,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ab90784f", + "metadata": { + "editable": true + }, "source": [ "with $u(x) = \\sin(\\pi x)$.\n", "\n", @@ -2301,9 +2811,6 @@ "The deep neural network will follow the same structure as discussed in the examples solving the ODEs.\n", "First, we will look into how Autograd could be used in a network tailored to solve for bivariate functions.\n", "\n", - "\n", - "\n", - "\n", "The only change to do here, is to extend our network such that\n", "functions of multiple parameters are correctly handled. In this case\n", "we have two variables in our function to solve for, that is time $t$\n", @@ -2315,7 +2822,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 7, + "id": "b07e858c", "metadata": { "collapsed": false, "editable": true @@ -2370,7 +2878,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9a167ec1", + "metadata": { + "editable": true + }, "source": [ "The cost function must then iterate through the given arrays\n", "containing values for $x$ and $t$, defines a point $(x,t)$ the deep\n", @@ -2392,8 +2903,6 @@ "$$\n", "since $(0) = u(1) = 0$ and $u(x) = \\sin(\\pi x)$.\n", "\n", - "\n", - "\n", "The Jacobian is used because the program must find the derivative of\n", "the trial solution with respect to $x$ and $t$.\n", "\n", @@ -2415,7 +2924,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 8, + "id": "921e5969", "metadata": { "collapsed": false, "editable": true @@ -2462,7 +2972,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0bc00b69", + "metadata": { + "editable": true + }, "source": [ "### Setting up the network using Autograd; The full program\n", "\n", @@ -2478,19 +2991,70 @@ "Be aware, though, that it is fairly slow for the parameters used.\n", "A better result is possible, but requires more iterations, and thus longer time to complete.\n", "\n", - "\n", "Indeed, the program below is not optimal in its implementation, but rather serves as an example on how to implement and use a neural network to solve a PDE.\n", "Using TensorFlow results in a much better execution time. Try it!" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 9, + "id": "20734418", "metadata": { "collapsed": false, "editable": true }, - "outputs": [], + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/Users/MortenImac/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray\n", + " return array(a, dtype, copy=False, order=order)\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Initial cost: 41.05505310046362\n" + ] + }, + { + "ename": "KeyboardInterrupt", + "evalue": "", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mKeyboardInterrupt\u001b[0m Traceback (most recent call last)", + "\u001b[0;32m/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47448/73752910.py\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 141\u001b[0m \u001b[0mlmb\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m0.01\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 142\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 143\u001b[0;31m \u001b[0mP\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msolve_pde_deep_neural_network\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mt\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnum_hidden_neurons\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnum_iter\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlmb\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 144\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 145\u001b[0m \u001b[0;31m## Store the results\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;32m/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47448/73752910.py\u001b[0m in \u001b[0;36msolve_pde_deep_neural_network\u001b[0;34m(x, t, num_neurons, num_iter, lmb)\u001b[0m\n\u001b[1;32m 118\u001b[0m \u001b[0;31m# Let the update be done num_iter times\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 119\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mnum_iter\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 120\u001b[0;31m \u001b[0mcost_grad\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcost_function_grad\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mP\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m \u001b[0;34m,\u001b[0m \u001b[0mt\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 121\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 122\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ml\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mN_hidden\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;32m~/anaconda3/lib/python3.8/site-packages/autograd/wrap_util.py\u001b[0m in \u001b[0;36mnary_f\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m 18\u001b[0m \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 19\u001b[0m \u001b[0mx\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mtuple\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[0;32min\u001b[0m \u001b[0margnum\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 20\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0munary_operator\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0munary_f\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0mnary_op_args\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mnary_op_kwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 21\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mnary_f\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 22\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mnary_operator\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;32m~/anaconda3/lib/python3.8/site-packages/autograd/differential_operators.py\u001b[0m in \u001b[0;36mgrad\u001b[0;34m(fun, x)\u001b[0m\n\u001b[1;32m 23\u001b[0m \u001b[0marguments\u001b[0m \u001b[0;32mas\u001b[0m\u001b[0;31m \u001b[0m\u001b[0;31m`\u001b[0m\u001b[0mfun\u001b[0m\u001b[0;31m`\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mbut\u001b[0m \u001b[0mreturns\u001b[0m \u001b[0mthe\u001b[0m \u001b[0mgradient\u001b[0m \u001b[0minstead\u001b[0m\u001b[0;34m.\u001b[0m \u001b[0mThe\u001b[0m \u001b[0mfunction\u001b[0m\u001b[0;31m \u001b[0m\u001b[0;31m`\u001b[0m\u001b[0mfun\u001b[0m\u001b[0;31m`\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 24\u001b[0m should be scalar-valued. 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np\n", "from autograd import jacobian,hessian,grad\n", @@ -2719,7 +3283,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "143f1c80", + "metadata": { + "editable": true + }, "source": [ "## Solving the wave equation with Neural Networks\n", "\n", @@ -2728,7 +3295,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "190bd4f2", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial^2 g(x,t)}{\\partial t^2} = c^2\\frac{\\partial^2 g(x,t)}{\\partial x^2}\n", @@ -2737,7 +3307,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9719cfc5", + "metadata": { + "editable": true + }, "source": [ "with $c$ being the specified wave speed.\n", "\n", @@ -2746,7 +3319,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f69ec7fd", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{align*}\n", @@ -2760,17 +3336,22 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6ce8c5a8", + "metadata": { + "editable": true + }, "source": [ "where $\\frac{\\partial g(x,t)}{\\partial t} \\Big |_{t = 0}$ means the derivative of $g(x,t)$ with respect to $t$ is evaluated at $t = 0$, and $u(x)$ and $v(x)$ being given functions.\n", "\n", - "\n", "The wave equation to solve for, is" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "4be700d7", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -2784,7 +3365,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "993f93ba", + "metadata": { + "editable": true + }, "source": [ "where $c$ is the given wave speed.\n", "The chosen conditions for this equation are" @@ -2792,7 +3376,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "2cb2a80f", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -2809,12 +3396,13 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "3f1dcfd4", + "metadata": { + "editable": true + }, "source": [ "In this example, let $c = 1$ and $u(x) = \\sin(\\pi x)$ and $v(x) = -\\pi\\sin(\\pi x)$.\n", "\n", - "\n", - "\n", "Setting up the network is done in similar matter as for the example of solving the diffusion equation.\n", "The only things we have to change, is the trial solution such that it satisfies the conditions from ([20](#condwave)) and the cost function.\n", "\n", @@ -2832,7 +3420,6 @@ "\n", "Note that this trial solution satisfies the conditions only if $u(0) = v(0) = u(1) = v(1) = 0$, which is the case in this example.\n", "\n", - "\n", "The analytical solution for our specific problem, is\n", "\n", "$$\n", @@ -2842,7 +3429,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 10, + "id": "230a9aef", "metadata": { "collapsed": false, "editable": true @@ -3073,7 +3661,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a23bd19a", + "metadata": { + "editable": true + }, "source": [ "## Resources on differential equations and deep learning\n", "\n", @@ -3102,5 +3693,5 @@ } }, "nbformat": 4, - "nbformat_minor": 4 + "nbformat_minor": 5 } \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter11.py b/doc/LectureNotes/_build/jupyter_execute/chapter11.py index e46d52a1f..ad19dc1c9 100644 --- a/doc/LectureNotes/_build/jupyter_execute/chapter11.py +++ b/doc/LectureNotes/_build/jupyter_execute/chapter11.py @@ -1,13 +1,15 @@ #!/usr/bin/env python # coding: utf-8 +# + # # Solving Differential Equations with Deep Learning # # The Universal Approximation Theorem states that a neural network can # approximate any function at a single hidden layer along with one input # and output layer to any given precision. # -# # An ordinary differential equation (ODE) is an equation involving functions having one variable. # # In general, an ordinary differential equation looks like @@ -29,8 +31,6 @@ # Along with ([1](#ode)), some additional conditions of the function $g(x)$ are typically given # for the solution to be unique. # -# -# # Let the trial solution $g_t(x)$ be # @@ -54,11 +54,8 @@ # # But what about the network $N(x,P)$? # -# # As described previously, an optimization method could be used to minimize the parameters of a neural network, that being its weights and biases, through backward propagation. # -# -# # For the minimization to be defined, we need to have a cost function at hand to minimize. # # It is given that $f\left(x, \, g(x), \, g'(x), \, g''(x), \, \dots \, , \, g^{(n)}(x)\right)$ should be equal to zero in ([1](#ode)). @@ -85,15 +82,12 @@ # The neural net should then find the parameters $P$ that minimizes the cost function in # ([3](#cost)) for a set of $N$ training samples $x_i$. # -# -# # To perform the minimization using gradient descent, the gradient of $C\left(\boldsymbol{x}, P\right)$ is needed. # It might happen so that finding an analytical expression of the gradient of $C(\boldsymbol{x}, P)$ from ([3](#cost)) gets too messy, depending on which cost function one desires to use. # # Luckily, there exists libraries that makes the job for us through automatic differentiation. # Automatic differentiation is a method of finding the derivatives numerically with very high precision. -# -# + # ### Example: Exponential decay # # An exponential decay of a quantity $g(x)$ is described by the equation @@ -123,8 +117,6 @@ # Having an analytical solution at hand, it is possible to use it to compare how well a neural network finds a solution of ([4](#solve_expdec)). # -# -# # The program will use a neural network to solve # @@ -140,7 +132,6 @@ # # In this example, $\gamma = 2$ and $g_0 = 10$. # -# # To begin with, a trial solution $g_t(t)$ must be chosen. A general trial solution for ordinary differential equations could be # $$ @@ -149,8 +140,6 @@ # with $h_1(x)$ ensuring that $g_t(x)$ satisfies some conditions and $h_2(x,N(x, P))$ an expression involving $x$ and the output from the neural network $N(x,P)$ with $P $ being the collection of the weights and biases for each layer. For now, it is assumed that the network consists of one input layer, one hidden layer, and one output layer. # -# -# # In this network, there are no weights and bias at the input layer, so $P = \{ P_{\text{hidden}}, P_{\text{output}} \}$. # If there are $N_{\text{hidden} }$ neurons in the hidden layer, then $P_{\text{hidden}}$ is a $N_{\text{hidden} } \times (1 + N_{\text{input}})$ matrix, given that there are $N_{\text{input}}$ neurons in the input layer. # @@ -198,7 +187,6 @@ # is fulfilled as *best as possible*. # -# # The left hand side and right hand side of ([8](#nnmin)) must be computed separately, and then the neural network must choose weights and biases, contained in $P$, such that the sides are equal as best as possible. # This means that the absolute or squared difference between the sides must be as close to zero, ideally equal to zero. # In this case, the difference squared shows to be an appropriate measurement of how erroneous the trial solution is with respect to $P$ of the neural network. @@ -219,8 +207,6 @@ # for an input value $x$. # -# -# # If the neural network evaluates $g_t(x, P)$ at more values for $x$, say $N$ values $x_i$ for $i = 1, \dots, N$, then the *total* error to minimize becomes # @@ -244,14 +230,12 @@ # \min_{P_{\text{hidden} }, \ P_{\text{output} }} C(\boldsymbol{x}, \{P_{\text{hidden} }, P_{\text{output} }\}) # $$ # -# # For simplicity, it is assumed that the input is an array $\boldsymbol{x} = (x_1, \dots, x_N)$ with $N$ elements. It is at these points the neural network should find $P$ such that it fulfills ([9](#min)). # # First, the neural network must feed forward the inputs. # This means that $\boldsymbol{x}s$ must be passed through an input layer, a hidden layer and a output layer. The input layer in this case, does not need to process the data any further. # The input layer will consist of $N_{\text{input} }$ neurons, passing its element to each neuron in the hidden layer. The number of neurons in the hidden layer will be $N_{\text{hidden} }$. # -# # For the $i$-th in the hidden layer with weight $w_i^{\text{hidden} }$ and bias $b_i^{\text{hidden} }$, the weighting from the $j$-th neuron at the input layer is: # $$ @@ -311,8 +295,6 @@ # and biases $b_i^{\text{output}}$. In this case, # it is assumes that the number of neurons in the output layer is one. # -# -# # The procedure of weighting the output neuron $j$ in the hidden layer to the $i$-th neuron in the output layer is similar as for the hidden layer described previously. # $$ @@ -343,7 +325,6 @@ # In this case we seek a continuous range of values since we are approximating a function. This means that after computing $\boldsymbol{z}_{1}^{\text{output}}$ the neural network has finished its feed forward step, and $\boldsymbol{z}_{1}^{\text{output}}$ is the final output of the network. # -# # The next step is to decide how the parameters should be changed such that they minimize the cost function. # # The chosen cost function for this problem is @@ -355,7 +336,7 @@ # In order to minimize the cost function, an optimization method must be chosen. # # Here, gradient descent with a constant step size has been chosen. -# + # ### Gradient descent # # The idea of the gradient descent algorithm is to update parameters in @@ -734,8 +715,6 @@ if __name__ == '__main__': # using a library like TensorFlow is recommended. # Here, we stay with a more simple approach and implement for comparison, the simple forward Euler method. # -# -# # Here, we will model a population $g(t)$ in an environment having carrying capacity $A$. # The population follows the model @@ -752,7 +731,6 @@ if __name__ == '__main__': # # In this example, we let $\alpha = 2$, $A = 1$, and $g_0 = 1.2$. # -# # We will get a slightly different trial solution, as the boundary conditions are different # compared to the case for exponential decay. # @@ -770,11 +748,9 @@ if __name__ == '__main__': # g(t) = \frac{Ag_0}{g_0 + (A - g_0)\exp(-\alpha A t)} # $$ # -# -# # The network will be the similar as for the exponential decay example, but with some small modifications for our problem. -# In[ ]: +# In[3]: import autograd.numpy as np @@ -996,7 +972,7 @@ if __name__ == '__main__': # Equation ([12](#odenum)) could be implemented in the following way, # extending the program that uses the network using Autograd: -# In[ ]: +# In[4]: # Assume that all function definitions from the example program using Autograd @@ -1094,7 +1070,6 @@ if __name__ == '__main__': # The results from the networks can then be compared to the analytical solution. # In addition, it could be interesting to see how a typical method for numerically solving second order ODEs compares to the neural networks. # -# # Here, the function $g(x)$ to solve for follows the equation # $$ @@ -1126,7 +1101,7 @@ if __name__ == '__main__': # g(x) = x(1 - x)\exp(x) # $$ -# In[ ]: +# In[5]: import autograd.numpy as np @@ -1372,10 +1347,9 @@ if __name__ == '__main__': # which makes it possible to solve for the vector $\boldsymbol{g}$. # -# # We can then compare the result from this numerical scheme with the output from our network using Autograd: -# In[ ]: +# In[6]: import autograd.numpy as np @@ -1589,7 +1563,7 @@ if __name__ == '__main__': # $$ # where $f$ is an expression involving all kinds of possible mixed derivatives of $g(x_1,\dots,x_N)$ up to an order $n$. In order for the solution to be unique, some additional conditions must also be given. -# + # ### Type of problem # # The problem our network must solve for, is similar to the ODE case. @@ -1607,9 +1581,7 @@ if __name__ == '__main__': # The neural network $N(x_1,\dots,x_N,P)$ has weights and biases described by $P$ and $h_2(x_1,\dots,x_N,N(x_1,\dots,x_N,P))$ is an expression using the output from the neural network in some way. # # The role of the function $h_2(x_1,\dots,x_N,N(x_1,\dots,x_N,P))$, is to ensure that the output of $N(x_1,\dots,x_N,P)$ is zero when $g_t(x_1,\dots,x_N)$ is evaluated at the values of $x_1,\dots,x_N$ where the given conditions must be satisfied. The function $h_1(x_1,\dots,x_N)$ should alone make $g_t(x_1,\dots,x_N)$ satisfy the conditions. -# -# -# + # ### Network requirements # # The network tries then the minimize the cost function following the @@ -1658,8 +1630,6 @@ if __name__ == '__main__': # with $u(x)$ being some given function. # -# -# # For this case, we want to find $g(x,t)$ such that # @@ -1687,9 +1657,6 @@ if __name__ == '__main__': # The deep neural network will follow the same structure as discussed in the examples solving the ODEs. # First, we will look into how Autograd could be used in a network tailored to solve for bivariate functions. # -# -# -# # The only change to do here, is to extend our network such that # functions of multiple parameters are correctly handled. In this case # we have two variables in our function to solve for, that is time $t$ @@ -1698,7 +1665,7 @@ if __name__ == '__main__': # network at each possible pair $(x,t)$, given an array for the desired # $x$-values and $t$-values to approximate the solution at. -# In[ ]: +# In[7]: def sigmoid(z): @@ -1767,8 +1734,6 @@ def deep_neural_network(deep_params, x): # $$ # since $(0) = u(1) = 0$ and $u(x) = \sin(\pi x)$. # -# -# # The Jacobian is used because the program must find the derivative of # the trial solution with respect to $x$ and $t$. # @@ -1787,7 +1752,7 @@ def deep_neural_network(deep_params, x): # matrix, which is the matrix containing all the possible second order # mixed derivatives of $g(x,t)$. -# In[ ]: +# In[8]: # Set up the trial function: @@ -1842,11 +1807,10 @@ def cost_function(P, x, t): # Be aware, though, that it is fairly slow for the parameters used. # A better result is possible, but requires more iterations, and thus longer time to complete. # -# # Indeed, the program below is not optimal in its implementation, but rather serves as an example on how to implement and use a neural network to solve a PDE. # Using TensorFlow results in a much better execution time. Try it! -# In[ ]: +# In[9]: import autograd.numpy as np @@ -2097,7 +2061,6 @@ if __name__ == '__main__': # where $\frac{\partial g(x,t)}{\partial t} \Big |_{t = 0}$ means the derivative of $g(x,t)$ with respect to $t$ is evaluated at $t = 0$, and $u(x)$ and $v(x)$ being given functions. # -# # The wave equation to solve for, is # @@ -2126,8 +2089,6 @@ if __name__ == '__main__': # In this example, let $c = 1$ and $u(x) = \sin(\pi x)$ and $v(x) = -\pi\sin(\pi x)$. # -# -# # Setting up the network is done in similar matter as for the example of solving the diffusion equation. # The only things we have to change, is the trial solution such that it satisfies the conditions from ([20](#condwave)) and the cost function. # @@ -2145,14 +2106,13 @@ if __name__ == '__main__': # # Note that this trial solution satisfies the conditions only if $u(0) = v(0) = u(1) = v(1) = 0$, which is the case in this example. # -# # The analytical solution for our specific problem, is # # $$ # g(x,t) = \sin(\pi x)\cos(\pi t) - \sin(\pi x)\sin(\pi t) # $$ -# In[ ]: +# In[10]: import autograd.numpy as np diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter11_50_2.png b/doc/LectureNotes/_build/jupyter_execute/chapter11_50_2.png new file mode 100644 index 000000000..5742598bc Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapter11_50_2.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter11_52_3.png b/doc/LectureNotes/_build/jupyter_execute/chapter11_52_3.png new file mode 100644 index 000000000..99f92bd8b Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapter11_52_3.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter11_58_3.png b/doc/LectureNotes/_build/jupyter_execute/chapter11_58_3.png new file mode 100644 index 000000000..6d89cdf1e Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapter11_58_3.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter11_66_3.png b/doc/LectureNotes/_build/jupyter_execute/chapter11_66_3.png new file mode 100644 index 000000000..6d89cdf1e Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapter11_66_3.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter11_66_4.png b/doc/LectureNotes/_build/jupyter_execute/chapter11_66_4.png new file mode 100644 index 000000000..4fdb738d2 Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapter11_66_4.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter11_79_3.png b/doc/LectureNotes/_build/jupyter_execute/chapter11_79_3.png new file mode 100644 index 000000000..4f89c99e9 Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapter11_79_3.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter11_91_3.png b/doc/LectureNotes/_build/jupyter_execute/chapter11_91_3.png new file mode 100644 index 000000000..4f89c99e9 Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapter11_91_3.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter11_91_4.png b/doc/LectureNotes/_build/jupyter_execute/chapter11_91_4.png new file mode 100644 index 000000000..5d106940e Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapter11_91_4.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter12.ipynb b/doc/LectureNotes/_build/jupyter_execute/chapter12.ipynb new file mode 100644 index 000000000..b8432d08a --- /dev/null +++ b/doc/LectureNotes/_build/jupyter_execute/chapter12.ipynb @@ -0,0 +1,1942 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "b1ad0b52", + "metadata": { + "editable": true + }, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "id": "9548423d", + "metadata": { + "editable": true + }, + "source": [ + "# Convolutional Neural Networks\n", + "\n", + "Convolutional neural networks (CNNs) were developed during the last\n", + "decade of the previous century, with a focus on character recognition\n", + "tasks. Nowadays, CNNs are a central element in the spectacular success\n", + "of deep learning methods. The success in for example image\n", + "classifications have made them a central tool for most machine\n", + "learning practitioners.\n", + "\n", + "CNNs are very similar to ordinary Neural Networks.\n", + "They are made up of neurons that have learnable weights and\n", + "biases. Each neuron receives some inputs, performs a dot product and\n", + "optionally follows it with a non-linearity. The whole network still\n", + "expresses a single differentiable score function: from the raw image\n", + "pixels on one end to class scores at the other. And they still have a\n", + "loss function (for example Softmax) on the last (fully-connected) layer\n", + "and all the tips/tricks we developed for learning regular Neural\n", + "Networks still apply (back propagation, gradient descent etc etc).\n", + "\n", + "**CNN architectures make the explicit assumption that\n", + "the inputs are images, which allows us to encode certain properties\n", + "into the architecture. These then make the forward function more\n", + "efficient to implement and vastly reduce the amount of parameters in\n", + "the network.**\n", + "\n", + "Here we provide only a superficial overview, for the more interested, we recommend highly the course\n", + "[IN5400 – Machine Learning for Image Analysis](https://www.uio.no/studier/emner/matnat/ifi/IN5400/index-eng.html)\n", + "and the slides of [CS231](http://cs231n.github.io/convolutional-networks/).\n", + "\n", + "Another good read is the article here ." + ] + }, + { + "cell_type": "markdown", + "id": "315b8308", + "metadata": { + "editable": true + }, + "source": [ + "## Neural Networks vs CNNs\n", + "\n", + "Neural networks are defined as **affine transformations**, that is \n", + "a vector is received as input and is multiplied with a matrix of so-called weights (our unknown paramters) to produce an\n", + "output (to which a bias vector is usually added before passing the result\n", + "through a nonlinear activation function). This is applicable to any type of input, be it an\n", + "image, a sound clip or an unordered collection of features: whatever their\n", + "dimensionality, their representation can always be flattened into a vector\n", + "before the transformation.\n", + "\n", + "However, when we consider images, sound clips and many other similar kinds of data, these data have an intrinsic\n", + "structure. More formally, they share these important properties:\n", + "* They are stored as multi-dimensional arrays (think of the pixels of a figure) .\n", + "\n", + "* They feature one or more axes for which ordering matters (e.g., width and height axes for an image, time axis for a sound clip).\n", + "\n", + "* One axis, called the channel axis, is used to access different views of the data (e.g., the red, green and blue channels of a color image, or the left and right channels of a stereo audio track).\n", + "\n", + "These properties are not exploited when an affine transformation is applied; in\n", + "fact, all the axes are treated in the same way and the topological information\n", + "is not taken into account. Still, taking advantage of the implicit structure of\n", + "the data may prove very handy in solving some tasks, like computer vision and\n", + "speech recognition, and in these cases it would be best to preserve it. This is\n", + "where discrete convolutions come into play.\n", + "\n", + "A discrete convolution is a linear transformation that preserves this notion of\n", + "ordering. It is sparse (only a few input units contribute to a given output\n", + "unit) and reuses parameters (the same weights are applied to multiple locations\n", + "in the input).\n", + "\n", + "As an example, consider\n", + "an image of size $32\\times 32\\times 3$ (32 wide, 32 high, 3 color channels), so a\n", + "single fully-connected neuron in a first hidden layer of a regular\n", + "Neural Network would have $32\\times 32\\times 3 = 3072$ weights. This amount still\n", + "seems manageable, but clearly this fully-connected structure does not\n", + "scale to larger images. For example, an image of more respectable\n", + "size, say $200\\times 200\\times 3$, would lead to neurons that have \n", + "$200\\times 200\\times 3 = 120,000$ weights. \n", + "\n", + "We could have\n", + "several such neurons, and the parameters would add up quickly! Clearly,\n", + "this full connectivity is wasteful and the huge number of parameters\n", + "would quickly lead to possible overfitting.\n", + "\n", + "\n", + "\n", + "\n", + "

    Figure 1: A regular 3-layer Neural Network.

    \n", + "\n", + "\n", + "Convolutional Neural Networks take advantage of the fact that the\n", + "input consists of images and they constrain the architecture in a more\n", + "sensible way. \n", + "\n", + "In particular, unlike a regular Neural Network, the\n", + "layers of a CNN have neurons arranged in 3 dimensions: width,\n", + "height, depth. (Note that the word depth here refers to the third\n", + "dimension of an activation volume, not to the depth of a full Neural\n", + "Network, which can refer to the total number of layers in a network.)\n", + "\n", + "To understand it better, the above example of an image \n", + "with an input volume of\n", + "activations has dimensions $32\\times 32\\times 3$ (width, height,\n", + "depth respectively). \n", + "\n", + "The neurons in a layer will\n", + "only be connected to a small region of the layer before it, instead of\n", + "all of the neurons in a fully-connected manner. Moreover, the final\n", + "output layer could for this specific image have dimensions $1\\times 1 \\times 10$, \n", + "because by the\n", + "end of the CNN architecture we will reduce the full image into a\n", + "single vector of class scores, arranged along the depth\n", + "dimension. \n", + "\n", + "\n", + "\n", + "\n", + "

    Figure 1: A CNN arranges its neurons in three dimensions (width, height, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D output volume of neuron activations. In this example, the red input layer holds the image, so its width and height would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels).

    \n", + "" + ] + }, + { + "cell_type": "markdown", + "id": "a142cd5d", + "metadata": { + "editable": true + }, + "source": [ + "## Layers used to build CNNs\n", + "\n", + "A simple CNN is a sequence of layers, and every layer of a CNN\n", + "transforms one volume of activations to another through a\n", + "differentiable function. We use three main types of layers to build\n", + "CNN architectures: Convolutional Layer, Pooling Layer, and\n", + "Fully-Connected Layer (exactly as seen in regular Neural Networks). We\n", + "will stack these layers to form a full CNN architecture.\n", + "\n", + "A simple CNN for image classification could have the architecture:\n", + "\n", + "* **INPUT** ($32\\times 32 \\times 3$) will hold the raw pixel values of the image, in this case an image of width 32, height 32, and with three color channels R,G,B.\n", + "\n", + "* **CONV** (convolutional )layer will compute the output of neurons that are connected to local regions in the input, each computing a dot product between their weights and a small region they are connected to in the input volume. This may result in volume such as $[32\\times 32\\times 12]$ if we decided to use 12 filters.\n", + "\n", + "* **RELU** layer will apply an elementwise activation function, such as the $max(0,x)$ thresholding at zero. This leaves the size of the volume unchanged ($[32\\times 32\\times 12]$).\n", + "\n", + "* **POOL** (pooling) layer will perform a downsampling operation along the spatial dimensions (width, height), resulting in volume such as $[16\\times 16\\times 12]$.\n", + "\n", + "* **FC** (i.e. fully-connected) layer will compute the class scores, resulting in volume of size $[1\\times 1\\times 10]$, where each of the 10 numbers correspond to a class score, such as among the 10 categories of the MNIST images we considered above . As with ordinary Neural Networks and as the name implies, each neuron in this layer will be connected to all the numbers in the previous volume.\n", + "\n", + "CNNs transform the original image layer by layer from the original\n", + "pixel values to the final class scores. \n", + "\n", + "Observe that some layers contain\n", + "parameters and other don’t. In particular, the CNN layers perform\n", + "transformations that are a function of not only the activations in the\n", + "input volume, but also of the parameters (the weights and biases of\n", + "the neurons). On the other hand, the RELU/POOL layers will implement a\n", + "fixed function. The parameters in the CONV/FC layers will be trained\n", + "with gradient descent so that the class scores that the CNN computes\n", + "are consistent with the labels in the training set for each image.\n", + "\n", + "In summary:\n", + "\n", + "* A CNN architecture is in the simplest case a list of Layers that transform the image volume into an output volume (e.g. holding the class scores)\n", + "\n", + "* There are a few distinct types of Layers (e.g. CONV/FC/RELU/POOL are by far the most popular)\n", + "\n", + "* Each Layer accepts an input 3D volume and transforms it to an output 3D volume through a differentiable function\n", + "\n", + "* Each Layer may or may not have parameters (e.g. CONV/FC do, RELU/POOL don’t)\n", + "\n", + "* Each Layer may or may not have additional hyperparameters (e.g. CONV/FC/POOL do, RELU doesn’t)\n", + "\n", + "A dense neural network is representd by an affine operation (like matrix-matrix multiplication) where all parameters are included.\n", + "\n", + "The key idea in CNNs for say imaging is that in images neighbor pixels tend to be related! So we connect\n", + "only neighboring neurons in the input instead of connecting all with the first hidden layer.\n", + "\n", + "We say we perform a filtering (convolution is the mathematical operation)." + ] + }, + { + "cell_type": "markdown", + "id": "bb1e3617", + "metadata": { + "editable": true + }, + "source": [ + "## Mathematics of CNNs\n", + "\n", + "The mathematics of CNNs is based on the mathematical operation of\n", + "**convolution**. In mathematics (in particular in functional analysis),\n", + "convolution is represented by matheematical operation (integration,\n", + "summation etc) on two function in order to produce a third function\n", + "that expresses how the shape of one gets modified by the other.\n", + "Convolution has a plethora of applications in a variety of disciplines, spanning from statistics to signal processing, computer vision, solutions of differential equations,linear algebra, engineering, and yes, machine learning.\n", + "\n", + "Mathematically, convolution is defined as follows (one-dimensional example):\n", + "Let us define a continuous function $y(t)$ given by" + ] + }, + { + "cell_type": "markdown", + "id": "051485e5", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "y(t) = \\int x(a) w(t-a) da,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "799b8e5f", + "metadata": { + "editable": true + }, + "source": [ + "where $x(a)$ represents a so-called input and $w(t-a)$ is normally called the weight function or kernel.\n", + "\n", + "The above integral is written in a more compact form as" + ] + }, + { + "cell_type": "markdown", + "id": "3353345f", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "y(t) = \\left(x * w\\right)(t).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "ef341ee6", + "metadata": { + "editable": true + }, + "source": [ + "The discretized version reads" + ] + }, + { + "cell_type": "markdown", + "id": "ca8f7582", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "y(t) = \\sum_{a=-\\infty}^{a=\\infty}x(a)w(t-a).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "216e011f", + "metadata": { + "editable": true + }, + "source": [ + "Computing the inverse of the above convolution operations is known as deconvolution.\n", + "\n", + "How can we use this? And what does it mean? Let us study some familiar examples first." + ] + }, + { + "cell_type": "markdown", + "id": "5545738c", + "metadata": { + "editable": true + }, + "source": [ + "### Convolution Examples: Polynomial multiplication\n", + "\n", + "We have already met such an example in project 1 when we tried to set\n", + "up the design matrix for a two-dimensional function. This was an\n", + "example of polynomial multiplication. Let us recast such a problem in terms of the convolution operation.\n", + "Let us look a the following polynomials to second and third order, respectively:" + ] + }, + { + "cell_type": "markdown", + "id": "9f317908", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(t) = \\alpha_0+\\alpha_1 t+\\alpha_2 t^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "eb6f1070", + "metadata": { + "editable": true + }, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "9fe61b3c", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "s(t) = \\beta_0+\\beta_1 t+\\beta_2 t^2+\\beta_3 t^3.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "38c68ea7", + "metadata": { + "editable": true + }, + "source": [ + "The polynomial multiplication gives us a new polynomial of degree $5$" + ] + }, + { + "cell_type": "markdown", + "id": "3e91a9f1", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "z(t) = \\delta_0+\\delta_1 t+\\delta_2 t^2+\\delta_3 t^3+\\delta_4 t^4+\\delta_5 t^5.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "5e24ac44", + "metadata": { + "editable": true + }, + "source": [ + "Computing polynomial products can be implemented efficiently if we rewrite the more brute force multiplications using convolution.\n", + "We note first that the new coefficients are given as" + ] + }, + { + "cell_type": "markdown", + "id": "ac215eb9", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{split}\n", + "\\delta_0=&\\alpha_0\\beta_0\\\\\n", + "\\delta_1=&\\alpha_1\\beta_0+\\alpha_1\\beta_0\\\\\n", + "\\delta_2=&\\alpha_0\\beta_2+\\alpha_1\\beta_1+\\alpha_2\\beta_0\\\\\n", + "\\delta_3=&\\alpha_1\\beta_2+\\alpha_2\\beta_1+\\alpha_0\\beta_3\\\\\n", + "\\delta_4=&\\alpha_2\\beta_2+\\alpha_1\\beta_3\\\\\n", + "\\delta_5=&\\alpha_2\\beta_3.\\\\\n", + "\\end{split}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "eca51025", + "metadata": { + "editable": true + }, + "source": [ + "We note that $\\alpha_i=0$ except for $i\\in \\left\\{0,1,2\\right\\}$ and $\\beta_i=0$ except for $i\\in\\left\\{0,1,2,3\\right\\}$.\n", + "\n", + "We can then rewrite the coefficients $\\delta_j$ using a discrete convolution as" + ] + }, + { + "cell_type": "markdown", + "id": "0d517816", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\delta_j = \\sum_{i=-\\infty}^{i=\\infty}\\alpha_i\\beta_{j-i}=(\\alpha * \\beta)_j,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "8cf1037e", + "metadata": { + "editable": true + }, + "source": [ + "or as a double sum with restriction $l=i+j$" + ] + }, + { + "cell_type": "markdown", + "id": "efe18d99", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\delta_l = \\sum_{ij}\\alpha_i\\beta_{j}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "14ae41db", + "metadata": { + "editable": true + }, + "source": [ + "Do you see a potential drawback with these equations?\n", + "\n", + "Since we only have a finite number of $\\alpha$ and $\\beta$ values\n", + "which are non-zero, we can rewrite the above convolution expressions\n", + "as a matrix-vector multiplication" + ] + }, + { + "cell_type": "markdown", + "id": "046e108b", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{\\delta}=\\begin{bmatrix}\\alpha_0 & 0 & 0 & 0 \\\\\n", + " \\alpha_1 & \\alpha_0 & 0 & 0 \\\\\n", + "\t\t\t \\alpha_2 & \\alpha_1 & \\alpha_0 & 0 \\\\\n", + "\t\t\t 0 & \\alpha_2 & \\alpha_1 & \\alpha_0 \\\\\n", + "\t\t\t 0 & 0 & \\alpha_2 & \\alpha_1 \\\\\n", + "\t\t\t 0 & 0 & 0 & \\alpha_2\n", + "\t\t\t \\end{bmatrix}\\begin{bmatrix} \\beta_0 \\\\ \\beta_1 \\\\ \\beta_2 \\\\ \\beta_3\\end{bmatrix}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "0c73a726", + "metadata": { + "editable": true + }, + "source": [ + "The process is commutative and we can easily see that we can rewrite the multiplication in terms of a matrix holding $\\beta$ and a vector holding $\\alpha$.\n", + "In this case we have" + ] + }, + { + "cell_type": "markdown", + "id": "81bc84d0", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{\\delta}=\\begin{bmatrix}\\beta_0 & 0 & 0 \\\\\n", + " \\beta_1 & \\beta_0 & 0 \\\\\n", + "\t\t\t \\beta_2 & \\beta_1 & \\beta_0 \\\\\n", + "\t\t\t \\beta_3 & \\beta_2 & \\beta_1 \\\\\n", + "\t\t\t 0 & \\beta_3 & \\beta_2 \\\\\n", + "\t\t\t 0 & 0 & \\beta_3\n", + "\t\t\t \\end{bmatrix}\\begin{bmatrix} \\alpha_0 \\\\ \\alpha_1 \\\\ \\alpha_2\\end{bmatrix}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "6f314582", + "metadata": { + "editable": true + }, + "source": [ + "Note that the use of these matrices is for mathematical purposes only and not implementation purposes.\n", + "When implementing the above equation we do not encode (and allocate memory) the matrices explicitely.\n", + "We rather code the convolutions in the minimal memory footprint that they require.\n", + "\n", + "Does the number of floating point operations change here when we use the commutative property?" + ] + }, + { + "cell_type": "markdown", + "id": "e01eb276", + "metadata": { + "editable": true + }, + "source": [ + "### Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)\n", + "\n", + "For problems with so-called harmonic oscillations, given by for example the following differential equation" + ] + }, + { + "cell_type": "markdown", + "id": "1016b422", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "m\\frac{d^2x}{dt^2}+\\eta\\frac{dx}{dt}+x(t)=F(t),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "c237cb74", + "metadata": { + "editable": true + }, + "source": [ + "where $F(t)$ is an applied external force acting on the system (often called a driving force), one can use the theory of Fourier transformations to find the solutions of this type of equations.\n", + "\n", + "If one has several driving forces, $F(t)=\\sum_n F_n(t)$, one can find\n", + "the particular solution to each $F_n$, $x_{pn}(t)$, and the particular\n", + "solution for the entire driving force is then given by a series like" + ] + }, + { + "cell_type": "markdown", + "id": "4f2f2089", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + "x_p(t)=\\sum_nx_{pn}(t).\n", + "\\label{_auto1} \\tag{1}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "a32fa928", + "metadata": { + "editable": true + }, + "source": [ + "This is known as the principle of superposition. It only applies when\n", + "the homogenous equation is linear. If there were an anharmonic term\n", + "such as $x^3$ in the homogenous equation, then when one summed various\n", + "solutions, $x=(\\sum_n x_n)^2$, one would get cross\n", + "terms. Superposition is especially useful when $F(t)$ can be written\n", + "as a sum of sinusoidal terms, because the solutions for each\n", + "sinusoidal (sine or cosine) term is analytic. \n", + "\n", + "Driving forces are often periodic, even when they are not\n", + "sinusoidal. Periodicity implies that for some time $\\tau$" + ] + }, + { + "cell_type": "markdown", + "id": "0b6ca2c5", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{eqnarray}\n", + "F(t+\\tau)=F(t). \n", + "\\end{eqnarray}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "789cd636", + "metadata": { + "editable": true + }, + "source": [ + "One example of a non-sinusoidal periodic force is a square wave. Many\n", + "components in electric circuits are non-linear, e.g. diodes, which\n", + "makes many wave forms non-sinusoidal even when the circuits are being\n", + "driven by purely sinusoidal sources.\n", + "\n", + "The code here shows a typical example of such a square wave generated using the functionality included in the **scipy** Python package. We have used a period of $\\tau=0.2$." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "2c90eb89", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [ + { + "data": { + "image/png": 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ivCS/luDfAVzf7da5BDiQmY9Vsl2PFsVPtHtjppZOfQJNUS3poyzFD31pqSR0n6wscZKI+AJwKbAmInYDHwNWAWTmVmAbcCWwC3gKuKGE3aWGbAoojXOqE3JUe9ItfjWYmbuqFqrHS/GLCH5mXneczxN4TwlbS5kISMHIr8U4p7wefnPi12g9/PHSe2falkQXwx/ar4/sWafdd9XzXTVOTpt41dwjDp1pa+ZDnnjVkhCIK4QqnZwE9b2tSrwas5COBb8g+mqZDU3165ucRj6Tk22/VaDMqPYI38yDrt6IdrSrraVT37Zuqi8skifaoiitpYP34Zt5UId0FOjqjYjFr7pV9QhftFCtDOmMoeJb8AvT2rS3XfFrcAGzvll5hVDH8M2chGiIrxwFjdqvarP73pT4iUe70tLf1S17l445DqrqetNbFAVXc0JeKrglJ9dVRZVUjmyxWqamz31iwS+Ivt6IKs6pCG8ow1jqh74IjKO6t9V1osZL8i34BZHVG5HGOdXhjfrEaAMqMmOxnXr4M/Y1dsdL7i34RdGP8OsT0eZW1CnhaFf2QO/mYvghuc59YsEviDoJqc16I6r6Qa2Jn3qXjuY6j9uqrQW/KKrYrjrOWd3stE3ZAqZw0bbFeviK6+yHmJt5US/yaOKcqt0bQvGT15Wpj7wefkNPNusTC35BROtK+ql+Y7Fd3Uyus95Q6E6bb+HEKzMPg8U85QKmKM4pQF9LR5hpKwrdKRYwFf9PQzzCN/OirzdSH/lCtWiqr0C5SwdVzSS5Y69vt08s+AWRl0duqN6Iuh5+cyWhZfvgdbNXVeiuTyz4BdHFs9Vlc6ubbbIevnS0S4sL1c60NfMQsh0rnf2Wtq41ONWX70xq7RGH9U32jgW/ILL62VLxU5UKVotffZrMPRBuy1T1uU8s+AWRFdVSJ+RUt6qupSOuh9/SQ1/kz3oYL8W34BdkEOdsa096iJRAOdVHXQ+/pYe+yGP4AsM9YsEvSJu7dNQjfFWNlepmZ0a79U3rnNyI/dqo1uT6xIJfEP1OBlVCji7ZTBXP1vS5eyGrllnfLtPXWdNpZeJXH1jwC9JsPXzltsymdiYpt9+Kd6AJbMucXI9Y8AuiH+HXR1ZvRCl+aJ2czrELY/iS3VgO6cxKRFweEd+LiF0R8cFZPr80Ig5ExP3d10dL2F1yyKoo6hS/SfGTV46sj7qWjmqEP26JVysXe4KIWAF8GrgM2A18NyLuyMyHjzn07sy8arH2ljItbtfTOblp89XRzeTU22+VI/zqpl0Pfw42Absy8/uZeQT4InB1gfMuO9QJOartegqU4idzcjPmqxOyh7509hsK3fVJCcFfBzw68n5397NjeX1EPBARX42IV8x1sojYHBHbI2L73r17CzSvHvKRX33TS6BUcH10Tq6z31LoThmudD38WZntUhz7V7oPeElmXgT8KXD7XCfLzJszczIzJ9euXVugefXQ1xtpMNPW9fDrIC4s05KT65MSgr8bWD/y/lxgz+gBmXkwMw91r7cBqyJiTQHbSwrXw6+HfKpf3ap6cb69TFtV6K5PSgj+d4GNEXF+RKwGrgXuGD0gIs6ObvgZEZs6u/sK2F5STExosxEVySkTsh0ryqm+Vgg0yWbqWjqae3vcWPQuncw8GhHvBb4GrABuycydEfGu7vOtwDXAuyPiKPA0cG2O234nANkIv13xa+uhL519P7i9Ct6WOQddmGbbMT/bOvJ6C7ClhK2lTIhT82SPvhNYbTOMpdyTrsoi7+x7W2YRnGlbkDYzbbXip5h2y8oMyBeq69udeQaAYq3G9fDNPOhrhqv2KusKibW0S2e6zw3Ww1egKgzYJxb8grS4k0GebNZQQo6ySJ5ufaqz7pBOESz4BWmzHr76EYfVTcudnAL5+pRo5WLMBvgW/JIEMCWoMDVTYKqhqb5S/ERObtq+KAlJUjxtSuvYx22Mb8EvSJOLeSyFkWd9m8rEK5ljb66EhhOvzPEYsxvkeAyKaukSr5qK4YuL5GkHM+3MXvvEgl8Q+chPlBjY3KxG5eSG9qtb1m+/leUejNkQ34JfEP0uHc0oSPNA785+fdM/Zr+qTeVoF/UTr6qb9gjfzE+zu3Tqm5VP9TVOTjjaVT/TVubkqpvtFQt+QeSx3fqmhYt56jID9VGOdgcNUNhU7oZySMfMw8SEqt6IrsyArIqieqqvjOHLqqLWJ9E5OId0zLzIR/iyBcz6drXiJxKCTK34iWL4qjLFwfgpvgW/JPIYvuphIBovpxM/nZNTRXNkD30hdX0eP7234JdENiLQpp1qMjBTK36KolpTmZqHtqOtlilz7Lh4mpkH5TNtdf8UGieX6MRP6dd1o11dPXzVQ+PVD/fpAwt+QWT1RlI97dXEdnUjfJWTU49269udEl5olZPrEwt+QZQLW7KpPqoMTPXuDZWTEyq+aIFKum4xXnpvwS+JbH82yqm+MrzRmpMTjnaFjziU5R140dbMR5sLW8J6+K3tz5bG8FX3dgod+/gpvgW/ILoESOE/RYviJ3Ny2hi+bCbXWOiuTyz4JenuzOpiIIzptBvDr490tCvMLnYMvxwW/IIMb8yG9F6mutqpfqOhO4Fd6YaE8YvoWPBLMrwva98kqcw6HWlDTZTiJ3NyNBjDV2bauh6+mY8JUUhHWW9kps917aqn+tDWdVaNslN4oWU1k3rEgl+QaSGobFc98hu2oSZtOjnltsyuDYIRr+o6I6qZ1CcW/IJMi19tIRAnXg3aUHm0K96WOWhDXbSlFWbaUJOlEK4cJ4oIfkRcHhHfi4hdEfHBWT6PiLip+/zBiLi4hN2lxlB0a2/lUlcUHLShLtrSCsM21B/+6Rz78N6uy5KYvY7RMH/Rgh8RK4BPA1cALweui4iXH3PYFcDG7msz8JnF2l3KKEb4ynojUL+qYIorR0L92jJT4nr4wzbURFohlOG9LTHfCysLnGMTsCszvw8QEV8ErgYeHjnmauDWHLjKeyLijIg4JzMfK2B/yTC8L3fuOchzVtaLlj1x6Bn59HPnnoOsXlGvz/sOH5HWwwfYuecAq2r3uZq1H2dod+eeg6ycqNeK/co+T/8/H6i+jrByRXDB2c8vf94C51gHPDryfjfwugUcsw4YK8E/dfXgz/nrn/l2ddtnP/+51W0CnLp6BQC/9mf1+/zin9L0+ZRVgz6/TdDndWecUt0mwPOeM7i3f/XT/1zd9rlnivrc3dtv3VK/z2tOew7bP/ym4uctIfizub5jJ0ELOWZwYMRmBmEfNmzYsLiWVebtr13P+hecwtFn688Bz19zanWbAG9/7QY2vPB5TfX5uk0bOG+Nps8vXavp8zs2beD81vr8ug2cv+ZUnhXEdFb3FCEoIfi7gfUj788F9pzEMQBk5s3AzQCTk5PLKnr23FUr+KULzlI3oyqnrHafW6DFPj9v9UreeOF49bmEG/kusDEizo+I1cC1wB3HHHMHcH23W+cS4MC4xe+NMWaps+gRfmYejYj3Al8DVgC3ZObOiHhX9/lWYBtwJbALeAq4YbF2jTHGnBglQjpk5jYGoj76s60jrxN4TwlbxhhjTg5n2hpjTCNY8I0xphEs+MYY0wgWfGOMaQQLvjHGNIIF3xhjGsGCb4wxjWDBN8aYRrDgG2NMI1jwjTGmESz4xhjTCBZ8Y4xpBAu+McY0ggXfGGMawYJvjDGNYME3xphGsOAbY0wjWPCNMaYRLPjGGNMIFnxjjGkEC74xxjSCBd8YYxrBgm+MMY1gwTfGmEaw4BtjTCNY8I0xphEs+MYY0wgWfGOMaYSVi/nliHgB8FfAecB/Ab+RmT+Y5bj/Ap4EngWOZubkYuwaY4w5cRY7wv8g8I3M3Ah8o3s/F7+Yma+y2BtjjIbFCv7VwOe7158HfnWR5zPGGNMTiwrpAGdl5mMAmflYRLxojuMS+PuISODPM/PmuU4YEZuBzd3bQxHxvZNs2xrgiZP83eWK+9wG7nMbnGyfXzLXB5GZ8/5mRPwDcPYsH30I+HxmnjFy7A8y88xZzvHizNzTOYSvA7+bmXctsPEnRURsby185D63gfvcBn30+bgj/Mx80zwN+t+IOKcb3Z8DPD7HOfZ03x+PiNuATUCvgm+MMebHWWwM/w7gt7rXvwV85dgDIuLUiDh9+Br4ZWDHIu0aY4w5QRYr+B8HLouIfwcu694TES+OiG3dMWcB34qIB4B/Bf42M/9ukXYXwpzrBGOM+9wG7nMbFO/zcWP4xhhjxgNn2hpjTCNY8I0xphGWteBHxOUR8b2I2BURP5HlGwNu6j5/MCIuVrSzJAvo8292fX0wIr4dERcp2lmS4/V55LjXRsSzEXFNzfb1wUL6HBGXRsT9EbEzIu6s3cbSLODe/qmI+JuIeKDr8w2KdpYkIm6JiMcjYtaNLMU1LDOX5RewAvgP4KXAauAB4OXHHHMl8FUggEuA76jbXaHPPwuc2b2+ooU+jxz3j8A24Bp1uytc5zOAh4EN3fsXqdtdoc83An/UvV4L7AdWq9u+yH7/AnAxsGOOz4tq2HIe4W8CdmXm9zPzCPBFBqUeRrkauDUH3AOc0eULLFeO2+fM/HbOFLC7Bzi3chtLs5DrDPC7wJeYIxdkmbGQPr8D+HJm/jcMclwqt7E0C+lzAqdHRACnMRD8o3WbWZYcJKDun+eQohq2nAV/HfDoyPvd3c9O9JjlxIn257cZjA6WM8ftc0SsA94GbK3Yrj5ZyHX+GeDMiPiniLg3Iq6v1rp+WEiftwAXAnuAh4D3ZeZUnebJKKphi62loyRm+dmxe0wXcsxyYsH9iYhfZCD4P9dri/pnIX3+E+ADmfnsYPC37FlIn1cCrwHeCJwC/EtE3JOZ/9Z343piIX1+M3A/8EvATwNfj4i7M/Ngz21TUlTDlrPg7wbWj7w/l4HnP9FjlhML6k9EvBL4LHBFZu6r1La+WEifJ4EvdmK/BrgyIo5m5u1VWliehd7bT2TmYeBwRNwFXAQsV8FfSJ9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    " + ] + }, + "metadata": { + "filenames": { + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter12_35_0.png" + }, + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "%matplotlib inline\n", + "\n", + "import numpy as np\n", + "import math\n", + "from scipy import signal\n", + "import matplotlib.pyplot as plt\n", + "\n", + "# number of points \n", + "n = 500\n", + "# start and final times \n", + "t0 = 0.0\n", + "tn = 1.0\n", + "# Period \n", + "t = np.linspace(t0, tn, n, endpoint=False)\n", + "SqrSignal = np.zeros(n)\n", + "SqrSignal = 1.0+signal.square(2*np.pi*5*t)\n", + "plt.plot(t, SqrSignal)\n", + "plt.ylim(-0.5, 2.5)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "8e7b8f5c", + "metadata": { + "editable": true + }, + "source": [ + "For the sinusoidal example the\n", + "period is $\\tau=2\\pi/\\omega$. However, higher harmonics can also\n", + "satisfy the periodicity requirement. In general, any force that\n", + "satisfies the periodicity requirement can be expressed as a sum over\n", + "harmonics," + ] + }, + { + "cell_type": "markdown", + "id": "4453c025", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + "F(t)=\\frac{f_0}{2}+\\sum_{n>0} f_n\\cos(2n\\pi t/\\tau)+g_n\\sin(2n\\pi t/\\tau).\n", + "\\label{_auto2} \\tag{2}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "32d5f410", + "metadata": { + "editable": true + }, + "source": [ + "We can write down the answer for\n", + "$x_{pn}(t)$, by substituting $f_n/m$ or $g_n/m$ for $F_0/m$. By\n", + "writing each factor $2n\\pi t/\\tau$ as $n\\omega t$, with $\\omega\\equiv\n", + "2\\pi/\\tau$," + ] + }, + { + "cell_type": "markdown", + "id": "4a58698d", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\\label{eq:fourierdef1} \\tag{3}\n", + "F(t)=\\frac{f_0}{2}+\\sum_{n>0}f_n\\cos(n\\omega t)+g_n\\sin(n\\omega t).\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "5664f63d", + "metadata": { + "editable": true + }, + "source": [ + "The solutions for $x(t)$ then come from replacing $\\omega$ with\n", + "$n\\omega$ for each term in the particular solution," + ] + }, + { + "cell_type": "markdown", + "id": "0d9a2811", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{eqnarray}\n", + "x_p(t)&=&\\frac{f_0}{2k}+\\sum_{n>0} \\alpha_n\\cos(n\\omega t-\\delta_n)+\\beta_n\\sin(n\\omega t-\\delta_n),\\\\\n", + "\\nonumber\n", + "\\alpha_n&=&\\frac{f_n/m}{\\sqrt{((n\\omega)^2-\\omega_0^2)+4\\beta^2n^2\\omega^2}},\\\\\n", + "\\nonumber\n", + "\\beta_n&=&\\frac{g_n/m}{\\sqrt{((n\\omega)^2-\\omega_0^2)+4\\beta^2n^2\\omega^2}},\\\\\n", + "\\nonumber\n", + "\\delta_n&=&\\tan^{-1}\\left(\\frac{2\\beta n\\omega}{\\omega_0^2-n^2\\omega^2}\\right).\n", + "\\end{eqnarray}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "ac8b5fff", + "metadata": { + "editable": true + }, + "source": [ + "Because the forces have been applied for a long time, any non-zero\n", + "damping eliminates the homogenous parts of the solution, so one need\n", + "only consider the particular solution for each $n$.\n", + "\n", + "The problem is considered solved if one can find expressions for the\n", + "coefficients $f_n$ and $g_n$, even though the solutions are expressed\n", + "as an infinite sum. The coefficients can be extracted from the\n", + "function $F(t)$ by" + ] + }, + { + "cell_type": "markdown", + "id": "5bb70ae9", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{eqnarray}\n", + "\\label{eq:fourierdef2} \\tag{4}\n", + "f_n&=&\\frac{2}{\\tau}\\int_{-\\tau/2}^{\\tau/2} dt~F(t)\\cos(2n\\pi t/\\tau),\\\\\n", + "\\nonumber\n", + "g_n&=&\\frac{2}{\\tau}\\int_{-\\tau/2}^{\\tau/2} dt~F(t)\\sin(2n\\pi t/\\tau).\n", + "\\end{eqnarray}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "4f64082e", + "metadata": { + "editable": true + }, + "source": [ + "To check the consistency of these expressions and to verify\n", + "Eq. ([4](#eq:fourierdef2)), one can insert the expansion of $F(t)$ in\n", + "Eq. ([3](#eq:fourierdef1)) into the expression for the coefficients in\n", + "Eq. ([4](#eq:fourierdef2)) and see whether" + ] + }, + { + "cell_type": "markdown", + "id": "bdd9e4e8", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{eqnarray}\n", + "f_n&=?&\\frac{2}{\\tau}\\int_{-\\tau/2}^{\\tau/2} dt~\\left\\{\n", + "\\frac{f_0}{2}+\\sum_{m>0}f_m\\cos(m\\omega t)+g_m\\sin(m\\omega t)\n", + "\\right\\}\\cos(n\\omega t).\n", + "\\end{eqnarray}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e64a4a4c", + "metadata": { + "editable": true + }, + "source": [ + "Immediately, one can throw away all the terms with $g_m$ because they\n", + "convolute an even and an odd function. The term with $f_0/2$\n", + "disappears because $\\cos(n\\omega t)$ is equally positive and negative\n", + "over the interval and will integrate to zero. For all the terms\n", + "$f_m\\cos(m\\omega t)$ appearing in the sum, one can use angle addition\n", + "formulas to see that $\\cos(m\\omega t)\\cos(n\\omega\n", + "t)=(1/2)(\\cos[(m+n)\\omega t]+\\cos[(m-n)\\omega t]$. This will integrate\n", + "to zero unless $m=n$. In that case the $m=n$ term gives" + ] + }, + { + "cell_type": "markdown", + "id": "d51a4f59", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\\int_{-\\tau/2}^{\\tau/2}dt~\\cos^2(m\\omega t)=\\frac{\\tau}{2},\n", + "\\label{_auto3} \\tag{5}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "44804e35", + "metadata": { + "editable": true + }, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "7de0dbe5", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{eqnarray}\n", + "f_n&=?&\\frac{2}{\\tau}\\int_{-\\tau/2}^{\\tau/2} dt~f_n/2\\\\\n", + "\\nonumber\n", + "&=&f_n~\\checkmark.\n", + "\\end{eqnarray}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "4d1a02e0", + "metadata": { + "editable": true + }, + "source": [ + "The same method can be used to check for the consistency of $g_n$.\n", + "\n", + "The code here uses the Fourier series applied to a \n", + "square wave signal. The code here\n", + "visualizes the various approximations given by Fourier series compared\n", + "with a square wave with period $T=0.2$ (dimensionless time), width $0.1$ and max value of the force $F=2$. We\n", + "see that when we increase the number of components in the Fourier\n", + "series, the Fourier series approximation gets closer and closer to the\n", + "square wave signal." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "4ae0ec42", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
    " + ] + }, + "metadata": { + "filenames": { + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter12_51_0.png" + }, + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "import numpy as np\n", + "import math\n", + "from scipy import signal\n", + "import matplotlib.pyplot as plt\n", + "\n", + "# number of points \n", + "n = 500\n", + "# start and final times \n", + "t0 = 0.0\n", + "tn = 1.0\n", + "# Period \n", + "T =0.2\n", + "# Max value of square signal \n", + "Fmax= 2.0\n", + "# Width of signal \n", + "Width = 0.1\n", + "t = np.linspace(t0, tn, n, endpoint=False)\n", + "SqrSignal = np.zeros(n)\n", + "FourierSeriesSignal = np.zeros(n)\n", + "SqrSignal = 1.0+signal.square(2*np.pi*5*t+np.pi*Width/T)\n", + "a0 = Fmax*Width/T\n", + "FourierSeriesSignal = a0\n", + "Factor = 2.0*Fmax/np.pi\n", + "for i in range(1,500):\n", + " FourierSeriesSignal += Factor/(i)*np.sin(np.pi*i*Width/T)*np.cos(i*t*2*np.pi/T)\n", + "plt.plot(t, SqrSignal)\n", + "plt.plot(t, FourierSeriesSignal)\n", + "plt.ylim(-0.5, 2.5)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "acd75ab0", + "metadata": { + "editable": true + }, + "source": [ + "## Two-dimensional Objects\n", + "\n", + "We often use convolutions over more than one dimension at a time. If\n", + "we have a two-dimensional image $I$ as input, we can have a **filter**\n", + "defined by a two-dimensional **kernel** $K$. This leads to an output $S$" + ] + }, + { + "cell_type": "markdown", + "id": "0cdf3af8", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "S_(i,j)=(I * K)(i,j) = \\sum_m\\sum_n I(m,n)K(i-m,j-n).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "1640aae2", + "metadata": { + "editable": true + }, + "source": [ + "Convolution is a commutatitave process, which means we can rewrite this equation as" + ] + }, + { + "cell_type": "markdown", + "id": "11491f4c", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "S_(i,j)=(I * K)(i,j) = \\sum_m\\sum_n I(i-m,j-n)K(m,n).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "a1e3ef9d", + "metadata": { + "editable": true + }, + "source": [ + "Normally the latter is more straightforward to implement in a machine elarning library since there is less variation in the range of values of $m$ and $n$.\n", + "\n", + "Many deep learning libraries implement cross-correlation instead of convolution" + ] + }, + { + "cell_type": "markdown", + "id": "919fb5e9", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "S_(i,j)=(I * K)(i,j) = \\sum_m\\sum_n I(i+m,j-+)K(m,n).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "c42d55ac", + "metadata": { + "editable": true + }, + "source": [ + "## More on Dimensionalities\n", + "\n", + "In fields like signal processing (and imaging as well), one designs\n", + "so-called filters. These filters are defined by the convolutions and\n", + "are often hand-crafted. One may specify filters for smoothing, edge\n", + "detection, frequency reshaping, and similar operations. However with\n", + "neural networks the idea is to automatically learn the filters and use\n", + "many of them in conjunction with non-linear operations (activation\n", + "functions).\n", + "\n", + "As an example consider a neural network operating on sound sequence\n", + "data. Assume that we an input vector $\\boldsymbol{x}$ of length $d=10^6$. We\n", + "construct then a neural network with onle hidden layer only with\n", + "$10^4$ nodes. This means that we will have a weight matrix with\n", + "$10^4\\times 10^6=10^{10}$ weights to be determined, together with $10^4$ biases.\n", + "\n", + "Assume furthermore that we have an output layer which is meant to train whether the sound sequence represents a human voice (true) or something else (false).\n", + "It means that we have only one output node. But since this output node connects to $10^4$ nodes in the hidden layer, there are in total $10^4$ weights to be determined for the output layer, plus one bias. In total we have" + ] + }, + { + "cell_type": "markdown", + "id": "4ebd8c85", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\mathrm{NumberParameters}=10^{10}+10^4+10^4+1 \\approx 10^{10},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "6d886fa7", + "metadata": { + "editable": true + }, + "source": [ + "that is ten billion parameters to determine." + ] + }, + { + "cell_type": "markdown", + "id": "cd3b731e", + "metadata": { + "editable": true + }, + "source": [ + "## Further Dimensionality Remarks\n", + "\n", + "In today’s architecture one can train such neural networks, however\n", + "this is a huge number of parameters for the task at hand. In general,\n", + "it is a very wasteful and inefficient use of dense matrices as\n", + "parameters. Just as importantly, such trained network parameters are\n", + "very specific for the type of input data on which they were trained\n", + "and the network is not likely to generalize easily to variations in\n", + "the input.\n", + "\n", + "The main principles that justify convolutions is locality of\n", + "information and repetion of patterns within the signal. Sound samples\n", + "of the input in adjacent spots are much more likely to affect each\n", + "other than those that are very far away. Similarly, sounds are\n", + "repeated in multiple times in the signal. While slightly simplistic,\n", + "reasoning about such a sound example demonstrates this. The same\n", + "principles then apply to images and other similar data." + ] + }, + { + "cell_type": "markdown", + "id": "fc38055c", + "metadata": { + "editable": true + }, + "source": [ + "## CNNs in more detail, building convolutional neural networks in Tensorflow and Keras\n", + "\n", + "As discussed above, CNNs are neural networks built from the assumption that the inputs\n", + "to the network are 2D images. This is important because the number of features or pixels in images\n", + "grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network. \n", + "\n", + "As before, we still have our input, a hidden layer and an output. What's novel about convolutional networks\n", + "are the **convolutional** and **pooling** layers stacked in pairs between the input and the hidden layer.\n", + "In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D\n", + "matrices, typically 1 for each color dimension (Red, Green, Blue). \n", + "\n", + "It means that to represent the entire\n", + "dataset of images, we require a 4D matrix or **tensor**. This tensor has the dimensions:" + ] + }, + { + "cell_type": "markdown", + "id": "c153f6eb", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "(n_{inputs},\\, n_{pixels, width},\\, n_{pixels, height},\\, depth) .\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "f45ab21d", + "metadata": { + "editable": true + }, + "source": [ + "### The MNIST dataset again\n", + "\n", + "The MNIST dataset consists of grayscale images with a pixel size of\n", + "$28\\times 28$, meaning we require $28 \\times 28 = 724$ weights to each\n", + "neuron in the first hidden layer.\n", + "\n", + "If we were to analyze images of size $128\\times 128$ we would require\n", + "$128 \\times 128 = 16384$ weights to each neuron. Even worse if we were\n", + "dealing with color images, as most images are, we have an image matrix\n", + "of size $128\\times 128$ for each color dimension (Red, Green, Blue),\n", + "meaning 3 times the number of weights $= 49152$ are required for every\n", + "single neuron in the first hidden layer.\n", + "\n", + "Images typically have strong local correlations, meaning that a small\n", + "part of the image varies little from its neighboring regions. If for\n", + "example we have an image of a blue car, we can roughly assume that a\n", + "small blue part of the image is surrounded by other blue regions.\n", + "\n", + "Therefore, instead of connecting every single pixel to a neuron in the\n", + "first hidden layer, as we have previously done with deep neural\n", + "networks, we can instead connect each neuron to a small part of the\n", + "image (in all 3 RGB depth dimensions). The size of each small area is\n", + "fixed, and known as a [receptive](https://en.wikipedia.org/wiki/Receptive_field).\n", + "\n", + "The layers of a convolutional neural network arrange neurons in 3D: width, height and depth. \n", + "The input image is typically a square matrix of depth 3. \n", + "\n", + "A **convolution** is performed on the image which outputs\n", + "a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as **filters**.\n", + "\n", + "Each filter slides along the input image, taking the dot product\n", + "between each small part of the image and the filter, in all depth\n", + "dimensions. This is then passed through a non-linear function,\n", + "typically the **Rectified Linear (ReLu)** function, which serves as the\n", + "activation of the neurons in the first convolutional layer. This is\n", + "further passed through a **pooling layer**, which reduces the size of the\n", + "convolutional layer, e.g. by taking the maximum or average across some\n", + "small regions, and this serves as input to the next convolutional\n", + "layer." + ] + }, + { + "cell_type": "markdown", + "id": "4af74ca7", + "metadata": { + "editable": true + }, + "source": [ + "### Systematic reduction\n", + "\n", + "By systematically reducing the size of the input volume, through\n", + "convolution and pooling, the network should create representations of\n", + "small parts of the input, and then from them assemble representations\n", + "of larger areas. The final pooling layer is flattened to serve as\n", + "input to a hidden layer, such that each neuron in the final pooling\n", + "layer is connected to every single neuron in the hidden layer. This\n", + "then serves as input to the output layer, e.g. a softmax output for\n", + "classification." + ] + }, + { + "cell_type": "markdown", + "id": "2b435dc0", + "metadata": { + "editable": true + }, + "source": [ + "### Prerequisites: Collect and pre-process data" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "ffb4904f", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "inputs = (n_inputs, pixel_width, pixel_height, depth) = (1797, 8, 8, 1)\n", + "labels = (n_inputs) = (1797,)\n" + ] + }, + { + "data": { + "image/png": "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\n", + "text/plain": [ + "
    " + ] + }, + "metadata": { + "filenames": { + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter12_67_1.png" + }, + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "# import necessary packages\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from sklearn import datasets\n", + "\n", + "\n", + "# ensure the same random numbers appear every time\n", + "np.random.seed(0)\n", + "\n", + "# display images in notebook\n", + "%matplotlib inline\n", + "plt.rcParams['figure.figsize'] = (12,12)\n", + "\n", + "\n", + "# download MNIST dataset\n", + "digits = datasets.load_digits()\n", + "\n", + "# define inputs and labels\n", + "inputs = digits.images\n", + "labels = digits.target\n", + "\n", + "# RGB images have a depth of 3\n", + "# our images are grayscale so they should have a depth of 1\n", + "inputs = inputs[:,:,:,np.newaxis]\n", + "\n", + "print(\"inputs = (n_inputs, pixel_width, pixel_height, depth) = \" + str(inputs.shape))\n", + "print(\"labels = (n_inputs) = \" + str(labels.shape))\n", + "\n", + "\n", + "# choose some random images to display\n", + "n_inputs = len(inputs)\n", + "indices = np.arange(n_inputs)\n", + "random_indices = np.random.choice(indices, size=5)\n", + "\n", + "for i, image in enumerate(digits.images[random_indices]):\n", + " plt.subplot(1, 5, i+1)\n", + " plt.axis('off')\n", + " plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')\n", + " plt.title(\"Label: %d\" % digits.target[random_indices[i]])\n", + "plt.show()" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "id": "bb2db46c", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "from tensorflow.keras import datasets, layers, models\n", + "from tensorflow.keras.layers import Input\n", + "from tensorflow.keras.models import Sequential #This allows appending layers to existing models\n", + "from tensorflow.keras.layers import Dense #This allows defining the characteristics of a particular layer\n", + "from tensorflow.keras import optimizers #This allows using whichever optimiser we want (sgd,adam,RMSprop)\n", + "from tensorflow.keras import regularizers #This allows using whichever regularizer we want (l1,l2,l1_l2)\n", + "from tensorflow.keras.utils import to_categorical #This allows using categorical cross entropy as the cost function\n", + "#from tensorflow.keras import Conv2D\n", + "#from tensorflow.keras import MaxPooling2D\n", + "#from tensorflow.keras import Flatten\n", + "\n", + "from sklearn.model_selection import train_test_split\n", + "\n", + "# representation of labels\n", + "labels = to_categorical(labels)\n", + "\n", + "# split into train and test data\n", + "# one-liner from scikit-learn library\n", + "train_size = 0.8\n", + "test_size = 1 - train_size\n", + "X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,\n", + " test_size=test_size)" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "id": "100a8d6d", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def create_convolutional_neural_network_keras(input_shape, receptive_field,\n", + " n_filters, n_neurons_connected, n_categories,\n", + " eta, lmbd):\n", + " model = Sequential()\n", + " model.add(layers.Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same',\n", + " activation='relu', kernel_regularizer=regularizers.l2(lmbd)))\n", + " model.add(layers.MaxPooling2D(pool_size=(2, 2)))\n", + " model.add(layers.Flatten())\n", + " model.add(layers.Dense(n_neurons_connected, activation='relu', kernel_regularizer=regularizers.l2(lmbd)))\n", + " model.add(layers.Dense(n_categories, activation='softmax', kernel_regularizer=regularizers.l2(lmbd)))\n", + " \n", + " sgd = optimizers.SGD(lr=eta)\n", + " model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])\n", + " \n", + " return model\n", + "\n", + "epochs = 100\n", + "batch_size = 100\n", + "input_shape = X_train.shape[1:4]\n", + "receptive_field = 3\n", + "n_filters = 10\n", + "n_neurons_connected = 50\n", + "n_categories = 10\n", + "\n", + "eta_vals = np.logspace(-5, 1, 7)\n", + "lmbd_vals = np.logspace(-5, 1, 7)" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "id": "635da5a7", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "2021-12-08 06:58:03.630224: I tensorflow/core/platform/cpu_feature_guard.cc:142] This TensorFlow binary is optimized with oneAPI Deep Neural Network Library (oneDNN) to use the following CPU instructions in performance-critical operations: SSE4.1 SSE4.2 AVX AVX2 FMA\n", + "To enable them in other operations, rebuild TensorFlow with the appropriate compiler flags.\n", + "/Users/MortenImac/anaconda3/lib/python3.8/site-packages/keras/optimizer_v2/optimizer_v2.py:355: UserWarning: The `lr` argument is deprecated, use `learning_rate` instead.\n", + " warnings.warn(\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "2021-12-08 06:58:04.114437: I tensorflow/compiler/mlir/mlir_graph_optimization_pass.cc:185] None of the MLIR Optimization Passes are enabled (registered 2)\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\r", + " 1/12 [=>............................] - ETA: 1s - loss: 2.4411 - accuracy: 0.2500" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\r", + "12/12 [==============================] - 0s 853us/step - loss: 2.4410 - accuracy: 0.1944\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Learning rate = 1e-05\n", + "Lambda = 1e-05\n", + "Test accuracy: 0.194\n", + "\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\r", + " 1/12 [=>............................] - ETA: 0s - loss: 4.2597 - accuracy: 0.0938" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\r", + "12/12 [==============================] - 0s 842us/step - loss: 3.4744 - accuracy: 0.1361\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Learning rate = 1e-05\n", + "Lambda = 0.0001\n", + "Test accuracy: 0.136\n", + "\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\r", + " 1/12 [=>............................] - ETA: 1s - loss: 3.0466 - accuracy: 0.1562" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\r", + "12/12 [==============================] - 0s 743us/step - loss: 2.9400 - accuracy: 0.1028\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Learning rate = 1e-05\n", + "Lambda = 0.001\n", + "Test accuracy: 0.103\n", + "\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\r", + " 1/12 [=>............................] - ETA: 1s - loss: 3.8064 - accuracy: 0.1250" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\r", + "12/12 [==============================] - 0s 912us/step - loss: 3.8175 - accuracy: 0.1056\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Learning rate = 1e-05\n", + "Lambda = 0.01\n", + "Test accuracy: 0.106\n", + "\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\r", + " 1/12 [=>............................] - ETA: 1s - loss: 12.1612 - accuracy: 0.0938" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\r", + "12/12 [==============================] - 0s 763us/step - loss: 12.3063 - accuracy: 0.1361\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Learning rate = 1e-05\n", + "Lambda = 0.1\n", + "Test accuracy: 0.136\n", + "\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\r", + " 1/12 [=>............................] - ETA: 0s - loss: 91.8768 - accuracy: 0.2812" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\r", + "12/12 [==============================] - 0s 922us/step - loss: 92.0923 - accuracy: 0.2972\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Learning rate = 1e-05\n", + "Lambda = 1.0\n", + "Test accuracy: 0.297\n", + "\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\r", + " 1/12 [=>............................] - ETA: 0s - loss: 529.5700 - accuracy: 0.2188" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\r", + "12/12 [==============================] - 0s 935us/step - loss: 529.7050 - accuracy: 0.1861\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Learning rate = 1e-05\n", + "Lambda = 10.0\n", + "Test accuracy: 0.186\n", + "\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\r", + " 1/12 [=>............................] - ETA: 1s - loss: 1.2495 - accuracy: 0.5312" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\r", + "12/12 [==============================] - 0s 750us/step - loss: 1.5138 - accuracy: 0.4694\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Learning rate = 0.0001\n", + "Lambda = 1e-05\n", + "Test accuracy: 0.469\n", + "\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\r", + " 1/12 [=>............................] - ETA: 0s - loss: 1.4077 - accuracy: 0.6562" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\r", + "12/12 [==============================] - 0s 954us/step - loss: 1.4837 - accuracy: 0.5611\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Learning rate = 0.0001\n", + "Lambda = 0.0001\n", + "Test accuracy: 0.561\n", + "\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\r", + " 1/12 [=>............................] - ETA: 0s - loss: 1.5539 - accuracy: 0.5625" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\r", + "12/12 [==============================] - 0s 932us/step - loss: 1.5615 - accuracy: 0.5639\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Learning rate = 0.0001\n", + "Lambda = 0.001\n", + "Test accuracy: 0.564\n", + "\n" + ] + }, + { + "ename": "KeyboardInterrupt", + "evalue": "", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mKeyboardInterrupt\u001b[0m Traceback (most recent call last)", + "\u001b[0;32m/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47647/2018906331.py\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 6\u001b[0m \u001b[0mn_filters\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mn_neurons_connected\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mn_categories\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 7\u001b[0m eta, lmbd)\n\u001b[0;32m----> 8\u001b[0;31m \u001b[0mCNN\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfit\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mX_train\u001b[0m\u001b[0;34m,\u001b[0m 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sample_weight, initial_epoch, steps_per_epoch, validation_steps, validation_batch_size, validation_freq, max_queue_size, workers, use_multiprocessing)\u001b[0m\n\u001b[1;32m 1182\u001b[0m _r=1):\n\u001b[1;32m 1183\u001b[0m \u001b[0mcallbacks\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mon_train_batch_begin\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mstep\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1184\u001b[0;31m \u001b[0mtmp_logs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtrain_function\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0miterator\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 1185\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mdata_handler\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mshould_sync\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 1186\u001b[0m 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variables.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 917\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_stateless_fn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;31m# pylint: disable=not-callable\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 918\u001b[0m \u001b[0;32melif\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_stateful_fn\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 919\u001b[0m \u001b[0;31m# Release the lock early so that multiple threads can perform the call\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;32m~/anaconda3/lib/python3.8/site-packages/tensorflow/python/eager/function.py\u001b[0m in \u001b[0;36m__call__\u001b[0;34m(self, *args, **kwargs)\u001b[0m\n\u001b[1;32m 3037\u001b[0m (graph_function,\n\u001b[1;32m 3038\u001b[0m filtered_flat_args) = self._maybe_define_function(args, kwargs)\n\u001b[0;32m-> 3039\u001b[0;31m return graph_function._call_flat(\n\u001b[0m\u001b[1;32m 3040\u001b[0m filtered_flat_args, captured_inputs=graph_function.captured_inputs) # pylint: disable=protected-access\n\u001b[1;32m 3041\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;32m~/anaconda3/lib/python3.8/site-packages/tensorflow/python/eager/function.py\u001b[0m in \u001b[0;36m_call_flat\u001b[0;34m(self, args, captured_inputs, cancellation_manager)\u001b[0m\n\u001b[1;32m 1961\u001b[0m and executing_eagerly):\n\u001b[1;32m 1962\u001b[0m \u001b[0;31m# No tape is watching; skip to running the function.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1963\u001b[0;31m return self._build_call_outputs(self._inference_function.call(\n\u001b[0m\u001b[1;32m 1964\u001b[0m ctx, args, cancellation_manager=cancellation_manager))\n\u001b[1;32m 1965\u001b[0m forward_backward = self._select_forward_and_backward_functions(\n", + "\u001b[0;32m~/anaconda3/lib/python3.8/site-packages/tensorflow/python/eager/function.py\u001b[0m in \u001b[0;36mcall\u001b[0;34m(self, ctx, args, cancellation_manager)\u001b[0m\n\u001b[1;32m 589\u001b[0m \u001b[0;32mwith\u001b[0m \u001b[0m_InterpolateFunctionError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 590\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mcancellation_manager\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 591\u001b[0;31m outputs = execute.execute(\n\u001b[0m\u001b[1;32m 592\u001b[0m \u001b[0mstr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msignature\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mname\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 593\u001b[0m \u001b[0mnum_outputs\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_num_outputs\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;32m~/anaconda3/lib/python3.8/site-packages/tensorflow/python/eager/execute.py\u001b[0m in \u001b[0;36mquick_execute\u001b[0;34m(op_name, num_outputs, inputs, attrs, ctx, name)\u001b[0m\n\u001b[1;32m 57\u001b[0m \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 58\u001b[0m \u001b[0mctx\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mensure_initialized\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 59\u001b[0;31m tensors = pywrap_tfe.TFE_Py_Execute(ctx._handle, device_name, op_name,\n\u001b[0m\u001b[1;32m 60\u001b[0m inputs, attrs, num_outputs)\n\u001b[1;32m 61\u001b[0m \u001b[0;32mexcept\u001b[0m \u001b[0mcore\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_NotOkStatusException\u001b[0m \u001b[0;32mas\u001b[0m \u001b[0me\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;31mKeyboardInterrupt\u001b[0m: " + ] + } + ], + "source": [ + "CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)\n", + " \n", + "for i, eta in enumerate(eta_vals):\n", + " for j, lmbd in enumerate(lmbd_vals):\n", + " CNN = create_convolutional_neural_network_keras(input_shape, receptive_field,\n", + " n_filters, n_neurons_connected, n_categories,\n", + " eta, lmbd)\n", + " CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)\n", + " scores = CNN.evaluate(X_test, Y_test)\n", + " \n", + " CNN_keras[i][j] = CNN\n", + " \n", + " print(\"Learning rate = \", eta)\n", + " print(\"Lambda = \", lmbd)\n", + " print(\"Test accuracy: %.3f\" % scores[1])\n", + " print()" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "id": "df145244", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "# visual representation of grid search\n", + "# uses seaborn heatmap, could probably do this in matplotlib\n", + "import seaborn as sns\n", + "\n", + "sns.set()\n", + "\n", + "train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", + "test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", + "\n", + "for i in range(len(eta_vals)):\n", + " for j in range(len(lmbd_vals)):\n", + " CNN = CNN_keras[i][j]\n", + "\n", + " train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1]\n", + " test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1]\n", + "\n", + " \n", + "fig, ax = plt.subplots(figsize = (10, 10))\n", + "sns.heatmap(train_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", + "ax.set_title(\"Training Accuracy\")\n", + "ax.set_ylabel(\"$\\eta$\")\n", + "ax.set_xlabel(\"$\\lambda$\")\n", + "plt.show()\n", + "\n", + "fig, ax = plt.subplots(figsize = (10, 10))\n", + "sns.heatmap(test_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", + "ax.set_title(\"Test Accuracy\")\n", + "ax.set_ylabel(\"$\\eta$\")\n", + "ax.set_xlabel(\"$\\lambda$\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "a6724f1c", + "metadata": { + "editable": true + }, + "source": [ + "## The CIFAR01 data set\n", + "\n", + "The CIFAR10 dataset contains 60,000 color images in 10 classes, with\n", + "6,000 images in each class. The dataset is divided into 50,000\n", + "training images and 10,000 testing images. The classes are mutually\n", + "exclusive and there is no overlap between them." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "id": "90c28005", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "import tensorflow as tf\n", + "\n", + "from tensorflow.keras import datasets, layers, models\n", + "import matplotlib.pyplot as plt\n", + "\n", + "# We import the data set\n", + "(train_images, train_labels), (test_images, test_labels) = datasets.cifar10.load_data()\n", + "\n", + "# Normalize pixel values to be between 0 and 1 by dividing by 255. \n", + "train_images, test_images = train_images / 255.0, test_images / 255.0" + ] + }, + { + "cell_type": "markdown", + "id": "759ea77a", + "metadata": { + "editable": true + }, + "source": [ + "To verify that the dataset looks correct, let's plot the first 25 images from the training set and display the class name below each image." + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "id": "494edadd", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "class_names = ['airplane', 'automobile', 'bird', 'cat', 'deer',\n", + " 'dog', 'frog', 'horse', 'ship', 'truck']\n", + "​\n", + "plt.figure(figsize=(10,10))\n", + "for i in range(25):\n", + " plt.subplot(5,5,i+1)\n", + " plt.xticks([])\n", + " plt.yticks([])\n", + " plt.grid(False)\n", + " plt.imshow(train_images[i], cmap=plt.cm.binary)\n", + " # The CIFAR labels happen to be arrays, \n", + " # which is why you need the extra index\n", + " plt.xlabel(class_names[train_labels[i][0]])\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "46d2ea5d", + "metadata": { + "editable": true + }, + "source": [ + "The six lines of code below define the convolutional base using a common pattern: a stack of Conv2D and MaxPooling2D layers.\n", + "\n", + "As input, a CNN takes tensors of shape (image_height, image_width, color_channels), ignoring the batch size. If you are new to these dimensions, color_channels refers to (R,G,B). In this example, you will configure our CNN to process inputs of shape (32, 32, 3), which is the format of CIFAR images. You can do this by passing the argument input_shape to our first layer." + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "id": "044310ec", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "model = models.Sequential()\n", + "model.add(layers.Conv2D(32, (3, 3), activation='relu', input_shape=(32, 32, 3)))\n", + "model.add(layers.MaxPooling2D((2, 2)))\n", + "model.add(layers.Conv2D(64, (3, 3), activation='relu'))\n", + "model.add(layers.MaxPooling2D((2, 2)))\n", + "model.add(layers.Conv2D(64, (3, 3), activation='relu'))\n", + "\n", + "# Let's display the architecture of our model so far.\n", + "\n", + "model.summary()" + ] + }, + { + "cell_type": "markdown", + "id": "738f34a7", + "metadata": { + "editable": true + }, + "source": [ + "You can see that the output of every Conv2D and MaxPooling2D layer is a 3D tensor of shape (height, width, channels). The width and height dimensions tend to shrink as you go deeper in the network. The number of output channels for each Conv2D layer is controlled by the first argument (e.g., 32 or 64). Typically, as the width and height shrink, you can afford (computationally) to add more output channels in each Conv2D layer.\n", + "\n", + "To complete our model, you will feed the last output tensor from the\n", + "convolutional base (of shape (4, 4, 64)) into one or more Dense layers\n", + "to perform classification. Dense layers take vectors as input (which\n", + "are 1D), while the current output is a 3D tensor. First, you will\n", + "flatten (or unroll) the 3D output to 1D, then add one or more Dense\n", + "layers on top. CIFAR has 10 output classes, so you use a final Dense\n", + "layer with 10 outputs and a softmax activation." + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "id": "e0fea435", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "model.add(layers.Flatten())\n", + "model.add(layers.Dense(64, activation='relu'))\n", + "model.add(layers.Dense(10))\n", + "Here's the complete architecture of our model.\n", + "\n", + "model.summary()" + ] + }, + { + "cell_type": "markdown", + "id": "62f1e50d", + "metadata": { + "editable": true + }, + "source": [ + "As you can see, our (4, 4, 64) outputs were flattened into vectors of shape (1024) before going through two Dense layers." + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "id": "e2d8f4f2", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "model.compile(optimizer='adam',\n", + " loss=tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True),\n", + " metrics=['accuracy'])\n", + "​\n", + "history = model.fit(train_images, train_labels, epochs=10, \n", + " validation_data=(test_images, test_labels))" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "id": "d2965f40", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "plt.plot(history.history['accuracy'], label='accuracy')\n", + "plt.plot(history.history['val_accuracy'], label = 'val_accuracy')\n", + "plt.xlabel('Epoch')\n", + "plt.ylabel('Accuracy')\n", + "plt.ylim([0.5, 1])\n", + "plt.legend(loc='lower right')\n", + "\n", + "test_loss, test_acc = model.evaluate(test_images, test_labels, verbose=2)\n", + "\n", + "print(test_acc)" + ] + } + ], + "metadata": { + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.8.12" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter12.py b/doc/LectureNotes/_build/jupyter_execute/chapter12.py new file mode 100644 index 000000000..f6700ecd8 --- /dev/null +++ b/doc/LectureNotes/_build/jupyter_execute/chapter12.py @@ -0,0 +1,921 @@ +#!/usr/bin/env python +# coding: utf-8 + +# + +# # Convolutional Neural Networks +# +# Convolutional neural networks (CNNs) were developed during the last +# decade of the previous century, with a focus on character recognition +# tasks. Nowadays, CNNs are a central element in the spectacular success +# of deep learning methods. The success in for example image +# classifications have made them a central tool for most machine +# learning practitioners. +# +# CNNs are very similar to ordinary Neural Networks. +# They are made up of neurons that have learnable weights and +# biases. Each neuron receives some inputs, performs a dot product and +# optionally follows it with a non-linearity. The whole network still +# expresses a single differentiable score function: from the raw image +# pixels on one end to class scores at the other. And they still have a +# loss function (for example Softmax) on the last (fully-connected) layer +# and all the tips/tricks we developed for learning regular Neural +# Networks still apply (back propagation, gradient descent etc etc). +# +# **CNN architectures make the explicit assumption that +# the inputs are images, which allows us to encode certain properties +# into the architecture. These then make the forward function more +# efficient to implement and vastly reduce the amount of parameters in +# the network.** +# +# Here we provide only a superficial overview, for the more interested, we recommend highly the course +# [IN5400 – Machine Learning for Image Analysis](https://www.uio.no/studier/emner/matnat/ifi/IN5400/index-eng.html) +# and the slides of [CS231](http://cs231n.github.io/convolutional-networks/). +# +# Another good read is the article here . + +# ## Neural Networks vs CNNs +# +# Neural networks are defined as **affine transformations**, that is +# a vector is received as input and is multiplied with a matrix of so-called weights (our unknown paramters) to produce an +# output (to which a bias vector is usually added before passing the result +# through a nonlinear activation function). This is applicable to any type of input, be it an +# image, a sound clip or an unordered collection of features: whatever their +# dimensionality, their representation can always be flattened into a vector +# before the transformation. +# +# However, when we consider images, sound clips and many other similar kinds of data, these data have an intrinsic +# structure. More formally, they share these important properties: +# * They are stored as multi-dimensional arrays (think of the pixels of a figure) . +# +# * They feature one or more axes for which ordering matters (e.g., width and height axes for an image, time axis for a sound clip). +# +# * One axis, called the channel axis, is used to access different views of the data (e.g., the red, green and blue channels of a color image, or the left and right channels of a stereo audio track). +# +# These properties are not exploited when an affine transformation is applied; in +# fact, all the axes are treated in the same way and the topological information +# is not taken into account. Still, taking advantage of the implicit structure of +# the data may prove very handy in solving some tasks, like computer vision and +# speech recognition, and in these cases it would be best to preserve it. This is +# where discrete convolutions come into play. +# +# A discrete convolution is a linear transformation that preserves this notion of +# ordering. It is sparse (only a few input units contribute to a given output +# unit) and reuses parameters (the same weights are applied to multiple locations +# in the input). +# +# As an example, consider +# an image of size $32\times 32\times 3$ (32 wide, 32 high, 3 color channels), so a +# single fully-connected neuron in a first hidden layer of a regular +# Neural Network would have $32\times 32\times 3 = 3072$ weights. This amount still +# seems manageable, but clearly this fully-connected structure does not +# scale to larger images. For example, an image of more respectable +# size, say $200\times 200\times 3$, would lead to neurons that have +# $200\times 200\times 3 = 120,000$ weights. +# +# We could have +# several such neurons, and the parameters would add up quickly! Clearly, +# this full connectivity is wasteful and the huge number of parameters +# would quickly lead to possible overfitting. +# +# +# +# +#

    Figure 1: A regular 3-layer Neural Network.

    +# +# +# Convolutional Neural Networks take advantage of the fact that the +# input consists of images and they constrain the architecture in a more +# sensible way. +# +# In particular, unlike a regular Neural Network, the +# layers of a CNN have neurons arranged in 3 dimensions: width, +# height, depth. (Note that the word depth here refers to the third +# dimension of an activation volume, not to the depth of a full Neural +# Network, which can refer to the total number of layers in a network.) +# +# To understand it better, the above example of an image +# with an input volume of +# activations has dimensions $32\times 32\times 3$ (width, height, +# depth respectively). +# +# The neurons in a layer will +# only be connected to a small region of the layer before it, instead of +# all of the neurons in a fully-connected manner. Moreover, the final +# output layer could for this specific image have dimensions $1\times 1 \times 10$, +# because by the +# end of the CNN architecture we will reduce the full image into a +# single vector of class scores, arranged along the depth +# dimension. +# +# +# +# +#

    Figure 1: A CNN arranges its neurons in three dimensions (width, height, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D output volume of neuron activations. In this example, the red input layer holds the image, so its width and height would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels).

    +# + +# ## Layers used to build CNNs +# +# A simple CNN is a sequence of layers, and every layer of a CNN +# transforms one volume of activations to another through a +# differentiable function. We use three main types of layers to build +# CNN architectures: Convolutional Layer, Pooling Layer, and +# Fully-Connected Layer (exactly as seen in regular Neural Networks). We +# will stack these layers to form a full CNN architecture. +# +# A simple CNN for image classification could have the architecture: +# +# * **INPUT** ($32\times 32 \times 3$) will hold the raw pixel values of the image, in this case an image of width 32, height 32, and with three color channels R,G,B. +# +# * **CONV** (convolutional )layer will compute the output of neurons that are connected to local regions in the input, each computing a dot product between their weights and a small region they are connected to in the input volume. This may result in volume such as $[32\times 32\times 12]$ if we decided to use 12 filters. +# +# * **RELU** layer will apply an elementwise activation function, such as the $max(0,x)$ thresholding at zero. This leaves the size of the volume unchanged ($[32\times 32\times 12]$). +# +# * **POOL** (pooling) layer will perform a downsampling operation along the spatial dimensions (width, height), resulting in volume such as $[16\times 16\times 12]$. +# +# * **FC** (i.e. fully-connected) layer will compute the class scores, resulting in volume of size $[1\times 1\times 10]$, where each of the 10 numbers correspond to a class score, such as among the 10 categories of the MNIST images we considered above . As with ordinary Neural Networks and as the name implies, each neuron in this layer will be connected to all the numbers in the previous volume. +# +# CNNs transform the original image layer by layer from the original +# pixel values to the final class scores. +# +# Observe that some layers contain +# parameters and other don’t. In particular, the CNN layers perform +# transformations that are a function of not only the activations in the +# input volume, but also of the parameters (the weights and biases of +# the neurons). On the other hand, the RELU/POOL layers will implement a +# fixed function. The parameters in the CONV/FC layers will be trained +# with gradient descent so that the class scores that the CNN computes +# are consistent with the labels in the training set for each image. +# +# In summary: +# +# * A CNN architecture is in the simplest case a list of Layers that transform the image volume into an output volume (e.g. holding the class scores) +# +# * There are a few distinct types of Layers (e.g. CONV/FC/RELU/POOL are by far the most popular) +# +# * Each Layer accepts an input 3D volume and transforms it to an output 3D volume through a differentiable function +# +# * Each Layer may or may not have parameters (e.g. CONV/FC do, RELU/POOL don’t) +# +# * Each Layer may or may not have additional hyperparameters (e.g. CONV/FC/POOL do, RELU doesn’t) +# +# A dense neural network is representd by an affine operation (like matrix-matrix multiplication) where all parameters are included. +# +# The key idea in CNNs for say imaging is that in images neighbor pixels tend to be related! So we connect +# only neighboring neurons in the input instead of connecting all with the first hidden layer. +# +# We say we perform a filtering (convolution is the mathematical operation). + +# ## Mathematics of CNNs +# +# The mathematics of CNNs is based on the mathematical operation of +# **convolution**. In mathematics (in particular in functional analysis), +# convolution is represented by matheematical operation (integration, +# summation etc) on two function in order to produce a third function +# that expresses how the shape of one gets modified by the other. +# Convolution has a plethora of applications in a variety of disciplines, spanning from statistics to signal processing, computer vision, solutions of differential equations,linear algebra, engineering, and yes, machine learning. +# +# Mathematically, convolution is defined as follows (one-dimensional example): +# Let us define a continuous function $y(t)$ given by + +# $$ +# y(t) = \int x(a) w(t-a) da, +# $$ + +# where $x(a)$ represents a so-called input and $w(t-a)$ is normally called the weight function or kernel. +# +# The above integral is written in a more compact form as + +# $$ +# y(t) = \left(x * w\right)(t). +# $$ + +# The discretized version reads + +# $$ +# y(t) = \sum_{a=-\infty}^{a=\infty}x(a)w(t-a). +# $$ + +# Computing the inverse of the above convolution operations is known as deconvolution. +# +# How can we use this? And what does it mean? Let us study some familiar examples first. + +# ### Convolution Examples: Polynomial multiplication +# +# We have already met such an example in project 1 when we tried to set +# up the design matrix for a two-dimensional function. This was an +# example of polynomial multiplication. Let us recast such a problem in terms of the convolution operation. +# Let us look a the following polynomials to second and third order, respectively: + +# $$ +# p(t) = \alpha_0+\alpha_1 t+\alpha_2 t^2, +# $$ + +# and + +# $$ +# s(t) = \beta_0+\beta_1 t+\beta_2 t^2+\beta_3 t^3. +# $$ + +# The polynomial multiplication gives us a new polynomial of degree $5$ + +# $$ +# z(t) = \delta_0+\delta_1 t+\delta_2 t^2+\delta_3 t^3+\delta_4 t^4+\delta_5 t^5. +# $$ + +# Computing polynomial products can be implemented efficiently if we rewrite the more brute force multiplications using convolution. +# We note first that the new coefficients are given as + +# $$ +# \begin{split} +# \delta_0=&\alpha_0\beta_0\\ +# \delta_1=&\alpha_1\beta_0+\alpha_1\beta_0\\ +# \delta_2=&\alpha_0\beta_2+\alpha_1\beta_1+\alpha_2\beta_0\\ +# \delta_3=&\alpha_1\beta_2+\alpha_2\beta_1+\alpha_0\beta_3\\ +# \delta_4=&\alpha_2\beta_2+\alpha_1\beta_3\\ +# \delta_5=&\alpha_2\beta_3.\\ +# \end{split} +# $$ + +# We note that $\alpha_i=0$ except for $i\in \left\{0,1,2\right\}$ and $\beta_i=0$ except for $i\in\left\{0,1,2,3\right\}$. +# +# We can then rewrite the coefficients $\delta_j$ using a discrete convolution as + +# $$ +# \delta_j = \sum_{i=-\infty}^{i=\infty}\alpha_i\beta_{j-i}=(\alpha * \beta)_j, +# $$ + +# or as a double sum with restriction $l=i+j$ + +# $$ +# \delta_l = \sum_{ij}\alpha_i\beta_{j}. +# $$ + +# Do you see a potential drawback with these equations? +# +# Since we only have a finite number of $\alpha$ and $\beta$ values +# which are non-zero, we can rewrite the above convolution expressions +# as a matrix-vector multiplication + +# $$ +# \boldsymbol{\delta}=\begin{bmatrix}\alpha_0 & 0 & 0 & 0 \\ +# \alpha_1 & \alpha_0 & 0 & 0 \\ +# \alpha_2 & \alpha_1 & \alpha_0 & 0 \\ +# 0 & \alpha_2 & \alpha_1 & \alpha_0 \\ +# 0 & 0 & \alpha_2 & \alpha_1 \\ +# 0 & 0 & 0 & \alpha_2 +# \end{bmatrix}\begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3\end{bmatrix}. +# $$ + +# The process is commutative and we can easily see that we can rewrite the multiplication in terms of a matrix holding $\beta$ and a vector holding $\alpha$. +# In this case we have + +# $$ +# \boldsymbol{\delta}=\begin{bmatrix}\beta_0 & 0 & 0 \\ +# \beta_1 & \beta_0 & 0 \\ +# \beta_2 & \beta_1 & \beta_0 \\ +# \beta_3 & \beta_2 & \beta_1 \\ +# 0 & \beta_3 & \beta_2 \\ +# 0 & 0 & \beta_3 +# \end{bmatrix}\begin{bmatrix} \alpha_0 \\ \alpha_1 \\ \alpha_2\end{bmatrix}. +# $$ + +# Note that the use of these matrices is for mathematical purposes only and not implementation purposes. +# When implementing the above equation we do not encode (and allocate memory) the matrices explicitely. +# We rather code the convolutions in the minimal memory footprint that they require. +# +# Does the number of floating point operations change here when we use the commutative property? + +# ### Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms) +# +# For problems with so-called harmonic oscillations, given by for example the following differential equation + +# $$ +# m\frac{d^2x}{dt^2}+\eta\frac{dx}{dt}+x(t)=F(t), +# $$ + +# where $F(t)$ is an applied external force acting on the system (often called a driving force), one can use the theory of Fourier transformations to find the solutions of this type of equations. +# +# If one has several driving forces, $F(t)=\sum_n F_n(t)$, one can find +# the particular solution to each $F_n$, $x_{pn}(t)$, and the particular +# solution for the entire driving force is then given by a series like + +# +#
    +# +# $$ +# \begin{equation} +# x_p(t)=\sum_nx_{pn}(t). +# \label{_auto1} \tag{1} +# \end{equation} +# $$ + +# This is known as the principle of superposition. It only applies when +# the homogenous equation is linear. If there were an anharmonic term +# such as $x^3$ in the homogenous equation, then when one summed various +# solutions, $x=(\sum_n x_n)^2$, one would get cross +# terms. Superposition is especially useful when $F(t)$ can be written +# as a sum of sinusoidal terms, because the solutions for each +# sinusoidal (sine or cosine) term is analytic. +# +# Driving forces are often periodic, even when they are not +# sinusoidal. Periodicity implies that for some time $\tau$ + +# $$ +# \begin{eqnarray} +# F(t+\tau)=F(t). +# \end{eqnarray} +# $$ + +# One example of a non-sinusoidal periodic force is a square wave. Many +# components in electric circuits are non-linear, e.g. diodes, which +# makes many wave forms non-sinusoidal even when the circuits are being +# driven by purely sinusoidal sources. +# +# The code here shows a typical example of such a square wave generated using the functionality included in the **scipy** Python package. We have used a period of $\tau=0.2$. + +# In[1]: + + +get_ipython().run_line_magic('matplotlib', 'inline') + +import numpy as np +import math +from scipy import signal +import matplotlib.pyplot as plt + +# number of points +n = 500 +# start and final times +t0 = 0.0 +tn = 1.0 +# Period +t = np.linspace(t0, tn, n, endpoint=False) +SqrSignal = np.zeros(n) +SqrSignal = 1.0+signal.square(2*np.pi*5*t) +plt.plot(t, SqrSignal) +plt.ylim(-0.5, 2.5) +plt.show() + + +# For the sinusoidal example the +# period is $\tau=2\pi/\omega$. However, higher harmonics can also +# satisfy the periodicity requirement. In general, any force that +# satisfies the periodicity requirement can be expressed as a sum over +# harmonics, + +# +#
    +# +# $$ +# \begin{equation} +# F(t)=\frac{f_0}{2}+\sum_{n>0} f_n\cos(2n\pi t/\tau)+g_n\sin(2n\pi t/\tau). +# \label{_auto2} \tag{2} +# \end{equation} +# $$ + +# We can write down the answer for +# $x_{pn}(t)$, by substituting $f_n/m$ or $g_n/m$ for $F_0/m$. By +# writing each factor $2n\pi t/\tau$ as $n\omega t$, with $\omega\equiv +# 2\pi/\tau$, + +# +#
    +# +# $$ +# \begin{equation} +# \label{eq:fourierdef1} \tag{3} +# F(t)=\frac{f_0}{2}+\sum_{n>0}f_n\cos(n\omega t)+g_n\sin(n\omega t). +# \end{equation} +# $$ + +# The solutions for $x(t)$ then come from replacing $\omega$ with +# $n\omega$ for each term in the particular solution, + +# $$ +# \begin{eqnarray} +# x_p(t)&=&\frac{f_0}{2k}+\sum_{n>0} \alpha_n\cos(n\omega t-\delta_n)+\beta_n\sin(n\omega t-\delta_n),\\ +# \nonumber +# \alpha_n&=&\frac{f_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ +# \nonumber +# \beta_n&=&\frac{g_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ +# \nonumber +# \delta_n&=&\tan^{-1}\left(\frac{2\beta n\omega}{\omega_0^2-n^2\omega^2}\right). +# \end{eqnarray} +# $$ + +# Because the forces have been applied for a long time, any non-zero +# damping eliminates the homogenous parts of the solution, so one need +# only consider the particular solution for each $n$. +# +# The problem is considered solved if one can find expressions for the +# coefficients $f_n$ and $g_n$, even though the solutions are expressed +# as an infinite sum. The coefficients can be extracted from the +# function $F(t)$ by + +# +#
    +# +# $$ +# \begin{eqnarray} +# \label{eq:fourierdef2} \tag{4} +# f_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\cos(2n\pi t/\tau),\\ +# \nonumber +# g_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\sin(2n\pi t/\tau). +# \end{eqnarray} +# $$ + +# To check the consistency of these expressions and to verify +# Eq. ([4](#eq:fourierdef2)), one can insert the expansion of $F(t)$ in +# Eq. ([3](#eq:fourierdef1)) into the expression for the coefficients in +# Eq. ([4](#eq:fourierdef2)) and see whether + +# $$ +# \begin{eqnarray} +# f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~\left\{ +# \frac{f_0}{2}+\sum_{m>0}f_m\cos(m\omega t)+g_m\sin(m\omega t) +# \right\}\cos(n\omega t). +# \end{eqnarray} +# $$ + +# Immediately, one can throw away all the terms with $g_m$ because they +# convolute an even and an odd function. The term with $f_0/2$ +# disappears because $\cos(n\omega t)$ is equally positive and negative +# over the interval and will integrate to zero. For all the terms +# $f_m\cos(m\omega t)$ appearing in the sum, one can use angle addition +# formulas to see that $\cos(m\omega t)\cos(n\omega +# t)=(1/2)(\cos[(m+n)\omega t]+\cos[(m-n)\omega t]$. This will integrate +# to zero unless $m=n$. In that case the $m=n$ term gives + +# +#
    +# +# $$ +# \begin{equation} +# \int_{-\tau/2}^{\tau/2}dt~\cos^2(m\omega t)=\frac{\tau}{2}, +# \label{_auto3} \tag{5} +# \end{equation} +# $$ + +# and + +# $$ +# \begin{eqnarray} +# f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~f_n/2\\ +# \nonumber +# &=&f_n~\checkmark. +# \end{eqnarray} +# $$ + +# The same method can be used to check for the consistency of $g_n$. +# +# The code here uses the Fourier series applied to a +# square wave signal. The code here +# visualizes the various approximations given by Fourier series compared +# with a square wave with period $T=0.2$ (dimensionless time), width $0.1$ and max value of the force $F=2$. We +# see that when we increase the number of components in the Fourier +# series, the Fourier series approximation gets closer and closer to the +# square wave signal. + +# In[2]: + + +import numpy as np +import math +from scipy import signal +import matplotlib.pyplot as plt + +# number of points +n = 500 +# start and final times +t0 = 0.0 +tn = 1.0 +# Period +T =0.2 +# Max value of square signal +Fmax= 2.0 +# Width of signal +Width = 0.1 +t = np.linspace(t0, tn, n, endpoint=False) +SqrSignal = np.zeros(n) +FourierSeriesSignal = np.zeros(n) +SqrSignal = 1.0+signal.square(2*np.pi*5*t+np.pi*Width/T) +a0 = Fmax*Width/T +FourierSeriesSignal = a0 +Factor = 2.0*Fmax/np.pi +for i in range(1,500): + FourierSeriesSignal += Factor/(i)*np.sin(np.pi*i*Width/T)*np.cos(i*t*2*np.pi/T) +plt.plot(t, SqrSignal) +plt.plot(t, FourierSeriesSignal) +plt.ylim(-0.5, 2.5) +plt.show() + + +# ## Two-dimensional Objects +# +# We often use convolutions over more than one dimension at a time. If +# we have a two-dimensional image $I$ as input, we can have a **filter** +# defined by a two-dimensional **kernel** $K$. This leads to an output $S$ + +# $$ +# S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(m,n)K(i-m,j-n). +# $$ + +# Convolution is a commutatitave process, which means we can rewrite this equation as + +# $$ +# S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(i-m,j-n)K(m,n). +# $$ + +# Normally the latter is more straightforward to implement in a machine elarning library since there is less variation in the range of values of $m$ and $n$. +# +# Many deep learning libraries implement cross-correlation instead of convolution + +# $$ +# S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(i+m,j-+)K(m,n). +# $$ + +# ## More on Dimensionalities +# +# In fields like signal processing (and imaging as well), one designs +# so-called filters. These filters are defined by the convolutions and +# are often hand-crafted. One may specify filters for smoothing, edge +# detection, frequency reshaping, and similar operations. However with +# neural networks the idea is to automatically learn the filters and use +# many of them in conjunction with non-linear operations (activation +# functions). +# +# As an example consider a neural network operating on sound sequence +# data. Assume that we an input vector $\boldsymbol{x}$ of length $d=10^6$. We +# construct then a neural network with onle hidden layer only with +# $10^4$ nodes. This means that we will have a weight matrix with +# $10^4\times 10^6=10^{10}$ weights to be determined, together with $10^4$ biases. +# +# Assume furthermore that we have an output layer which is meant to train whether the sound sequence represents a human voice (true) or something else (false). +# It means that we have only one output node. But since this output node connects to $10^4$ nodes in the hidden layer, there are in total $10^4$ weights to be determined for the output layer, plus one bias. In total we have + +# $$ +# \mathrm{NumberParameters}=10^{10}+10^4+10^4+1 \approx 10^{10}, +# $$ + +# that is ten billion parameters to determine. + +# ## Further Dimensionality Remarks +# +# In today’s architecture one can train such neural networks, however +# this is a huge number of parameters for the task at hand. In general, +# it is a very wasteful and inefficient use of dense matrices as +# parameters. Just as importantly, such trained network parameters are +# very specific for the type of input data on which they were trained +# and the network is not likely to generalize easily to variations in +# the input. +# +# The main principles that justify convolutions is locality of +# information and repetion of patterns within the signal. Sound samples +# of the input in adjacent spots are much more likely to affect each +# other than those that are very far away. Similarly, sounds are +# repeated in multiple times in the signal. While slightly simplistic, +# reasoning about such a sound example demonstrates this. The same +# principles then apply to images and other similar data. + +# ## CNNs in more detail, building convolutional neural networks in Tensorflow and Keras +# +# As discussed above, CNNs are neural networks built from the assumption that the inputs +# to the network are 2D images. This is important because the number of features or pixels in images +# grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network. +# +# As before, we still have our input, a hidden layer and an output. What's novel about convolutional networks +# are the **convolutional** and **pooling** layers stacked in pairs between the input and the hidden layer. +# In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D +# matrices, typically 1 for each color dimension (Red, Green, Blue). +# +# It means that to represent the entire +# dataset of images, we require a 4D matrix or **tensor**. This tensor has the dimensions: + +# $$ +# (n_{inputs},\, n_{pixels, width},\, n_{pixels, height},\, depth) . +# $$ + +# ### The MNIST dataset again +# +# The MNIST dataset consists of grayscale images with a pixel size of +# $28\times 28$, meaning we require $28 \times 28 = 724$ weights to each +# neuron in the first hidden layer. +# +# If we were to analyze images of size $128\times 128$ we would require +# $128 \times 128 = 16384$ weights to each neuron. Even worse if we were +# dealing with color images, as most images are, we have an image matrix +# of size $128\times 128$ for each color dimension (Red, Green, Blue), +# meaning 3 times the number of weights $= 49152$ are required for every +# single neuron in the first hidden layer. +# +# Images typically have strong local correlations, meaning that a small +# part of the image varies little from its neighboring regions. If for +# example we have an image of a blue car, we can roughly assume that a +# small blue part of the image is surrounded by other blue regions. +# +# Therefore, instead of connecting every single pixel to a neuron in the +# first hidden layer, as we have previously done with deep neural +# networks, we can instead connect each neuron to a small part of the +# image (in all 3 RGB depth dimensions). The size of each small area is +# fixed, and known as a [receptive](https://en.wikipedia.org/wiki/Receptive_field). +# +# The layers of a convolutional neural network arrange neurons in 3D: width, height and depth. +# The input image is typically a square matrix of depth 3. +# +# A **convolution** is performed on the image which outputs +# a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as **filters**. +# +# Each filter slides along the input image, taking the dot product +# between each small part of the image and the filter, in all depth +# dimensions. This is then passed through a non-linear function, +# typically the **Rectified Linear (ReLu)** function, which serves as the +# activation of the neurons in the first convolutional layer. This is +# further passed through a **pooling layer**, which reduces the size of the +# convolutional layer, e.g. by taking the maximum or average across some +# small regions, and this serves as input to the next convolutional +# layer. + +# ### Systematic reduction +# +# By systematically reducing the size of the input volume, through +# convolution and pooling, the network should create representations of +# small parts of the input, and then from them assemble representations +# of larger areas. The final pooling layer is flattened to serve as +# input to a hidden layer, such that each neuron in the final pooling +# layer is connected to every single neuron in the hidden layer. This +# then serves as input to the output layer, e.g. a softmax output for +# classification. + +# ### Prerequisites: Collect and pre-process data + +# In[3]: + + +# 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 +get_ipython().run_line_magic('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 + +# RGB images have a depth of 3 +# our images are grayscale so they should have a depth of 1 +inputs = inputs[:,:,:,np.newaxis] + +print("inputs = (n_inputs, pixel_width, pixel_height, depth) = " + str(inputs.shape)) +print("labels = (n_inputs) = " + str(labels.shape)) + + +# choose some random images to display +n_inputs = len(inputs) +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() + + +# In[4]: + + +from tensorflow.keras import datasets, layers, models +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 +#from tensorflow.keras import Conv2D +#from tensorflow.keras import MaxPooling2D +#from tensorflow.keras import Flatten + +from sklearn.model_selection import train_test_split + +# representation of labels +labels = to_categorical(labels) + +# split into train and test data +# 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) + + +# In[5]: + + +def create_convolutional_neural_network_keras(input_shape, receptive_field, + n_filters, n_neurons_connected, n_categories, + eta, lmbd): + model = Sequential() + model.add(layers.Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same', + activation='relu', kernel_regularizer=regularizers.l2(lmbd))) + model.add(layers.MaxPooling2D(pool_size=(2, 2))) + model.add(layers.Flatten()) + model.add(layers.Dense(n_neurons_connected, activation='relu', kernel_regularizer=regularizers.l2(lmbd))) + model.add(layers.Dense(n_categories, activation='softmax', kernel_regularizer=regularizers.l2(lmbd))) + + sgd = optimizers.SGD(lr=eta) + model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy']) + + return model + +epochs = 100 +batch_size = 100 +input_shape = X_train.shape[1:4] +receptive_field = 3 +n_filters = 10 +n_neurons_connected = 50 +n_categories = 10 + +eta_vals = np.logspace(-5, 1, 7) +lmbd_vals = np.logspace(-5, 1, 7) + + +# In[6]: + + +CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object) + +for i, eta in enumerate(eta_vals): + for j, lmbd in enumerate(lmbd_vals): + CNN = create_convolutional_neural_network_keras(input_shape, receptive_field, + n_filters, n_neurons_connected, n_categories, + eta, lmbd) + CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0) + scores = CNN.evaluate(X_test, Y_test) + + CNN_keras[i][j] = CNN + + print("Learning rate = ", eta) + print("Lambda = ", lmbd) + print("Test accuracy: %.3f" % scores[1]) + print() + + +# In[7]: + + +# 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)): + CNN = CNN_keras[i][j] + + train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1] + test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1] + + +fig, ax = plt.subplots(figsize = (10, 10)) +sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis") +ax.set_title("Training Accuracy") +ax.set_ylabel("$\eta$") +ax.set_xlabel("$\lambda$") +plt.show() + +fig, ax = plt.subplots(figsize = (10, 10)) +sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis") +ax.set_title("Test Accuracy") +ax.set_ylabel("$\eta$") +ax.set_xlabel("$\lambda$") +plt.show() + + +# ## The CIFAR01 data set +# +# The CIFAR10 dataset contains 60,000 color images in 10 classes, with +# 6,000 images in each class. The dataset is divided into 50,000 +# training images and 10,000 testing images. The classes are mutually +# exclusive and there is no overlap between them. + +# In[8]: + + +import tensorflow as tf + +from tensorflow.keras import datasets, layers, models +import matplotlib.pyplot as plt + +# We import the data set +(train_images, train_labels), (test_images, test_labels) = datasets.cifar10.load_data() + +# Normalize pixel values to be between 0 and 1 by dividing by 255. +train_images, test_images = train_images / 255.0, test_images / 255.0 + + +# To verify that the dataset looks correct, let's plot the first 25 images from the training set and display the class name below each image. + +# In[9]: + + +class_names = ['airplane', 'automobile', 'bird', 'cat', 'deer', + 'dog', 'frog', 'horse', 'ship', 'truck'] +​ +plt.figure(figsize=(10,10)) +for i in range(25): + plt.subplot(5,5,i+1) + plt.xticks([]) + plt.yticks([]) + plt.grid(False) + plt.imshow(train_images[i], cmap=plt.cm.binary) + # The CIFAR labels happen to be arrays, + # which is why you need the extra index + plt.xlabel(class_names[train_labels[i][0]]) +plt.show() + + +# The six lines of code below define the convolutional base using a common pattern: a stack of Conv2D and MaxPooling2D layers. +# +# As input, a CNN takes tensors of shape (image_height, image_width, color_channels), ignoring the batch size. If you are new to these dimensions, color_channels refers to (R,G,B). In this example, you will configure our CNN to process inputs of shape (32, 32, 3), which is the format of CIFAR images. You can do this by passing the argument input_shape to our first layer. + +# In[10]: + + +model = models.Sequential() +model.add(layers.Conv2D(32, (3, 3), activation='relu', input_shape=(32, 32, 3))) +model.add(layers.MaxPooling2D((2, 2))) +model.add(layers.Conv2D(64, (3, 3), activation='relu')) +model.add(layers.MaxPooling2D((2, 2))) +model.add(layers.Conv2D(64, (3, 3), activation='relu')) + +# Let's display the architecture of our model so far. + +model.summary() + + +# You can see that the output of every Conv2D and MaxPooling2D layer is a 3D tensor of shape (height, width, channels). The width and height dimensions tend to shrink as you go deeper in the network. The number of output channels for each Conv2D layer is controlled by the first argument (e.g., 32 or 64). Typically, as the width and height shrink, you can afford (computationally) to add more output channels in each Conv2D layer. +# +# To complete our model, you will feed the last output tensor from the +# convolutional base (of shape (4, 4, 64)) into one or more Dense layers +# to perform classification. Dense layers take vectors as input (which +# are 1D), while the current output is a 3D tensor. First, you will +# flatten (or unroll) the 3D output to 1D, then add one or more Dense +# layers on top. CIFAR has 10 output classes, so you use a final Dense +# layer with 10 outputs and a softmax activation. + +# In[11]: + + +model.add(layers.Flatten()) +model.add(layers.Dense(64, activation='relu')) +model.add(layers.Dense(10)) +Here's the complete architecture of our model. + +model.summary() + + +# As you can see, our (4, 4, 64) outputs were flattened into vectors of shape (1024) before going through two Dense layers. + +# In[12]: + + +model.compile(optimizer='adam', + loss=tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True), + metrics=['accuracy']) +​ +history = model.fit(train_images, train_labels, epochs=10, + validation_data=(test_images, test_labels)) + + +# In[13]: + + +plt.plot(history.history['accuracy'], label='accuracy') +plt.plot(history.history['val_accuracy'], label = 'val_accuracy') +plt.xlabel('Epoch') +plt.ylabel('Accuracy') +plt.ylim([0.5, 1]) +plt.legend(loc='lower right') + +test_loss, test_acc = model.evaluate(test_images, test_labels, verbose=2) + +print(test_acc) + diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter12_35_0.png b/doc/LectureNotes/_build/jupyter_execute/chapter12_35_0.png new file mode 100644 index 000000000..282fbd25e Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapter12_35_0.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter12_51_0.png b/doc/LectureNotes/_build/jupyter_execute/chapter12_51_0.png new file mode 100644 index 000000000..cd49564e7 Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapter12_51_0.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter12_67_1.png b/doc/LectureNotes/_build/jupyter_execute/chapter12_67_1.png new file mode 100644 index 000000000..91a4496c5 Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapter12_67_1.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter13.ipynb b/doc/LectureNotes/_build/jupyter_execute/chapter13.ipynb new file mode 100644 index 000000000..87a579b19 --- /dev/null +++ b/doc/LectureNotes/_build/jupyter_execute/chapter13.ipynb @@ -0,0 +1,3294 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "6d2db899", + "metadata": { + "editable": true + }, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "id": "f4908a97", + "metadata": { + "editable": true + }, + "source": [ + "# Recurrent neural networks: Overarching view\n", + "\n", + "Till now our focus has been, including convolutional neural networks\n", + "as well, on feedforward neural networks. The output or the activations\n", + "flow only in one direction, from the input layer to the output layer.\n", + "\n", + "A recurrent neural network (RNN) looks very much like a feedforward\n", + "neural network, except that it also has connections pointing\n", + "backward. \n", + "\n", + "RNNs are used to analyze time series data such as stock prices, and\n", + "tell you when to buy or sell. In autonomous driving systems, they can\n", + "anticipate car trajectories and help avoid accidents. More generally,\n", + "they can work on sequences of arbitrary lengths, rather than on\n", + "fixed-sized inputs like all the nets we have discussed so far. For\n", + "example, they can take sentences, documents, or audio samples as\n", + "input, making them extremely useful for natural language processing\n", + "systems such as automatic translation and speech-to-text.\n", + "\n", + "More to text to be added" + ] + }, + { + "cell_type": "markdown", + "id": "43cb4913", + "metadata": { + "editable": true + }, + "source": [ + "## A simple example" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "cf6b9dab", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [ + { + "data": { + "image/png": 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    " + ] + }, + "metadata": { + "filenames": { + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter13_3_0.png" + }, + "needs_background": "light" + }, + "output_type": "display_data" + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "2021-12-08 06:58:41.160539: I tensorflow/core/platform/cpu_feature_guard.cc:142] This TensorFlow binary is optimized with oneAPI Deep Neural Network Library (oneDNN) to use the following CPU instructions in performance-critical operations: SSE4.1 SSE4.2 AVX AVX2 FMA\n", + "To enable them in other operations, rebuild TensorFlow with the appropriate compiler flags.\n", + "2021-12-08 06:58:41.346810: I tensorflow/compiler/mlir/mlir_graph_optimization_pass.cc:185] None of the MLIR Optimization Passes are enabled (registered 2)\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Model: \"sequential\"\n", + "_________________________________________________________________\n", + "Layer (type) Output Shape Param # \n", + "=================================================================\n", + "simple_rnn (SimpleRNN) (None, 32) 1184 \n", + "_________________________________________________________________\n", + "dense (Dense) (None, 8) 264 \n", + "_________________________________________________________________\n", + "dense_1 (Dense) (None, 1) 9 \n", + "=================================================================\n", + "Total params: 1,457\n", + "Trainable params: 1,457\n", + "Non-trainable params: 0\n", + "_________________________________________________________________\n", + "Epoch 1/100\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "50/50 - 1s - loss: 0.4255\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Epoch 2/100\n", + "50/50 - 0s - loss: 0.4030\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Epoch 3/100\n", + "50/50 - 0s - loss: 0.3958\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Epoch 4/100\n", + "50/50 - 0s - loss: 0.3931\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Epoch 5/100\n", + "50/50 - 0s - loss: 0.3914\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Epoch 6/100\n", + "50/50 - 0s - loss: 0.3898\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Epoch 7/100\n", + "50/50 - 0s - loss: 0.3882\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Epoch 8/100\n", + "50/50 - 0s - loss: 0.3885\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Epoch 9/100\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "50/50 - 0s - loss: 0.3871\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Epoch 10/100\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "50/50 - 0s - loss: 0.3876\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Epoch 11/100\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "50/50 - 0s - loss: 0.3867\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Epoch 12/100\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "50/50 - 0s - loss: 0.3850\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Epoch 13/100\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "50/50 - 0s - loss: 0.3854\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Epoch 14/100\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "50/50 - 0s - loss: 0.3802\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Epoch 15/100\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "50/50 - 0s - loss: 0.3842\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Epoch 16/100\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "50/50 - 0s - loss: 0.3820\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Epoch 17/100\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "50/50 - 0s - loss: 0.3842\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Epoch 18/100\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "50/50 - 0s - loss: 0.3827\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Epoch 19/100\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "50/50 - 0s - loss: 0.3819\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Epoch 20/100\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "50/50 - 0s 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\u001b[0merr\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 387\u001b[0m \u001b[0;32mraise\u001b[0m \u001b[0mKeyError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mkey\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfrom\u001b[0m \u001b[0merr\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 388\u001b[0;31m \u001b[0;32mraise\u001b[0m \u001b[0mKeyError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mkey\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 389\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0msuper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget_loc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mkey\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mmethod\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mmethod\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtolerance\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mtolerance\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 390\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;31mKeyError\u001b[0m: (slice(None, None, None), None)" + ] + }, + { + "data": { + "image/png": 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    " + ] + }, + "metadata": { + "filenames": { + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter13_3_192.png" + }, + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "%matplotlib inline\n", + "\n", + "# Start importing packages\n", + "import pandas as pd\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "import tensorflow as tf\n", + "from tensorflow.keras import datasets, layers, models\n", + "from tensorflow.keras.layers import Input\n", + "from tensorflow.keras.models import Model, Sequential \n", + "from tensorflow.keras.layers import Dense, SimpleRNN, LSTM, GRU\n", + "from tensorflow.keras import optimizers \n", + "from tensorflow.keras import regularizers \n", + "from tensorflow.keras.utils import to_categorical \n", + "\n", + "\n", + "\n", + "# convert into dataset matrix\n", + "def convertToMatrix(data, step):\n", + " X, Y =[], []\n", + " for i in range(len(data)-step):\n", + " d=i+step \n", + " X.append(data[i:d,])\n", + " Y.append(data[d,])\n", + " return np.array(X), np.array(Y)\n", + "\n", + "step = 4\n", + "N = 1000 \n", + "Tp = 800 \n", + "\n", + "t=np.arange(0,N)\n", + "x=np.sin(0.02*t)+2*np.random.rand(N)\n", + "df = pd.DataFrame(x)\n", + "df.head()\n", + "\n", + "plt.plot(df)\n", + "plt.show()\n", + "\n", + "values=df.values\n", + "train,test = values[0:Tp,:], values[Tp:N,:]\n", + "\n", + "# add step elements into train and test\n", + "test = np.append(test,np.repeat(test[-1,],step))\n", + "train = np.append(train,np.repeat(train[-1,],step))\n", + " \n", + "trainX,trainY =convertToMatrix(train,step)\n", + "testX,testY =convertToMatrix(test,step)\n", + "trainX = np.reshape(trainX, (trainX.shape[0], 1, trainX.shape[1]))\n", + "testX = np.reshape(testX, (testX.shape[0], 1, testX.shape[1]))\n", + "\n", + "model = Sequential()\n", + "model.add(SimpleRNN(units=32, input_shape=(1,step), activation=\"relu\"))\n", + "model.add(Dense(8, activation=\"relu\")) \n", + "model.add(Dense(1))\n", + "model.compile(loss='mean_squared_error', optimizer='rmsprop')\n", + "model.summary()\n", + "\n", + "model.fit(trainX,trainY, epochs=100, batch_size=16, verbose=2)\n", + "trainPredict = model.predict(trainX)\n", + "testPredict= model.predict(testX)\n", + "predicted=np.concatenate((trainPredict,testPredict),axis=0)\n", + "\n", + "trainScore = model.evaluate(trainX, trainY, verbose=0)\n", + "print(trainScore)\n", + "\n", + "index = df.index.values\n", + "plt.plot(index,df)\n", + "plt.plot(index,predicted)\n", + "plt.axvline(df.index[Tp], c=\"r\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "897a47c7", + "metadata": { + "editable": true + }, + "source": [ + "## An extrapolation example\n", + "\n", + "The following code provides an example of how recurrent neural\n", + "networks can be used to extrapolate to unknown values of physics data\n", + "sets. Specifically, the data sets used in this program come from\n", + "a quantum mechanical many-body calculation of energies as functions of the number of particles." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "6776ae2a", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "\n", + "# For matrices and calculations\n", + "import numpy as np\n", + "# For machine learning (backend for keras)\n", + "import tensorflow as tf\n", + "# User-friendly machine learning library\n", + "# Front end for TensorFlow\n", + "import tensorflow.keras\n", + "# Different methods from Keras needed to create an RNN\n", + "# This is not necessary but it shortened function calls \n", + "# that need to be used in the code.\n", + "from tensorflow.keras import datasets, layers, models\n", + "from tensorflow.keras.layers import Input\n", + "from tensorflow.keras import regularizers\n", + "from tensorflow.keras.models import Model, Sequential\n", + "from tensorflow.keras.layers import Dense, SimpleRNN, LSTM, GRU\n", + "# For timing the code\n", + "from timeit import default_timer as timer\n", + "# For plotting\n", + "import matplotlib.pyplot as plt\n", + "\n", + "\n", + "# The data set\n", + "datatype='VaryDimension'\n", + "X_tot = np.arange(2, 42, 2)\n", + "y_tot = np.array([-0.03077640549, -0.08336233266, -0.1446729567, -0.2116753732, -0.2830637392, -0.3581341341, -0.436462435, -0.5177783846,\n", + "\t-0.6019067271, -0.6887363571, -0.7782028952, -0.8702784034, -0.9649652536, -1.062292565, -1.16231451, \n", + "\t-1.265109911, -1.370782966, -1.479465113, -1.591317992, -1.70653767])" + ] + }, + { + "cell_type": "markdown", + "id": "35227d36", + "metadata": { + "editable": true + }, + "source": [ + "The way the recurrent neural networks are trained in this program\n", + "differs from how machine learning algorithms are usually trained.\n", + "Typically a machine learning algorithm is trained by learning the\n", + "relationship between the x data and the y data. In this program, the\n", + "recurrent neural network will be trained to recognize the relationship\n", + "in a sequence of y values. This is type of data formatting is\n", + "typically used time series forcasting, but it can also be used in any\n", + "extrapolation (time series forecasting is just a specific type of\n", + "extrapolation along the time axis). This method of data formatting\n", + "does not use the x data and assumes that the y data are evenly spaced.\n", + "\n", + "For a standard machine learning algorithm, the training data has the\n", + "form of (x,y) so the machine learning algorithm learns to assiciate a\n", + "y value with a given x value. This is useful when the test data has x\n", + "values within the same range as the training data. However, for this\n", + "application, the x values of the test data are outside of the x values\n", + "of the training data and the traditional method of training a machine\n", + "learning algorithm does not work as well. For this reason, the\n", + "recurrent neural network is trained on sequences of y values of the\n", + "form ((y1, y2), y3), so that the network is concerned with learning\n", + "the pattern of the y data and not the relation between the x and y\n", + "data. As long as the pattern of y data outside of the training region\n", + "stays relatively stable compared to what was inside the training\n", + "region, this method of training can produce accurate extrapolations to\n", + "y values far removed from the training data set." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "7dc577d1", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "# FORMAT_DATA\n", + "def format_data(data, length_of_sequence = 2): \n", + " \"\"\"\n", + " Inputs:\n", + " data(a numpy array): the data that will be the inputs to the recurrent neural\n", + " network\n", + " length_of_sequence (an int): the number of elements in one iteration of the\n", + " sequence patter. For a function approximator use length_of_sequence = 2.\n", + " Returns:\n", + " rnn_input (a 3D numpy array): the input data for the recurrent neural network. Its\n", + " dimensions are length of data - length of sequence, length of sequence, \n", + " dimnsion of data\n", + " rnn_output (a numpy array): the training data for the neural network\n", + " Formats data to be used in a recurrent neural network.\n", + " \"\"\"\n", + "\n", + " X, Y = [], []\n", + " for i in range(len(data)-length_of_sequence):\n", + " # Get the next length_of_sequence elements\n", + " a = data[i:i+length_of_sequence]\n", + " # Get the element that immediately follows that\n", + " b = data[i+length_of_sequence]\n", + " # Reshape so that each data point is contained in its own array\n", + " a = np.reshape (a, (len(a), 1))\n", + " X.append(a)\n", + " Y.append(b)\n", + " rnn_input = np.array(X)\n", + " rnn_output = np.array(Y)\n", + "\n", + " return rnn_input, rnn_output\n", + "\n", + "\n", + "# ## Defining the Recurrent Neural Network Using Keras\n", + "# \n", + "# The following method defines a simple recurrent neural network in keras consisting of one input layer, one hidden layer, and one output layer.\n", + "\n", + "def rnn(length_of_sequences, batch_size = None, stateful = False):\n", + " \"\"\"\n", + " Inputs:\n", + " length_of_sequences (an int): the number of y values in \"x data\". This is determined\n", + " when the data is formatted\n", + " batch_size (an int): Default value is None. See Keras documentation of SimpleRNN.\n", + " stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN.\n", + " Returns:\n", + " model (a Keras model): The recurrent neural network that is built and compiled by this\n", + " method\n", + " Builds and compiles a recurrent neural network with one hidden layer and returns the model.\n", + " \"\"\"\n", + " # Number of neurons in the input and output layers\n", + " in_out_neurons = 1\n", + " # Number of neurons in the hidden layer\n", + " hidden_neurons = 200\n", + " # Define the input layer\n", + " inp = Input(batch_shape=(batch_size, \n", + " length_of_sequences, \n", + " in_out_neurons)) \n", + " # Define the hidden layer as a simple RNN layer with a set number of neurons and add it to \n", + " # the network immediately after the input layer\n", + " rnn = SimpleRNN(hidden_neurons, \n", + " return_sequences=False,\n", + " stateful = stateful,\n", + " name=\"RNN\")(inp)\n", + " # Define the output layer as a dense neural network layer (standard neural network layer)\n", + " #and add it to the network immediately after the hidden layer.\n", + " dens = Dense(in_out_neurons,name=\"dense\")(rnn)\n", + " # Create the machine learning model starting with the input layer and ending with the \n", + " # output layer\n", + " model = Model(inputs=[inp],outputs=[dens])\n", + " # Compile the machine learning model using the mean squared error function as the loss \n", + " # function and an Adams optimizer.\n", + " model.compile(loss=\"mean_squared_error\", optimizer=\"adam\") \n", + " return model" + ] + }, + { + "cell_type": "markdown", + "id": "1978b6a1", + "metadata": { + "editable": true + }, + "source": [ + "## Predicting New Points With A Trained Recurrent Neural Network" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "id": "d4ad417a", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def test_rnn (x1, y_test, plot_min, plot_max):\n", + " \"\"\"\n", + " Inputs:\n", + " x1 (a list or numpy array): The complete x component of the data set\n", + " y_test (a list or numpy array): The complete y component of the data set\n", + " plot_min (an int or float): the smallest x value used in the training data\n", + " plot_max (an int or float): the largest x valye used in the training data\n", + " Returns:\n", + " None.\n", + " Uses a trained recurrent neural network model to predict future points in the \n", + " series. Computes the MSE of the predicted data set from the true data set, saves\n", + " the predicted data set to a csv file, and plots the predicted and true data sets w\n", + " while also displaying the data range used for training.\n", + " \"\"\"\n", + " # Add the training data as the first dim points in the predicted data array as these\n", + " # are known values.\n", + " y_pred = y_test[:dim].tolist()\n", + " # Generate the first input to the trained recurrent neural network using the last two \n", + " # points of the training data. Based on how the network was trained this means that it\n", + " # will predict the first point in the data set after the training data. All of the \n", + " # brackets are necessary for Tensorflow.\n", + " next_input = np.array([[[y_test[dim-2]], [y_test[dim-1]]]])\n", + " # Save the very last point in the training data set. This will be used later.\n", + " last = [y_test[dim-1]]\n", + "\n", + " # Iterate until the complete data set is created.\n", + " for i in range (dim, len(y_test)):\n", + " # Predict the next point in the data set using the previous two points.\n", + " next = model.predict(next_input)\n", + " # Append just the number of the predicted data set\n", + " y_pred.append(next[0][0])\n", + " # Create the input that will be used to predict the next data point in the data set.\n", + " next_input = np.array([[last, next[0]]], dtype=np.float64)\n", + " last = next\n", + "\n", + " # Print the mean squared error between the known data set and the predicted data set.\n", + " print('MSE: ', np.square(np.subtract(y_test, y_pred)).mean())\n", + " # Save the predicted data set as a csv file for later use\n", + " name = datatype + 'Predicted'+str(dim)+'.csv'\n", + " np.savetxt(name, y_pred, delimiter=',')\n", + " # Plot the known data set and the predicted data set. The red box represents the region that was used\n", + " # for the training data.\n", + " fig, ax = plt.subplots()\n", + " ax.plot(x1, y_test, label=\"true\", linewidth=3)\n", + " ax.plot(x1, y_pred, 'g-.',label=\"predicted\", linewidth=4)\n", + " ax.legend()\n", + " # Created a red region to represent the points used in the training data.\n", + " ax.axvspan(plot_min, plot_max, alpha=0.25, color='red')\n", + " plt.show()\n", + "\n", + "# Check to make sure the data set is complete\n", + "assert len(X_tot) == len(y_tot)\n", + "\n", + "# This is the number of points that will be used in as the training data\n", + "dim=12\n", + "\n", + "# Separate the training data from the whole data set\n", + "X_train = X_tot[:dim]\n", + "y_train = y_tot[:dim]\n", + "\n", + "\n", + "# Generate the training data for the RNN, using a sequence of 2\n", + "rnn_input, rnn_training = format_data(y_train, 2)\n", + "\n", + "\n", + "# Create a recurrent neural network in Keras and produce a summary of the \n", + "# machine learning model\n", + "model = rnn(length_of_sequences = rnn_input.shape[1])\n", + "model.summary()\n", + "\n", + "# Start the timer. Want to time training+testing\n", + "start = timer()\n", + "# Fit the model using the training data genenerated above using 150 training iterations and a 5%\n", + "# validation split. Setting verbose to True prints information about each training iteration.\n", + "hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, \n", + " verbose=True,validation_split=0.05)\n", + "\n", + "for label in [\"loss\",\"val_loss\"]:\n", + " plt.plot(hist.history[label],label=label)\n", + "\n", + "plt.ylabel(\"loss\")\n", + "plt.xlabel(\"epoch\")\n", + "plt.title(\"The final validation loss: {}\".format(hist.history[\"val_loss\"][-1]))\n", + "plt.legend()\n", + "plt.show()\n", + "\n", + "# Use the trained neural network to predict more points of the data set\n", + "test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1])\n", + "# Stop the timer and calculate the total time needed.\n", + "end = timer()\n", + "print('Time: ', end-start)" + ] + }, + { + "cell_type": "markdown", + "id": "e2cad4fc", + "metadata": { + "editable": true + }, + "source": [ + "Changing the size of the recurrent neural network and its parameters\n", + "can drastically change the results you get from the model. The below\n", + "code takes the simple recurrent neural network from above and adds a\n", + "second hidden layer, changes the number of neurons in the hidden\n", + "layer, and explicitly declares the activation function of the hidden\n", + "layers to be a sigmoid function. The loss function and optimizer can\n", + "also be changed but are kept the same as the above network. These\n", + "parameters can be tuned to provide the optimal result from the\n", + "network. For some ideas on how to improve the performance of a\n", + "[recurrent neural network](https://danijar.com/tips-for-training-recurrent-neural-networks)." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "id": "c39f1516", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def rnn_2layers(length_of_sequences, batch_size = None, stateful = False):\n", + " \"\"\"\n", + " Inputs:\n", + " length_of_sequences (an int): the number of y values in \"x data\". This is determined\n", + " when the data is formatted\n", + " batch_size (an int): Default value is None. See Keras documentation of SimpleRNN.\n", + " stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN.\n", + " Returns:\n", + " model (a Keras model): The recurrent neural network that is built and compiled by this\n", + " method\n", + " Builds and compiles a recurrent neural network with two hidden layers and returns the model.\n", + " \"\"\"\n", + " # Number of neurons in the input and output layers\n", + " in_out_neurons = 1\n", + " # Number of neurons in the hidden layer, increased from the first network\n", + " hidden_neurons = 500\n", + " # Define the input layer\n", + " inp = Input(batch_shape=(batch_size, \n", + " length_of_sequences, \n", + " in_out_neurons)) \n", + " # Create two hidden layers instead of one hidden layer. Explicitly set the activation\n", + " # function to be the sigmoid function (the default value is hyperbolic tangent)\n", + " rnn1 = SimpleRNN(hidden_neurons, \n", + " return_sequences=True, # This needs to be True if another hidden layer is to follow\n", + " stateful = stateful, activation = 'sigmoid',\n", + " name=\"RNN1\")(inp)\n", + " rnn2 = SimpleRNN(hidden_neurons, \n", + " return_sequences=False, activation = 'sigmoid',\n", + " stateful = stateful,\n", + " name=\"RNN2\")(rnn1)\n", + " # Define the output layer as a dense neural network layer (standard neural network layer)\n", + " #and add it to the network immediately after the hidden layer.\n", + " dens = Dense(in_out_neurons,name=\"dense\")(rnn2)\n", + " # Create the machine learning model starting with the input layer and ending with the \n", + " # output layer\n", + " model = Model(inputs=[inp],outputs=[dens])\n", + " # Compile the machine learning model using the mean squared error function as the loss \n", + " # function and an Adams optimizer.\n", + " model.compile(loss=\"mean_squared_error\", optimizer=\"adam\") \n", + " return model\n", + "\n", + "# Check to make sure the data set is complete\n", + "assert len(X_tot) == len(y_tot)\n", + "\n", + "# This is the number of points that will be used in as the training data\n", + "dim=12\n", + "\n", + "# Separate the training data from the whole data set\n", + "X_train = X_tot[:dim]\n", + "y_train = y_tot[:dim]\n", + "\n", + "\n", + "# Generate the training data for the RNN, using a sequence of 2\n", + "rnn_input, rnn_training = format_data(y_train, 2)\n", + "\n", + "\n", + "# Create a recurrent neural network in Keras and produce a summary of the \n", + "# machine learning model\n", + "model = rnn_2layers(length_of_sequences = 2)\n", + "model.summary()\n", + "\n", + "# Start the timer. Want to time training+testing\n", + "start = timer()\n", + "# Fit the model using the training data genenerated above using 150 training iterations and a 5%\n", + "# validation split. Setting verbose to True prints information about each training iteration.\n", + "hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, \n", + " verbose=True,validation_split=0.05)\n", + "\n", + "\n", + "# This section plots the training loss and the validation loss as a function of training iteration.\n", + "# This is not required for analyzing the couple cluster data but can help determine if the network is\n", + "# being overtrained.\n", + "for label in [\"loss\",\"val_loss\"]:\n", + " plt.plot(hist.history[label],label=label)\n", + "\n", + "plt.ylabel(\"loss\")\n", + "plt.xlabel(\"epoch\")\n", + "plt.title(\"The final validation loss: {}\".format(hist.history[\"val_loss\"][-1]))\n", + "plt.legend()\n", + "plt.show()\n", + "\n", + "# Use the trained neural network to predict more points of the data set\n", + "test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1])\n", + "# Stop the timer and calculate the total time needed.\n", + "end = timer()\n", + "print('Time: ', end-start)" + ] + }, + { + "cell_type": "markdown", + "id": "842c7602", + "metadata": { + "editable": true + }, + "source": [ + "## Other Types of Recurrent Neural Networks\n", + "\n", + "Besides a simple recurrent neural network layer, there are two other\n", + "commonly used types of recurrent neural network layers: Long Short\n", + "Term Memory (LSTM) and Gated Recurrent Unit (GRU). For a short\n", + "introduction to these layers see \n", + "and .\n", + "\n", + "The first network created below is similar to the previous network,\n", + "but it replaces the SimpleRNN layers with LSTM layers. The second\n", + "network below has two hidden layers made up of GRUs, which are\n", + "preceeded by two dense (feeddorward) neural network layers. These\n", + "dense layers \"preprocess\" the data before it reaches the recurrent\n", + "layers. This architecture has been shown to improve the performance\n", + "of recurrent neural networks (see the link above and also\n", + "." + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "id": "6f0e9b62", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def lstm_2layers(length_of_sequences, batch_size = None, stateful = False):\n", + " \"\"\"\n", + " Inputs:\n", + " length_of_sequences (an int): the number of y values in \"x data\". This is determined\n", + " when the data is formatted\n", + " batch_size (an int): Default value is None. See Keras documentation of SimpleRNN.\n", + " stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN.\n", + " Returns:\n", + " model (a Keras model): The recurrent neural network that is built and compiled by this\n", + " method\n", + " Builds and compiles a recurrent neural network with two LSTM hidden layers and returns the model.\n", + " \"\"\"\n", + " # Number of neurons on the input/output layer and the number of neurons in the hidden layer\n", + " in_out_neurons = 1\n", + " hidden_neurons = 250\n", + " # Input Layer\n", + " inp = Input(batch_shape=(batch_size, \n", + " length_of_sequences, \n", + " in_out_neurons)) \n", + " # Hidden layers (in this case they are LSTM layers instead if SimpleRNN layers)\n", + " rnn= LSTM(hidden_neurons, \n", + " return_sequences=True,\n", + " stateful = stateful,\n", + " name=\"RNN\", use_bias=True, activation='tanh')(inp)\n", + " rnn1 = LSTM(hidden_neurons, \n", + " return_sequences=False,\n", + " stateful = stateful,\n", + " name=\"RNN1\", use_bias=True, activation='tanh')(rnn)\n", + " # Output layer\n", + " dens = Dense(in_out_neurons,name=\"dense\")(rnn1)\n", + " # Define the midel\n", + " model = Model(inputs=[inp],outputs=[dens])\n", + " # Compile the model\n", + " model.compile(loss='mean_squared_error', optimizer='adam') \n", + " # Return the model\n", + " return model\n", + "\n", + "def dnn2_gru2(length_of_sequences, batch_size = None, stateful = False):\n", + " \"\"\"\n", + " Inputs:\n", + " length_of_sequences (an int): the number of y values in \"x data\". This is determined\n", + " when the data is formatted\n", + " batch_size (an int): Default value is None. See Keras documentation of SimpleRNN.\n", + " stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN.\n", + " Returns:\n", + " model (a Keras model): The recurrent neural network that is built and compiled by this\n", + " method\n", + " Builds and compiles a recurrent neural network with four hidden layers (two dense followed by\n", + " two GRU layers) and returns the model.\n", + " \"\"\" \n", + " # Number of neurons on the input/output layers and hidden layers\n", + " in_out_neurons = 1\n", + " hidden_neurons = 250\n", + " # Input layer\n", + " inp = Input(batch_shape=(batch_size, \n", + " length_of_sequences, \n", + " in_out_neurons)) \n", + " # Hidden Dense (feedforward) layers\n", + " dnn = Dense(hidden_neurons/2, activation='relu', name='dnn')(inp)\n", + " dnn1 = Dense(hidden_neurons/2, activation='relu', name='dnn1')(dnn)\n", + " # Hidden GRU layers\n", + " rnn1 = GRU(hidden_neurons, \n", + " return_sequences=True,\n", + " stateful = stateful,\n", + " name=\"RNN1\", use_bias=True)(dnn1)\n", + " rnn = GRU(hidden_neurons, \n", + " return_sequences=False,\n", + " stateful = stateful,\n", + " name=\"RNN\", use_bias=True)(rnn1)\n", + " # Output layer\n", + " dens = Dense(in_out_neurons,name=\"dense\")(rnn)\n", + " # Define the model\n", + " model = Model(inputs=[inp],outputs=[dens])\n", + " # Compile the mdoel\n", + " model.compile(loss='mean_squared_error', optimizer='adam') \n", + " # Return the model\n", + " return model\n", + "\n", + "# Check to make sure the data set is complete\n", + "assert len(X_tot) == len(y_tot)\n", + "\n", + "# This is the number of points that will be used in as the training data\n", + "dim=12\n", + "\n", + "# Separate the training data from the whole data set\n", + "X_train = X_tot[:dim]\n", + "y_train = y_tot[:dim]\n", + "\n", + "\n", + "# Generate the training data for the RNN, using a sequence of 2\n", + "rnn_input, rnn_training = format_data(y_train, 2)\n", + "\n", + "\n", + "# Create a recurrent neural network in Keras and produce a summary of the \n", + "# machine learning model\n", + "# Change the method name to reflect which network you want to use\n", + "model = dnn2_gru2(length_of_sequences = 2)\n", + "model.summary()\n", + "\n", + "# Start the timer. Want to time training+testing\n", + "start = timer()\n", + "# Fit the model using the training data genenerated above using 150 training iterations and a 5%\n", + "# validation split. Setting verbose to True prints information about each training iteration.\n", + "hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, \n", + " verbose=True,validation_split=0.05)\n", + "\n", + "\n", + "# This section plots the training loss and the validation loss as a function of training iteration.\n", + "# This is not required for analyzing the couple cluster data but can help determine if the network is\n", + "# being overtrained.\n", + "for label in [\"loss\",\"val_loss\"]:\n", + " plt.plot(hist.history[label],label=label)\n", + "\n", + "plt.ylabel(\"loss\")\n", + "plt.xlabel(\"epoch\")\n", + "plt.title(\"The final validation loss: {}\".format(hist.history[\"val_loss\"][-1]))\n", + "plt.legend()\n", + "plt.show()\n", + "\n", + "# Use the trained neural network to predict more points of the data set\n", + "test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1])\n", + "# Stop the timer and calculate the total time needed.\n", + "end = timer()\n", + "print('Time: ', end-start)\n", + "\n", + "\n", + "# ### Training Recurrent Neural Networks in the Standard Way (i.e. learning the relationship between the X and Y data)\n", + "# \n", + "# Finally, comparing the performace of a recurrent neural network using the standard data formatting to the performance of the network with time sequence data formatting shows the benefit of this type of data formatting with extrapolation.\n", + "\n", + "# Check to make sure the data set is complete\n", + "assert len(X_tot) == len(y_tot)\n", + "\n", + "# This is the number of points that will be used in as the training data\n", + "dim=12\n", + "\n", + "# Separate the training data from the whole data set\n", + "X_train = X_tot[:dim]\n", + "y_train = y_tot[:dim]\n", + "\n", + "# Reshape the data for Keras specifications\n", + "X_train = X_train.reshape((dim, 1))\n", + "y_train = y_train.reshape((dim, 1))\n", + "\n", + "\n", + "# Create a recurrent neural network in Keras and produce a summary of the \n", + "# machine learning model\n", + "# Set the sequence length to 1 for regular data formatting \n", + "model = rnn(length_of_sequences = 1)\n", + "model.summary()\n", + "\n", + "# Start the timer. Want to time training+testing\n", + "start = timer()\n", + "# Fit the model using the training data genenerated above using 150 training iterations and a 5%\n", + "# validation split. Setting verbose to True prints information about each training iteration.\n", + "hist = model.fit(X_train, y_train, batch_size=None, epochs=150, \n", + " verbose=True,validation_split=0.05)\n", + "\n", + "\n", + "# This section plots the training loss and the validation loss as a function of training iteration.\n", + "# This is not required for analyzing the couple cluster data but can help determine if the network is\n", + "# being overtrained.\n", + "for label in [\"loss\",\"val_loss\"]:\n", + " plt.plot(hist.history[label],label=label)\n", + "\n", + "plt.ylabel(\"loss\")\n", + "plt.xlabel(\"epoch\")\n", + "plt.title(\"The final validation loss: {}\".format(hist.history[\"val_loss\"][-1]))\n", + "plt.legend()\n", + "plt.show()\n", + "\n", + "# Use the trained neural network to predict the remaining data points\n", + "X_pred = X_tot[dim:]\n", + "X_pred = X_pred.reshape((len(X_pred), 1))\n", + "y_model = model.predict(X_pred)\n", + "y_pred = np.concatenate((y_tot[:dim], y_model.flatten()))\n", + "\n", + "# Plot the known data set and the predicted data set. The red box represents the region that was used\n", + "# for the training data.\n", + "fig, ax = plt.subplots()\n", + "ax.plot(X_tot, y_tot, label=\"true\", linewidth=3)\n", + "ax.plot(X_tot, y_pred, 'g-.',label=\"predicted\", linewidth=4)\n", + "ax.legend()\n", + "# Created a red region to represent the points used in the training data.\n", + "ax.axvspan(X_tot[0], X_tot[dim], alpha=0.25, color='red')\n", + "plt.show()\n", + "\n", + "# Stop the timer and calculate the total time needed.\n", + "end = timer()\n", + "print('Time: ', end-start)" + ] + }, + { + "cell_type": "markdown", + "id": "0752ba7f", + "metadata": { + "editable": true + }, + "source": [ + "# Generative Models\n", + "\n", + "**Generative models** describe a class of statistical models that are a contrast\n", + "to **discriminative models**. Informally we say that generative models can\n", + "generate new data instances while discriminative models discriminate between\n", + "different kinds of data instances. A generative model could generate new photos\n", + "of animals that look like 'real' animals while a discriminative model could tell\n", + "a dog from a cat. More formally, given a data set $x$ and a set of labels /\n", + "targets $y$. Generative models capture the joint probability $p(x, y)$, or\n", + "just $p(x)$ if there are no labels, while discriminative models capture the\n", + "conditional probability $p(y | x)$. Discriminative models generally try to draw\n", + "boundaries in the data space (often high dimensional), while generative models\n", + "try to model how data is placed throughout the space.\n", + "\n", + "**Note**: this material is thanks to Linus Ekstrøm." + ] + }, + { + "cell_type": "markdown", + "id": "784138f8", + "metadata": { + "editable": true + }, + "source": [ + "## Generative Adversarial Networks\n", + "\n", + "**Generative Adversarial Networks** are a type of unsupervised machine learning\n", + "algorithm proposed by [Goodfellow et. al](https://arxiv.org/pdf/1406.2661.pdf)\n", + "in 2014 (short and good article).\n", + "\n", + "The simplest formulation of\n", + "the model is based on a game theoretic approach, *zero sum game*, where we pit\n", + "two neural networks against one another. We define two rival networks, one\n", + "generator $g$, and one discriminator $d$. The generator directly produces\n", + "samples" + ] + }, + { + "cell_type": "markdown", + "id": "a42f89ee", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " x = g(z; \\theta^{(g)})\n", + "\\label{_auto1} \\tag{1}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "abe7212f", + "metadata": { + "editable": true + }, + "source": [ + "The discriminator attempts to distinguish between samples drawn from the\n", + "training data and samples drawn from the generator. In other words, it tries to\n", + "tell the difference between the fake data produced by $g$ and the actual data\n", + "samples we want to do prediction on. The discriminator outputs a probability\n", + "value given by" + ] + }, + { + "cell_type": "markdown", + "id": "0821eaee", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " d(x; \\theta^{(d)})\n", + "\\label{_auto2} \\tag{2}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "55e0ccf6", + "metadata": { + "editable": true + }, + "source": [ + "indicating the probability that $x$ is a real training example rather than a\n", + "fake sample the generator has generated. The simplest way to formulate the\n", + "learning process in a generative adversarial network is a zero-sum game, in\n", + "which a function" + ] + }, + { + "cell_type": "markdown", + "id": "f37ece14", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " v(\\theta^{(g)}, \\theta^{(d)})\n", + "\\label{_auto3} \\tag{3}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "d6d6d5fa", + "metadata": { + "editable": true + }, + "source": [ + "determines the reward for the discriminator, while the generator gets the\n", + "conjugate reward" + ] + }, + { + "cell_type": "markdown", + "id": "3c74b45c", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " -v(\\theta^{(g)}, \\theta^{(d)})\n", + "\\label{_auto4} \\tag{4}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "605ac8c9", + "metadata": { + "editable": true + }, + "source": [ + "During learning both of the networks maximize their own reward function, so that\n", + "the generator gets better and better at tricking the discriminator, while the\n", + "discriminator gets better and better at telling the difference between the fake\n", + "and real data. The generator and discriminator alternate on which one trains at\n", + "one time (i.e. for one epoch). In other words, we keep the generator constant\n", + "and train the discriminator, then we keep the discriminator constant to train\n", + "the generator and repeat. It is this back and forth dynamic which lets GANs\n", + "tackle otherwise intractable generative problems. As the generator improves with\n", + " training, the discriminator's performance gets worse because it cannot easily\n", + " tell the difference between real and fake. If the generator ends up succeeding\n", + " perfectly, the the discriminator will do no better than random guessing i.e.\n", + " 50\\%. This progression in the training poses a problem for the convergence\n", + " criteria for GANs. The discriminator feedback gets less meaningful over time,\n", + " if we continue training after this point then the generator is effectively\n", + " training on junk data which can undo the learning up to that point. Therefore,\n", + " we stop training when the discriminator starts outputting $1/2$ everywhere.\n", + "\n", + "At convergence we have" + ] + }, + { + "cell_type": "markdown", + "id": "cfec7462", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " g^* = \\underset{g}{\\mathrm{argmin}}\\hspace{2pt}\n", + " \\underset{d}{\\mathrm{max}}v(\\theta^{(g)}, \\theta^{(d)})\n", + "\\label{_auto5} \\tag{5}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "37c65d4c", + "metadata": { + "editable": true + }, + "source": [ + "The default choice for $v$ is" + ] + }, + { + "cell_type": "markdown", + "id": "c868a092", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " v(\\theta^{(g)}, \\theta^{(d)}) = \\mathbb{E}_{x\\sim p_\\mathrm{data}}\\log d(x)\n", + " + \\mathbb{E}_{x\\sim p_\\mathrm{model}}\n", + " \\log (1 - d(x))\n", + "\\label{_auto6} \\tag{6}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "ad465af3", + "metadata": { + "editable": true + }, + "source": [ + "The main motivation for the design of GANs is that the learning process requires\n", + "neither approximate inference (variational autoencoders for example) nor\n", + "approximation of a partition function. In the case where" + ] + }, + { + "cell_type": "markdown", + "id": "27858a4e", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " \\underset{d}{\\mathrm{max}}v(\\theta^{(g)}, \\theta^{(d)})\n", + "\\label{_auto7} \\tag{7}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "86006023", + "metadata": { + "editable": true + }, + "source": [ + "is convex in $\\theta^{(g)} then the procedure is guaranteed to converge and is\n", + "asymptotically consistent\n", + "( [Seth Lloyd on QuGANs](https://arxiv.org/pdf/1804.09139.pdf) ).\n", + "\n", + "This is in\n", + "general not the case and it is possible to get situations where the training\n", + "process never converges because the generator and discriminator chase one\n", + "another around in the parameter space indefinitely. A much deeper discussion on\n", + "the currently open research problem of GAN convergence is available\n", + "[here](https://www.deeplearningbook.org/contents/generative_models.html). To\n", + "anyone interested in learning more about GANs it is a highly recommended read.\n", + "Direct quote: \"In this best-performing formulation, the generator aims to\n", + "increase the log probability that the discriminator makes a mistake, rather than\n", + "aiming to decrease the log probability that the discriminator makes the correct\n", + "prediction.\" [Another interesting read](https://arxiv.org/abs/1701.00160)" + ] + }, + { + "cell_type": "markdown", + "id": "2fee38bd", + "metadata": { + "editable": true + }, + "source": [ + "## Writing Our First Generative Adversarial Network\n", + "Let us now move on to actually implementing a GAN in tensorflow. We will study\n", + "the performance of our GAN on the MNIST dataset. This code is based on and\n", + "adapted from the\n", + "[google tutorial](https://www.tensorflow.org/tutorials/generative/dcgan)\n", + "\n", + "First we import our libraries" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "id": "004a0b53", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "import os\n", + "import time\n", + "import numpy as np\n", + "import tensorflow as tf\n", + "import matplotlib.pyplot as plt\n", + "from tensorflow.keras import layers\n", + "from tensorflow.keras.utils import plot_model" + ] + }, + { + "cell_type": "markdown", + "id": "353af161", + "metadata": { + "editable": true + }, + "source": [ + "Next we define our hyperparameters and import our data the usual way" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "id": "8cbaf16a", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "BUFFER_SIZE = 60000\n", + "BATCH_SIZE = 256\n", + "EPOCHS = 30\n", + "\n", + "data = tf.keras.datasets.mnist.load_data()\n", + "(train_images, train_labels), (test_images, test_labels) = data\n", + "train_images = np.reshape(train_images, (train_images.shape[0],\n", + " 28,\n", + " 28,\n", + " 1)).astype('float32')\n", + "\n", + "# we normalize between -1 and 1\n", + "train_images = (train_images - 127.5) / 127.5\n", + "training_dataset = tf.data.Dataset.from_tensor_slices(\n", + " train_images).shuffle(BUFFER_SIZE).batch(BATCH_SIZE)" + ] + }, + { + "cell_type": "markdown", + "id": "822b8cc7", + "metadata": { + "editable": true + }, + "source": [ + "### MNIST and GANs\n", + "\n", + "Let's have a quick look" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "id": "52b5965c", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "plt.imshow(train_images[0], cmap='Greys')\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "21c5199c", + "metadata": { + "editable": true + }, + "source": [ + "Now we define our two models. This is where the 'magic' happens. There are a\n", + "huge amount of possible formulations for both models. A lot of engineering and\n", + "trial and error can be done here to try to produce better performing models. For\n", + "more advanced GANs this is by far the step where you can 'make or break' a\n", + "model.\n", + "\n", + "We start with the generator. As stated in the introductory text the generator\n", + "$g$ upsamples from a random sample to the shape of what we want to predict. In\n", + "our case we are trying to predict MNIST images ($28\\times 28$ pixels)." + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "id": "356759c7", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def generator_model():\n", + " \"\"\"\n", + " The generator uses upsampling layers tf.keras.layers.Conv2DTranspose() to\n", + " produce an image from a random seed. We start with a Dense layer taking this\n", + " random sample as an input and subsequently upsample through multiple\n", + " convolutional layers.\n", + " \"\"\"\n", + "\n", + " # we define our model\n", + " model = tf.keras.Sequential()\n", + "\n", + "\n", + " # adding our input layer. Dense means that every neuron is connected and\n", + " # the input shape is the shape of our random noise. The units need to match\n", + " # in some sense the upsampling strides to reach our desired output shape.\n", + " # we are using 100 random numbers as our seed\n", + " model.add(layers.Dense(units=7*7*BATCH_SIZE,\n", + " use_bias=False,\n", + " input_shape=(100, )))\n", + " # we normalize the output form the Dense layer\n", + " model.add(layers.BatchNormalization())\n", + " # and add an activation function to our 'layer'. LeakyReLU avoids vanishing\n", + " # gradient problem\n", + " model.add(layers.LeakyReLU())\n", + " model.add(layers.Reshape((7, 7, BATCH_SIZE)))\n", + " assert model.output_shape == (None, 7, 7, BATCH_SIZE)\n", + " # even though we just added four keras layers we think of everything above\n", + " # as 'one' layer\n", + "\n", + " # next we add our upscaling convolutional layers\n", + " model.add(layers.Conv2DTranspose(filters=128,\n", + " kernel_size=(5, 5),\n", + " strides=(1, 1),\n", + " padding='same',\n", + " use_bias=False))\n", + " model.add(layers.BatchNormalization())\n", + " model.add(layers.LeakyReLU())\n", + " assert model.output_shape == (None, 7, 7, 128)\n", + "\n", + " model.add(layers.Conv2DTranspose(filters=64,\n", + " kernel_size=(5, 5),\n", + " strides=(2, 2),\n", + " padding='same',\n", + " use_bias=False))\n", + " model.add(layers.BatchNormalization())\n", + " model.add(layers.LeakyReLU())\n", + " assert model.output_shape == (None, 14, 14, 64)\n", + "\n", + " model.add(layers.Conv2DTranspose(filters=1,\n", + " kernel_size=(5, 5),\n", + " strides=(2, 2),\n", + " padding='same',\n", + " use_bias=False,\n", + " activation='tanh'))\n", + " assert model.output_shape == (None, 28, 28, 1)\n", + "\n", + " return model" + ] + }, + { + "cell_type": "markdown", + "id": "854bcd6b", + "metadata": { + "editable": true + }, + "source": [ + "And there we have our 'simple' generator model. Now we move on to defining our\n", + "discriminator model $d$, which is a convolutional neural network based image\n", + "classifier." + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "id": "41473304", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def discriminator_model():\n", + " \"\"\"\n", + " The discriminator is a convolutional neural network based image classifier\n", + " \"\"\"\n", + "\n", + " # we define our model\n", + " model = tf.keras.Sequential()\n", + " model.add(layers.Conv2D(filters=64,\n", + " kernel_size=(5, 5),\n", + " strides=(2, 2),\n", + " padding='same',\n", + " input_shape=[28, 28, 1]))\n", + " model.add(layers.LeakyReLU())\n", + " # adding a dropout layer as you do in conv-nets\n", + " model.add(layers.Dropout(0.3))\n", + "\n", + "\n", + " model.add(layers.Conv2D(filters=128,\n", + " kernel_size=(5, 5),\n", + " strides=(2, 2),\n", + " padding='same'))\n", + " model.add(layers.LeakyReLU())\n", + " # adding a dropout layer as you do in conv-nets\n", + " model.add(layers.Dropout(0.3))\n", + "\n", + " model.add(layers.Flatten())\n", + " model.add(layers.Dense(1))\n", + "\n", + " return model" + ] + }, + { + "cell_type": "markdown", + "id": "353af567", + "metadata": { + "editable": true + }, + "source": [ + "Let us take a look at our models." + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "id": "f899d4e3", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "generator = generator_model()\n", + "plot_model(generator, show_shapes=True, rankdir='LR')" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "id": "87ef384b", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "discriminator = discriminator_model()\n", + "plot_model(discriminator, show_shapes=True, rankdir='LR')" + ] + }, + { + "cell_type": "markdown", + "id": "b2bef82d", + "metadata": { + "editable": true + }, + "source": [ + "Next we need a few helper objects we will use in training" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "id": "e397847a", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "cross_entropy = tf.keras.losses.BinaryCrossentropy(from_logits=True)\n", + "generator_optimizer = tf.keras.optimizers.Adam(1e-4)\n", + "discriminator_optimizer = tf.keras.optimizers.Adam(1e-4)" + ] + }, + { + "cell_type": "markdown", + "id": "db3396cc", + "metadata": { + "editable": true + }, + "source": [ + "The first object, *cross_entropy* is our loss function and the two others are\n", + "our optimizers. Notice we use the same learning rate for both $g$ and $d$. This\n", + "is because they need to improve their accuracy at approximately equal speeds to\n", + "get convergence (not necessarily exactly equal). Now we define our loss\n", + "functions" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "id": "931eaced", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def generator_loss(fake_output):\n", + " loss = cross_entropy(tf.ones_like(fake_output), fake_output)\n", + "\n", + " return loss" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "id": "0c4a44bb", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def discriminator_loss(real_output, fake_output):\n", + " real_loss = cross_entropy(tf.ones_like(real_output), real_output)\n", + " fake_loss = cross_entropy(tf.zeros_liks(fake_output), fake_output)\n", + " total_loss = real_loss + fake_loss\n", + "\n", + " return total_loss" + ] + }, + { + "cell_type": "markdown", + "id": "fcf8f066", + "metadata": { + "editable": true + }, + "source": [ + "Next we define a kind of seed to help us compare the learning process over\n", + "multiple training epochs." + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "id": "eea2bbee", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "noise_dimension = 100\n", + "n_examples_to_generate = 16\n", + "seed_images = tf.random.normal([n_examples_to_generate, noise_dimension])" + ] + }, + { + "cell_type": "markdown", + "id": "94a6e341", + "metadata": { + "editable": true + }, + "source": [ + "Now we have everything we need to define our training step, which we will apply\n", + "for every step in our training loop. Notice the @tf.function flag signifying\n", + "that the function is tensorflow 'compiled'. Removing this flag doubles the\n", + "computation time." + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "id": "8d48470b", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "@tf.function\n", + "def train_step(images):\n", + " noise = tf.random.normal([BATCH_SIZE, noise_dimension])\n", + "\n", + " with tf.GradientTape() as gen_tape, tf.GradientTape() as disc_tape:\n", + " generated_images = generator(noise, training=True)\n", + "\n", + " real_output = discriminator(images, training=True)\n", + " fake_output = discriminator(generated_images, training=True)\n", + "\n", + " gen_loss = generator_loss(fake_output)\n", + " disc_loss = discriminator_loss(real_output, fake_output)\n", + "\n", + " gradients_of_generator = gen_tape.gradient(gen_loss,\n", + " generator.trainable_variables)\n", + " gradients_of_discriminator = disc_tape.gradient(disc_loss,\n", + " discriminator.trainable_variables)\n", + " generator_optimizer.apply_gradients(zip(gradients_of_generator,\n", + " generator.trainable_variables))\n", + " discriminator_optimizer.apply_gradients(zip(gradients_of_discriminator,\n", + " discriminator.trainable_variables))\n", + "\n", + " return gen_loss, disc_loss" + ] + }, + { + "cell_type": "markdown", + "id": "62015b88", + "metadata": { + "editable": true + }, + "source": [ + "Next we define a helper function to produce an output over our training epochs\n", + "to see the predictive progression of our generator model. **Note**: I am including\n", + "this code here, but comment it out in the training loop." + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "id": "b189ed96", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def generate_and_save_images(model, epoch, test_input):\n", + " # we're making inferences here\n", + " predictions = model(test_input, training=False)\n", + "\n", + " fig = plt.figure(figsize=(4, 4))\n", + "\n", + " for i in range(predictions.shape[0]):\n", + " plt.subplot(4, 4, i+1)\n", + " plt.imshow(predictions[i, :, :, 0] * 127.5 + 127.5, cmap='gray')\n", + " plt.axis('off')\n", + "\n", + " plt.savefig(f'./images_from_seed_images/image_at_epoch_{str(epoch).zfill(3)}.png')\n", + " plt.close()\n", + " #plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "ba2c82d5", + "metadata": { + "editable": true + }, + "source": [ + "Setting up checkpoints to periodically save our model during training so that\n", + "everything is not lost even if the program were to somehow terminate while\n", + "training." + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "id": "a0e2fc8a", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "# Setting up checkpoints to save model during training\n", + "checkpoint_dir = './training_checkpoints'\n", + "checkpoint_prefix = os.path.join(checkpoint_dir, 'ckpt')\n", + "checkpoint = tf.train.Checkpoint(generator_optimizer=generator_optimizer,\n", + " discriminator_optimizer=discriminator_optimizer,\n", + " generator=generator,\n", + " discriminator=discriminator)" + ] + }, + { + "cell_type": "markdown", + "id": "4d93a0f7", + "metadata": { + "editable": true + }, + "source": [ + "Now we define our training loop" + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "id": "a1275556", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def train(dataset, epochs):\n", + " generator_loss_list = []\n", + " discriminator_loss_list = []\n", + "\n", + " for epoch in range(epochs):\n", + " start = time.time()\n", + "\n", + " for image_batch in dataset:\n", + " gen_loss, disc_loss = train_step(image_batch)\n", + " generator_loss_list.append(gen_loss.numpy())\n", + " discriminator_loss_list.append(disc_loss.numpy())\n", + "\n", + " #generate_and_save_images(generator, epoch + 1, seed_images)\n", + "\n", + " if (epoch + 1) % 15 == 0:\n", + " checkpoint.save(file_prefix=checkpoint_prefix)\n", + "\n", + " print(f'Time for epoch {epoch} is {time.time() - start}')\n", + "\n", + " #generate_and_save_images(generator, epochs, seed_images)\n", + "\n", + " loss_file = './data/lossfile.txt'\n", + " with open(loss_file, 'w') as outfile:\n", + " outfile.write(str(generator_loss_list))\n", + " outfile.write('\\n')\n", + " outfile.write('\\n')\n", + " outfile.write(str(discriminator_loss_list))\n", + " outfile.write('\\n')\n", + " outfile.write('\\n')" + ] + }, + { + "cell_type": "markdown", + "id": "6ff3a75a", + "metadata": { + "editable": true + }, + "source": [ + "To train simply call this function. **Warning**: this might take a long time so\n", + "there is a folder of a pretrained network already included in the repository." + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "id": "371ed41a", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "train(train_dataset, EPOCHS)" + ] + }, + { + "cell_type": "markdown", + "id": "654399f1", + "metadata": { + "editable": true + }, + "source": [ + "Now to avoid having to train and everything, which will take a while depending\n", + "on your computer setup we now load in the model which produced the above gif." + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "id": "dec4b560", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "checkpoint.restore(tf.train.latest_checkpoint(checkpoint_dir))\n", + "restored_generator = checkpoint.generator\n", + "restored_discriminator = checkpoint.discriminator\n", + "\n", + "print(restored_generator)\n", + "print(restored_discriminator)" + ] + }, + { + "cell_type": "markdown", + "id": "296bfa5c", + "metadata": { + "editable": true + }, + "source": [ + "We have successfully loaded in our latest model. Let us now play around a bit\n", + "and see what kind of things we can learn about this model. Our generator takes\n", + "an array of 100 numbers. One idea can be to try to systematically change our\n", + "input. Let us try and see what we get" + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "id": "eecfbb1f", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def generate_latent_points(number=100, scale_means=1, scale_stds=1):\n", + " latent_dim = 100\n", + " means = scale_means * tf.linspace(-1, 1, num=latent_dim)\n", + " stds = scale_stds * tf.linspace(-1, 1, num=latent_dim)\n", + " latent_space_value_range = tf.random.normal([number, latent_dim],\n", + " means,\n", + " stds,\n", + " dtype=tf.float64)\n", + "\n", + " return latent_space_value_range\n", + "\n", + "def generate_images(latent_points):\n", + " # notice we set training to false because we are making inferences\n", + " generated_images = restored_generator.predict(latent_points)\n", + "\n", + " return generated_images" + ] + }, + { + "cell_type": "code", + "execution_count": 25, + "id": "333a593d", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def plot_result(generated_images, number=100):\n", + " # obviously this assumes sqrt number is an int\n", + " fig, axs = plt.subplots(int(np.sqrt(number)), int(np.sqrt(number)),\n", + " figsize=(10, 10))\n", + "\n", + " for i in range(int(np.sqrt(number))):\n", + " for j in range(int(np.sqrt(number))):\n", + " axs[i, j].imshow(generated_images[i*j], cmap='Greys')\n", + " axs[i, j].axis('off')\n", + "\n", + " plt.show()" + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "id": "2f5f0154", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "generated_images = generate_images(generate_latent_points())\n", + "plot_result(generated_images)" + ] + }, + { + "cell_type": "markdown", + "id": "ff581bf2", + "metadata": { + "editable": true + }, + "source": [ + "We see that the generator generates images that look like MNIST\n", + "numbers: $1, 4, 7, 9$. Let's try to tweak it a bit more to see if we are able\n", + "to generate a similar plot where we generate every MNIST number. Let us now try\n", + "to 'move' a bit around in the latent space. **Note**: decrease the plot number if\n", + "these following cells take too long to run on your computer." + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "id": "d3617ad8", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "plot_number = 225\n", + "\n", + "generated_images = generate_images(generate_latent_points(number=plot_number,\n", + " scale_means=5,\n", + " scale_stds=1))\n", + "plot_result(generated_images, number=plot_number)\n", + "\n", + "generated_images = generate_images(generate_latent_points(number=plot_number,\n", + " scale_means=-5,\n", + " scale_stds=1))\n", + "plot_result(generated_images, number=plot_number)\n", + "\n", + "generated_images = generate_images(generate_latent_points(number=plot_number,\n", + " scale_means=1,\n", + " scale_stds=5))\n", + "plot_result(generated_images, number=plot_number)" + ] + }, + { + "cell_type": "markdown", + "id": "1a074f93", + "metadata": { + "editable": true + }, + "source": [ + "Again, we have found something interesting. *Moving* around using our means\n", + "takes us from digit to digit, while *moving* around using our standard\n", + "deviations seem to increase the number of different digits! In the last image\n", + "above, we can barely make out every MNIST digit. Let us make on last plot using\n", + "this information by upping the standard deviation of our Gaussian noises." + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "id": "4ef8937d", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "plot_number = 400\n", + "generated_images = generate_images(generate_latent_points(number=plot_number,\n", + " scale_means=1,\n", + " scale_stds=10))\n", + "plot_result(generated_images, number=plot_number)" + ] + }, + { + "cell_type": "markdown", + "id": "385a2d0a", + "metadata": { + "editable": true + }, + "source": [ + "A pretty cool result! We see that our generator indeed has learned a\n", + "distribution which qualitatively looks a whole lot like the MNIST dataset.\n", + "\n", + "Another interesting way to explore the latent space of our generator model is by\n", + "interpolating between the MNIST digits. This section is largely based on\n", + "[this excellent blogpost](https://machinelearningmastery.com/how-to-interpolate-and-perform-vector-arithmetic-with-faces-using-a-generative-adversarial-network/)\n", + "by Jason Brownlee.\n", + "\n", + "So let us start by defining a function to interpolate between two points in the\n", + "latent space." + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "id": "57de87b8", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def interpolation(point_1, point_2, n_steps=10):\n", + " ratios = np.linspace(0, 1, num=n_steps)\n", + " vectors = []\n", + " for i, ratio in enumerate(ratios):\n", + " vectors.append(((1.0 - ratio) * point_1 + ratio * point_2))\n", + "\n", + " return tf.stack(vectors)" + ] + }, + { + "cell_type": "markdown", + "id": "cfb76bb6", + "metadata": { + "editable": true + }, + "source": [ + "Now we have all we need to do our interpolation analysis." + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "id": "e25decef", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "plot_number = 100\n", + "latent_points = generate_latent_points(number=plot_number)\n", + "results = None\n", + "for i in range(0, 2*np.sqrt(plot_number), 2):\n", + " interpolated = interpolation(latent_points[i], latent_points[i+1])\n", + " generated_images = generate_images(interpolated)\n", + "\n", + " if results is None:\n", + " results = generated_images\n", + " else:\n", + " results = tf.stack((results, generated_images))\n", + "\n", + "plot_results(results, plot_number)" + ] + } + ], + "metadata": { + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.8.12" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter13.py b/doc/LectureNotes/_build/jupyter_execute/chapter13.py new file mode 100644 index 000000000..97cd2c4bd --- /dev/null +++ b/doc/LectureNotes/_build/jupyter_execute/chapter13.py @@ -0,0 +1,1306 @@ +#!/usr/bin/env python +# coding: utf-8 + +# + +# # Recurrent neural networks: Overarching view +# +# Till now our focus has been, including convolutional neural networks +# as well, on feedforward neural networks. The output or the activations +# flow only in one direction, from the input layer to the output layer. +# +# A recurrent neural network (RNN) looks very much like a feedforward +# neural network, except that it also has connections pointing +# backward. +# +# RNNs are used to analyze time series data such as stock prices, and +# tell you when to buy or sell. In autonomous driving systems, they can +# anticipate car trajectories and help avoid accidents. More generally, +# they can work on sequences of arbitrary lengths, rather than on +# fixed-sized inputs like all the nets we have discussed so far. For +# example, they can take sentences, documents, or audio samples as +# input, making them extremely useful for natural language processing +# systems such as automatic translation and speech-to-text. +# +# More to text to be added + +# ## A simple example + +# In[1]: + + +get_ipython().run_line_magic('matplotlib', 'inline') + +# Start importing packages +import pandas as pd +import numpy as np +import matplotlib.pyplot as plt +import tensorflow as tf +from tensorflow.keras import datasets, layers, models +from tensorflow.keras.layers import Input +from tensorflow.keras.models import Model, Sequential +from tensorflow.keras.layers import Dense, SimpleRNN, LSTM, GRU +from tensorflow.keras import optimizers +from tensorflow.keras import regularizers +from tensorflow.keras.utils import to_categorical + + + +# convert into dataset matrix +def convertToMatrix(data, step): + X, Y =[], [] + for i in range(len(data)-step): + d=i+step + X.append(data[i:d,]) + Y.append(data[d,]) + return np.array(X), np.array(Y) + +step = 4 +N = 1000 +Tp = 800 + +t=np.arange(0,N) +x=np.sin(0.02*t)+2*np.random.rand(N) +df = pd.DataFrame(x) +df.head() + +plt.plot(df) +plt.show() + +values=df.values +train,test = values[0:Tp,:], values[Tp:N,:] + +# add step elements into train and test +test = np.append(test,np.repeat(test[-1,],step)) +train = np.append(train,np.repeat(train[-1,],step)) + +trainX,trainY =convertToMatrix(train,step) +testX,testY =convertToMatrix(test,step) +trainX = np.reshape(trainX, (trainX.shape[0], 1, trainX.shape[1])) +testX = np.reshape(testX, (testX.shape[0], 1, testX.shape[1])) + +model = Sequential() +model.add(SimpleRNN(units=32, input_shape=(1,step), activation="relu")) +model.add(Dense(8, activation="relu")) +model.add(Dense(1)) +model.compile(loss='mean_squared_error', optimizer='rmsprop') +model.summary() + +model.fit(trainX,trainY, epochs=100, batch_size=16, verbose=2) +trainPredict = model.predict(trainX) +testPredict= model.predict(testX) +predicted=np.concatenate((trainPredict,testPredict),axis=0) + +trainScore = model.evaluate(trainX, trainY, verbose=0) +print(trainScore) + +index = df.index.values +plt.plot(index,df) +plt.plot(index,predicted) +plt.axvline(df.index[Tp], c="r") +plt.show() + + +# ## An extrapolation example +# +# The following code provides an example of how recurrent neural +# networks can be used to extrapolate to unknown values of physics data +# sets. Specifically, the data sets used in this program come from +# a quantum mechanical many-body calculation of energies as functions of the number of particles. + +# In[2]: + + + +# For matrices and calculations +import numpy as np +# For machine learning (backend for keras) +import tensorflow as tf +# User-friendly machine learning library +# Front end for TensorFlow +import tensorflow.keras +# Different methods from Keras needed to create an RNN +# This is not necessary but it shortened function calls +# that need to be used in the code. +from tensorflow.keras import datasets, layers, models +from tensorflow.keras.layers import Input +from tensorflow.keras import regularizers +from tensorflow.keras.models import Model, Sequential +from tensorflow.keras.layers import Dense, SimpleRNN, LSTM, GRU +# For timing the code +from timeit import default_timer as timer +# For plotting +import matplotlib.pyplot as plt + + +# The data set +datatype='VaryDimension' +X_tot = np.arange(2, 42, 2) +y_tot = np.array([-0.03077640549, -0.08336233266, -0.1446729567, -0.2116753732, -0.2830637392, -0.3581341341, -0.436462435, -0.5177783846, + -0.6019067271, -0.6887363571, -0.7782028952, -0.8702784034, -0.9649652536, -1.062292565, -1.16231451, + -1.265109911, -1.370782966, -1.479465113, -1.591317992, -1.70653767]) + + +# The way the recurrent neural networks are trained in this program +# differs from how machine learning algorithms are usually trained. +# Typically a machine learning algorithm is trained by learning the +# relationship between the x data and the y data. In this program, the +# recurrent neural network will be trained to recognize the relationship +# in a sequence of y values. This is type of data formatting is +# typically used time series forcasting, but it can also be used in any +# extrapolation (time series forecasting is just a specific type of +# extrapolation along the time axis). This method of data formatting +# does not use the x data and assumes that the y data are evenly spaced. +# +# For a standard machine learning algorithm, the training data has the +# form of (x,y) so the machine learning algorithm learns to assiciate a +# y value with a given x value. This is useful when the test data has x +# values within the same range as the training data. However, for this +# application, the x values of the test data are outside of the x values +# of the training data and the traditional method of training a machine +# learning algorithm does not work as well. For this reason, the +# recurrent neural network is trained on sequences of y values of the +# form ((y1, y2), y3), so that the network is concerned with learning +# the pattern of the y data and not the relation between the x and y +# data. As long as the pattern of y data outside of the training region +# stays relatively stable compared to what was inside the training +# region, this method of training can produce accurate extrapolations to +# y values far removed from the training data set. + +# In[3]: + + +# FORMAT_DATA +def format_data(data, length_of_sequence = 2): + """ + Inputs: + data(a numpy array): the data that will be the inputs to the recurrent neural + network + length_of_sequence (an int): the number of elements in one iteration of the + sequence patter. For a function approximator use length_of_sequence = 2. + Returns: + rnn_input (a 3D numpy array): the input data for the recurrent neural network. Its + dimensions are length of data - length of sequence, length of sequence, + dimnsion of data + rnn_output (a numpy array): the training data for the neural network + Formats data to be used in a recurrent neural network. + """ + + X, Y = [], [] + for i in range(len(data)-length_of_sequence): + # Get the next length_of_sequence elements + a = data[i:i+length_of_sequence] + # Get the element that immediately follows that + b = data[i+length_of_sequence] + # Reshape so that each data point is contained in its own array + a = np.reshape (a, (len(a), 1)) + X.append(a) + Y.append(b) + rnn_input = np.array(X) + rnn_output = np.array(Y) + + return rnn_input, rnn_output + + +# ## Defining the Recurrent Neural Network Using Keras +# +# The following method defines a simple recurrent neural network in keras consisting of one input layer, one hidden layer, and one output layer. + +def rnn(length_of_sequences, batch_size = None, stateful = False): + """ + Inputs: + length_of_sequences (an int): the number of y values in "x data". This is determined + when the data is formatted + batch_size (an int): Default value is None. See Keras documentation of SimpleRNN. + stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN. + Returns: + model (a Keras model): The recurrent neural network that is built and compiled by this + method + Builds and compiles a recurrent neural network with one hidden layer and returns the model. + """ + # Number of neurons in the input and output layers + in_out_neurons = 1 + # Number of neurons in the hidden layer + hidden_neurons = 200 + # Define the input layer + inp = Input(batch_shape=(batch_size, + length_of_sequences, + in_out_neurons)) + # Define the hidden layer as a simple RNN layer with a set number of neurons and add it to + # the network immediately after the input layer + rnn = SimpleRNN(hidden_neurons, + return_sequences=False, + stateful = stateful, + name="RNN")(inp) + # Define the output layer as a dense neural network layer (standard neural network layer) + #and add it to the network immediately after the hidden layer. + dens = Dense(in_out_neurons,name="dense")(rnn) + # Create the machine learning model starting with the input layer and ending with the + # output layer + model = Model(inputs=[inp],outputs=[dens]) + # Compile the machine learning model using the mean squared error function as the loss + # function and an Adams optimizer. + model.compile(loss="mean_squared_error", optimizer="adam") + return model + + +# ## Predicting New Points With A Trained Recurrent Neural Network + +# In[4]: + + +def test_rnn (x1, y_test, plot_min, plot_max): + """ + Inputs: + x1 (a list or numpy array): The complete x component of the data set + y_test (a list or numpy array): The complete y component of the data set + plot_min (an int or float): the smallest x value used in the training data + plot_max (an int or float): the largest x valye used in the training data + Returns: + None. + Uses a trained recurrent neural network model to predict future points in the + series. Computes the MSE of the predicted data set from the true data set, saves + the predicted data set to a csv file, and plots the predicted and true data sets w + while also displaying the data range used for training. + """ + # Add the training data as the first dim points in the predicted data array as these + # are known values. + y_pred = y_test[:dim].tolist() + # Generate the first input to the trained recurrent neural network using the last two + # points of the training data. Based on how the network was trained this means that it + # will predict the first point in the data set after the training data. All of the + # brackets are necessary for Tensorflow. + next_input = np.array([[[y_test[dim-2]], [y_test[dim-1]]]]) + # Save the very last point in the training data set. This will be used later. + last = [y_test[dim-1]] + + # Iterate until the complete data set is created. + for i in range (dim, len(y_test)): + # Predict the next point in the data set using the previous two points. + next = model.predict(next_input) + # Append just the number of the predicted data set + y_pred.append(next[0][0]) + # Create the input that will be used to predict the next data point in the data set. + next_input = np.array([[last, next[0]]], dtype=np.float64) + last = next + + # Print the mean squared error between the known data set and the predicted data set. + print('MSE: ', np.square(np.subtract(y_test, y_pred)).mean()) + # Save the predicted data set as a csv file for later use + name = datatype + 'Predicted'+str(dim)+'.csv' + np.savetxt(name, y_pred, delimiter=',') + # Plot the known data set and the predicted data set. The red box represents the region that was used + # for the training data. + fig, ax = plt.subplots() + ax.plot(x1, y_test, label="true", linewidth=3) + ax.plot(x1, y_pred, 'g-.',label="predicted", linewidth=4) + ax.legend() + # Created a red region to represent the points used in the training data. + ax.axvspan(plot_min, plot_max, alpha=0.25, color='red') + plt.show() + +# Check to make sure the data set is complete +assert len(X_tot) == len(y_tot) + +# This is the number of points that will be used in as the training data +dim=12 + +# Separate the training data from the whole data set +X_train = X_tot[:dim] +y_train = y_tot[:dim] + + +# Generate the training data for the RNN, using a sequence of 2 +rnn_input, rnn_training = format_data(y_train, 2) + + +# Create a recurrent neural network in Keras and produce a summary of the +# machine learning model +model = rnn(length_of_sequences = rnn_input.shape[1]) +model.summary() + +# Start the timer. Want to time training+testing +start = timer() +# Fit the model using the training data genenerated above using 150 training iterations and a 5% +# validation split. Setting verbose to True prints information about each training iteration. +hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, + verbose=True,validation_split=0.05) + +for label in ["loss","val_loss"]: + plt.plot(hist.history[label],label=label) + +plt.ylabel("loss") +plt.xlabel("epoch") +plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1])) +plt.legend() +plt.show() + +# Use the trained neural network to predict more points of the data set +test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1]) +# Stop the timer and calculate the total time needed. +end = timer() +print('Time: ', end-start) + + +# Changing the size of the recurrent neural network and its parameters +# can drastically change the results you get from the model. The below +# code takes the simple recurrent neural network from above and adds a +# second hidden layer, changes the number of neurons in the hidden +# layer, and explicitly declares the activation function of the hidden +# layers to be a sigmoid function. The loss function and optimizer can +# also be changed but are kept the same as the above network. These +# parameters can be tuned to provide the optimal result from the +# network. For some ideas on how to improve the performance of a +# [recurrent neural network](https://danijar.com/tips-for-training-recurrent-neural-networks). + +# In[5]: + + +def rnn_2layers(length_of_sequences, batch_size = None, stateful = False): + """ + Inputs: + length_of_sequences (an int): the number of y values in "x data". This is determined + when the data is formatted + batch_size (an int): Default value is None. See Keras documentation of SimpleRNN. + stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN. + Returns: + model (a Keras model): The recurrent neural network that is built and compiled by this + method + Builds and compiles a recurrent neural network with two hidden layers and returns the model. + """ + # Number of neurons in the input and output layers + in_out_neurons = 1 + # Number of neurons in the hidden layer, increased from the first network + hidden_neurons = 500 + # Define the input layer + inp = Input(batch_shape=(batch_size, + length_of_sequences, + in_out_neurons)) + # Create two hidden layers instead of one hidden layer. Explicitly set the activation + # function to be the sigmoid function (the default value is hyperbolic tangent) + rnn1 = SimpleRNN(hidden_neurons, + return_sequences=True, # This needs to be True if another hidden layer is to follow + stateful = stateful, activation = 'sigmoid', + name="RNN1")(inp) + rnn2 = SimpleRNN(hidden_neurons, + return_sequences=False, activation = 'sigmoid', + stateful = stateful, + name="RNN2")(rnn1) + # Define the output layer as a dense neural network layer (standard neural network layer) + #and add it to the network immediately after the hidden layer. + dens = Dense(in_out_neurons,name="dense")(rnn2) + # Create the machine learning model starting with the input layer and ending with the + # output layer + model = Model(inputs=[inp],outputs=[dens]) + # Compile the machine learning model using the mean squared error function as the loss + # function and an Adams optimizer. + model.compile(loss="mean_squared_error", optimizer="adam") + return model + +# Check to make sure the data set is complete +assert len(X_tot) == len(y_tot) + +# This is the number of points that will be used in as the training data +dim=12 + +# Separate the training data from the whole data set +X_train = X_tot[:dim] +y_train = y_tot[:dim] + + +# Generate the training data for the RNN, using a sequence of 2 +rnn_input, rnn_training = format_data(y_train, 2) + + +# Create a recurrent neural network in Keras and produce a summary of the +# machine learning model +model = rnn_2layers(length_of_sequences = 2) +model.summary() + +# Start the timer. Want to time training+testing +start = timer() +# Fit the model using the training data genenerated above using 150 training iterations and a 5% +# validation split. Setting verbose to True prints information about each training iteration. +hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, + verbose=True,validation_split=0.05) + + +# This section plots the training loss and the validation loss as a function of training iteration. +# This is not required for analyzing the couple cluster data but can help determine if the network is +# being overtrained. +for label in ["loss","val_loss"]: + plt.plot(hist.history[label],label=label) + +plt.ylabel("loss") +plt.xlabel("epoch") +plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1])) +plt.legend() +plt.show() + +# Use the trained neural network to predict more points of the data set +test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1]) +# Stop the timer and calculate the total time needed. +end = timer() +print('Time: ', end-start) + + +# ## Other Types of Recurrent Neural Networks +# +# Besides a simple recurrent neural network layer, there are two other +# commonly used types of recurrent neural network layers: Long Short +# Term Memory (LSTM) and Gated Recurrent Unit (GRU). For a short +# introduction to these layers see +# and . +# +# The first network created below is similar to the previous network, +# but it replaces the SimpleRNN layers with LSTM layers. The second +# network below has two hidden layers made up of GRUs, which are +# preceeded by two dense (feeddorward) neural network layers. These +# dense layers "preprocess" the data before it reaches the recurrent +# layers. This architecture has been shown to improve the performance +# of recurrent neural networks (see the link above and also +# . + +# In[6]: + + +def lstm_2layers(length_of_sequences, batch_size = None, stateful = False): + """ + Inputs: + length_of_sequences (an int): the number of y values in "x data". This is determined + when the data is formatted + batch_size (an int): Default value is None. See Keras documentation of SimpleRNN. + stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN. + Returns: + model (a Keras model): The recurrent neural network that is built and compiled by this + method + Builds and compiles a recurrent neural network with two LSTM hidden layers and returns the model. + """ + # Number of neurons on the input/output layer and the number of neurons in the hidden layer + in_out_neurons = 1 + hidden_neurons = 250 + # Input Layer + inp = Input(batch_shape=(batch_size, + length_of_sequences, + in_out_neurons)) + # Hidden layers (in this case they are LSTM layers instead if SimpleRNN layers) + rnn= LSTM(hidden_neurons, + return_sequences=True, + stateful = stateful, + name="RNN", use_bias=True, activation='tanh')(inp) + rnn1 = LSTM(hidden_neurons, + return_sequences=False, + stateful = stateful, + name="RNN1", use_bias=True, activation='tanh')(rnn) + # Output layer + dens = Dense(in_out_neurons,name="dense")(rnn1) + # Define the midel + model = Model(inputs=[inp],outputs=[dens]) + # Compile the model + model.compile(loss='mean_squared_error', optimizer='adam') + # Return the model + return model + +def dnn2_gru2(length_of_sequences, batch_size = None, stateful = False): + """ + Inputs: + length_of_sequences (an int): the number of y values in "x data". This is determined + when the data is formatted + batch_size (an int): Default value is None. See Keras documentation of SimpleRNN. + stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN. + Returns: + model (a Keras model): The recurrent neural network that is built and compiled by this + method + Builds and compiles a recurrent neural network with four hidden layers (two dense followed by + two GRU layers) and returns the model. + """ + # Number of neurons on the input/output layers and hidden layers + in_out_neurons = 1 + hidden_neurons = 250 + # Input layer + inp = Input(batch_shape=(batch_size, + length_of_sequences, + in_out_neurons)) + # Hidden Dense (feedforward) layers + dnn = Dense(hidden_neurons/2, activation='relu', name='dnn')(inp) + dnn1 = Dense(hidden_neurons/2, activation='relu', name='dnn1')(dnn) + # Hidden GRU layers + rnn1 = GRU(hidden_neurons, + return_sequences=True, + stateful = stateful, + name="RNN1", use_bias=True)(dnn1) + rnn = GRU(hidden_neurons, + return_sequences=False, + stateful = stateful, + name="RNN", use_bias=True)(rnn1) + # Output layer + dens = Dense(in_out_neurons,name="dense")(rnn) + # Define the model + model = Model(inputs=[inp],outputs=[dens]) + # Compile the mdoel + model.compile(loss='mean_squared_error', optimizer='adam') + # Return the model + return model + +# Check to make sure the data set is complete +assert len(X_tot) == len(y_tot) + +# This is the number of points that will be used in as the training data +dim=12 + +# Separate the training data from the whole data set +X_train = X_tot[:dim] +y_train = y_tot[:dim] + + +# Generate the training data for the RNN, using a sequence of 2 +rnn_input, rnn_training = format_data(y_train, 2) + + +# Create a recurrent neural network in Keras and produce a summary of the +# machine learning model +# Change the method name to reflect which network you want to use +model = dnn2_gru2(length_of_sequences = 2) +model.summary() + +# Start the timer. Want to time training+testing +start = timer() +# Fit the model using the training data genenerated above using 150 training iterations and a 5% +# validation split. Setting verbose to True prints information about each training iteration. +hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, + verbose=True,validation_split=0.05) + + +# This section plots the training loss and the validation loss as a function of training iteration. +# This is not required for analyzing the couple cluster data but can help determine if the network is +# being overtrained. +for label in ["loss","val_loss"]: + plt.plot(hist.history[label],label=label) + +plt.ylabel("loss") +plt.xlabel("epoch") +plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1])) +plt.legend() +plt.show() + +# Use the trained neural network to predict more points of the data set +test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1]) +# Stop the timer and calculate the total time needed. +end = timer() +print('Time: ', end-start) + + +# ### Training Recurrent Neural Networks in the Standard Way (i.e. learning the relationship between the X and Y data) +# +# Finally, comparing the performace of a recurrent neural network using the standard data formatting to the performance of the network with time sequence data formatting shows the benefit of this type of data formatting with extrapolation. + +# Check to make sure the data set is complete +assert len(X_tot) == len(y_tot) + +# This is the number of points that will be used in as the training data +dim=12 + +# Separate the training data from the whole data set +X_train = X_tot[:dim] +y_train = y_tot[:dim] + +# Reshape the data for Keras specifications +X_train = X_train.reshape((dim, 1)) +y_train = y_train.reshape((dim, 1)) + + +# Create a recurrent neural network in Keras and produce a summary of the +# machine learning model +# Set the sequence length to 1 for regular data formatting +model = rnn(length_of_sequences = 1) +model.summary() + +# Start the timer. Want to time training+testing +start = timer() +# Fit the model using the training data genenerated above using 150 training iterations and a 5% +# validation split. Setting verbose to True prints information about each training iteration. +hist = model.fit(X_train, y_train, batch_size=None, epochs=150, + verbose=True,validation_split=0.05) + + +# This section plots the training loss and the validation loss as a function of training iteration. +# This is not required for analyzing the couple cluster data but can help determine if the network is +# being overtrained. +for label in ["loss","val_loss"]: + plt.plot(hist.history[label],label=label) + +plt.ylabel("loss") +plt.xlabel("epoch") +plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1])) +plt.legend() +plt.show() + +# Use the trained neural network to predict the remaining data points +X_pred = X_tot[dim:] +X_pred = X_pred.reshape((len(X_pred), 1)) +y_model = model.predict(X_pred) +y_pred = np.concatenate((y_tot[:dim], y_model.flatten())) + +# Plot the known data set and the predicted data set. The red box represents the region that was used +# for the training data. +fig, ax = plt.subplots() +ax.plot(X_tot, y_tot, label="true", linewidth=3) +ax.plot(X_tot, y_pred, 'g-.',label="predicted", linewidth=4) +ax.legend() +# Created a red region to represent the points used in the training data. +ax.axvspan(X_tot[0], X_tot[dim], alpha=0.25, color='red') +plt.show() + +# Stop the timer and calculate the total time needed. +end = timer() +print('Time: ', end-start) + + +# # Generative Models +# +# **Generative models** describe a class of statistical models that are a contrast +# to **discriminative models**. Informally we say that generative models can +# generate new data instances while discriminative models discriminate between +# different kinds of data instances. A generative model could generate new photos +# of animals that look like 'real' animals while a discriminative model could tell +# a dog from a cat. More formally, given a data set $x$ and a set of labels / +# targets $y$. Generative models capture the joint probability $p(x, y)$, or +# just $p(x)$ if there are no labels, while discriminative models capture the +# conditional probability $p(y | x)$. Discriminative models generally try to draw +# boundaries in the data space (often high dimensional), while generative models +# try to model how data is placed throughout the space. +# +# **Note**: this material is thanks to Linus Ekstrøm. + +# ## Generative Adversarial Networks +# +# **Generative Adversarial Networks** are a type of unsupervised machine learning +# algorithm proposed by [Goodfellow et. al](https://arxiv.org/pdf/1406.2661.pdf) +# in 2014 (short and good article). +# +# The simplest formulation of +# the model is based on a game theoretic approach, *zero sum game*, where we pit +# two neural networks against one another. We define two rival networks, one +# generator $g$, and one discriminator $d$. The generator directly produces +# samples + +# +#
    +# +# $$ +# \begin{equation} +# x = g(z; \theta^{(g)}) +# \label{_auto1} \tag{1} +# \end{equation} +# $$ + +# The discriminator attempts to distinguish between samples drawn from the +# training data and samples drawn from the generator. In other words, it tries to +# tell the difference between the fake data produced by $g$ and the actual data +# samples we want to do prediction on. The discriminator outputs a probability +# value given by + +# +#
    +# +# $$ +# \begin{equation} +# d(x; \theta^{(d)}) +# \label{_auto2} \tag{2} +# \end{equation} +# $$ + +# indicating the probability that $x$ is a real training example rather than a +# fake sample the generator has generated. The simplest way to formulate the +# learning process in a generative adversarial network is a zero-sum game, in +# which a function + +# +#
    +# +# $$ +# \begin{equation} +# v(\theta^{(g)}, \theta^{(d)}) +# \label{_auto3} \tag{3} +# \end{equation} +# $$ + +# determines the reward for the discriminator, while the generator gets the +# conjugate reward + +# +#
    +# +# $$ +# \begin{equation} +# -v(\theta^{(g)}, \theta^{(d)}) +# \label{_auto4} \tag{4} +# \end{equation} +# $$ + +# During learning both of the networks maximize their own reward function, so that +# the generator gets better and better at tricking the discriminator, while the +# discriminator gets better and better at telling the difference between the fake +# and real data. The generator and discriminator alternate on which one trains at +# one time (i.e. for one epoch). In other words, we keep the generator constant +# and train the discriminator, then we keep the discriminator constant to train +# the generator and repeat. It is this back and forth dynamic which lets GANs +# tackle otherwise intractable generative problems. As the generator improves with +# training, the discriminator's performance gets worse because it cannot easily +# tell the difference between real and fake. If the generator ends up succeeding +# perfectly, the the discriminator will do no better than random guessing i.e. +# 50\%. This progression in the training poses a problem for the convergence +# criteria for GANs. The discriminator feedback gets less meaningful over time, +# if we continue training after this point then the generator is effectively +# training on junk data which can undo the learning up to that point. Therefore, +# we stop training when the discriminator starts outputting $1/2$ everywhere. +# +# At convergence we have + +# +#
    +# +# $$ +# \begin{equation} +# g^* = \underset{g}{\mathrm{argmin}}\hspace{2pt} +# \underset{d}{\mathrm{max}}v(\theta^{(g)}, \theta^{(d)}) +# \label{_auto5} \tag{5} +# \end{equation} +# $$ + +# The default choice for $v$ is + +# +#
    +# +# $$ +# \begin{equation} +# v(\theta^{(g)}, \theta^{(d)}) = \mathbb{E}_{x\sim p_\mathrm{data}}\log d(x) +# + \mathbb{E}_{x\sim p_\mathrm{model}} +# \log (1 - d(x)) +# \label{_auto6} \tag{6} +# \end{equation} +# $$ + +# The main motivation for the design of GANs is that the learning process requires +# neither approximate inference (variational autoencoders for example) nor +# approximation of a partition function. In the case where + +# +#
    +# +# $$ +# \begin{equation} +# \underset{d}{\mathrm{max}}v(\theta^{(g)}, \theta^{(d)}) +# \label{_auto7} \tag{7} +# \end{equation} +# $$ + +# is convex in $\theta^{(g)} then the procedure is guaranteed to converge and is +# asymptotically consistent +# ( [Seth Lloyd on QuGANs](https://arxiv.org/pdf/1804.09139.pdf) ). +# +# This is in +# general not the case and it is possible to get situations where the training +# process never converges because the generator and discriminator chase one +# another around in the parameter space indefinitely. A much deeper discussion on +# the currently open research problem of GAN convergence is available +# [here](https://www.deeplearningbook.org/contents/generative_models.html). To +# anyone interested in learning more about GANs it is a highly recommended read. +# Direct quote: "In this best-performing formulation, the generator aims to +# increase the log probability that the discriminator makes a mistake, rather than +# aiming to decrease the log probability that the discriminator makes the correct +# prediction." [Another interesting read](https://arxiv.org/abs/1701.00160) + +# ## Writing Our First Generative Adversarial Network +# Let us now move on to actually implementing a GAN in tensorflow. We will study +# the performance of our GAN on the MNIST dataset. This code is based on and +# adapted from the +# [google tutorial](https://www.tensorflow.org/tutorials/generative/dcgan) +# +# First we import our libraries + +# In[7]: + + +import os +import time +import numpy as np +import tensorflow as tf +import matplotlib.pyplot as plt +from tensorflow.keras import layers +from tensorflow.keras.utils import plot_model + + +# Next we define our hyperparameters and import our data the usual way + +# In[8]: + + +BUFFER_SIZE = 60000 +BATCH_SIZE = 256 +EPOCHS = 30 + +data = tf.keras.datasets.mnist.load_data() +(train_images, train_labels), (test_images, test_labels) = data +train_images = np.reshape(train_images, (train_images.shape[0], + 28, + 28, + 1)).astype('float32') + +# we normalize between -1 and 1 +train_images = (train_images - 127.5) / 127.5 +training_dataset = tf.data.Dataset.from_tensor_slices( + train_images).shuffle(BUFFER_SIZE).batch(BATCH_SIZE) + + +# ### MNIST and GANs +# +# Let's have a quick look + +# In[9]: + + +plt.imshow(train_images[0], cmap='Greys') +plt.show() + + +# Now we define our two models. This is where the 'magic' happens. There are a +# huge amount of possible formulations for both models. A lot of engineering and +# trial and error can be done here to try to produce better performing models. For +# more advanced GANs this is by far the step where you can 'make or break' a +# model. +# +# We start with the generator. As stated in the introductory text the generator +# $g$ upsamples from a random sample to the shape of what we want to predict. In +# our case we are trying to predict MNIST images ($28\times 28$ pixels). + +# In[10]: + + +def generator_model(): + """ + The generator uses upsampling layers tf.keras.layers.Conv2DTranspose() to + produce an image from a random seed. We start with a Dense layer taking this + random sample as an input and subsequently upsample through multiple + convolutional layers. + """ + + # we define our model + model = tf.keras.Sequential() + + + # adding our input layer. Dense means that every neuron is connected and + # the input shape is the shape of our random noise. The units need to match + # in some sense the upsampling strides to reach our desired output shape. + # we are using 100 random numbers as our seed + model.add(layers.Dense(units=7*7*BATCH_SIZE, + use_bias=False, + input_shape=(100, ))) + # we normalize the output form the Dense layer + model.add(layers.BatchNormalization()) + # and add an activation function to our 'layer'. LeakyReLU avoids vanishing + # gradient problem + model.add(layers.LeakyReLU()) + model.add(layers.Reshape((7, 7, BATCH_SIZE))) + assert model.output_shape == (None, 7, 7, BATCH_SIZE) + # even though we just added four keras layers we think of everything above + # as 'one' layer + + # next we add our upscaling convolutional layers + model.add(layers.Conv2DTranspose(filters=128, + kernel_size=(5, 5), + strides=(1, 1), + padding='same', + use_bias=False)) + model.add(layers.BatchNormalization()) + model.add(layers.LeakyReLU()) + assert model.output_shape == (None, 7, 7, 128) + + model.add(layers.Conv2DTranspose(filters=64, + kernel_size=(5, 5), + strides=(2, 2), + padding='same', + use_bias=False)) + model.add(layers.BatchNormalization()) + model.add(layers.LeakyReLU()) + assert model.output_shape == (None, 14, 14, 64) + + model.add(layers.Conv2DTranspose(filters=1, + kernel_size=(5, 5), + strides=(2, 2), + padding='same', + use_bias=False, + activation='tanh')) + assert model.output_shape == (None, 28, 28, 1) + + return model + + +# And there we have our 'simple' generator model. Now we move on to defining our +# discriminator model $d$, which is a convolutional neural network based image +# classifier. + +# In[11]: + + +def discriminator_model(): + """ + The discriminator is a convolutional neural network based image classifier + """ + + # we define our model + model = tf.keras.Sequential() + model.add(layers.Conv2D(filters=64, + kernel_size=(5, 5), + strides=(2, 2), + padding='same', + input_shape=[28, 28, 1])) + model.add(layers.LeakyReLU()) + # adding a dropout layer as you do in conv-nets + model.add(layers.Dropout(0.3)) + + + model.add(layers.Conv2D(filters=128, + kernel_size=(5, 5), + strides=(2, 2), + padding='same')) + model.add(layers.LeakyReLU()) + # adding a dropout layer as you do in conv-nets + model.add(layers.Dropout(0.3)) + + model.add(layers.Flatten()) + model.add(layers.Dense(1)) + + return model + + +# Let us take a look at our models. + +# In[12]: + + +generator = generator_model() +plot_model(generator, show_shapes=True, rankdir='LR') + + +# In[13]: + + +discriminator = discriminator_model() +plot_model(discriminator, show_shapes=True, rankdir='LR') + + +# Next we need a few helper objects we will use in training + +# In[14]: + + +cross_entropy = tf.keras.losses.BinaryCrossentropy(from_logits=True) +generator_optimizer = tf.keras.optimizers.Adam(1e-4) +discriminator_optimizer = tf.keras.optimizers.Adam(1e-4) + + +# The first object, *cross_entropy* is our loss function and the two others are +# our optimizers. Notice we use the same learning rate for both $g$ and $d$. This +# is because they need to improve their accuracy at approximately equal speeds to +# get convergence (not necessarily exactly equal). Now we define our loss +# functions + +# In[15]: + + +def generator_loss(fake_output): + loss = cross_entropy(tf.ones_like(fake_output), fake_output) + + return loss + + +# In[16]: + + +def discriminator_loss(real_output, fake_output): + real_loss = cross_entropy(tf.ones_like(real_output), real_output) + fake_loss = cross_entropy(tf.zeros_liks(fake_output), fake_output) + total_loss = real_loss + fake_loss + + return total_loss + + +# Next we define a kind of seed to help us compare the learning process over +# multiple training epochs. + +# In[17]: + + +noise_dimension = 100 +n_examples_to_generate = 16 +seed_images = tf.random.normal([n_examples_to_generate, noise_dimension]) + + +# Now we have everything we need to define our training step, which we will apply +# for every step in our training loop. Notice the @tf.function flag signifying +# that the function is tensorflow 'compiled'. Removing this flag doubles the +# computation time. + +# In[18]: + + +@tf.function +def train_step(images): + noise = tf.random.normal([BATCH_SIZE, noise_dimension]) + + with tf.GradientTape() as gen_tape, tf.GradientTape() as disc_tape: + generated_images = generator(noise, training=True) + + real_output = discriminator(images, training=True) + fake_output = discriminator(generated_images, training=True) + + gen_loss = generator_loss(fake_output) + disc_loss = discriminator_loss(real_output, fake_output) + + gradients_of_generator = gen_tape.gradient(gen_loss, + generator.trainable_variables) + gradients_of_discriminator = disc_tape.gradient(disc_loss, + discriminator.trainable_variables) + generator_optimizer.apply_gradients(zip(gradients_of_generator, + generator.trainable_variables)) + discriminator_optimizer.apply_gradients(zip(gradients_of_discriminator, + discriminator.trainable_variables)) + + return gen_loss, disc_loss + + +# Next we define a helper function to produce an output over our training epochs +# to see the predictive progression of our generator model. **Note**: I am including +# this code here, but comment it out in the training loop. + +# In[19]: + + +def generate_and_save_images(model, epoch, test_input): + # we're making inferences here + predictions = model(test_input, training=False) + + fig = plt.figure(figsize=(4, 4)) + + for i in range(predictions.shape[0]): + plt.subplot(4, 4, i+1) + plt.imshow(predictions[i, :, :, 0] * 127.5 + 127.5, cmap='gray') + plt.axis('off') + + plt.savefig(f'./images_from_seed_images/image_at_epoch_{str(epoch).zfill(3)}.png') + plt.close() + #plt.show() + + +# Setting up checkpoints to periodically save our model during training so that +# everything is not lost even if the program were to somehow terminate while +# training. + +# In[20]: + + +# Setting up checkpoints to save model during training +checkpoint_dir = './training_checkpoints' +checkpoint_prefix = os.path.join(checkpoint_dir, 'ckpt') +checkpoint = tf.train.Checkpoint(generator_optimizer=generator_optimizer, + discriminator_optimizer=discriminator_optimizer, + generator=generator, + discriminator=discriminator) + + +# Now we define our training loop + +# In[21]: + + +def train(dataset, epochs): + generator_loss_list = [] + discriminator_loss_list = [] + + for epoch in range(epochs): + start = time.time() + + for image_batch in dataset: + gen_loss, disc_loss = train_step(image_batch) + generator_loss_list.append(gen_loss.numpy()) + discriminator_loss_list.append(disc_loss.numpy()) + + #generate_and_save_images(generator, epoch + 1, seed_images) + + if (epoch + 1) % 15 == 0: + checkpoint.save(file_prefix=checkpoint_prefix) + + print(f'Time for epoch {epoch} is {time.time() - start}') + + #generate_and_save_images(generator, epochs, seed_images) + + loss_file = './data/lossfile.txt' + with open(loss_file, 'w') as outfile: + outfile.write(str(generator_loss_list)) + outfile.write('\n') + outfile.write('\n') + outfile.write(str(discriminator_loss_list)) + outfile.write('\n') + outfile.write('\n') + + +# To train simply call this function. **Warning**: this might take a long time so +# there is a folder of a pretrained network already included in the repository. + +# In[22]: + + +train(train_dataset, EPOCHS) + + +# Now to avoid having to train and everything, which will take a while depending +# on your computer setup we now load in the model which produced the above gif. + +# In[23]: + + +checkpoint.restore(tf.train.latest_checkpoint(checkpoint_dir)) +restored_generator = checkpoint.generator +restored_discriminator = checkpoint.discriminator + +print(restored_generator) +print(restored_discriminator) + + +# We have successfully loaded in our latest model. Let us now play around a bit +# and see what kind of things we can learn about this model. Our generator takes +# an array of 100 numbers. One idea can be to try to systematically change our +# input. Let us try and see what we get + +# In[24]: + + +def generate_latent_points(number=100, scale_means=1, scale_stds=1): + latent_dim = 100 + means = scale_means * tf.linspace(-1, 1, num=latent_dim) + stds = scale_stds * tf.linspace(-1, 1, num=latent_dim) + latent_space_value_range = tf.random.normal([number, latent_dim], + means, + stds, + dtype=tf.float64) + + return latent_space_value_range + +def generate_images(latent_points): + # notice we set training to false because we are making inferences + generated_images = restored_generator.predict(latent_points) + + return generated_images + + +# In[25]: + + +def plot_result(generated_images, number=100): + # obviously this assumes sqrt number is an int + fig, axs = plt.subplots(int(np.sqrt(number)), int(np.sqrt(number)), + figsize=(10, 10)) + + for i in range(int(np.sqrt(number))): + for j in range(int(np.sqrt(number))): + axs[i, j].imshow(generated_images[i*j], cmap='Greys') + axs[i, j].axis('off') + + plt.show() + + +# In[26]: + + +generated_images = generate_images(generate_latent_points()) +plot_result(generated_images) + + +# We see that the generator generates images that look like MNIST +# numbers: $1, 4, 7, 9$. Let's try to tweak it a bit more to see if we are able +# to generate a similar plot where we generate every MNIST number. Let us now try +# to 'move' a bit around in the latent space. **Note**: decrease the plot number if +# these following cells take too long to run on your computer. + +# In[27]: + + +plot_number = 225 + +generated_images = generate_images(generate_latent_points(number=plot_number, + scale_means=5, + scale_stds=1)) +plot_result(generated_images, number=plot_number) + +generated_images = generate_images(generate_latent_points(number=plot_number, + scale_means=-5, + scale_stds=1)) +plot_result(generated_images, number=plot_number) + +generated_images = generate_images(generate_latent_points(number=plot_number, + scale_means=1, + scale_stds=5)) +plot_result(generated_images, number=plot_number) + + +# Again, we have found something interesting. *Moving* around using our means +# takes us from digit to digit, while *moving* around using our standard +# deviations seem to increase the number of different digits! In the last image +# above, we can barely make out every MNIST digit. Let us make on last plot using +# this information by upping the standard deviation of our Gaussian noises. + +# In[28]: + + +plot_number = 400 +generated_images = generate_images(generate_latent_points(number=plot_number, + scale_means=1, + scale_stds=10)) +plot_result(generated_images, number=plot_number) + + +# A pretty cool result! We see that our generator indeed has learned a +# distribution which qualitatively looks a whole lot like the MNIST dataset. +# +# Another interesting way to explore the latent space of our generator model is by +# interpolating between the MNIST digits. This section is largely based on +# [this excellent blogpost](https://machinelearningmastery.com/how-to-interpolate-and-perform-vector-arithmetic-with-faces-using-a-generative-adversarial-network/) +# by Jason Brownlee. +# +# So let us start by defining a function to interpolate between two points in the +# latent space. + +# In[29]: + + +def interpolation(point_1, point_2, n_steps=10): + ratios = np.linspace(0, 1, num=n_steps) + vectors = [] + for i, ratio in enumerate(ratios): + vectors.append(((1.0 - ratio) * point_1 + ratio * point_2)) + + return tf.stack(vectors) + + +# Now we have all we need to do our interpolation analysis. + +# In[30]: + + +plot_number = 100 +latent_points = generate_latent_points(number=plot_number) +results = None +for i in range(0, 2*np.sqrt(plot_number), 2): + interpolated = interpolation(latent_points[i], latent_points[i+1]) + generated_images = generate_images(interpolated) + + if results is None: + results = generated_images + else: + results = tf.stack((results, generated_images)) + +plot_results(results, plot_number) + diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter13_3_0.png b/doc/LectureNotes/_build/jupyter_execute/chapter13_3_0.png new file mode 100644 index 000000000..2619cfba8 Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapter13_3_0.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter13_3_192.png b/doc/LectureNotes/_build/jupyter_execute/chapter13_3_192.png new file mode 100644 index 000000000..8e81d8d78 Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapter13_3_192.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapteroptimization.ipynb b/doc/LectureNotes/_build/jupyter_execute/chapteroptimization.ipynb index db15b1b80..0e8e35acd 100644 --- a/doc/LectureNotes/_build/jupyter_execute/chapteroptimization.ipynb +++ b/doc/LectureNotes/_build/jupyter_execute/chapteroptimization.ipynb @@ -2,7 +2,21 @@ "cells": [ { "cell_type": "markdown", - "metadata": {}, + "id": "4d72e1df", + "metadata": { + "editable": true + }, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "id": "fb6e8fcd", + "metadata": { + "editable": true + }, "source": [ "# Optimization, the central part of any Machine Learning algortithm\n", "\n", @@ -15,9 +29,6 @@ "analytically, however this is not possible in general and we must use\n", "some approximative/numerical method to compute the minimum.\n", "\n", - "\n", - "\n", - "\n", "In our discussion on Logistic Regression we studied the \n", "case of\n", "two classes, with $y_i$ either\n", @@ -28,7 +39,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a507b82a", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{align*}\n", @@ -40,12 +54,13 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "4071429e", + "metadata": { + "editable": true + }, "source": [ "where $\\boldsymbol{\\beta}$ are the weights we wish to extract from data, in our case $\\beta_0$ and $\\beta_1$. \n", "\n", - "\n", - "\n", "Our compact equations used a definition of a vector $\\boldsymbol{y}$ with $n$\n", "elements $y_i$, an $n\\times p$ matrix $\\boldsymbol{X}$ which contains the\n", "$x_i$ values and a vector $\\boldsymbol{p}$ of fitted probabilities\n", @@ -55,7 +70,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a3cca0b2", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}} = -\\boldsymbol{X}^T\\left(\\boldsymbol{y}-\\boldsymbol{p}\\right).\n", @@ -64,7 +82,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "12a4ad28", + "metadata": { + "editable": true + }, "source": [ "If we in addition define a diagonal matrix $\\boldsymbol{W}$ with elements \n", "$p(y_i\\vert x_i,\\boldsymbol{\\beta})(1-p(y_i\\vert x_i,\\boldsymbol{\\beta})$, we can obtain a compact expression of the second derivative as" @@ -72,7 +93,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5d6e7796", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial^2 \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}\\partial \\boldsymbol{\\beta}^T} = \\boldsymbol{X}^T\\boldsymbol{W}\\boldsymbol{X}.\n", @@ -81,12 +105,13 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b324ae78", + "metadata": { + "editable": true + }, "source": [ "This defines what is called the Hessian matrix.\n", "\n", - "\n", - "\n", "If we can set up these equations, Newton-Raphson's iterative method is normally the method of choice. It requires however that we can compute in an efficient way the matrices that define the first and second derivatives. \n", "\n", "Our iterative scheme is then given by" @@ -94,7 +119,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "7c39cc3f", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{\\beta}^{\\mathrm{new}} = \\boldsymbol{\\beta}^{\\mathrm{old}}-\\left(\\frac{\\partial^2 \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}\\partial \\boldsymbol{\\beta}^T}\\right)^{-1}_{\\boldsymbol{\\beta}^{\\mathrm{old}}}\\times \\left(\\frac{\\partial \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}}\\right)_{\\boldsymbol{\\beta}^{\\mathrm{old}}},\n", @@ -103,14 +131,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "10749f5d", + "metadata": { + "editable": true + }, "source": [ "or in matrix form as" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "3ec50136", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{\\beta}^{\\mathrm{new}} = \\boldsymbol{\\beta}^{\\mathrm{old}}-\\left(\\boldsymbol{X}^T\\boldsymbol{W}\\boldsymbol{X} \\right)^{-1}\\times \\left(-\\boldsymbol{X}^T(\\boldsymbol{y}-\\boldsymbol{p}) \\right)_{\\boldsymbol{\\beta}^{\\mathrm{old}}}.\n", @@ -119,13 +153,15 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0145d9eb", + "metadata": { + "editable": true + }, "source": [ "The right-hand side is computed with the old values of $\\beta$. \n", "\n", "If we can compute these matrices, in particular the Hessian, the above is often the easiest method to implement. \n", "\n", - "\n", "Let us quickly remind ourselves how we derive the above method.\n", "\n", "Perhaps the most celebrated of all one-dimensional root-finding\n", @@ -136,8 +172,6 @@ "numerically and/or your function is not of the smooth type, we\n", "normally discourage the use of this method.\n", "\n", - "\n", - "\n", "The Newton-Raphson formula consists geometrically of extending the\n", "tangent line at a current point until it crosses zero, then setting\n", "the next guess to the abscissa of that zero-crossing. The mathematics\n", @@ -147,7 +181,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "43c17534", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -160,7 +197,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "32ef04f0", + "metadata": { + "editable": true + }, "source": [ "For small enough values of the function and for well-behaved\n", "functions, the terms beyond linear are unimportant, hence we obtain" @@ -168,7 +208,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6700017c", + "metadata": { + "editable": true + }, "source": [ "$$\n", "f(x)+(s-x)f'(x)\\approx 0,\n", @@ -177,14 +220,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "8791dec4", + "metadata": { + "editable": true + }, "source": [ "yielding" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "69008872", + "metadata": { + "editable": true + }, "source": [ "$$\n", "s\\approx x-\\frac{f(x)}{f'(x)}.\n", @@ -193,14 +242,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "fce4ef1f", + "metadata": { + "editable": true + }, "source": [ "Having in mind an iterative procedure, it is natural to start iterating with" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "e64aef7e", + "metadata": { + "editable": true + }, "source": [ "$$\n", "x_{n+1}=x_n-\\frac{f(x_n)}{f'(x_n)}.\n", @@ -209,7 +264,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9acecc44", + "metadata": { + "editable": true + }, "source": [ "The above is Newton-Raphson's method. It has a simple geometric\n", "interpretation, namely $x_{n+1}$ is the point where the tangent from\n", @@ -223,16 +281,16 @@ "guess near such a local extremum, so that the first derivative nearly\n", "vanishes, then Newton-Raphson may fail totally\n", "\n", - "\n", - "\n", - "\n", "Newton's method can be generalized to systems of several non-linear equations\n", "and variables. Consider the case with two equations" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "18bc9fd6", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{array}{cc} f_1(x_1,x_2) &=0\\\\\n", @@ -242,14 +300,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "7dcad370", + "metadata": { + "editable": true + }, "source": [ "which we Taylor expand to obtain" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "f1724121", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{array}{cc} 0=f_1(x_1+h_1,x_2+h_2)=&f_1(x_1,x_2)+h_1\n", @@ -264,14 +328,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d04bfa9f", + "metadata": { + "editable": true + }, "source": [ "Defining the Jacobian matrix $\\boldsymbol{J}$ we have" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "a4d3a9e3", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{J}=\\left( \\begin{array}{cc}\n", @@ -283,14 +353,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d9d6ff69", + "metadata": { + "editable": true + }, "source": [ "we can rephrase Newton's method as" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "362a86a3", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\left(\\begin{array}{c} x_1^{n+1} \\\\ x_2^{n+1} \\end{array} \\right)=\n", @@ -301,14 +377,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "4b8c2937", + "metadata": { + "editable": true + }, "source": [ "where we have defined" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "7580aa9a", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\left(\\begin{array}{c} h_1^{n} \\\\ h_2^{n} \\end{array} \\right)=\n", @@ -319,18 +401,26 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a2e3adc3", + "metadata": { + "editable": true + }, "source": [ "We need thus to compute the inverse of the Jacobian matrix and it\n", "is to understand that difficulties may\n", "arise in case $\\boldsymbol{J}$ is nearly singular.\n", "\n", "It is rather straightforward to extend the above scheme to systems of\n", - "more than two non-linear equations. In our case, the Jacobian matrix is given by the Hessian that represents the second derivative of cost function. \n", - "\n", - "\n", - "\n", - "\n", + "more than two non-linear equations. In our case, the Jacobian matrix is given by the Hessian that represents the second derivative of cost function." + ] + }, + { + "cell_type": "markdown", + "id": "83a585c9", + "metadata": { + "editable": true + }, + "source": [ "## Steepest descent\n", "\n", "The basic idea of gradient descent is\n", @@ -343,7 +433,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e127ea11", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathbf{x}_{k+1} = \\mathbf{x}_k - \\gamma_k \\nabla F(\\mathbf{x}_k),\n", @@ -352,7 +445,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "bce435bc", + "metadata": { + "editable": true + }, "source": [ "with $\\gamma_k > 0$.\n", "\n", @@ -360,7 +456,6 @@ "F(\\mathbf{x}_k)$. This means that for a sufficiently small $\\gamma_k$\n", "we are always moving towards smaller function values, i.e a minimum.\n", "\n", - "\n", "The previous observation is the basis of the method of steepest\n", "descent, which is also referred to as just gradient descent (GD). One\n", "starts with an initial guess $\\mathbf{x}_0$ for a minimum of $F$ and\n", @@ -369,7 +464,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "2691da5f", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathbf{x}_{k+1} = \\mathbf{x}_k - \\gamma_k \\nabla F(\\mathbf{x}_k), \\ \\ k \\geq 0.\n", @@ -378,12 +476,14 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e7ae7322", + "metadata": { + "editable": true + }, "source": [ "The parameter $\\gamma_k$ is often referred to as the step length or\n", "the learning rate within the context of Machine Learning.\n", "\n", - "\n", "Ideally the sequence $\\{\\mathbf{x}_k \\}_{k=0}$ converges to a global\n", "minimum of the function $F$. In general we do not know if we are in a\n", "global or local minimum. In the special case when $F$ is a convex\n", @@ -402,9 +502,6 @@ "Note that the gradient is a function of $\\mathbf{x} =\n", "(x_1,\\cdots,x_n)$ which makes it expensive to compute numerically.\n", "\n", - "\n", - "\n", - "\n", "The gradient descent method \n", "is sensitive to the choice of learning rate $\\gamma_k$. This is due\n", "to the fact that we are only guaranteed that $F(\\mathbf{x}_{k+1}) \\leq\n", @@ -415,10 +512,16 @@ "\n", "Many of these shortcomings can be alleviated by introducing\n", "randomness. One such method is that of Stochastic Gradient Descent\n", - "(SGD), see below.\n", - "\n", - "\n", - "\n", + "(SGD), see below." + ] + }, + { + "cell_type": "markdown", + "id": "a6a44ea7", + "metadata": { + "editable": true + }, + "source": [ "## Convex functions\n", "\n", "Ideally we want our cost/loss function to be convex(concave).\n", @@ -433,11 +536,8 @@ "$\\mathbb{R}$. Examples of convex sets of $\\mathbb{R}^2$ are the\n", "regular polygons (triangles, rectangles, pentagons, etc...).\n", "\n", - "\n", - "\n", "**Convex function**: Let $X \\subset \\mathbb{R}^n$ be a convex set. Assume that the function $f: X \\rightarrow \\mathbb{R}$ is continuous, then $f$ is said to be convex if $$f(tx_1 + (1-t)x_2) \\leq tf(x_1) + (1-t)f(x_2) $$ for all $x_1, x_2 \\in X$ and for all $t \\in [0,1]$. If $\\leq$ is replaced with a strict inequaltiy in the definition, we demand $x_1 \\neq x_2$ and $t\\in(0,1)$ then $f$ is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting $f(x_1)$ and $f(x_2)$, the value of the function on the interval $[x_1,x_2]$ is always below the line as illustrated below.\n", "\n", - "\n", "In the following we state first and second-order conditions which\n", "ensures convexity of a function $f$. We write $D_f$ to denote the\n", "domain of $f$, i.e the subset of $R^n$ where $f$ is defined. For more\n", @@ -454,8 +554,6 @@ "make a drawing of $f(x) = x^2+1$ and draw the tangent line to $f(x)$ and\n", "note that it is always below the graph.\n", "\n", - "\n", - "\n", "**Second order condition.**\n", "\n", "Assume that $f$ is twice\n", @@ -465,12 +563,8 @@ "single-variable function this reduces to $f''(x) \\geq 0$. Geometrically this means that $f$ has nonnegative curvature\n", "everywhere.\n", "\n", - "\n", - "\n", "This condition is particularly useful since it gives us an procedure for determining if the function under consideration is convex, apart from using the definition.\n", "\n", - "\n", - "\n", "The next result is of great importance to us and the reason why we are\n", "going on about convex functions. In machine learning we frequently\n", "have to minimize a loss/cost function in order to find the best\n", @@ -487,11 +581,16 @@ "is minimal, where $f$ is convex and differentiable. Then, any point\n", "$x^*$ that satisfies $\\nabla f(x^*) = 0$ is a global minimum.\n", "\n", - "\n", - "\n", - "This result means that if we know that the cost/loss function is convex and we are able to find a minimum, we are guaranteed that it is a global minimum.\n", - "\n", - "\n", + "This result means that if we know that the cost/loss function is convex and we are able to find a minimum, we are guaranteed that it is a global minimum." + ] + }, + { + "cell_type": "markdown", + "id": "809f8f01", + "metadata": { + "editable": true + }, + "source": [ "### Some simple problems\n", "\n", "1. Show that $f(x)=x^2$ is convex for $x \\in \\mathbb{R}$ using the definition of convexity. Hint: If you re-write the definition, $f$ is convex if the following holds for all $x,y \\in D_f$ and any $\\lambda \\in [0,1]$ $\\lambda f(x)+(1-\\lambda)f(y)-f(\\lambda x + (1-\\lambda) y ) \\geq 0$.\n", @@ -502,7 +601,6 @@ "\n", " * $g(x) = -\\ln(x)$ is convex for $x \\in (0,\\infty)$.\n", "\n", - "\n", "3. Let $f(x) = x^2$ and $g(x) = e^x$. Show that $f(g(x))$ and $g(f(x))$ is convex for $x \\in \\mathbb{R}$. Also show that if $f(x)$ is any convex function than $h(x) = e^{f(x)}$ is convex.\n", "\n", "4. A norm is any function that satisfy the following properties\n", @@ -513,13 +611,18 @@ "\n", " * $f(x) \\leq 0$ for all $x \\in \\mathbb{R}^n$ with equality if and only if $x = 0$\n", "\n", - "\n", - "Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this).\n", - "\n", - "\n", + "Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this)." + ] + }, + { + "cell_type": "markdown", + "id": "f3b91277", + "metadata": { + "editable": true + }, + "source": [ "## Standard steepest descent\n", "\n", - "\n", "Before we proceed, we would like to discuss the approach called the\n", "**standard Steepest descent** (different from the above steepest descent discussion), which again leads to us having to be able\n", "to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).\n", @@ -533,7 +636,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ec752109", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{A}\\boldsymbol{x} = \\boldsymbol{b}.\n", @@ -542,14 +648,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "eb64c8e7", + "metadata": { + "editable": true + }, "source": [ "In the iterative process we end up with a problem like" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "7e99eb7f", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{r}= \\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x},\n", @@ -558,20 +670,24 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "3a3f6414", + "metadata": { + "editable": true + }, "source": [ "where $\\boldsymbol{r}$ is the so-called residual or error in the iterative process.\n", "\n", "When we have found the exact solution, $\\boldsymbol{r}=0$.\n", "\n", - "\n", - "\n", "The residual is zero when we reach the minimum of the quadratic equation" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "a88175a4", + "metadata": { + "editable": true + }, "source": [ "$$\n", "P(\\boldsymbol{x})=\\frac{1}{2}\\boldsymbol{x}^T\\boldsymbol{A}\\boldsymbol{x} - \\boldsymbol{x}^T\\boldsymbol{b},\n", @@ -580,19 +696,24 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6992cc4f", + "metadata": { + "editable": true + }, "source": [ "with the constraint that the matrix $\\boldsymbol{A}$ is positive definite and\n", "symmetric. This defines also the Hessian and we want it to be positive definite. \n", "\n", - "\n", "We denote the initial guess for $\\boldsymbol{x}$ as $\\boldsymbol{x}_0$. \n", "We can assume without loss of generality that" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "00e22e2b", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{x}_0=0,\n", @@ -601,14 +722,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "faab896d", + "metadata": { + "editable": true + }, "source": [ "or consider the system" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "02cb8061", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{A}\\boldsymbol{z} = \\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_0,\n", @@ -617,17 +744,22 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "fe52c448", + "metadata": { + "editable": true + }, "source": [ "instead.\n", "\n", - "\n", "One can show that the solution $\\boldsymbol{x}$ is also the unique minimizer of the quadratic form" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "f698b7fe", + "metadata": { + "editable": true + }, "source": [ "$$\n", "f(\\boldsymbol{x}) = \\frac{1}{2}\\boldsymbol{x}^T\\boldsymbol{A}\\boldsymbol{x} - \\boldsymbol{x}^T \\boldsymbol{x} , \\quad \\boldsymbol{x}\\in\\mathbf{R}^n.\n", @@ -636,7 +768,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "022bda7f", + "metadata": { + "editable": true + }, "source": [ "This suggests taking the first basis vector $\\boldsymbol{r}_1$ (see below for definition) \n", "to be the gradient of $f$ at $\\boldsymbol{x}=\\boldsymbol{x}_0$, \n", @@ -645,7 +780,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b64077bc", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{A}\\boldsymbol{x}_0-\\boldsymbol{b},\n", @@ -654,18 +792,23 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "dec06904", + "metadata": { + "editable": true + }, "source": [ "and \n", "$\\boldsymbol{x}_0=0$ it is equal $-\\boldsymbol{b}$.\n", "\n", - "\n", "We can compute the residual iteratively as" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "b566de75", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{r}_{k+1}=\\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_{k+1},\n", @@ -674,14 +817,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "2c2d16e7", + "metadata": { + "editable": true + }, "source": [ "which equals" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "6c97f03d", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{b}-\\boldsymbol{A}(\\boldsymbol{x}_k+\\alpha_k\\boldsymbol{r}_k),\n", @@ -690,14 +839,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "2b40818b", + "metadata": { + "editable": true + }, "source": [ "or" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "6619d064", + "metadata": { + "editable": true + }, "source": [ "$$\n", "(\\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_k)-\\alpha_k\\boldsymbol{A}\\boldsymbol{r}_k,\n", @@ -706,14 +861,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "467c71be", + "metadata": { + "editable": true + }, "source": [ "which gives" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "d58fd1af", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\alpha_k = \\frac{\\boldsymbol{r}_k^T\\boldsymbol{r}_k}{\\boldsymbol{r}_k^T\\boldsymbol{A}\\boldsymbol{r}_k}\n", @@ -722,14 +883,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "38e32957", + "metadata": { + "editable": true + }, "source": [ "leading to the iterative scheme" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "98043fd6", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{x}_{k+1}=\\boldsymbol{x}_k-\\alpha_k\\boldsymbol{r}_{k},\n", @@ -739,6 +906,7 @@ { "cell_type": "code", "execution_count": 1, + "id": "8c7efe84", "metadata": { "collapsed": false, "editable": true @@ -748,14 +916,14 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42573/483257001.py:18: MatplotlibDeprecationWarning: Calling gca() with keyword arguments was deprecated in Matplotlib 3.4. Starting two minor releases later, gca() will take no keyword arguments. The gca() function should only be used to get the current axes, or if no axes exist, create new axes with default keyword arguments. To create a new axes with non-default arguments, use plt.axes() or plt.subplot().\n", + "/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47735/483257001.py:18: MatplotlibDeprecationWarning: Calling gca() with keyword arguments was deprecated in Matplotlib 3.4. Starting two minor releases later, gca() will take no keyword arguments. The gca() function should only be used to get the current axes, or if no axes exist, create new axes with default keyword arguments. To create a new axes with non-default arguments, use plt.axes() or plt.subplot().\n", " ax = fig.gca(projection=\"3d\")\n" ] }, { "data": { "text/plain": [ - "" + "" ] }, "execution_count": 1, @@ -771,7 +939,7 @@ }, "metadata": { "filenames": { - "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_56_2.png" + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_61_2.png" }, "needs_background": "light" }, @@ -805,7 +973,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "3bdeb3c1", + "metadata": { + "editable": true + }, "source": [ "And then as countor plot" ] @@ -813,6 +984,7 @@ { "cell_type": "code", "execution_count": 2, + "id": "e6e460db", "metadata": { "collapsed": false, "editable": true @@ -827,7 +999,7 @@ }, "metadata": { "filenames": { - "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_58_0.png" + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_63_0.png" }, "needs_background": "light" }, @@ -842,7 +1014,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d165add7", + "metadata": { + "editable": true + }, "source": [ "Find guesses" ] @@ -850,6 +1025,7 @@ { "cell_type": "code", "execution_count": 3, + "id": "236615ae", "metadata": { "collapsed": false, "editable": true @@ -862,7 +1038,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "4b90442c", + "metadata": { + "editable": true + }, "source": [ "Run it!" ] @@ -870,6 +1049,7 @@ { "cell_type": "code", "execution_count": 4, + "id": "0346fd1d", "metadata": { "collapsed": false, "editable": true @@ -895,7 +1075,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ecff8b5b", + "metadata": { + "editable": true + }, "source": [ "What happened?" ] @@ -903,6 +1086,7 @@ { "cell_type": "code", "execution_count": 5, + "id": "0d9b7732", "metadata": { "collapsed": false, "editable": true @@ -911,7 +1095,7 @@ { "data": { "text/plain": [ - "[]" + "[]" ] }, "execution_count": 5, @@ -927,7 +1111,7 @@ }, "metadata": { "filenames": { - "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_64_1.png" + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_69_1.png" }, "needs_background": "light" }, @@ -943,7 +1127,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b0f8920b", + "metadata": { + "editable": true + }, "source": [ "## Conjugate gradient method\n", "In the CG method we define so-called conjugate directions and two vectors \n", @@ -954,7 +1141,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "67b215c6", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{s}^T\\boldsymbol{A}\\boldsymbol{t}= 0.\n", @@ -963,7 +1153,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "efc1c192", + "metadata": { + "editable": true + }, "source": [ "The philosophy of the CG method is to perform searches in various conjugate directions\n", "of our vectors $\\boldsymbol{x}_i$ obeying the above criterion, namely" @@ -971,7 +1164,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "eb4e1832", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{x}_i^T\\boldsymbol{A}\\boldsymbol{x}_j= 0.\n", @@ -980,7 +1176,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "775cf999", + "metadata": { + "editable": true + }, "source": [ "Two vectors are conjugate if they are orthogonal with respect to \n", "this inner product. Being conjugate is a symmetric relation: if $\\boldsymbol{s}$ is conjugate to $\\boldsymbol{t}$, then $\\boldsymbol{t}$ is conjugate to $\\boldsymbol{s}$.\n", @@ -990,7 +1189,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "95893950", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{v}_i^T\\boldsymbol{A}\\boldsymbol{v}_j= \\lambda\\boldsymbol{v}_i^T\\boldsymbol{v}_j,\n", @@ -999,7 +1201,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "16bfad5c", + "metadata": { + "editable": true + }, "source": [ "which is zero unless $i=j$. \n", "\n", @@ -1009,7 +1214,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e2577f1c", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{x}_{i+1}=\\boldsymbol{x}_{i}+\\alpha_i\\boldsymbol{p}_{i}.\n", @@ -1018,7 +1226,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e4defaaf", + "metadata": { + "editable": true + }, "source": [ "We assume that $\\boldsymbol{p}_{i}$ is a sequence of $n$ mutually conjugate directions. \n", "Then the $\\boldsymbol{p}_{i}$ form a basis of $R^n$ and we can expand the solution \n", @@ -1027,7 +1238,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b4632612", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{x} = \\sum^{n}_{i=1} \\alpha_i \\boldsymbol{p}_i.\n", @@ -1036,14 +1250,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a0a06bf2", + "metadata": { + "editable": true + }, "source": [ "The coefficients are given by" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "1db7cc6a", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathbf{A}\\mathbf{x} = \\sum^{n}_{i=1} \\alpha_i \\mathbf{A} \\mathbf{p}_i = \\mathbf{b}.\n", @@ -1052,14 +1272,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e7aa5384", + "metadata": { + "editable": true + }, "source": [ "Multiplying with $\\boldsymbol{p}_k^T$ from the left gives" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "884580b0", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{p}_k^T \\boldsymbol{A}\\boldsymbol{x} = \\sum^{n}_{i=1} \\alpha_i\\boldsymbol{p}_k^T \\boldsymbol{A}\\boldsymbol{p}_i= \\boldsymbol{p}_k^T \\boldsymbol{b},\n", @@ -1068,14 +1294,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "934990e4", + "metadata": { + "editable": true + }, "source": [ "and we can define the coefficients $\\alpha_k$ as" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "8be96384", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\alpha_k = \\frac{\\boldsymbol{p}_k^T \\boldsymbol{b}}{\\boldsymbol{p}_k^T \\boldsymbol{A} \\boldsymbol{p}_k}\n", @@ -1084,7 +1316,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "44f4d8d5", + "metadata": { + "editable": true + }, "source": [ "If we choose the conjugate vectors $\\boldsymbol{p}_k$ carefully, \n", "then we may not need all of them to obtain a good approximation to the solution \n", @@ -1099,7 +1334,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a8739d7d", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{x}_0=0,\n", @@ -1108,14 +1346,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d871e171", + "metadata": { + "editable": true + }, "source": [ "or consider the system" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "7ed84e84", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{A}\\boldsymbol{z} = \\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_0,\n", @@ -1124,7 +1368,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b690f75b", + "metadata": { + "editable": true + }, "source": [ "instead.\n", "\n", @@ -1133,7 +1380,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "86ce9e8a", + "metadata": { + "editable": true + }, "source": [ "$$\n", "f(\\boldsymbol{x}) = \\frac{1}{2}\\boldsymbol{x}^T\\boldsymbol{A}\\boldsymbol{x} - \\boldsymbol{x}^T \\boldsymbol{x} , \\quad \\boldsymbol{x}\\in\\mathbf{R}^n.\n", @@ -1142,7 +1392,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "7968787f", + "metadata": { + "editable": true + }, "source": [ "This suggests taking the first basis vector $\\boldsymbol{p}_1$ \n", "to be the gradient of $f$ at $\\boldsymbol{x}=\\boldsymbol{x}_0$, \n", @@ -1151,7 +1404,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0d510b11", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{A}\\boldsymbol{x}_0-\\boldsymbol{b},\n", @@ -1160,7 +1416,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "c7c47a8e", + "metadata": { + "editable": true + }, "source": [ "and \n", "$\\boldsymbol{x}_0=0$ it is equal $-\\boldsymbol{b}$.\n", @@ -1172,7 +1431,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5dc0e4ac", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{r}_k=\\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_k.\n", @@ -1181,7 +1443,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0aecb4b3", + "metadata": { + "editable": true + }, "source": [ "Note that $\\boldsymbol{r}_k$ is the negative gradient of $f$ at \n", "$\\boldsymbol{x}=\\boldsymbol{x}_k$, \n", @@ -1194,7 +1459,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f3dfc701", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{p}_{k+1}=\\boldsymbol{r}_k-\\frac{\\boldsymbol{p}_k^T \\boldsymbol{A}\\boldsymbol{r}_k}{\\boldsymbol{p}_k^T\\boldsymbol{A}\\boldsymbol{p}_k} \\boldsymbol{p}_k.\n", @@ -1203,14 +1471,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "12e92892", + "metadata": { + "editable": true + }, "source": [ "We can also compute the residual iteratively as" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "1514b03f", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{r}_{k+1}=\\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_{k+1},\n", @@ -1219,14 +1493,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "7b09db3d", + "metadata": { + "editable": true + }, "source": [ "which equals" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "4a0638f8", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{b}-\\boldsymbol{A}(\\boldsymbol{x}_k+\\alpha_k\\boldsymbol{p}_k),\n", @@ -1235,14 +1515,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "298f5e46", + "metadata": { + "editable": true + }, "source": [ "or" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "acd35abb", + "metadata": { + "editable": true + }, "source": [ "$$\n", "(\\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_k)-\\alpha_k\\boldsymbol{A}\\boldsymbol{p}_k,\n", @@ -1251,14 +1537,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "79625f01", + "metadata": { + "editable": true + }, "source": [ "which gives" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "dc5888e2", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{r}_{k+1}=\\boldsymbol{r}_k-\\boldsymbol{A}\\boldsymbol{p}_{k},\n", @@ -1267,7 +1559,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d11b96a9", + "metadata": { + "editable": true + }, "source": [ "## Revisiting our Linear Regression Solvers\n", "\n", @@ -1288,6 +1583,7 @@ { "cell_type": "code", "execution_count": 6, + "id": "cf7c349e", "metadata": { "collapsed": false, "editable": true @@ -1301,7 +1597,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "adf34219", + "metadata": { + "editable": true + }, "source": [ "with $x_i \\in [0,1] $ is chosen randomly using a uniform distribution. Additionally we have a stochastic noise chosen according to a normal distribution $\\cal {N}(0,1)$. \n", "The linear regression model is given by" @@ -1309,7 +1608,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d28eb4e0", + "metadata": { + "editable": true + }, "source": [ "$$\n", "h_\\beta(x) = \\boldsymbol{y} = \\beta_0 + \\beta_1 x,\n", @@ -1318,14 +1620,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "60060fcb", + "metadata": { + "editable": true + }, "source": [ "such that" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "82983f16", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{y}_i = \\beta_0 + \\beta_1 x_i.\n", @@ -1334,7 +1642,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "60fc4bc0", + "metadata": { + "editable": true + }, "source": [ "Let $\\mathbf{y} = (y_1,\\cdots,y_n)^T$, $\\mathbf{\\boldsymbol{y}} = (\\boldsymbol{y}_1,\\cdots,\\boldsymbol{y}_n)^T$ and $\\beta = (\\beta_0, \\beta_1)^T$\n", "\n", @@ -1343,7 +1654,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0c5dac72", + "metadata": { + "editable": true + }, "source": [ "$$\n", "X \\equiv \\begin{bmatrix}\n", @@ -1356,14 +1670,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "002d8c9b", + "metadata": { + "editable": true + }, "source": [ "The cost/loss/risk function is given by (" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "b91d8b5a", + "metadata": { + "editable": true + }, "source": [ "$$\n", "C(\\beta) = \\frac{1}{n}||X\\beta-\\mathbf{y}||_{2}^{2} = \\frac{1}{n}\\sum_{i=1}^{100}\\left[ (\\beta_0 + \\beta_1 x_i)^2 - 2 y_i (\\beta_0 + \\beta_1 x_i) + y_i^2\\right]\n", @@ -1372,17 +1692,22 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d0fc6c20", + "metadata": { + "editable": true + }, "source": [ "and we want to find $\\beta$ such that $C(\\beta)$ is minimized.\n", "\n", - "\n", "Computing $\\partial C(\\beta) / \\partial \\beta_0$ and $\\partial C(\\beta) / \\partial \\beta_1$ we can show that the gradient can be written as" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "3f99dab5", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\nabla_{\\beta} C(\\beta) = \\frac{2}{n}\\begin{bmatrix} \\sum_{i=1}^{100} \\left(\\beta_0+\\beta_1x_i-y_i\\right) \\\\\n", @@ -1393,17 +1718,22 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e1305a22", + "metadata": { + "editable": true + }, "source": [ "where $X$ is the design matrix defined above.\n", "\n", - "\n", "The Hessian matrix of $C(\\beta)$ is given by" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "ef62073f", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{H} \\equiv \\begin{bmatrix}\n", @@ -1415,18 +1745,22 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "dfb78235", + "metadata": { + "editable": true + }, "source": [ "This result implies that $C(\\beta)$ is a convex function since the matrix $X^T X$ always is positive semi-definite.\n", "\n", - "\n", - "\n", "We can now write a program that minimizes $C(\\beta)$ using the gradient descent method with a constant learning rate $\\gamma$ according to" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "56cee86c", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\beta_{k+1} = \\beta_k - \\gamma \\nabla_\\beta C(\\beta_k), \\ k=0,1,\\cdots\n", @@ -1435,7 +1769,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6cc71454", + "metadata": { + "editable": true + }, "source": [ "We can use the expression we computed for the gradient and let use a\n", "$\\beta_0$ be chosen randomly and let $\\gamma = 0.001$. Stop iterating\n", @@ -1444,14 +1781,13 @@ "And finally we can compare our solution for $\\beta$ with the analytic result given by \n", "$\\beta= (X^TX)^{-1} X^T \\mathbf{y}$.\n", "\n", - "\n", - "\n", "Here is our simple example" ] }, { "cell_type": "code", "execution_count": 7, + "id": "90fab6b7", "metadata": { "collapsed": false, "editable": true @@ -1461,29 +1797,23 @@ "name": "stdout", "output_type": "stream", "text": [ - "[0.32903042 4.1484256 ]\n", - "[[4.04621521]\n", - " [3.00415763]]\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "[[4.04621521]\n", - " [3.00415763]]\n" + "[0.28001319 4.21265216]\n", + "[[3.96987657]\n", + " [3.02493054]]\n", + "[[3.96987657]\n", + " [3.02493054]]\n" ] }, { "data": { - "image/png": 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\n", + "image/png": 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"text/plain": [ "
    " ] }, "metadata": { "filenames": { - "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_118_2.png" + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_123_1.png" }, "needs_background": "light" }, @@ -1541,7 +1871,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "48ba87fe", + "metadata": { + "editable": true + }, "source": [ "Alternatively, we can use **Scikit-Learn** as done here" ] @@ -1549,6 +1882,7 @@ { "cell_type": "code", "execution_count": 8, + "id": "7522775d", "metadata": { "collapsed": false, "editable": true @@ -1558,9 +1892,9 @@ "name": "stdout", "output_type": "stream", "text": [ - "[[3.97230501]\n", - " [3.14741468]]\n", - "[3.94735055] [3.17084902]\n" + "[[4.00275135]\n", + " [2.99724883]]\n", + "[3.97065296] [3.07656896]\n" ] } ], @@ -1585,14 +1919,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "90552e2f", + "metadata": { + "editable": true + }, "source": [ "We have also discussed Ridge regression where the loss function contains a regularized term given by the $L_2$ norm of $\\beta$," ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "5d7c032c", + "metadata": { + "editable": true + }, "source": [ "$$\n", "C_{\\text{ridge}}(\\beta) = \\frac{1}{n}||X\\beta -\\mathbf{y}||^2 + \\lambda ||\\beta||^2, \\ \\lambda \\geq 0.\n", @@ -1601,14 +1941,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ed86f1ba", + "metadata": { + "editable": true + }, "source": [ "In order to minimize $C_{\\text{ridge}}(\\beta)$ using GD we only have adjust the gradient as follows" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "0386e23c", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\nabla_\\beta C_{\\text{ridge}}(\\beta) = \\frac{2}{n}\\begin{bmatrix} \\sum_{i=1}^{100} \\left(\\beta_0+\\beta_1x_i-y_i\\right) \\\\\n", @@ -1619,14 +1965,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b65523fc", + "metadata": { + "editable": true + }, "source": [ "We can easily extend our program to minimize $C_{\\text{ridge}}(\\beta)$ using gradient descent and compare with the analytical solution given by" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "c62584ee", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\beta_{\\text{ridge}} = \\left(X^T X + \\lambda I_{2 \\times 2} \\right)^{-1} X^T \\mathbf{y}.\n", @@ -1636,6 +1988,7 @@ { "cell_type": "code", "execution_count": 9, + "id": "e8d43667", "metadata": { "collapsed": false, "editable": true @@ -1645,22 +1998,22 @@ "name": "stdout", "output_type": "stream", "text": [ - "[[3.92343595]\n", - " [3.15258907]]\n", - "[[3.815563 ]\n", - " [3.23522201]]\n" + "[[4.1533795 ]\n", + " [2.92819235]]\n", + "[[4.06858699]\n", + " [2.99829953]]\n" ] }, { "data": { - "image/png": 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\n", 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rL+u4zJs3M+fR6IjXWmzV+P2XLi3/96e8gPGgmR0OEP99qKR0yFxVp/L45cth/XrYsyf6W1brqIrd7eaqiJuHrOO1e3f79XbsgIsvjgKEGY+dvZKHNu6kEqNyuHvhL2AMuDPx+UPARfH7i4C/DtnOCSec4CK5GB11j0JF82t0tOyUVdvERHSMzKK/ExNlp2j2Jibch4ebz4Ph4Xy+W9rxyjr3Eq/dmIP7MiZ8G7NPGzDleVzL89hI2x3ANcD9wC+ATcB5wAhR66j/iP8eErItBQzJjVn6P6tZ2SmrhkEMDFlGRnq/eejlOKUFqJbXOkYd3NcxmsuNTW0CRp4vBQzJjXIY2Yq8466aiYnsC3enm4duj9PChe7gWznYlzHh6xj13XFuIrmNbQz7Mib8hBM8txsbBQyR2ZhLF8VuzaVg2q54qNP37XSc3vAGd/Ad7Ne2BGo6eKTkUnL6LfIKGJVvJSVSiCq1PipatxW6c6n5bLvv1KkyP2Nd37Ahqq/+h2sxnGF2pi4H0dX/al/OmK9nyFMaNVSsoYEChsxdVWl9VKReWoNlNQc95JBi0limrO86MtL5fFiyJHXyBrKHH9+9uzmr0FHFbmwUMEQGWS+9s1evhgULZk5/4olqNjuejaw7+Msum7ms+96mrmbwxk0fYDvN625nmIuZvvvfvr05QAz1csWt0I2NAobIIOuleGn5cnja02ZOf+qpwRsGpNMdfCJA2JBhOEaUNbiG5byVNaxnlD0Y6xll18fXcLUv3xsgWmPRXnXoNJpCj2gVGWRjY1ExVKvR0ehuNcvQUHqZiVl0pzuo4seiNoJCJ3feCccc0+U+WoeshyiyFFjUpEe0ikhnvVaazpVhQJI5CKMpB5Hm619vLmLqOlhArQdxVMAQGWS9VppWrHVObl73uq4CxJVXNgeIV7wihzTUuBWaAobIoOul0jTv1jllldl/9avNAeL6z7cNEBde2Bwg3vzmAtJU49ybAoaIpMurdU4/B3p87LHmAPHqU9sGiFNOaQ4QH/5wTulIBshFi6JXI1iedlptc2+q9BaRYvVa8R7KLLiSep994MknZ7/LttIqtZOGh+Hcc+HLX46KoZYujYJFgc1lVektIvXQ7kFCvRRPmXGE3ddUD9FOMgdReLCA9ErtpB07omBRkb4V3VDAEJH2Zlv/0K5svvG0ube9LXsZM86wLzUFiPs5InPx1oGX+i6k8nrDhtr1wQAFDBFpJ4/6h7QWV0nucPnl09s04132V00B4gbOyF59YhIfHcNtCB8d65y2oivgQyuvi67PKUIeIxj266XRaitiLj0rYa7La+TagAcHNZ4B0ek1Y7vdjDrcj1GKA5530e+RgMlptFpVekt3SuilKiXKu8d3VgU4sAeYl1If0fYS1W2FetEV8A2Tk1FdxsaN04M2Pvxw9pcpuAe9Kr2lHDXupdo3NR0nKFUefQZ27OA2OzEasG/DavZgGQsay5jsrg6i205w/eo0l2ySvGVL9NqzJwpMaWrQBwMUMKRbNe6l2hf97HPQD2n1DwsWwLZtbQPiRls6XQexcJiTuA2IBuz7OOentmsawrl6tMsbj24DWtmd5uregz6Pcq1+vVSHUQFz6WlsvRjE45OssxoZcd9nnxl1ADtZEFxc7+7ZM7t9pnoV6zBC0tznOkD0iFYpRRX+4aosp2cwt9WvC07afjICYrsK6z17UradZ2Dt9njMwUYbAxEwgP8B3AXcCVwD7NdueQWMipiD/3DBur0Q9nKx60fATtnPNoZ9d0ZU2I3t/fjzn/e2fd14FKf2AQNYAqwD9o8/fxZ4U7t1FDCk8rq5EPZy0exXkdf++6fu5xfM633/rcFx5UrdePTJoASMe4FDgPnADcBvt1tHAUNqITTX0MvFP7TIq9ucyy23NG1uN+n72Q2+e78ecgbKUZSq9gEj+g5cAGwDNgOTGcusAKaAqaVLl+Z6EEVK1Ut9R0iQCbk479qVWecA7utos59eiiQHsTFAjeQVMEprVmtmBwNnAs8EjgAWmtnZrcu5+xp3H3f38cWLF/c7mVJnVe8P0UsTz5BmmRl9ZdafvWq6qev8eZm7+PM/h7GJNvvpZdhzNcceDHlEnV5ewOuBTyY+/z7w8XbrqEhKgtWhCKTXNHa6w8/IuSQrppOvU165q7f9dEM5jFJR9yIp4ESiFlLDgAGfBv643ToKGAOiH62s6nKByvFYdCpOajR93W+fjABRpDoE8AGWV8AorUjK3W8DrgO+D/yYqNf5mrLSI33Sr57QdSkC6aZ4p6WI7Y02GQ23YZOstzF2M8Q6xriB09jOzOKksYnVuMPOJ7OLowqT9yNfpRx5RJ1+vZTDGAD9uvOvSw4jqV1uY2LCtzGzX8THWJk6vZZNVtW/pzD0q0gK+Drwa3nsbLYvBYwB0I+e0O71KwLJ6Ci3jIm2xUyz6hdRJXX7vWomr4ARUiT1DuAjZvYpMzu8wMyOzAX9Gvyt1yKQslpWpbRsWsgO/pJoML6lpBelzWd3+vaqVvTWiUZBroWOAcPdv+/uv0XUse6rZvZuM9u/+KTJQOrnaJ3dNv/s80izZ57J3mauezakX+AbgWJoNCOgzsuoj2gXgPMOinlsry51TnNdSDaEqBXTscD5wBZgE3BOHlmcbl4qkhoQVS2rLrje4z3vSd98245yIyPRyllFNitXljtaa17bq2OdU43QxzqMm4H7gBuB9wGnA88BPgasySMRoS8FDCnMxET21bzH+pV//MfpTSxjwtcx6rsxX8fo3rqJxsvf8paZw4aD+4IF0xffrEDbTQDO+8Kc5yNcVYdRmH4GjGMhepRryry1eSQi9KWAIYXo9AzmwIvf976XvvoyZrZwcph5MRwZyfdinibvRgd5bq+qOc8BkFfACKnDuDPeYZrX9FgSJpKt3xXPaRWuDW3qV+67b7oOwgxe+ML0Tfwlq1hIyvZbK3Qffjh9A3mW4+fd6CDP7fUy5Ij01aw67rn7z/JKiAjQn4rn1oC0YUP2somWVTt3NgeIJUuyV3Ns7+32mAVW6OZ9MU8LvHk3Oqj7I0elO3lkU/r1UpHUHFB05Wda8VNGscqepaOZpVStrz0w+++UZzl+u23lXfSjoqTKo+5jSfXyUsDoQd3+mYvu2Jd18W7Zb7LTXNprB/tlPHs0RbcPVcrj91KrI0lQwJDOym550s3Fr83zonO90GWO5ErbVkybOML9nnt6+269LD9b/epRL7WggCGdtbvLzOMC1mHso1k9qrSgINdpJNfGa4rj3S+5JH0jK1fOvCBXpQlovwKv1IoChnSWdZfZuMDN5oLXKSB0UyTS6QKXw5DfjVdaE9dtDPu3OSn+dwj43lnHteyLcR8Dr9SLAkYd9btYIutCPC+HAes6BYRuikRyLD7Jula2Bo2tHNxbQGoX3PpZ3JN2LhUYeHNVt3q1AaCAUTdl1Cdk7TOPC16ni3weOYyAABYSIMD9FdzoQTmIXr93Mr1FXxCL/F2LVna92hylgFE33V4U87rodHMnmmcOI+TCkExLYJ1AaIA4gk35BIjQ72023WS16AtikTnHoqn1VikUMOqmm2KXoi86eWy/m4AQWineOEaJZbMCQtrYTA7uu3YVe4efle6VK6P5/bggFlk3VTS13iqFAkbdtLsrbP2H7sdFp+hWUp1kfMfW1kpZwWLG2EzJTmlFXzTbfe9+XBCLbv1WJOUwSqGAUTftWrC0XtDaXS0HRWZ/CMv+6uB+/vntLzplX5D6FeyrnpPIUue015gCRh1NTLgPDXW+oGSVRc+bV1rS83L66dFXCekPkRkk293Fl13k0a8LYtVzEu3UOe01lVfAmNXgg7NlZgeZ2XVm9hMzW2tmLy4zPX2xZ0/69ORAdLszHruZNb2Tsh47SjQga3LAvhtuiKZfzGq2Mzxj+TE27L3STr9p0W6Qvn49AjZLr4+G7WU/dR3Ztc5pn+vyiDq9voBPA2+J3+8DHNRu+drnMEJ74OZZrNHnIoDPfCb7K6YWMTXu/ru50+w0sJ6KPKSOCsx5UfciKeBpwDoyHs6U9qp9wGjXuqXXYTU6KbhM/Vvf6iFATEw0PyxoZGT6u4X+03RqgaUiD6mTgm90BiFgHAd8F7gK+AFwBbAwZbkVwBQwtXTp0lwOXmmyLt6N5zYn5XXRy7lM/6c/7SFA7NjR/L2yHkXa7fOpRQZFwTd2eQUMi7bVf2Y2DtwKvNTdbzOzy4DH3f0vstYZHx/3qampvqUxd42HAyWf7jY8XEwZd0PWA4JGR6Py4w4eeQQOOSRsV45Fb266CV7+8u7SAzBvXno9TWBaRWpraCi9vs4su96zC2Z2u7uPz3Y7ZVZ6bwI2uftt8efrgONLTE/x0ipEzz03qhkuqkK6yyeiPfVUcyV1u2DhWPR64/LoXG/cF2UFC2j/uNGsSv08H1GalxIbEsgAKruxRqg8sim9voBvAc+L318KfKjd8ic0smiDUkRRgSaYe/aEFzHtJlG81at2Ff91GNrCXRXrkj/VYQTXY0wBPwK+ABzcbvkTBu2fs6ROZqEBoqk3dV4GoQ6j7M6BMphq0EqqtDqMXoyb+d4ajEEo1y643DK5uRCbWMIS7os+FHleTE7CBRfA1q3R55ERuOyyqMhucjIqotu4McqOr15dvXb6ffrdRPIyCHUYs1PFcu1u5VFumVKWnqyDaBcsfsBx0/UQDkv8P6fvl4u0fDls2TK9ry1bpoNCslPX6tXF1u/0qi7lzSI5q2/AGIR/zi4rpGdotLrasCG68G7YwPazV7CM9AvrVzh1OkBsfZjj/I7+BIhepHw3VqyoRtCY7e8mUld5lGv16zVwdRjuPZVbNg5Dp/GYPsF509Ovv77gL5KzqtcTqHOg1Ahztg5jdLT4cu20cnQorWw9q1hpN0MMkfH7nXEGfPGLxSWqaKonEMlNXnUY9QoY/ei4l9a5bsGC6EL11FPT0wrscBdaSf0Az+BQNs+cMQgNAmbZ4VBEpqnSuyirVjUHC4Bf/KI5WEC0zKpVuewytJL6MO6froPAOHTiI4Nblq56ApHKUcBo1U3rqx5bah17bFiAAJoCxP1+eHOJfruhtOveE7lfw4SLSDAVSbVqN9ZRq8Dikbe+Fa64ImyTe8djgqisPrR8KqmMMatEpLJUJFWUtKKQBQtgn32ap7UpHvngB5tzEO2CRTIH4fc/0JyD6CVYQHqxWo5FaCIyNylgtEorCvnUp+DKKzOLR669tjlAXHRR9uabAsRn/19zgDjssHy+Q1ZR2SB0dhSR0ihgpEl7hGRi2s0T67Gzl+8NEMuWZW+qKUAQB5uJiShAvP71xaRfPZGz1b1uR6REczdgdHHhuPvu5hzEr/969mZnBIiJieYirn70WFYLo3RV7j0uUgd59P7r12tWj2hN9swdGZk5Ymqi9/jWremdjNNeMye0KKvHsnoiz1T13uMiBWFO9vTutZVUWquhFOsZ5Zmsb7tMUysm6DwOk3osV0dWIwL9FjLg1EqqG2mthlIsZWal8Iwipl27mu9PO25U9QmVMDmZHTD0W4gEmRsBI7B10EaWspuh5gCxfn1zgJg3r7t9qz6hGlatys7p6bcQCTKwAePww6crqTd7mwdTJ4yxgaHPXdccIEZHZ5cQ9ViuhqybhkaPeRHpaGACxvnnN7dkeuCBaPoyJnkaT7RfOdnU9XWvyz9xac10pb+yip1me0MgMofUNmDcfHNzgHj87yZZxxi7GWIdY3sfIvT3nMO+PDVzAyMj07kIXcQHn4oGRWatVq2kDjts3B98cGYrqWVM8glWsJDOFdt7qWXM3FOH54WLFGBgWkmZ2Twz+4GZ3dBp2QcfbP78Mr7FIxzE1ZydHixGR7OLHKrYMka9kIulokGRWZlfdgKAC4C1wNM6LXg0d/E9hhlm5/TEZz0L1j2e3gKmMeqsWfP8KhZFtPYVafRCBl3YRKQSSs1hmNmRwGuAoMG/9z/8YIav+Fj0MKNG/cM993TOLSRHfq1qKyWNMCsiFVd2kdRHgXcAmZUJZrbCzKbMbGrzggVw3nnRcONJaRWarRpNZKtaFKERZoul4j6RWSstYJjZ6cBD7n57u+XcfY27j7v7+OLFi9MXau3rkKXKF1/1CC+OBh0UyUWZOYyXAr9jZuuBa4HfMrOJnreWrNCsU0V3g5p9FkfFfSK5KC1guPu73P1Idx8DzgK+4e5n57LxrKfmbdtW3SIJ9Qgvjor7RHJRdh1G79qVSbdefEdGor9bt1a7SELNPouh4j6RXFQiYLj7Te5+evAKIWXSyYvvAQdELauSVCQxd6i4TyQXlQgYXeu2TFpFEnObivtEclGFjnvd6zYALF063YmvdbrMDY3nsotIz+qZw+i2TLpdkYTa54uIBKlnwOi2TDqrSALUPl9EJFCtRqtteqZ3HiOPjo2lF1U1eoSLiAyAvEarrW/AyMPQUPZjOzX0uYgMiIEZ3rxUap8vIhJsbgcMtc8XEQk2twOG2ueLiASrZz+MPKl9vohIkMHPYaifRTF0XEXmnMHOYeixp8XQcRWZkwa7Wa36WRRDx1WkVtSsNoQGHSyGjqvInDTYAUP9LIqh4yoyJ9UzYIRWuKqfRTF0XEXmpPoFjJCHJzWon0UxdFxF5qT6VXpv2aIKVxGRLszdSm9VuIqIlKJ+AUMVriIipSgtYJjZUWb2L2a21szuMrMLglZUhauISCnKzGHsAi50918BTgL+0MyO7riWKlxFREpR2tAg7n4/cH/8/gkzWwssAf6t48oaMFBEpO8qUYdhZmPAC4DbUuatMLMpM5vavHlz39MmIiKR0gOGmR0AfA54u7s/3jrf3de4+7i7jy9evLj/CRQREaDkgGFmC4iCxaS7f77MtIiISHtltpIy4JPAWnf/m7LSISIiYcrMYbwUOAf4LTO7I36dVmJ6RESkjTJbSd0MWFn7FxGR7pRe6S0iIvWggCEiIkEUMEREJIgChoiIBFHAEBGRIAoYIiISRAFDRESCKGCIiEgQBQwREQmigCEiIkEUMEREJIgChoiIBFHAEBGRIAoYIiISRAFDRESCKGCIiEgQBQwREQmigCEiIkEUMEREJIgChoiIBCk1YJjZqWb272Z2t5ldVGZaRESkvdIChpnNA/4P8GrgaGCZmR1dVnpERKS9MnMYLwLudvefuftTwLXAmSWmR0RE2phf4r6XAPcmPm8CTmxdyMxWACvij0+a2Z19SNtsLQK2lJ2IAEpnfuqQRlA681aXdD4vj42UGTAsZZrPmOC+BlgDYGZT7j5edMJmS+nMVx3SWYc0gtKZtzqlM4/tlFkktQk4KvH5SOC+ktIiIiIdlBkwvgf8kpk908z2Ac4CvlhiekREpI3SiqTcfZeZ/RHwT8A84Ep3v6vDamuKT1kulM581SGddUgjKJ15m1PpNPcZ1QYiIiIzqKe3iIgEUcAQEZEglQgYnYYIscjfxvN/ZGbHh67b53Quj9P3IzP7jpn9WmLeejP7sZndkVcTt1mk82QzeyxOyx1mdknoun1O558l0ninme02s0PieX05nmZ2pZk9lNX/p0LnZqd0VuXc7JTOqpybndJZhXPzKDP7FzNba2Z3mdkFKcvke366e6kvogrve4BnAfsAPwSOblnmNOArRH03TgJuC123z+l8CXBw/P7VjXTGn9cDiypyPE8Gbuhl3X6ms2X5M4BvlHA8fwM4HrgzY37p52ZgOks/NwPTWfq5GZLOipybhwPHx+8PBH5a9LWzCjmMkCFCzgQ+45FbgYPM7PDAdfuWTnf/jrs/En+8lahvSb/N5phU6ni2WAZcU1BaMrn7N4GH2yxShXOzYzorcm6GHM8slTqeLco6N+939+/H758A1hKNoJGU6/lZhYCRNkRI65fOWiZk3bx0u6/ziCJ7gwNfM7PbLRrupCih6Xyxmf3QzL5iZsd0uW4egvdlZsPAqcDnEpP7dTw7qcK52a2yzs1QZZ+bwapybprZGPAC4LaWWbmen2UODdIQMkRI1jJBw4vkJHhfZvabRP+UL0tMfqm732dmzwBuNLOfxHcxZaTz+8Cou28zs9OALwC/FLhuXrrZ1xnAt909ecfXr+PZSRXOzWAln5shqnBudqP0c9PMDiAKWG9398dbZ6es0vP5WYUcRsgQIVnL9HN4kaB9mdnzgSuAM919a2O6u98X/30IuJ4oS1hKOt39cXffFr//MrDAzBaFrNvPdCacRUuWv4/Hs5MqnJtBKnBudlSRc7MbpZ6bZraAKFhMuvvnUxbJ9/wsumImoOJmPvAz4JlMV74c07LMa2iuuPlu6Lp9TudS4G7gJS3TFwIHJt5/Bzi1xHQexnSnzRcBG+NjW6njGS/3dKKy5IVlHM94H2NkV9KWfm4GprP0czMwnaWfmyHprMK5GR+XzwAfbbNMrudn6UVSnjFEiJmdH8+/HPgyUW3/3cAO4M3t1i0xnZcAI8DHzQxgl0cjWR4KXB9Pmw9c7e5fLTGdvwusNLNdwE7gLI/OoqodT4D/BnzN3bcnVu/b8TSza4ha7iwys03Au4EFiTSWfm4GprP0czMwnaWfm4HphJLPTeClwDnAj83sjnjaxUQ3B4WcnxoaREREglShDkNERGpAAUNERIIoYIiISBAFDBERCaKAISIiQRQwREQkiAKGiIgEUcAQmYX4eQSvit+/38z+tuw0iRSl9J7eIjX3buC98UBzLwB+p+T0iBRGPb1FZsnM/hU4ADjZo+cSiAwkFUmJzIKZ/SrRk8+eVLCQQaeAIdKj+Mllk0RPKttuZqeUnCSRQilgiPQgftLa54EL3X0t8D7g0lITJVIw1WGIiEgQ5TBERCSIAoaIiARRwBARkSAKGCIiEkQBQ0REgihgiIhIEAUMEREJ8l/Q+Hs0O9UNZwAAAABJRU5ErkJggg==\n", 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    " ] }, "metadata": { "filenames": { - "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_127_1.png" + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_132_1.png" }, "needs_background": "light" }, @@ -1715,7 +2068,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "1ca83847", + "metadata": { + "editable": true + }, "source": [ "## Using gradient descent methods, limitations\n", "\n", @@ -1729,9 +2085,39 @@ "\n", "* **GD treats all directions in parameter space uniformly.** Another major drawback of GD is that unlike Newton's method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive. \n", "\n", - "* GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points.\n", + "* GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points." + ] + }, + { + "cell_type": "markdown", + "id": "dcf3e808", + "metadata": { + "editable": true + }, + "source": [ + "## Stochastic Gradient Descent (SGD)\n", "\n", - "## Stochastic Gradient Descent\n", + "In stochastic gradient descent, the extreme case is the case where we\n", + "have only one batch, that is we include the whole data set.\n", + "\n", + "This process is called Stochastic Gradient\n", + "Descent (SGD) (or also sometimes on-line gradient descent). This is\n", + "relatively less common to see because in practice due to vectorized\n", + "code optimizations it can be computationally much more efficient to\n", + "evaluate the gradient for 100 examples, than the gradient for one\n", + "example 100 times. Even though SGD technically refers to using a\n", + "single example at a time to evaluate the gradient, you will hear\n", + "people use the term SGD even when referring to mini-batch gradient\n", + "descent (i.e. mentions of MGD for “Minibatch Gradient Descent”, or BGD\n", + "for “Batch gradient descent” are rare to see), where it is usually\n", + "assumed that mini-batches are used. The size of the mini-batch is a\n", + "hyperparameter but it is not very common to cross-validate or bootstrap it. It is\n", + "usually based on memory constraints (if any), or set to some value,\n", + "e.g. 32, 64 or 128. We use powers of 2 in practice because many\n", + "vectorized operation implementations work faster when their inputs are\n", + "sized in powers of 2.\n", + "\n", + "In our notes with SGD we mean stochastic gradient descent with mini-batches.\n", "\n", "Stochastic gradient descent (SGD) and variants thereof address some of\n", "the shortcomings of the Gradient descent method discussed above.\n", @@ -1743,7 +2129,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "473b1af6", + "metadata": { + "editable": true + }, "source": [ "$$\n", "C(\\mathbf{\\beta}) = \\sum_{i=1}^n c_i(\\mathbf{x}_i,\n", @@ -1753,7 +2142,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "3353fe2a", + "metadata": { + "editable": true + }, "source": [ "This in turn means that the gradient can be\n", "computed as a sum over $i$-gradients" @@ -1761,7 +2153,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6e8e47c3", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\nabla_\\beta C(\\mathbf{\\beta}) = \\sum_i^n \\nabla_\\beta c_i(\\mathbf{x}_i,\n", @@ -1771,7 +2166,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a2eeb6ad", + "metadata": { + "editable": true + }, "source": [ "Stochasticity/randomness is introduced by only taking the\n", "gradient on a subset of the data called minibatches. If there are $n$\n", @@ -1779,8 +2177,6 @@ "minibatches. We denote these minibatches by $B_k$ where\n", "$k=1,\\cdots,n/M$.\n", "\n", - "\n", - "\n", "As an example, suppose we have $10$ data points $(\\mathbf{x}_1,\\cdots, \\mathbf{x}_{10})$ \n", "and we choose to have $M=5$ minibathces,\n", "then each minibatch contains two data points. In particular we have\n", @@ -1797,7 +2193,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5a7a0f8b", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\nabla_{\\beta}\n", @@ -1809,14 +2208,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b7b5884f", + "metadata": { + "editable": true + }, "source": [ "Thus a gradient descent step now looks like" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "6492d660", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\beta_{j+1} = \\beta_j - \\gamma_j \\sum_{i \\in B_k}^n \\nabla_\\beta c_i(\\mathbf{x}_i,\n", @@ -1826,7 +2231,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "584164f4", + "metadata": { + "editable": true + }, "source": [ "where $k$ is picked at random with equal\n", "probability from $[1,n/M]$. An iteration over the number of\n", @@ -1838,6 +2246,7 @@ { "cell_type": "code", "execution_count": 10, + "id": "42d97cf8", "metadata": { "collapsed": false, "editable": true @@ -1847,7 +2256,7 @@ "import numpy as np \n", "\n", "n = 100 #100 datapoints \n", - "M = 5 #size of each minibatch\n", + "M = 5 #size of each mini-batche\n", "m = int(n/M) #number of minibatches\n", "n_epochs = 10 #number of epochs\n", "\n", @@ -1862,7 +2271,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ca9c6c58", + "metadata": { + "editable": true + }, "source": [ "Taking the gradient only on a subset of the data has two important\n", "benefits. First, it introduces randomness which decreases the chance\n", @@ -1872,8 +2284,6 @@ "cheaper since we sum over the datapoints in the $k-th$ minibatch and not\n", "all $n$ datapoints.\n", "\n", - "\n", - "\n", "A natural question is when do we stop the search for a new minimum?\n", "One possibility is to compute the full gradient after a given number\n", "of epochs and check if the norm of the gradient is smaller than some\n", @@ -1885,8 +2295,6 @@ "compare the values of the cost function and keep the $\\beta$ that\n", "gave the lowest value.\n", "\n", - "\n", - "\n", "Another approach is to let the step length $\\gamma_j$ depend on the\n", "number of epochs in such a way that it becomes very small after a\n", "reasonable time such that we do not move at all.\n", @@ -1903,6 +2311,7 @@ { "cell_type": "code", "execution_count": 11, + "id": "d2921658", "metadata": { "collapsed": false, "editable": true @@ -1945,7 +2354,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "84469eb8", + "metadata": { + "editable": true + }, + "source": [ + "We note that we have defined several hyperparameters. These are now the number of epochs, the number of mini-batches and the parameters $t_0$ and $t_1$." + ] + }, + { + "cell_type": "markdown", + "id": "b4b94e7a", + "metadata": { + "editable": true + }, "source": [ "### Program for stochastic gradient" ] @@ -1953,6 +2375,7 @@ { "cell_type": "code", "execution_count": 12, + "id": "71dcdb35", "metadata": { "collapsed": false, "editable": true @@ -1963,34 +2386,27 @@ "output_type": "stream", "text": [ "Own inversion\n", - "[[4.15629539]\n", - " [2.7293182 ]]\n", - "sgdreg from scikit\n", - "[4.0728785] [2.66410989]\n", + "[[3.99775949]\n", + " [2.94659383]]\n", + "Eigenvalues of Hessian Matrix:[0.36102113 4.18276924]\n", "theta from own gd\n", - "[[4.15629539]\n", - " [2.7293182 ]]\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ + "[[3.99775949]\n", + " [2.94659383]]\n", "theta from own sdg\n", - "[[4.15454301]\n", - " [2.72522848]]\n" + "[[3.96489434]\n", + " [2.98399675]]\n" ] }, { "data": { - "image/png": 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\n", 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\n", 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    " ] }, "metadata": { "filenames": { - "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_141_2.png" + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_148_1.png" }, "needs_background": "light" }, @@ -1998,34 +2414,34 @@ } ], "source": [ + "# Importing various packages\n", "# Importing various packages\n", "from math import exp, sqrt\n", "from random import random, seed\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", - "from sklearn.linear_model import SGDRegressor\n", "\n", - "m = 100\n", - "x = 2*np.random.rand(m,1)\n", - "y = 4+3*x+np.random.randn(m,1)\n", + "n = 100\n", + "x = 2*np.random.rand(n,1)\n", + "y = 4+3*x+np.random.randn(n,1)\n", "\n", - "X = np.c_[np.ones((m,1)), x]\n", + "X = np.c_[np.ones((n,1)), x]\n", + "XT_X = X.T @ X\n", "theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)\n", "print(\"Own inversion\")\n", "print(theta_linreg)\n", - "sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1)\n", - "sgdreg.fit(x,y.ravel())\n", - "print(\"sgdreg from scikit\")\n", - "print(sgdreg.intercept_, sgdreg.coef_)\n", - "\n", + "# Hessian matrix\n", + "H = (2.0/n)* XT_X\n", + "EigValues, EigVectors = np.linalg.eig(H)\n", + "print(f\"Eigenvalues of Hessian Matrix:{EigValues}\")\n", "\n", "theta = np.random.randn(2,1)\n", - "eta = 0.1\n", + "eta = 1.0/np.max(EigValues)\n", "Niterations = 1000\n", "\n", "\n", "for iter in range(Niterations):\n", - " gradients = 2.0/m*X.T @ ((X @ theta)-y)\n", + " gradients = 2.0/n*X.T @ ((X @ theta)-y)\n", " theta -= eta*gradients\n", "print(\"theta from own gd\")\n", "print(theta)\n", @@ -2035,8 +2451,9 @@ "ypredict = Xnew.dot(theta)\n", "ypredict2 = Xnew.dot(theta_linreg)\n", "\n", - "\n", "n_epochs = 50\n", + "M = 5 #size of each minibatch\n", + "m = int(n/M) #number of minibatches\n", "t0, t1 = 5, 50\n", "def learning_schedule(t):\n", " return t0/(t+t1)\n", @@ -2044,16 +2461,20 @@ "theta = np.random.randn(2,1)\n", "\n", "for epoch in range(n_epochs):\n", + "# Can you figure out a better way of setting up the contributions to each batch?\n", " for i in range(m):\n", - " random_index = np.random.randint(m)\n", - " xi = X[random_index:random_index+1]\n", - " yi = y[random_index:random_index+1]\n", - " gradients = 2 * xi.T @ ((xi @ theta)-yi)\n", + " random_index = M*np.random.randint(m)\n", + " xi = X[random_index:random_index+M]\n", + " yi = y[random_index:random_index+M]\n", + " gradients = (2.0/M)* xi.T @ ((xi @ theta)-yi)\n", " eta = learning_schedule(epoch*m+i)\n", " theta = theta - eta*gradients\n", "print(\"theta from own sdg\")\n", "print(theta)\n", "\n", + "\n", + "\n", + "\n", "plt.plot(xnew, ypredict, \"r-\")\n", "plt.plot(xnew, ypredict2, \"b-\")\n", "plt.plot(x, y ,'ro')\n", @@ -2066,7 +2487,23 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6231b86f", + "metadata": { + "editable": true + }, + "source": [ + "In the above code, we have use replacement in setting up the\n", + "mini-batches. The discussion\n", + "[here](https://sebastianraschka.com/faq/docs/sgd-methods.html) may be\n", + "useful. More material will be added later." + ] + }, + { + "cell_type": "markdown", + "id": "8d50214c", + "metadata": { + "editable": true + }, "source": [ "## Momentum based GD\n", "\n", @@ -2078,7 +2515,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "45d43ca3", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathbf{v}_{t}=\\gamma \\mathbf{v}_{t-1}+\\eta_{t}\\nabla_\\theta E(\\boldsymbol{\\theta}_t) \\nonumber\n", @@ -2087,7 +2527,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "dcdd91bf", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -2102,7 +2545,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "c7519d1e", + "metadata": { + "editable": true + }, "source": [ "where we have introduced a momentum parameter $\\gamma$, with\n", "$0\\le\\gamma\\le 1$, and for brevity we dropped the explicit notation to\n", @@ -2118,7 +2564,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f4d4340d", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\Delta \\boldsymbol{\\theta}_{t+1} = \\gamma \\Delta \\boldsymbol{\\theta}_t -\\ \\eta_{t}\\nabla_\\theta E(\\boldsymbol{\\theta}_t),\n", @@ -2127,12 +2576,13 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "41ad532a", + "metadata": { + "editable": true + }, "source": [ "where we have defined $\\Delta \\boldsymbol{\\theta}_{t}= \\boldsymbol{\\theta}_t-\\boldsymbol{\\theta}_{t-1}$.\n", "\n", - "\n", - "\n", "Let us try to get more intuition from these equations. It is helpful\n", "to consider a simple physical analogy with a particle of mass $m$\n", "moving in a viscous medium with drag coefficient $\\mu$ and potential\n", @@ -2142,7 +2592,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "eef8ae92", + "metadata": { + "editable": true + }, "source": [ "$$\n", "m {d^2 \\mathbf{w} \\over dt^2} + \\mu {d \\mathbf{w} \\over dt }= -\\nabla_w E(\\mathbf{w}).\n", @@ -2151,14 +2604,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "80481a6d", + "metadata": { + "editable": true + }, "source": [ "We can discretize this equation in the usual way to get" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "6aab8db7", + "metadata": { + "editable": true + }, "source": [ "$$\n", "m { \\mathbf{w}_{t+\\Delta t}-2 \\mathbf{w}_{t} +\\mathbf{w}_{t-\\Delta t} \\over (\\Delta t)^2}+\\mu {\\mathbf{w}_{t+\\Delta t}- \\mathbf{w}_{t} \\over \\Delta t} = -\\nabla_w E(\\mathbf{w}).\n", @@ -2167,14 +2626,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "dd6414d2", + "metadata": { + "editable": true + }, "source": [ "Rearranging this equation, we can rewrite this as" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "b76a8372", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\Delta \\mathbf{w}_{t +\\Delta t}= - { (\\Delta t)^2 \\over m +\\mu \\Delta t} \\nabla_w E(\\mathbf{w})+ {m \\over m +\\mu \\Delta t} \\Delta \\mathbf{w}_t.\n", @@ -2183,7 +2648,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "07bbe6db", + "metadata": { + "editable": true + }, "source": [ "Notice that this equation is identical to previous one if we identify\n", "the position of the particle, $\\mathbf{w}$, with the parameters\n", @@ -2194,7 +2662,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "7a9a2ecf", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\gamma= {m \\over m +\\mu \\Delta t }, \\qquad \\eta = {(\\Delta t)^2 \\over m +\\mu \\Delta t}.\n", @@ -2203,7 +2674,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "fe5a2bbc", + "metadata": { + "editable": true + }, "source": [ "Thus, as the name suggests, the momentum parameter is proportional to\n", "the mass of the particle and effectively provides inertia.\n", @@ -2233,7 +2707,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "04540023", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathbf{v}_{t}=\\gamma \\mathbf{v}_{t-1}+\\eta_{t}\\nabla_\\theta E(\\boldsymbol{\\theta}_t +\\gamma \\mathbf{v}_{t-1}) \\nonumber\n", @@ -2242,7 +2719,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "dfbf53a5", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -2257,12 +2737,13 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "32ac3869", + "metadata": { + "editable": true + }, "source": [ "One of the major advantages of NAG is that it allows for the use of a larger learning rate than GDM for the same choice of $\\gamma$.\n", "\n", - "\n", - "\n", "In stochastic gradient descent, with and without momentum, we still\n", "have to specify a schedule for tuning the learning rates $\\eta_t$\n", "as a function of time. As discussed in the context of Newton's\n", @@ -2281,10 +2762,17 @@ "\n", "Recently, a number of methods have been introduced that accomplish\n", "this by tracking not only the gradient, but also the second moment of\n", - "the gradient. These methods include AdaGrad, AdaDelta, RMS-Prop, and\n", - "ADAM.\n", - "\n", - "\n", + "the gradient. These methods include AdaGrad, AdaDelta, Root Mean Squared Propagation (RMS-Prop), and\n", + "ADAM." + ] + }, + { + "cell_type": "markdown", + "id": "6f575b16", + "metadata": { + "editable": true + }, + "source": [ "### RMS prop\n", "\n", "In RMS prop, in addition to keeping a running average of the first\n", @@ -2295,7 +2783,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "274604c4", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -2310,7 +2801,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "cd679323", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathbf{s}_t =\\beta \\mathbf{s}_{t-1} +(1-\\beta)\\mathbf{g}_t^2 \\nonumber\n", @@ -2319,7 +2813,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0a7f8e9d", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{\\theta}_{t+1}=\\boldsymbol{\\theta}_t - \\eta_t { \\mathbf{g}_t \\over \\sqrt{\\mathbf{s}_t +\\epsilon}}, \\nonumber\n", @@ -2328,7 +2825,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "48ba8fff", + "metadata": { + "editable": true + }, "source": [ "where $\\beta$ controls the averaging time of the second moment and is\n", "typically taken to be about $\\beta=0.9$, $\\eta_t$ is a learning rate\n", @@ -2338,8 +2838,16 @@ "is clear from this formula that the learning rate is reduced in\n", "directions where the norm of the gradient is consistently large. This\n", "greatly speeds up the convergence by allowing us to use a larger\n", - "learning rate for flat directions.\n", - "\n", + "learning rate for flat directions." + ] + }, + { + "cell_type": "markdown", + "id": "83140ab2", + "metadata": { + "editable": true + }, + "source": [ "### ADAM optimizer\n", "\n", "A related algorithm is the ADAM optimizer. In ADAM, we keep a running\n", @@ -2358,7 +2866,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "61874dc3", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -2373,7 +2884,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "24e19d86", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathbf{m}_t = \\beta_1 \\mathbf{m}_{t-1} + (1-\\beta_1) \\mathbf{g}_t \\nonumber\n", @@ -2382,7 +2896,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "506f78ea", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathbf{s}_t =\\beta_2 \\mathbf{s}_{t-1} +(1-\\beta_2)\\mathbf{g}_t^2 \\nonumber\n", @@ -2391,7 +2908,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "cb4b8585", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{\\mathbf{m}}_t={\\mathbf{m}_t \\over 1-\\beta_1^t} \\nonumber\n", @@ -2400,7 +2920,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9b5b11b1", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{\\mathbf{s}}_t ={\\mathbf{s}_t \\over1-\\beta_2^t} \\nonumber\n", @@ -2409,7 +2932,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "92292a73", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{\\theta}_{t+1}=\\boldsymbol{\\theta}_t - \\eta_t { \\boldsymbol{\\mathbf{m}}_t \\over \\sqrt{\\boldsymbol{\\mathbf{s}}_t} +\\epsilon}, \\nonumber\n", @@ -2418,7 +2944,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "1a264832", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -2432,7 +2961,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6a307202", + "metadata": { + "editable": true + }, "source": [ "where $\\beta_1$ and $\\beta_2$ set the memory lifetime of the first and\n", "second moment and are typically taken to be $0.9$ and $0.99$\n", @@ -2448,7 +2980,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "3285f010", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\Delta \\theta_{t+1}= -\\eta_t { \\boldsymbol{m}_t \\over \\sqrt{\\sigma_t^2 + m_t^2 }+\\epsilon}.\n", @@ -2457,7 +2992,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "657349da", + "metadata": { + "editable": true + }, "source": [ "## Practical tips\n", "\n", @@ -2467,8 +3005,16 @@ "\n", "* **Monitor the out-of-sample performance.** Always monitor the performance of your model on a validation set (a small portion of the training data that is held out of the training process to serve as a proxy for the test set. If the validation error starts increasing, then the model is beginning to overfit. Terminate the learning process. This *early stopping* significantly improves performance in many settings.\n", "\n", - "* **Adaptive optimization methods don't always have good generalization.** Recent studies have shown that adaptive methods such as ADAM, RMSPorp, and AdaGrad tend to have poor generalization compared to SGD or SGD with momentum, particularly in the high-dimensional limit (i.e. the number of parameters exceeds the number of data points). Although it is not clear at this stage why these methods perform so well in training deep neural networks, simpler procedures like properly-tuned SGD may work as well or better in these applications.\n", - "\n", + "* **Adaptive optimization methods don't always have good generalization.** Recent studies have shown that adaptive methods such as ADAM, RMSPorp, and AdaGrad tend to have poor generalization compared to SGD or SGD with momentum, particularly in the high-dimensional limit (i.e. the number of parameters exceeds the number of data points). Although it is not clear at this stage why these methods perform so well in training deep neural networks, simpler procedures like properly-tuned SGD may work as well or better in these applications." + ] + }, + { + "cell_type": "markdown", + "id": "65044ac7", + "metadata": { + "editable": true + }, + "source": [ "## Automatic differentiation\n", "\n", "[Automatic differentiation (AD)](https://en.wikipedia.org/wiki/Automatic_differentiation), \n", @@ -2496,15 +3042,16 @@ "while numerical differentiation can introduce round-off errors in the\n", "discretization process and cancellation\n", "\n", - "\n", - "\n", "Python has tools for so-called **automatic differentiation**.\n", "Consider the following example" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "dacf05cf", + "metadata": { + "editable": true + }, "source": [ "$$\n", "f(x) = \\sin\\left(2\\pi x + x^2\\right)\n", @@ -2513,14 +3060,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "da4ad36e", + "metadata": { + "editable": true + }, "source": [ "which has the following derivative" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "f4c3e6c4", + "metadata": { + "editable": true + }, "source": [ "$$\n", "f'(x) = \\cos\\left(2\\pi x + x^2\\right)\\left(2\\pi + 2x\\right)\n", @@ -2529,7 +3082,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "1c8bfd4a", + "metadata": { + "editable": true + }, "source": [ "Using **autograd** we have" ] @@ -2537,6 +3093,7 @@ { "cell_type": "code", "execution_count": 13, + "id": "e1d91b8b", "metadata": { "collapsed": false, "editable": true @@ -2551,7 +3108,7 @@ }, "metadata": { "filenames": { - "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_177_0.png" + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_188_0.png" }, "needs_background": "light" }, @@ -2604,7 +3161,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "cd1158a5", + "metadata": { + "editable": true + }, "source": [ "Here we\n", "experiment with what kind of functions Autograd is capable\n", @@ -2616,6 +3176,7 @@ { "cell_type": "code", "execution_count": 14, + "id": "e2e9faff", "metadata": { "collapsed": false, "editable": true @@ -2652,7 +3213,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e4a83059", + "metadata": { + "editable": true + }, "source": [ "To differentiate with respect to two (or more) arguments of a Python\n", "function, Autograd need to know at which variable the function if\n", @@ -2662,6 +3226,7 @@ { "cell_type": "code", "execution_count": 15, + "id": "f65983d8", "metadata": { "collapsed": false, "editable": true @@ -2719,7 +3284,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "1d369f97", + "metadata": { + "editable": true + }, "source": [ "Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable." ] @@ -2727,6 +3295,7 @@ { "cell_type": "code", "execution_count": 16, + "id": "46ea0652", "metadata": { "collapsed": false, "editable": true @@ -2763,7 +3332,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9cc6674a", + "metadata": { + "editable": true + }, "source": [ "Note that in this case, when sending an array as input argument, the\n", "output from Autograd is another array. This is the true gradient of\n", @@ -2776,6 +3348,7 @@ { "cell_type": "code", "execution_count": 17, + "id": "17813055", "metadata": { "collapsed": false, "editable": true @@ -2813,6 +3386,7 @@ { "cell_type": "code", "execution_count": 18, + "id": "6da49540", "metadata": { "collapsed": false, "editable": true @@ -2843,40 +3417,10 @@ "print(\"The computed derivative of f5 at x = %g is: %g\"%(x,f5_grad(x)))" ] }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "1\n", - "8\n", - " \n", - "<\n", - "<\n", - "<\n", - "!\n", - "!\n", - "C\n", - "O\n", - "D\n", - "E\n", - "_\n", - "B\n", - "L\n", - "O\n", - "C\n", - "K\n", - " \n", - " \n", - "p\n", - "y\n", - "c\n", - "o\n", - "d" - ] - }, { "cell_type": "code", "execution_count": 19, + "id": "a8ff2c17", "metadata": { "collapsed": false, "editable": true @@ -2886,7 +3430,52 @@ "name": "stdout", "output_type": "stream", "text": [ - "The analytical derivative of f6 at x = 2.7 is: 37732.5\n" + "The computed derivative of f6_for at x = 0.5 is: 3.95703\n", + "The computed derivative of f6_while at x = 0.5 is: 3.95703\n" + ] + } + ], + "source": [ + "import autograd.numpy as np\n", + "from autograd import grad\n", + "def f6_for(x):\n", + " val = 0\n", + " for i in range(10):\n", + " val = val + x**i\n", + " return val\n", + "\n", + "def f6_while(x):\n", + " val = 0\n", + " i = 0\n", + " while i < 10:\n", + " val = val + x**i\n", + " i = i + 1\n", + " return val\n", + "\n", + "f6_for_grad = grad(f6_for)\n", + "f6_while_grad = grad(f6_while)\n", + "\n", + "x = 0.5\n", + "\n", + "# Print the computed derivaties of f6_for and f6_while\n", + "print(\"The computed derivative of f6_for at x = %g is: %g\"%(x,f6_for_grad(x)))\n", + "print(\"The computed derivative of f6_while at x = %g is: %g\"%(x,f6_while_grad(x)))" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "id": "ec67d4a3", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The analytical derivative of f6 at x = 0.5 is: 3.95703\n" ] } ], @@ -2904,7 +3493,8 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 21, + "id": "742a2d68", "metadata": { "collapsed": false, "editable": true @@ -2951,71 +3541,78 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e7be6348", + "metadata": { + "editable": true + }, "source": [ "Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input.\n", "\n", + "Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.\n", "\n", - "Autograd supports many features. However, there are some functions that are not supported (yet) by Autograd.\n", - "\n", - "Assigning a value to the variable being differentiated with respect to is an example thereof." - ] - }, - { - "cell_type": "code", - "execution_count": 21, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "#import autograd.numpy as np\n", - "#from autograd import grad\n", - "#def f8(x): # Assume x is an array\n", - "# x[2] = 3\n", - "# return x*2\n", - "\n", - "#f8_grad = grad(f8)\n", - "\n", - "#x = 8.4\n", - "\n", - "#print(\"The derivative of f8 is:\",f8_grad(x))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible." + "Assigning a value to the variable being differentiated with respect to" ] }, { "cell_type": "code", "execution_count": 22, + "id": "c551058c", "metadata": { "collapsed": false, "editable": true }, "outputs": [ { - "ename": "AttributeError", - "evalue": "'ArrayBox' object has no attribute 'dot'", + "ename": "TypeError", + "evalue": "'ArrayBox' object does not support item assignment", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", - "\u001b[0;31mAttributeError\u001b[0m Traceback (most recent call last)", - "\u001b[0;32m/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_42573/546166676.py\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 9\u001b[0m \u001b[0mx\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marray\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1.0\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m0.0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 10\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 11\u001b[0;31m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"The derivative of f9 is:\"\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mf9_grad\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m", + "\u001b[0;31mTypeError\u001b[0m Traceback (most recent call last)", + "\u001b[0;32m/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47735/1122558214.py\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 9\u001b[0m \u001b[0mx\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m8.4\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 10\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 11\u001b[0;31m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"The derivative of f8 is:\"\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mf8_grad\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m", "\u001b[0;32m~/anaconda3/lib/python3.8/site-packages/autograd/wrap_util.py\u001b[0m in \u001b[0;36mnary_f\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m 18\u001b[0m \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 19\u001b[0m \u001b[0mx\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mtuple\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[0;32min\u001b[0m \u001b[0margnum\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 20\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0munary_operator\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0munary_f\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0mnary_op_args\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mnary_op_kwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 21\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mnary_f\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 22\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mnary_operator\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;32m~/anaconda3/lib/python3.8/site-packages/autograd/differential_operators.py\u001b[0m in \u001b[0;36mgrad\u001b[0;34m(fun, x)\u001b[0m\n\u001b[1;32m 23\u001b[0m \u001b[0marguments\u001b[0m \u001b[0;32mas\u001b[0m\u001b[0;31m \u001b[0m\u001b[0;31m`\u001b[0m\u001b[0mfun\u001b[0m\u001b[0;31m`\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mbut\u001b[0m \u001b[0mreturns\u001b[0m \u001b[0mthe\u001b[0m \u001b[0mgradient\u001b[0m \u001b[0minstead\u001b[0m\u001b[0;34m.\u001b[0m \u001b[0mThe\u001b[0m \u001b[0mfunction\u001b[0m\u001b[0;31m \u001b[0m\u001b[0;31m`\u001b[0m\u001b[0mfun\u001b[0m\u001b[0;31m`\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 24\u001b[0m should be scalar-valued. The gradient has the same type as the argument.\"\"\"\n\u001b[0;32m---> 25\u001b[0;31m \u001b[0mvjp\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mans\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_make_vjp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfun\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 26\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mvspace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mans\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msize\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 27\u001b[0m raise TypeError(\"Grad only applies to real scalar-output functions. \"\n", "\u001b[0;32m~/anaconda3/lib/python3.8/site-packages/autograd/core.py\u001b[0m in \u001b[0;36mmake_vjp\u001b[0;34m(fun, x)\u001b[0m\n\u001b[1;32m 8\u001b[0m \u001b[0;32mdef\u001b[0m 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\u001b[0mf9\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0;31m# Assume a is an array with 2 elements\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 4\u001b[0m \u001b[0mb\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marray\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1.0\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m2.0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 5\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 6\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 7\u001b[0m \u001b[0mf9_grad\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mgrad\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf9\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", - "\u001b[0;31mAttributeError\u001b[0m: 'ArrayBox' object has no attribute 'dot'" + "\u001b[0;32m/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47735/1122558214.py\u001b[0m in \u001b[0;36mf8\u001b[0;34m(x)\u001b[0m\n\u001b[1;32m 2\u001b[0m \u001b[0;32mfrom\u001b[0m \u001b[0mautograd\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mgrad\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 3\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mf8\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0;31m# Assume x is an array\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 4\u001b[0;31m \u001b[0mx\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m3\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 5\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 6\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;31mTypeError\u001b[0m: 'ArrayBox' object does not support item assignment" ] } ], + "source": [ + "import autograd.numpy as np\n", + "from autograd import grad\n", + "def f8(x): # Assume x is an array\n", + " x[2] = 3\n", + " return x*2\n", + "\n", + "f8_grad = grad(f8)\n", + "\n", + "x = 8.4\n", + "\n", + "print(\"The derivative of f8 is:\",f8_grad(x))" + ] + }, + { + "cell_type": "markdown", + "id": "7a13b21d", + "metadata": { + "editable": true + }, + "source": [ + "Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible." + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "id": "19c7502b", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], "source": [ "import autograd.numpy as np\n", "from autograd import grad\n", @@ -3032,7 +3629,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "4450885d", + "metadata": { + "editable": true + }, "source": [ "Here we are told that the 'dot' function does not belong to Autograd's\n", "version of a Numpy array. To overcome this, an alternative syntax\n", @@ -3041,7 +3641,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 24, + "id": "013fc7f8", "metadata": { "collapsed": false, "editable": true @@ -3066,14 +3667,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "8d360f7d", + "metadata": { + "editable": true + }, "source": [ "The documentation recommends to avoid inplace operations such as" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 25, + "id": "e29a24eb", "metadata": { "collapsed": false, "editable": true @@ -3088,9 +3693,229 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ad8fbbb7", + "metadata": { + "editable": true + }, "source": [ - "More examples will be added, in particular how to compare autograd with own codes for the gradients." + "## Using Autograd with OLS\n", + "\n", + "We conclude the part on optmization by showing how we can make codes\n", + "for linear regression and logistic regression using **autograd**. The\n", + "first example shows results with ordinary leats squares." + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "id": "904f65dc", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "# Using Autograd to calculate gradients for OLS\n", + "from random import random, seed\n", + "import numpy as np\n", + "import autograd.numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from autograd import grad\n", + "\n", + "def CostOLS(beta):\n", + " return (1.0/n)*np.sum((y-X @ beta)**2)\n", + "\n", + "n = 100\n", + "x = 2*np.random.rand(n,1)\n", + "y = 4+3*x+np.random.randn(n,1)\n", + "\n", + "X = np.c_[np.ones((n,1)), x]\n", + "XT_X = X.T @ X\n", + "theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)\n", + "print(\"Own inversion\")\n", + "print(theta_linreg)\n", + "# Hessian matrix\n", + "H = (2.0/n)* XT_X\n", + "EigValues, EigVectors = np.linalg.eig(H)\n", + "print(f\"Eigenvalues of Hessian Matrix:{EigValues}\")\n", + "\n", + "theta = np.random.randn(2,1)\n", + "eta = 1.0/np.max(EigValues)\n", + "Niterations = 1000\n", + "# define the gradient\n", + "training_gradient = grad(CostOLS)\n", + "\n", + "for iter in range(Niterations):\n", + " gradients = training_gradient(theta)\n", + " theta -= eta*gradients\n", + "print(\"theta from own gd\")\n", + "print(theta)\n", + "\n", + "xnew = np.array([[0],[2]])\n", + "Xnew = np.c_[np.ones((2,1)), xnew]\n", + "ypredict = Xnew.dot(theta)\n", + "ypredict2 = Xnew.dot(theta_linreg)\n", + "\n", + "plt.plot(xnew, ypredict, \"r-\")\n", + "plt.plot(xnew, ypredict2, \"b-\")\n", + "plt.plot(x, y ,'ro')\n", + "plt.axis([0,2.0,0, 15.0])\n", + "plt.xlabel(r'$x$')\n", + "plt.ylabel(r'$y$')\n", + "plt.title(r'Random numbers ')\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "ce338980", + "metadata": { + "editable": true + }, + "source": [ + "### Including Stochastic Gradient Descent with Autograd\n", + "\n", + "In this code we include the stochastic gradient descent approach discussed above. Note here that we specify which argument we are taking the derivative with respect to when using **autograd**." + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "id": "de261f10", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "# Using Autograd to calculate gradients using SGD\n", + "# OLS example\n", + "from random import random, seed\n", + "import numpy as np\n", + "import autograd.numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from autograd import grad\n", + "\n", + "# Note change from previous example\n", + "def CostOLS(y,X,theta):\n", + " return np.sum((y-X @ theta)**2)\n", + "\n", + "n = 100\n", + "x = 2*np.random.rand(n,1)\n", + "y = 4+3*x+np.random.randn(n,1)\n", + "\n", + "X = np.c_[np.ones((n,1)), x]\n", + "XT_X = X.T @ X\n", + "theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)\n", + "print(\"Own inversion\")\n", + "print(theta_linreg)\n", + "# Hessian matrix\n", + "H = (2.0/n)* XT_X\n", + "EigValues, EigVectors = np.linalg.eig(H)\n", + "print(f\"Eigenvalues of Hessian Matrix:{EigValues}\")\n", + "\n", + "theta = np.random.randn(2,1)\n", + "eta = 1.0/np.max(EigValues)\n", + "Niterations = 1000\n", + "\n", + "# Note that we request the derivative wrt third argument (theta, 2 here)\n", + "training_gradient = grad(CostOLS,2)\n", + "\n", + "for iter in range(Niterations):\n", + " gradients = (1.0/n)*training_gradient(y, X, theta)\n", + " theta -= eta*gradients\n", + "print(\"theta from own gd\")\n", + "print(theta)\n", + "\n", + "xnew = np.array([[0],[2]])\n", + "Xnew = np.c_[np.ones((2,1)), xnew]\n", + "ypredict = Xnew.dot(theta)\n", + "ypredict2 = Xnew.dot(theta_linreg)\n", + "\n", + "plt.plot(xnew, ypredict, \"r-\")\n", + "plt.plot(xnew, ypredict2, \"b-\")\n", + "plt.plot(x, y ,'ro')\n", + "plt.axis([0,2.0,0, 15.0])\n", + "plt.xlabel(r'$x$')\n", + "plt.ylabel(r'$y$')\n", + "plt.title(r'Random numbers ')\n", + "plt.show()\n", + "\n", + "n_epochs = 50\n", + "M = 5 #size of each minibatch\n", + "m = int(n/M) #number of minibatches\n", + "t0, t1 = 5, 50\n", + "def learning_schedule(t):\n", + " return t0/(t+t1)\n", + "\n", + "theta = np.random.randn(2,1)\n", + "\n", + "for epoch in range(n_epochs):\n", + "# Can you figure out a better way of setting up the contributions to each batch?\n", + " for i in range(m):\n", + " random_index = M*np.random.randint(m)\n", + " xi = X[random_index:random_index+M]\n", + " yi = y[random_index:random_index+M]\n", + " gradients = (1.0/M)*training_gradient(yi, xi, theta)\n", + " eta = learning_schedule(epoch*m+i)\n", + " theta = theta - eta*gradients\n", + "print(\"theta from own sdg\")\n", + "print(theta)" + ] + }, + { + "cell_type": "markdown", + "id": "ccd8829e", + "metadata": { + "editable": true + }, + "source": [ + "### And Logistic Regression" + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "id": "cc5811d1", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "import autograd.numpy as np\n", + "from autograd import grad\n", + "\n", + "def sigmoid(x):\n", + " return 0.5 * (np.tanh(x / 2.) + 1)\n", + "\n", + "def logistic_predictions(weights, inputs):\n", + " # Outputs probability of a label being true according to logistic model.\n", + " return sigmoid(np.dot(inputs, weights))\n", + "\n", + "def training_loss(weights):\n", + " # Training loss is the negative log-likelihood of the training labels.\n", + " preds = logistic_predictions(weights, inputs)\n", + " label_probabilities = preds * targets + (1 - preds) * (1 - targets)\n", + " return -np.sum(np.log(label_probabilities))\n", + "\n", + "# Build a toy dataset.\n", + "inputs = np.array([[0.52, 1.12, 0.77],\n", + " [0.88, -1.08, 0.15],\n", + " [0.52, 0.06, -1.30],\n", + " [0.74, -2.49, 1.39]])\n", + "targets = np.array([True, True, False, True])\n", + "\n", + "# Define a function that returns gradients of training loss using Autograd.\n", + "training_gradient_fun = grad(training_loss)\n", + "\n", + "# Optimize weights using gradient descent.\n", + "weights = np.array([0.0, 0.0, 0.0])\n", + "print(\"Initial loss:\", training_loss(weights))\n", + "for i in range(100):\n", + " weights -= training_gradient_fun(weights) * 0.01\n", + "\n", + "print(\"Trained loss:\", training_loss(weights))" ] } ], @@ -3109,5 +3934,5 @@ } }, "nbformat": 4, - "nbformat_minor": 4 + "nbformat_minor": 5 } \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapteroptimization.py b/doc/LectureNotes/_build/jupyter_execute/chapteroptimization.py index e368e574f..cb4459c01 100644 --- a/doc/LectureNotes/_build/jupyter_execute/chapteroptimization.py +++ b/doc/LectureNotes/_build/jupyter_execute/chapteroptimization.py @@ -1,6 +1,9 @@ #!/usr/bin/env python # coding: utf-8 +# + # # Optimization, the central part of any Machine Learning algortithm # # Almost every problem in machine learning and data science starts with @@ -12,9 +15,6 @@ # analytically, however this is not possible in general and we must use # some approximative/numerical method to compute the minimum. # -# -# -# # In our discussion on Logistic Regression we studied the # case of # two classes, with $y_i$ either @@ -31,8 +31,6 @@ # where $\boldsymbol{\beta}$ are the weights we wish to extract from data, in our case $\beta_0$ and $\beta_1$. # -# -# # Our compact equations used a definition of a vector $\boldsymbol{y}$ with $n$ # elements $y_i$, an $n\times p$ matrix $\boldsymbol{X}$ which contains the # $x_i$ values and a vector $\boldsymbol{p}$ of fitted probabilities @@ -52,8 +50,6 @@ # This defines what is called the Hessian matrix. # -# -# # If we can set up these equations, Newton-Raphson's iterative method is normally the method of choice. It requires however that we can compute in an efficient way the matrices that define the first and second derivatives. # # Our iterative scheme is then given by @@ -72,7 +68,6 @@ # # If we can compute these matrices, in particular the Hessian, the above is often the easiest method to implement. # -# # Let us quickly remind ourselves how we derive the above method. # # Perhaps the most celebrated of all one-dimensional root-finding @@ -83,8 +78,6 @@ # numerically and/or your function is not of the smooth type, we # normally discourage the use of this method. # -# -# # The Newton-Raphson formula consists geometrically of extending the # tangent line at a current point until it crosses zero, then setting # the next guess to the abscissa of that zero-crossing. The mathematics @@ -130,9 +123,6 @@ # guess near such a local extremum, so that the first derivative nearly # vanishes, then Newton-Raphson may fail totally # -# -# -# # Newton's method can be generalized to systems of several non-linear equations # and variables. Consider the case with two equations @@ -183,11 +173,8 @@ # arise in case $\boldsymbol{J}$ is nearly singular. # # It is rather straightforward to extend the above scheme to systems of -# more than two non-linear equations. In our case, the Jacobian matrix is given by the Hessian that represents the second derivative of cost function. -# -# -# -# +# more than two non-linear equations. In our case, the Jacobian matrix is given by the Hessian that represents the second derivative of cost function. + # ## Steepest descent # # The basic idea of gradient descent is @@ -207,7 +194,6 @@ # F(\mathbf{x}_k)$. This means that for a sufficiently small $\gamma_k$ # we are always moving towards smaller function values, i.e a minimum. # -# # The previous observation is the basis of the method of steepest # descent, which is also referred to as just gradient descent (GD). One # starts with an initial guess $\mathbf{x}_0$ for a minimum of $F$ and @@ -220,7 +206,6 @@ # The parameter $\gamma_k$ is often referred to as the step length or # the learning rate within the context of Machine Learning. # -# # Ideally the sequence $\{\mathbf{x}_k \}_{k=0}$ converges to a global # minimum of the function $F$. In general we do not know if we are in a # global or local minimum. In the special case when $F$ is a convex @@ -239,9 +224,6 @@ # Note that the gradient is a function of $\mathbf{x} = # (x_1,\cdots,x_n)$ which makes it expensive to compute numerically. # -# -# -# # The gradient descent method # is sensitive to the choice of learning rate $\gamma_k$. This is due # to the fact that we are only guaranteed that $F(\mathbf{x}_{k+1}) \leq @@ -253,9 +235,7 @@ # Many of these shortcomings can be alleviated by introducing # randomness. One such method is that of Stochastic Gradient Descent # (SGD), see below. -# -# -# + # ## Convex functions # # Ideally we want our cost/loss function to be convex(concave). @@ -270,11 +250,8 @@ # $\mathbb{R}$. Examples of convex sets of $\mathbb{R}^2$ are the # regular polygons (triangles, rectangles, pentagons, etc...). # -# -# # **Convex function**: Let $X \subset \mathbb{R}^n$ be a convex set. Assume that the function $f: X \rightarrow \mathbb{R}$ is continuous, then $f$ is said to be convex if $$f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2) $$ for all $x_1, x_2 \in X$ and for all $t \in [0,1]$. If $\leq$ is replaced with a strict inequaltiy in the definition, we demand $x_1 \neq x_2$ and $t\in(0,1)$ then $f$ is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting $f(x_1)$ and $f(x_2)$, the value of the function on the interval $[x_1,x_2]$ is always below the line as illustrated below. # -# # In the following we state first and second-order conditions which # ensures convexity of a function $f$. We write $D_f$ to denote the # domain of $f$, i.e the subset of $R^n$ where $f$ is defined. For more @@ -291,8 +268,6 @@ # make a drawing of $f(x) = x^2+1$ and draw the tangent line to $f(x)$ and # note that it is always below the graph. # -# -# # **Second order condition.** # # Assume that $f$ is twice @@ -302,12 +277,8 @@ # single-variable function this reduces to $f''(x) \geq 0$. Geometrically this means that $f$ has nonnegative curvature # everywhere. # -# -# # This condition is particularly useful since it gives us an procedure for determining if the function under consideration is convex, apart from using the definition. # -# -# # The next result is of great importance to us and the reason why we are # going on about convex functions. In machine learning we frequently # have to minimize a loss/cost function in order to find the best @@ -324,11 +295,8 @@ # is minimal, where $f$ is convex and differentiable. Then, any point # $x^*$ that satisfies $\nabla f(x^*) = 0$ is a global minimum. # -# -# # This result means that if we know that the cost/loss function is convex and we are able to find a minimum, we are guaranteed that it is a global minimum. -# -# + # ### Some simple problems # # 1. Show that $f(x)=x^2$ is convex for $x \in \mathbb{R}$ using the definition of convexity. Hint: If you re-write the definition, $f$ is convex if the following holds for all $x,y \in D_f$ and any $\lambda \in [0,1]$ $\lambda f(x)+(1-\lambda)f(y)-f(\lambda x + (1-\lambda) y ) \geq 0$. @@ -339,7 +307,6 @@ # # * $g(x) = -\ln(x)$ is convex for $x \in (0,\infty)$. # -# # 3. Let $f(x) = x^2$ and $g(x) = e^x$. Show that $f(g(x))$ and $g(f(x))$ is convex for $x \in \mathbb{R}$. Also show that if $f(x)$ is any convex function than $h(x) = e^{f(x)}$ is convex. # # 4. A norm is any function that satisfy the following properties @@ -350,13 +317,10 @@ # # * $f(x) \leq 0$ for all $x \in \mathbb{R}^n$ with equality if and only if $x = 0$ # -# # Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this). -# -# + # ## Standard steepest descent # -# # Before we proceed, we would like to discuss the approach called the # **standard Steepest descent** (different from the above steepest descent discussion), which again leads to us having to be able # to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG). @@ -381,8 +345,6 @@ # # When we have found the exact solution, $\boldsymbol{r}=0$. # -# -# # The residual is zero when we reach the minimum of the quadratic equation # $$ @@ -392,7 +354,6 @@ # with the constraint that the matrix $\boldsymbol{A}$ is positive definite and # symmetric. This defines also the Hessian and we want it to be positive definite. # -# # We denote the initial guess for $\boldsymbol{x}$ as $\boldsymbol{x}_0$. # We can assume without loss of generality that @@ -408,7 +369,6 @@ # instead. # -# # One can show that the solution $\boldsymbol{x}$ is also the unique minimizer of the quadratic form # $$ @@ -426,7 +386,6 @@ # and # $\boldsymbol{x}_0=0$ it is equal $-\boldsymbol{b}$. # -# # We can compute the residual iteratively as # $$ @@ -728,7 +687,6 @@ y = 4+3*x+np.random.randn(m,1) # and we want to find $\beta$ such that $C(\beta)$ is minimized. # -# # Computing $\partial C(\beta) / \partial \beta_0$ and $\partial C(\beta) / \partial \beta_1$ we can show that the gradient can be written as # $$ @@ -739,7 +697,6 @@ y = 4+3*x+np.random.randn(m,1) # where $X$ is the design matrix defined above. # -# # The Hessian matrix of $C(\beta)$ is given by # $$ @@ -751,8 +708,6 @@ y = 4+3*x+np.random.randn(m,1) # This result implies that $C(\beta)$ is a convex function since the matrix $X^T X$ always is positive semi-definite. # -# -# # We can now write a program that minimizes $C(\beta)$ using the gradient descent method with a constant learning rate $\gamma$ according to # $$ @@ -766,8 +721,6 @@ y = 4+3*x+np.random.randn(m,1) # And finally we can compare our solution for $\beta$ with the analytic result given by # $\beta= (X^TX)^{-1} X^T \mathbf{y}$. # -# -# # Here is our simple example # In[7]: @@ -925,8 +878,30 @@ plt.show() # * **GD treats all directions in parameter space uniformly.** Another major drawback of GD is that unlike Newton's method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive. # # * GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points. + +# ## Stochastic Gradient Descent (SGD) # -# ## Stochastic Gradient Descent +# In stochastic gradient descent, the extreme case is the case where we +# have only one batch, that is we include the whole data set. +# +# This process is called Stochastic Gradient +# Descent (SGD) (or also sometimes on-line gradient descent). This is +# relatively less common to see because in practice due to vectorized +# code optimizations it can be computationally much more efficient to +# evaluate the gradient for 100 examples, than the gradient for one +# example 100 times. Even though SGD technically refers to using a +# single example at a time to evaluate the gradient, you will hear +# people use the term SGD even when referring to mini-batch gradient +# descent (i.e. mentions of MGD for “Minibatch Gradient Descent”, or BGD +# for “Batch gradient descent” are rare to see), where it is usually +# assumed that mini-batches are used. The size of the mini-batch is a +# hyperparameter but it is not very common to cross-validate or bootstrap it. It is +# usually based on memory constraints (if any), or set to some value, +# e.g. 32, 64 or 128. We use powers of 2 in practice because many +# vectorized operation implementations work faster when their inputs are +# sized in powers of 2. +# +# In our notes with SGD we mean stochastic gradient descent with mini-batches. # # Stochastic gradient descent (SGD) and variants thereof address some of # the shortcomings of the Gradient descent method discussed above. @@ -954,8 +929,6 @@ plt.show() # minibatches. We denote these minibatches by $B_k$ where # $k=1,\cdots,n/M$. # -# -# # As an example, suppose we have $10$ data points $(\mathbf{x}_1,\cdots, \mathbf{x}_{10})$ # and we choose to have $M=5$ minibathces, # then each minibatch contains two data points. In particular we have @@ -995,7 +968,7 @@ plt.show() import numpy as np n = 100 #100 datapoints -M = 5 #size of each minibatch +M = 5 #size of each mini-batche m = int(n/M) #number of minibatches n_epochs = 10 #number of epochs @@ -1016,8 +989,6 @@ for epoch in range(1,n_epochs+1): # cheaper since we sum over the datapoints in the $k-th$ minibatch and not # all $n$ datapoints. # -# -# # A natural question is when do we stop the search for a new minimum? # One possibility is to compute the full gradient after a given number # of epochs and check if the norm of the gradient is smaller than some @@ -1029,8 +1000,6 @@ for epoch in range(1,n_epochs+1): # compare the values of the cost function and keep the $\beta$ that # gave the lowest value. # -# -# # Another approach is to let the step length $\gamma_j$ depend on the # number of epochs in such a way that it becomes very small after a # reasonable time such that we do not move at all. @@ -1072,39 +1041,41 @@ for epoch in range(1,n_epochs+1): print("gamma_j after %d epochs: %g" % (n_epochs,gamma_j)) +# We note that we have defined several hyperparameters. These are now the number of epochs, the number of mini-batches and the parameters $t_0$ and $t_1$. + # ### Program for stochastic gradient # In[12]: +# Importing various packages # Importing various packages from math import exp, sqrt from random import random, seed import numpy as np import matplotlib.pyplot as plt -from sklearn.linear_model import SGDRegressor -m = 100 -x = 2*np.random.rand(m,1) -y = 4+3*x+np.random.randn(m,1) +n = 100 +x = 2*np.random.rand(n,1) +y = 4+3*x+np.random.randn(n,1) -X = np.c_[np.ones((m,1)), x] +X = np.c_[np.ones((n,1)), x] +XT_X = X.T @ X theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y) print("Own inversion") print(theta_linreg) -sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1) -sgdreg.fit(x,y.ravel()) -print("sgdreg from scikit") -print(sgdreg.intercept_, sgdreg.coef_) - +# Hessian matrix +H = (2.0/n)* XT_X +EigValues, EigVectors = np.linalg.eig(H) +print(f"Eigenvalues of Hessian Matrix:{EigValues}") theta = np.random.randn(2,1) -eta = 0.1 +eta = 1.0/np.max(EigValues) Niterations = 1000 for iter in range(Niterations): - gradients = 2.0/m*X.T @ ((X @ theta)-y) + gradients = 2.0/n*X.T @ ((X @ theta)-y) theta -= eta*gradients print("theta from own gd") print(theta) @@ -1114,8 +1085,9 @@ Xnew = np.c_[np.ones((2,1)), xnew] ypredict = Xnew.dot(theta) ypredict2 = Xnew.dot(theta_linreg) - n_epochs = 50 +M = 5 #size of each minibatch +m = int(n/M) #number of minibatches t0, t1 = 5, 50 def learning_schedule(t): return t0/(t+t1) @@ -1123,16 +1095,20 @@ def learning_schedule(t): theta = np.random.randn(2,1) for epoch in range(n_epochs): +# Can you figure out a better way of setting up the contributions to each batch? for i in range(m): - random_index = np.random.randint(m) - xi = X[random_index:random_index+1] - yi = y[random_index:random_index+1] - gradients = 2 * xi.T @ ((xi @ theta)-yi) + random_index = M*np.random.randint(m) + xi = X[random_index:random_index+M] + yi = y[random_index:random_index+M] + gradients = (2.0/M)* xi.T @ ((xi @ theta)-yi) eta = learning_schedule(epoch*m+i) theta = theta - eta*gradients print("theta from own sdg") print(theta) + + + plt.plot(xnew, ypredict, "r-") plt.plot(xnew, ypredict2, "b-") plt.plot(x, y ,'ro') @@ -1143,6 +1119,11 @@ plt.title(r'Random numbers ') plt.show() +# In the above code, we have use replacement in setting up the +# mini-batches. The discussion +# [here](https://sebastianraschka.com/faq/docs/sgd-methods.html) may be +# useful. More material will be added later. + # ## Momentum based GD # # The stochastic gradient descent (SGD) is almost always used with a @@ -1181,8 +1162,6 @@ plt.show() # where we have defined $\Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\boldsymbol{\theta}_{t-1}$. # -# -# # Let us try to get more intuition from these equations. It is helpful # to consider a simple physical analogy with a particle of mass $m$ # moving in a viscous medium with drag coefficient $\mu$ and potential @@ -1256,8 +1235,6 @@ plt.show() # One of the major advantages of NAG is that it allows for the use of a larger learning rate than GDM for the same choice of $\gamma$. # -# -# # In stochastic gradient descent, with and without momentum, we still # have to specify a schedule for tuning the learning rates $\eta_t$ # as a function of time. As discussed in the context of Newton's @@ -1276,10 +1253,9 @@ plt.show() # # Recently, a number of methods have been introduced that accomplish # this by tracking not only the gradient, but also the second moment of -# the gradient. These methods include AdaGrad, AdaDelta, RMS-Prop, and +# the gradient. These methods include AdaGrad, AdaDelta, Root Mean Squared Propagation (RMS-Prop), and # ADAM. -# -# + # ### RMS prop # # In RMS prop, in addition to keeping a running average of the first @@ -1314,7 +1290,7 @@ plt.show() # directions where the norm of the gradient is consistently large. This # greatly speeds up the convergence by allowing us to use a larger # learning rate for flat directions. -# + # ### ADAM optimizer # # A related algorithm is the ADAM optimizer. In ADAM, we keep a running @@ -1393,7 +1369,7 @@ plt.show() # * **Monitor the out-of-sample performance.** Always monitor the performance of your model on a validation set (a small portion of the training data that is held out of the training process to serve as a proxy for the test set. If the validation error starts increasing, then the model is beginning to overfit. Terminate the learning process. This *early stopping* significantly improves performance in many settings. # # * **Adaptive optimization methods don't always have good generalization.** Recent studies have shown that adaptive methods such as ADAM, RMSPorp, and AdaGrad tend to have poor generalization compared to SGD or SGD with momentum, particularly in the high-dimensional limit (i.e. the number of parameters exceeds the number of data points). Although it is not clear at this stage why these methods perform so well in training deep neural networks, simpler procedures like properly-tuned SGD may work as well or better in these applications. -# + # ## Automatic differentiation # # [Automatic differentiation (AD)](https://en.wikipedia.org/wiki/Automatic_differentiation), @@ -1421,8 +1397,6 @@ plt.show() # while numerical differentiation can introduce round-off errors in the # discretization process and cancellation # -# -# # Python has tools for so-called **automatic differentiation**. # Consider the following example @@ -1619,35 +1593,38 @@ x = 2.7 print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x))) -# 1 -# 8 -# -# < -# < -# < -# ! -# ! -# C -# O -# D -# E -# _ -# B -# L -# O -# C -# K -# -# -# p -# y -# c -# o -# d - # In[19]: +import autograd.numpy as np +from autograd import grad +def f6_for(x): + val = 0 + for i in range(10): + val = val + x**i + return val + +def f6_while(x): + val = 0 + i = 0 + while i < 10: + val = val + x**i + i = i + 1 + return val + +f6_for_grad = grad(f6_for) +f6_while_grad = grad(f6_while) + +x = 0.5 + +# Print the computed derivaties of f6_for and f6_while +print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x))) +print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x))) + + +# In[20]: + + import autograd.numpy as np from autograd import grad # Both of the functions are implementation of the sum: sum(x**i) for i = 0, ..., 9 @@ -1659,7 +1636,7 @@ for i in range(10): print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical)) -# In[20]: +# In[21]: import autograd.numpy as np @@ -1693,30 +1670,29 @@ print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical)) # Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input. # +# Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd. # -# Autograd supports many features. However, there are some functions that are not supported (yet) by Autograd. -# -# Assigning a value to the variable being differentiated with respect to is an example thereof. +# Assigning a value to the variable being differentiated with respect to -# In[21]: +# In[22]: -#import autograd.numpy as np -#from autograd import grad -#def f8(x): # Assume x is an array -# x[2] = 3 -# return x*2 +import autograd.numpy as np +from autograd import grad +def f8(x): # Assume x is an array + x[2] = 3 + return x*2 -#f8_grad = grad(f8) +f8_grad = grad(f8) -#x = 8.4 +x = 8.4 -#print("The derivative of f8 is:",f8_grad(x)) +print("The derivative of f8 is:",f8_grad(x)) # Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible. -# In[22]: +# In[23]: import autograd.numpy as np @@ -1736,7 +1712,7 @@ print("The derivative of f9 is:",f9_grad(x)) # version of a Numpy array. To overcome this, an alternative syntax # which also computed the dot product can be used: -# In[ ]: +# In[24]: import autograd.numpy as np @@ -1757,7 +1733,7 @@ print("The gradient of f9 is:",f9_alternative_grad(x)) # The documentation recommends to avoid inplace operations such as -# In[ ]: +# In[25]: a += b @@ -1766,4 +1742,184 @@ a*= b a /=b -# More examples will be added, in particular how to compare autograd with own codes for the gradients. +# ## Using Autograd with OLS +# +# We conclude the part on optmization by showing how we can make codes +# for linear regression and logistic regression using **autograd**. The +# first example shows results with ordinary leats squares. + +# In[26]: + + +# Using Autograd to calculate gradients for OLS +from random import random, seed +import numpy as np +import autograd.numpy as np +import matplotlib.pyplot as plt +from autograd import grad + +def CostOLS(beta): + return (1.0/n)*np.sum((y-X @ beta)**2) + +n = 100 +x = 2*np.random.rand(n,1) +y = 4+3*x+np.random.randn(n,1) + +X = np.c_[np.ones((n,1)), x] +XT_X = X.T @ X +theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y) +print("Own inversion") +print(theta_linreg) +# Hessian matrix +H = (2.0/n)* XT_X +EigValues, EigVectors = np.linalg.eig(H) +print(f"Eigenvalues of Hessian Matrix:{EigValues}") + +theta = np.random.randn(2,1) +eta = 1.0/np.max(EigValues) +Niterations = 1000 +# define the gradient +training_gradient = grad(CostOLS) + +for iter in range(Niterations): + gradients = training_gradient(theta) + theta -= eta*gradients +print("theta from own gd") +print(theta) + +xnew = np.array([[0],[2]]) +Xnew = np.c_[np.ones((2,1)), xnew] +ypredict = Xnew.dot(theta) +ypredict2 = Xnew.dot(theta_linreg) + +plt.plot(xnew, ypredict, "r-") +plt.plot(xnew, ypredict2, "b-") +plt.plot(x, y ,'ro') +plt.axis([0,2.0,0, 15.0]) +plt.xlabel(r'$x$') +plt.ylabel(r'$y$') +plt.title(r'Random numbers ') +plt.show() + + +# ### Including Stochastic Gradient Descent with Autograd +# +# In this code we include the stochastic gradient descent approach discussed above. Note here that we specify which argument we are taking the derivative with respect to when using **autograd**. + +# In[27]: + + +# Using Autograd to calculate gradients using SGD +# OLS example +from random import random, seed +import numpy as np +import autograd.numpy as np +import matplotlib.pyplot as plt +from autograd import grad + +# Note change from previous example +def CostOLS(y,X,theta): + return np.sum((y-X @ theta)**2) + +n = 100 +x = 2*np.random.rand(n,1) +y = 4+3*x+np.random.randn(n,1) + +X = np.c_[np.ones((n,1)), x] +XT_X = X.T @ X +theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y) +print("Own inversion") +print(theta_linreg) +# Hessian matrix +H = (2.0/n)* XT_X +EigValues, EigVectors = np.linalg.eig(H) +print(f"Eigenvalues of Hessian Matrix:{EigValues}") + +theta = np.random.randn(2,1) +eta = 1.0/np.max(EigValues) +Niterations = 1000 + +# Note that we request the derivative wrt third argument (theta, 2 here) +training_gradient = grad(CostOLS,2) + +for iter in range(Niterations): + gradients = (1.0/n)*training_gradient(y, X, theta) + theta -= eta*gradients +print("theta from own gd") +print(theta) + +xnew = np.array([[0],[2]]) +Xnew = np.c_[np.ones((2,1)), xnew] +ypredict = Xnew.dot(theta) +ypredict2 = Xnew.dot(theta_linreg) + +plt.plot(xnew, ypredict, "r-") +plt.plot(xnew, ypredict2, "b-") +plt.plot(x, y ,'ro') +plt.axis([0,2.0,0, 15.0]) +plt.xlabel(r'$x$') +plt.ylabel(r'$y$') +plt.title(r'Random numbers ') +plt.show() + +n_epochs = 50 +M = 5 #size of each minibatch +m = int(n/M) #number of minibatches +t0, t1 = 5, 50 +def learning_schedule(t): + return t0/(t+t1) + +theta = np.random.randn(2,1) + +for epoch in range(n_epochs): +# Can you figure out a better way of setting up the contributions to each batch? + for i in range(m): + random_index = M*np.random.randint(m) + xi = X[random_index:random_index+M] + yi = y[random_index:random_index+M] + gradients = (1.0/M)*training_gradient(yi, xi, theta) + eta = learning_schedule(epoch*m+i) + theta = theta - eta*gradients +print("theta from own sdg") +print(theta) + + +# ### And Logistic Regression + +# In[28]: + + +import autograd.numpy as np +from autograd import grad + +def sigmoid(x): + return 0.5 * (np.tanh(x / 2.) + 1) + +def logistic_predictions(weights, inputs): + # Outputs probability of a label being true according to logistic model. + return sigmoid(np.dot(inputs, weights)) + +def training_loss(weights): + # Training loss is the negative log-likelihood of the training labels. + preds = logistic_predictions(weights, inputs) + label_probabilities = preds * targets + (1 - preds) * (1 - targets) + return -np.sum(np.log(label_probabilities)) + +# Build a toy dataset. +inputs = np.array([[0.52, 1.12, 0.77], + [0.88, -1.08, 0.15], + [0.52, 0.06, -1.30], + [0.74, -2.49, 1.39]]) +targets = np.array([True, True, False, True]) + +# Define a function that returns gradients of training loss using Autograd. +training_gradient_fun = grad(training_loss) + +# Optimize weights using gradient descent. +weights = np.array([0.0, 0.0, 0.0]) +print("Initial loss:", training_loss(weights)) +for i in range(100): + weights -= training_gradient_fun(weights) * 0.01 + +print("Trained loss:", training_loss(weights)) + diff --git a/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_123_1.png b/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_123_1.png new file mode 100644 index 000000000..fcf4c6806 Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_123_1.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_132_1.png b/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_132_1.png new file mode 100644 index 000000000..bca109752 Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_132_1.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_148_1.png b/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_148_1.png new file mode 100644 index 000000000..ac481f86e Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_148_1.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_188_0.png b/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_188_0.png new file mode 100644 index 000000000..1e983bdb8 Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_188_0.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_61_2.png b/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_61_2.png new file mode 100644 index 000000000..7d78856da Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_61_2.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_63_0.png b/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_63_0.png new file mode 100644 index 000000000..b1441ecf4 Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_63_0.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_69_1.png b/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_69_1.png new file mode 100644 index 000000000..eeca55933 Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_69_1.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/clustering.ipynb b/doc/LectureNotes/_build/jupyter_execute/clustering.ipynb new file mode 100644 index 000000000..bf67f2669 --- /dev/null +++ b/doc/LectureNotes/_build/jupyter_execute/clustering.ipynb @@ -0,0 +1,718 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "c3edcae5", + "metadata": { + "editable": true + }, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "id": "4102577e", + "metadata": { + "editable": true + }, + "source": [ + "# Clustering and Unsupervised Learning\n", + "\n", + "In general terms cluster analysis, or clustering, is the task of grouping a\n", + "data-set into different distinct categories based on some measure of equality of\n", + "the data. This measure is often referred to as a **metric** or **similarity\n", + "measure** in the literature (note: sometimes we deal with a **dissimilarity\n", + "measure** instead). Usually, these metrics are formulated as some kind of\n", + "distance function between points in a high-dimensional space.\n", + "\n", + "The simplest, and also the most\n", + "common is the **Euclidean distance**.\n", + "\n", + "The simplest of all clustering algorithms is the **k-means algorithm**\n", + ", sometimes also referred to as *Lloyds algorithm*. It is the simplest and also\n", + "the most common. From its simplicity it obtains both strengths and weaknesses.\n", + "These will be discussed in more detail later. The $k$-means algorithm is a\n", + "**centroid based** clustering algorithm.\n", + "\n", + "Assume, we are given $n$ data points and we wish to split the data into $K < n$\n", + "different categories, or clusters. We label each cluster by an integer" + ] + }, + { + "cell_type": "markdown", + "id": "0deb3255", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "k\\in\\{1, \\cdots, K \\}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "cfd8fe00", + "metadata": { + "editable": true + }, + "source": [ + "In the basic k-means algorithm each point is assigned to only\n", + "one cluster $k$, and these assignments are *non-injective* i.e. many-to-one. We\n", + "can think of these mappings as an encoder $k = C(i)$, which assigns the $i$-th\n", + "data-point $\\bf x_i$ to the $k$-th cluster.\n", + "\n", + "$k$-means algorithm in words:\n", + "1. We start with guesses / random initializations of our $k$ cluster centers/centroids\n", + "\n", + "2. For each centroid the points that are most similar are identified\n", + "\n", + "3. Then we move / replace each centroid with a coordinate average of all the points that were assigned to that centroid.\n", + "\n", + "4. Iterate 2-3 until the centroids no longer move (to some tolerance)\n", + "\n", + "We assume we have $n$ data-points" + ] + }, + { + "cell_type": "markdown", + "id": "a29b7459", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\\label{eq:kmeanspoints} \\tag{1}\n", + " \\boldsymbol{x_i} = \\{x_{i, 1}, \\cdots, x_{i, p}\\}\\in\\mathbb{R}^p.\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "98f9e37b", + "metadata": { + "editable": true + }, + "source": [ + "which we wish to group into $K < n$ clusters. For our dissimilarity measure we\n", + "use the *squared Euclidean distance*" + ] + }, + { + "cell_type": "markdown", + "id": "d3c32572", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\\label{eq:squaredeuclidean} \\tag{2}\n", + " d(\\boldsymbol{x_i}, \\boldsymbol{x_i'}) = \\sum_{j=1}^p(x_{ij} - x_{i'j})^2\n", + " = ||\\boldsymbol{x_i} - \\boldsymbol{x_{i'}}||^2\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "29d24648", + "metadata": { + "editable": true + }, + "source": [ + "We define the so called *within-cluster point scatter* which gives us a\n", + "measure of how close each data point assigned to the same cluster tends to be to\n", + "the all the others." + ] + }, + { + "cell_type": "markdown", + "id": "fce5c797", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\\label{eq:withincluster} \\tag{3}\n", + " W(C) = \\frac{1}{2}\\sum_{k=1}^K\\sum_{C(i)=k}\n", + " \\sum_{C(i')=k}d(\\boldsymbol{x_i}, \\boldsymbol{x_{i'}}) =\n", + " \\sum_{k=1}^KN_k\\sum_{C(i)=k}||\\boldsymbol{x_i} - \\boldsymbol{\\overline{x_k}}||^2\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "674a26b7", + "metadata": { + "editable": true + }, + "source": [ + "where $\\boldsymbol{\\overline{x_k}}$ is the mean vector associated with the $k$-th\n", + "cluster, and $N_k = \\sum_{i=1}^nI(C(i) = k)$, where the $I()$ notation is\n", + "similar to the Kronecker delta (*Commonly used in statistics, it just means that\n", + "when $i = k$ we have the encoder $C(i)$*). In other words, the within-cluster\n", + "scatter measures the compactness of each cluster with respect to the data points\n", + "assigned to each cluster. This is the quantity that the $k$-means algorithm aims\n", + "to minimize. We refer to this quantity $W(C)$ as the within cluster scatter\n", + "because of its relation to the *total scatter*.\n", + "\n", + "We have" + ] + }, + { + "cell_type": "markdown", + "id": "f200e7ff", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\\label{eq:totalscatter} \\tag{4}\n", + " T = W(C) + B(C) = \\frac{1}{2}\\sum_{i=1}^n\n", + " \\sum_{i'=1}^nd(\\boldsymbol{x_i}, \\boldsymbol{x_{i'}})\n", + " = \\frac{1}{2}\\sum_{k=1}^K\\sum_{C(i)=k}\n", + " \\Big(\\sum_{C(i') = k}d(\\boldsymbol{x_i}, \\boldsymbol{x_{i'}})\n", + " + \\sum_{C(i')\\neq k}d(\\boldsymbol{x_i}, \\boldsymbol{x_{i'}})\\Big).\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "5471a94d", + "metadata": { + "editable": true + }, + "source": [ + "This is a quantity that is conserved throughout the $k$-means algorithm. It can\n", + "be thought of as the total amount of information in the data, and it is composed\n", + "of the aforementioned within-cluster scatter and the *between-cluster scatter*\n", + "$B(C)$. In methods such as principle component analysis the total scatter is not\n", + "conserved.\n", + "\n", + "Given a cluster mean $\\boldsymbol{m_k}$ we define the **total cluster variance**" + ] + }, + { + "cell_type": "markdown", + "id": "299a99ce", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\\label{eq:totalclustervariance} \\tag{5}\n", + " \\min_{C, \\{\\boldsymbol{m_k}\\}_1^K}\\sum_{k=1}^KN_k\\sum||\\boldsymbol{x_i} - \\boldsymbol{m_k}||^2\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "bdfa54ee", + "metadata": { + "editable": true + }, + "source": [ + "Now we have all the pieces necessary to formally revisit the $k$-means algorithm.\n", + "\n", + "The $k$-means clustering algorithm goes as follows \n", + "\n", + "1. For a given cluster assignment $C$, and $k$ cluster means $\\left\\{m_1, \\cdots, m_k\\right\\}$. We minimize the total cluster variance with respect to the cluster means $\\{m_k\\}$ yielding the means of the currently assigned clusters.\n", + "\n", + "2. Given a current set of $k$ means $\\{m_k\\}$ the total cluster variance is minimized by assigning each observation to the closest (current) cluster mean. That is $$C(i) = \\underset{1\\leq k\\leq K}{\\mathrm{argmin}} ||\\boldsymbol{x_i} - \\boldsymbol{m_k}||^2$$\n", + "\n", + "3. Steps 1 and 2 are repeated until the assignments do not change." + ] + }, + { + "cell_type": "markdown", + "id": "f5def86c", + "metadata": { + "editable": true + }, + "source": [ + "## Codes and Approaches\n", + "\n", + "1. Before we start we specify a number $k$ which is the number of clusters we want to try to separate our data into.\n", + "\n", + "2. We initially choose $k$ random data points in our data as our initial centroids, *or means* (this is where the name comes from).\n", + "\n", + "3. Assign each data point to their closest centroid, based on the squared Euclidean distance.\n", + "\n", + "4. For each of the $k$ cluster we update the centroid by calculating new mean values for all the data points in the cluster.\n", + "\n", + "5. Iteratively minimize the within cluster scatter by performing steps (3, 4) until the new assignments stop changing (can be to some tolerance) or until a maximum number of iterations have passed.\n", + "\n", + "Let us now program the most basic version of the algorithm using nothing but\n", + "Python with numpy arrays. This code is kept intentionally simple to gradually\n", + "progress our understanding. There is no vectorization of any kind, and even most\n", + "helper functions are not utilized.\n", + "\n", + "We need first a dataset to do our cluster analysis on. In our case\n", + "this is a plain *vanilla* data set using random numbers using a\n", + "Gaussian distribution." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "b0260188", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "%matplotlib inline\n", + "\n", + "import time\n", + "import numpy as np\n", + "import tensorflow as tf\n", + "from matplotlib import image\n", + "import matplotlib.pyplot as plt\n", + "from sklearn.cluster import KMeans\n", + "from IPython.display import display\n", + "\n", + "np.random.seed(2021)" + ] + }, + { + "cell_type": "markdown", + "id": "fe680e35", + "metadata": { + "editable": true + }, + "source": [ + "Next we define functions, for ease of use later, to generate Gaussians and to\n", + "set up our toy data set." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "9db2bbce", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [ + { + "data": { + 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+ "text/plain": [ + "
    " + ] + }, + "metadata": { + "filenames": { + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/clustering_17_0.png" + }, + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "def gaussian_points(dim=2, n_points=1000, mean_vector=np.array([0, 0]),\n", + " sample_variance=1):\n", + " \"\"\"\n", + " Very simple custom function to generate gaussian distributed point clusters\n", + " with variable dimension, number of points, means in each direction\n", + " (must match dim) and sample variance.\n", + "\n", + " Inputs:\n", + " dim (int)\n", + " n_points (int)\n", + " mean_vector (np.array) (where index 0 is x, index 1 is y etc.)\n", + " sample_variance (float)\n", + "\n", + " Returns:\n", + " data (np.array): with dimensions (dim x n_points)\n", + " \"\"\"\n", + "\n", + " mean_matrix = np.zeros(dim) + mean_vector\n", + " covariance_matrix = np.eye(dim) * sample_variance\n", + " data = np.random.multivariate_normal(mean_matrix, covariance_matrix,\n", + " n_points)\n", + " return data\n", + "\n", + "\n", + "\n", + "def generate_simple_clustering_dataset(dim=2, n_points=1000, plotting=True,\n", + " return_data=True):\n", + " \"\"\"\n", + " Toy model to illustrate k-means clustering\n", + " \"\"\"\n", + "\n", + " data1 = gaussian_points(mean_vector=np.array([5, 5]))\n", + " data2 = gaussian_points()\n", + " data3 = gaussian_points(mean_vector=np.array([1, 4.5]))\n", + " data4 = gaussian_points(mean_vector=np.array([5, 1]))\n", + " data = np.concatenate((data1, data2, data3, data4), axis=0)\n", + "\n", + " if plotting:\n", + " fig, ax = plt.subplots()\n", + " ax.scatter(data[:, 0], data[:, 1], alpha=0.2)\n", + " ax.set_title('Toy Model Dataset')\n", + " plt.show()\n", + "\n", + "\n", + " if return_data:\n", + " return data\n", + "\n", + "\n", + "data = generate_simple_clustering_dataset()" + ] + }, + { + "cell_type": "markdown", + "id": "c0bb8c76", + "metadata": { + "editable": true + }, + "source": [ + "With the above dataset we start\n", + "implementing the $k$-means algorithm." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "29a75065", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "\n", + "n_samples, dimensions = data.shape\n", + "n_clusters = 4\n", + "\n", + "# we randomly initialize our centroids\n", + "np.random.seed(2021)\n", + "centroids = data[np.random.choice(n_samples, n_clusters, replace=False), :]\n", + "distances = np.zeros((n_samples, n_clusters))\n", + "\n", + "# first we need to calculate the distance to each centroid from our data\n", + "for k in range(n_clusters):\n", + " for n in range(n_samples):\n", + " dist = 0\n", + " for d in range(dimensions):\n", + " dist += np.abs(data[n, d] - centroids[k, d])**2\n", + " distances[n, k] = dist\n", + "\n", + "# we initialize an array to keep track of to which cluster each point belongs\n", + "# the way we set it up here the index tracks which point and the value which\n", + "# cluster the point belongs to\n", + "cluster_labels = np.zeros(n_samples, dtype='int')\n", + "\n", + "# next we loop through our samples and for every point assign it to the cluster\n", + "# to which it has the smallest distance to\n", + "for n in range(n_samples):\n", + " # tracking variables (all of this is basically just an argmin)\n", + " smallest = 1e10\n", + " smallest_row_index = 1e10\n", + " for k in range(n_clusters):\n", + " if distances[n, k] < smallest:\n", + " smallest = distances[n, k]\n", + " smallest_row_index = k\n", + "\n", + " cluster_labels[n] = smallest_row_index" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "id": "9fae7fc9", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
    " + ] + }, + "metadata": { + "filenames": { + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/clustering_20_0.png" + }, + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "fig = plt.figure()\n", + "ax = fig.add_subplot()\n", + "unique_cluster_labels = np.unique(cluster_labels)\n", + "for i in unique_cluster_labels:\n", + " ax.scatter(data[cluster_labels == i, 0],\n", + " data[cluster_labels == i, 1],\n", + " label = i,\n", + " alpha = 0.2)\n", + " ax.scatter(centroids[:, 0], centroids[:, 1], c='black')\n", + "\n", + "ax.set_title(\"First Grouping of Points to Centroids\")\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "90d2a873", + "metadata": { + "editable": true + }, + "source": [ + "So what do we have so far? We have 'picked' $k$ centroids at random from our\n", + "data points. There are other ways of more intelligently choosing their\n", + "initializations, however for our purposes randomly is fine. Then we have\n", + "initialized an array 'distances' which holds the information of the distance,\n", + "*or dissimilarity*, of every point to of our centroids. Finally, we have\n", + "initialized an array 'cluster_labels' which according to our distances array\n", + "holds the information of to which centroid every point is assigned. This was the\n", + "first pass of our algorithm. Essentially, all we need to do now is repeat the\n", + "distance and assignment steps above until we have reached a desired convergence\n", + "or a maximum amount of iterations." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "id": "378c29fc", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Converged at iteration 5\n" + ] + } + ], + "source": [ + "\n", + "max_iterations = 100\n", + "tolerance = 1e-8\n", + "\n", + "for iteration in range(max_iterations):\n", + " prev_centroids = centroids.copy()\n", + " for k in range(n_clusters):\n", + " # this array will be used to update our centroid positions\n", + " vector_mean = np.zeros(dimensions)\n", + " mean_divisor = 0\n", + " for n in range(n_samples):\n", + " if cluster_labels[n] == k:\n", + " vector_mean += data[n, :]\n", + " mean_divisor += 1\n", + "\n", + " # update according to the k means\n", + " centroids[k, :] = vector_mean / mean_divisor\n", + "\n", + " # we find the dissimilarity\n", + " for k in range(n_clusters):\n", + " for n in range(n_samples):\n", + " dist = 0\n", + " for d in range(dimensions):\n", + " dist += np.abs(data[n, d] - centroids[k, d])**2\n", + " distances[n, k] = dist\n", + "\n", + " # assign each point\n", + " for n in range(n_samples):\n", + " smallest = 1e10\n", + " smallest_row_index = 1e10\n", + " for k in range(n_clusters):\n", + " if distances[n, k] < smallest:\n", + " smallest = distances[n, k]\n", + " smallest_row_index = k\n", + "\n", + " cluster_labels[n] = smallest_row_index\n", + "\n", + " # convergence criteria\n", + " centroid_difference = np.sum(np.abs(centroids - prev_centroids))\n", + " if centroid_difference < tolerance:\n", + " print(f'Converged at iteration {iteration}')\n", + " break\n", + "\n", + " elif iteration == max_iterations:\n", + " print(f'Did not converge in {max_iterations} iterations')" + ] + }, + { + "cell_type": "markdown", + "id": "545a6742", + "metadata": { + "editable": true + }, + "source": [ + "We now have a simple , un-optimized $k$-means\n", + "clustering implementation. Lets plot the final result" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "id": "d9d3973b", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [ + { + "data": { + "image/png": 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+ "text/plain": [ + "
    " + ] + }, + "metadata": { + "filenames": { + "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/clustering_24_0.png" + }, + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "fig = plt.figure()\n", + "ax = fig.add_subplot()\n", + "unique_cluster_labels = np.unique(cluster_labels)\n", + "for i in unique_cluster_labels:\n", + " ax.scatter(data[cluster_labels == i, 0],\n", + " data[cluster_labels == i, 1],\n", + " label = i,\n", + " alpha = 0.2)\n", + " ax.scatter(centroids[:, 0], centroids[:, 1], c='black')\n", + "\n", + "ax.set_title(\"Final Result of K-means Clustering\")\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "id": "ee6a145f", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "def naive_kmeans(data, n_clusters=4, max_iterations=100, tolerance=1e-8):\n", + " start_time = time.time()\n", + "\n", + " n_samples, dimensions = data.shape\n", + " n_clusters = 4\n", + " #np.random.seed(2021)\n", + " centroids = data[np.random.choice(n_samples, n_clusters, replace=False), :]\n", + " distances = np.zeros((n_samples, n_clusters))\n", + "\n", + " for k in range(n_clusters):\n", + " for n in range(n_samples):\n", + " dist = 0\n", + " for d in range(dimensions):\n", + " dist += np.abs(data[n, d] - centroids[k, d])**2\n", + " distances[n, k] = dist\n", + "\n", + " cluster_labels = np.zeros(n_samples, dtype='int')\n", + "\n", + " for n in range(n_samples):\n", + " smallest = 1e10\n", + " smallest_row_index = 1e10\n", + " for k in range(n_clusters):\n", + " if distances[n, k] < smallest:\n", + " smallest = distances[n, k]\n", + " smallest_row_index = k\n", + "\n", + " cluster_labels[n] = smallest_row_index\n", + "\n", + " for iteration in range(max_iterations):\n", + " prev_centroids = centroids.copy()\n", + " for k in range(n_clusters):\n", + " vector_mean = np.zeros(dimensions)\n", + " mean_divisor = 0\n", + " for n in range(n_samples):\n", + " if cluster_labels[n] == k:\n", + " vector_mean += data[n, :]\n", + " mean_divisor += 1\n", + "\n", + " centroids[k, :] = vector_mean / mean_divisor\n", + "\n", + " for k in range(n_clusters):\n", + " for n in range(n_samples):\n", + " dist = 0\n", + " for d in range(dimensions):\n", + " dist += np.abs(data[n, d] - centroids[k, d])**2\n", + " distances[n, k] = dist\n", + "\n", + " for n in range(n_samples):\n", + " smallest = 1e10\n", + " smallest_row_index = 1e10\n", + " for k in range(n_clusters):\n", + " if distances[n, k] < smallest:\n", + " smallest = distances[n, k]\n", + " smallest_row_index = k\n", + "\n", + " cluster_labels[n] = smallest_row_index\n", + "\n", + " centroid_difference = np.sum(np.abs(centroids - prev_centroids))\n", + " if centroid_difference < tolerance:\n", + " print(f'Converged at iteration {iteration}')\n", + " print(f'Runtime: {time.time() - start_time} seconds')\n", + "\n", + " return cluster_labels, centroids\n", + "\n", + " print(f'Did not converge in {max_iterations} iterations')\n", + " print(f'Runtime: {time.time() - start_time} seconds')\n", + "\n", + " return cluster_labels, centroids" + ] + } + ], + "metadata": { + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.8.12" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/clustering.py b/doc/LectureNotes/_build/jupyter_execute/clustering.py new file mode 100644 index 000000000..ddd1ea383 --- /dev/null +++ b/doc/LectureNotes/_build/jupyter_execute/clustering.py @@ -0,0 +1,443 @@ +#!/usr/bin/env python +# coding: utf-8 + +# + +# # Clustering and Unsupervised Learning +# +# In general terms cluster analysis, or clustering, is the task of grouping a +# data-set into different distinct categories based on some measure of equality of +# the data. This measure is often referred to as a **metric** or **similarity +# measure** in the literature (note: sometimes we deal with a **dissimilarity +# measure** instead). Usually, these metrics are formulated as some kind of +# distance function between points in a high-dimensional space. +# +# The simplest, and also the most +# common is the **Euclidean distance**. +# +# The simplest of all clustering algorithms is the **k-means algorithm** +# , sometimes also referred to as *Lloyds algorithm*. It is the simplest and also +# the most common. From its simplicity it obtains both strengths and weaknesses. +# These will be discussed in more detail later. The $k$-means algorithm is a +# **centroid based** clustering algorithm. +# +# Assume, we are given $n$ data points and we wish to split the data into $K < n$ +# different categories, or clusters. We label each cluster by an integer + +# $$ +# k\in\{1, \cdots, K \}. +# $$ + +# In the basic k-means algorithm each point is assigned to only +# one cluster $k$, and these assignments are *non-injective* i.e. many-to-one. We +# can think of these mappings as an encoder $k = C(i)$, which assigns the $i$-th +# data-point $\bf x_i$ to the $k$-th cluster. +# +# $k$-means algorithm in words: +# 1. We start with guesses / random initializations of our $k$ cluster centers/centroids +# +# 2. For each centroid the points that are most similar are identified +# +# 3. Then we move / replace each centroid with a coordinate average of all the points that were assigned to that centroid. +# +# 4. Iterate 2-3 until the centroids no longer move (to some tolerance) +# +# We assume we have $n$ data-points + +# +#
    +# +# $$ +# \begin{equation}\label{eq:kmeanspoints} \tag{1} +# \boldsymbol{x_i} = \{x_{i, 1}, \cdots, x_{i, p}\}\in\mathbb{R}^p. +# \end{equation} +# $$ + +# which we wish to group into $K < n$ clusters. For our dissimilarity measure we +# use the *squared Euclidean distance* + +# +#
    +# +# $$ +# \begin{equation}\label{eq:squaredeuclidean} \tag{2} +# d(\boldsymbol{x_i}, \boldsymbol{x_i'}) = \sum_{j=1}^p(x_{ij} - x_{i'j})^2 +# = ||\boldsymbol{x_i} - \boldsymbol{x_{i'}}||^2 +# \end{equation} +# $$ + +# We define the so called *within-cluster point scatter* which gives us a +# measure of how close each data point assigned to the same cluster tends to be to +# the all the others. + +# +#
    +# +# $$ +# \begin{equation}\label{eq:withincluster} \tag{3} +# W(C) = \frac{1}{2}\sum_{k=1}^K\sum_{C(i)=k} +# \sum_{C(i')=k}d(\boldsymbol{x_i}, \boldsymbol{x_{i'}}) = +# \sum_{k=1}^KN_k\sum_{C(i)=k}||\boldsymbol{x_i} - \boldsymbol{\overline{x_k}}||^2 +# \end{equation} +# $$ + +# where $\boldsymbol{\overline{x_k}}$ is the mean vector associated with the $k$-th +# cluster, and $N_k = \sum_{i=1}^nI(C(i) = k)$, where the $I()$ notation is +# similar to the Kronecker delta (*Commonly used in statistics, it just means that +# when $i = k$ we have the encoder $C(i)$*). In other words, the within-cluster +# scatter measures the compactness of each cluster with respect to the data points +# assigned to each cluster. This is the quantity that the $k$-means algorithm aims +# to minimize. We refer to this quantity $W(C)$ as the within cluster scatter +# because of its relation to the *total scatter*. +# +# We have + +# +#
    +# +# $$ +# \begin{equation}\label{eq:totalscatter} \tag{4} +# T = W(C) + B(C) = \frac{1}{2}\sum_{i=1}^n +# \sum_{i'=1}^nd(\boldsymbol{x_i}, \boldsymbol{x_{i'}}) +# = \frac{1}{2}\sum_{k=1}^K\sum_{C(i)=k} +# \Big(\sum_{C(i') = k}d(\boldsymbol{x_i}, \boldsymbol{x_{i'}}) +# + \sum_{C(i')\neq k}d(\boldsymbol{x_i}, \boldsymbol{x_{i'}})\Big). +# \end{equation} +# $$ + +# This is a quantity that is conserved throughout the $k$-means algorithm. It can +# be thought of as the total amount of information in the data, and it is composed +# of the aforementioned within-cluster scatter and the *between-cluster scatter* +# $B(C)$. In methods such as principle component analysis the total scatter is not +# conserved. +# +# Given a cluster mean $\boldsymbol{m_k}$ we define the **total cluster variance** + +# +#
    +# +# $$ +# \begin{equation}\label{eq:totalclustervariance} \tag{5} +# \min_{C, \{\boldsymbol{m_k}\}_1^K}\sum_{k=1}^KN_k\sum||\boldsymbol{x_i} - \boldsymbol{m_k}||^2 +# \end{equation} +# $$ + +# Now we have all the pieces necessary to formally revisit the $k$-means algorithm. +# +# The $k$-means clustering algorithm goes as follows +# +# 1. For a given cluster assignment $C$, and $k$ cluster means $\left\{m_1, \cdots, m_k\right\}$. We minimize the total cluster variance with respect to the cluster means $\{m_k\}$ yielding the means of the currently assigned clusters. +# +# 2. Given a current set of $k$ means $\{m_k\}$ the total cluster variance is minimized by assigning each observation to the closest (current) cluster mean. That is $$C(i) = \underset{1\leq k\leq K}{\mathrm{argmin}} ||\boldsymbol{x_i} - \boldsymbol{m_k}||^2$$ +# +# 3. Steps 1 and 2 are repeated until the assignments do not change. + +# ## Codes and Approaches +# +# 1. Before we start we specify a number $k$ which is the number of clusters we want to try to separate our data into. +# +# 2. We initially choose $k$ random data points in our data as our initial centroids, *or means* (this is where the name comes from). +# +# 3. Assign each data point to their closest centroid, based on the squared Euclidean distance. +# +# 4. For each of the $k$ cluster we update the centroid by calculating new mean values for all the data points in the cluster. +# +# 5. Iteratively minimize the within cluster scatter by performing steps (3, 4) until the new assignments stop changing (can be to some tolerance) or until a maximum number of iterations have passed. +# +# Let us now program the most basic version of the algorithm using nothing but +# Python with numpy arrays. This code is kept intentionally simple to gradually +# progress our understanding. There is no vectorization of any kind, and even most +# helper functions are not utilized. +# +# We need first a dataset to do our cluster analysis on. In our case +# this is a plain *vanilla* data set using random numbers using a +# Gaussian distribution. + +# In[1]: + + +get_ipython().run_line_magic('matplotlib', 'inline') + +import time +import numpy as np +import tensorflow as tf +from matplotlib import image +import matplotlib.pyplot as plt +from sklearn.cluster import KMeans +from IPython.display import display + +np.random.seed(2021) + + +# Next we define functions, for ease of use later, to generate Gaussians and to +# set up our toy data set. + +# In[2]: + + +def gaussian_points(dim=2, n_points=1000, mean_vector=np.array([0, 0]), + sample_variance=1): + """ + Very simple custom function to generate gaussian distributed point clusters + with variable dimension, number of points, means in each direction + (must match dim) and sample variance. + + Inputs: + dim (int) + n_points (int) + mean_vector (np.array) (where index 0 is x, index 1 is y etc.) + sample_variance (float) + + Returns: + data (np.array): with dimensions (dim x n_points) + """ + + mean_matrix = np.zeros(dim) + mean_vector + covariance_matrix = np.eye(dim) * sample_variance + data = np.random.multivariate_normal(mean_matrix, covariance_matrix, + n_points) + return data + + + +def generate_simple_clustering_dataset(dim=2, n_points=1000, plotting=True, + return_data=True): + """ + Toy model to illustrate k-means clustering + """ + + data1 = gaussian_points(mean_vector=np.array([5, 5])) + data2 = gaussian_points() + data3 = gaussian_points(mean_vector=np.array([1, 4.5])) + data4 = gaussian_points(mean_vector=np.array([5, 1])) + data = np.concatenate((data1, data2, data3, data4), axis=0) + + if plotting: + fig, ax = plt.subplots() + ax.scatter(data[:, 0], data[:, 1], alpha=0.2) + ax.set_title('Toy Model Dataset') + plt.show() + + + if return_data: + return data + + +data = generate_simple_clustering_dataset() + + +# With the above dataset we start +# implementing the $k$-means algorithm. + +# In[3]: + + + +n_samples, dimensions = data.shape +n_clusters = 4 + +# we randomly initialize our centroids +np.random.seed(2021) +centroids = data[np.random.choice(n_samples, n_clusters, replace=False), :] +distances = np.zeros((n_samples, n_clusters)) + +# first we need to calculate the distance to each centroid from our data +for k in range(n_clusters): + for n in range(n_samples): + dist = 0 + for d in range(dimensions): + dist += np.abs(data[n, d] - centroids[k, d])**2 + distances[n, k] = dist + +# we initialize an array to keep track of to which cluster each point belongs +# the way we set it up here the index tracks which point and the value which +# cluster the point belongs to +cluster_labels = np.zeros(n_samples, dtype='int') + +# next we loop through our samples and for every point assign it to the cluster +# to which it has the smallest distance to +for n in range(n_samples): + # tracking variables (all of this is basically just an argmin) + smallest = 1e10 + smallest_row_index = 1e10 + for k in range(n_clusters): + if distances[n, k] < smallest: + smallest = distances[n, k] + smallest_row_index = k + + cluster_labels[n] = smallest_row_index + + +# In[4]: + + +fig = plt.figure() +ax = fig.add_subplot() +unique_cluster_labels = np.unique(cluster_labels) +for i in unique_cluster_labels: + ax.scatter(data[cluster_labels == i, 0], + data[cluster_labels == i, 1], + label = i, + alpha = 0.2) + ax.scatter(centroids[:, 0], centroids[:, 1], c='black') + +ax.set_title("First Grouping of Points to Centroids") + +plt.show() + + +# So what do we have so far? We have 'picked' $k$ centroids at random from our +# data points. There are other ways of more intelligently choosing their +# initializations, however for our purposes randomly is fine. Then we have +# initialized an array 'distances' which holds the information of the distance, +# *or dissimilarity*, of every point to of our centroids. Finally, we have +# initialized an array 'cluster_labels' which according to our distances array +# holds the information of to which centroid every point is assigned. This was the +# first pass of our algorithm. Essentially, all we need to do now is repeat the +# distance and assignment steps above until we have reached a desired convergence +# or a maximum amount of iterations. + +# In[5]: + + + +max_iterations = 100 +tolerance = 1e-8 + +for iteration in range(max_iterations): + prev_centroids = centroids.copy() + for k in range(n_clusters): + # this array will be used to update our centroid positions + vector_mean = np.zeros(dimensions) + mean_divisor = 0 + for n in range(n_samples): + if cluster_labels[n] == k: + vector_mean += data[n, :] + mean_divisor += 1 + + # update according to the k means + centroids[k, :] = vector_mean / mean_divisor + + # we find the dissimilarity + for k in range(n_clusters): + for n in range(n_samples): + dist = 0 + for d in range(dimensions): + dist += np.abs(data[n, d] - centroids[k, d])**2 + distances[n, k] = dist + + # assign each point + for n in range(n_samples): + smallest = 1e10 + smallest_row_index = 1e10 + for k in range(n_clusters): + if distances[n, k] < smallest: + smallest = distances[n, k] + smallest_row_index = k + + cluster_labels[n] = smallest_row_index + + # convergence criteria + centroid_difference = np.sum(np.abs(centroids - prev_centroids)) + if centroid_difference < tolerance: + print(f'Converged at iteration {iteration}') + break + + elif iteration == max_iterations: + print(f'Did not converge in {max_iterations} iterations') + + +# We now have a simple , un-optimized $k$-means +# clustering implementation. Lets plot the final result + +# In[6]: + + +fig = plt.figure() +ax = fig.add_subplot() +unique_cluster_labels = np.unique(cluster_labels) +for i in unique_cluster_labels: + ax.scatter(data[cluster_labels == i, 0], + data[cluster_labels == i, 1], + label = i, + alpha = 0.2) + ax.scatter(centroids[:, 0], centroids[:, 1], c='black') + +ax.set_title("Final Result of K-means Clustering") + +plt.show() + + +# In[7]: + + +def naive_kmeans(data, n_clusters=4, max_iterations=100, tolerance=1e-8): + start_time = time.time() + + n_samples, dimensions = data.shape + n_clusters = 4 + #np.random.seed(2021) + centroids = data[np.random.choice(n_samples, n_clusters, replace=False), :] + distances = np.zeros((n_samples, n_clusters)) + + for k in range(n_clusters): + for n in range(n_samples): + dist = 0 + for d in range(dimensions): + dist += np.abs(data[n, d] - centroids[k, d])**2 + distances[n, k] = dist + + cluster_labels = np.zeros(n_samples, dtype='int') + + for n in range(n_samples): + smallest = 1e10 + smallest_row_index = 1e10 + for k in range(n_clusters): + if distances[n, k] < smallest: + smallest = distances[n, k] + smallest_row_index = k + + cluster_labels[n] = smallest_row_index + + for iteration in range(max_iterations): + prev_centroids = centroids.copy() + for k in range(n_clusters): + vector_mean = np.zeros(dimensions) + mean_divisor = 0 + for n in range(n_samples): + if cluster_labels[n] == k: + vector_mean += data[n, :] + mean_divisor += 1 + + centroids[k, :] = vector_mean / mean_divisor + + for k in range(n_clusters): + for n in range(n_samples): + dist = 0 + for d in range(dimensions): + dist += np.abs(data[n, d] - centroids[k, d])**2 + distances[n, k] = dist + + for n in range(n_samples): + smallest = 1e10 + smallest_row_index = 1e10 + for k in range(n_clusters): + if distances[n, k] < smallest: + smallest = distances[n, k] + smallest_row_index = k + + cluster_labels[n] = smallest_row_index + + centroid_difference = np.sum(np.abs(centroids - prev_centroids)) + if centroid_difference < tolerance: + print(f'Converged at iteration {iteration}') + print(f'Runtime: {time.time() - start_time} seconds') + + return cluster_labels, centroids + + print(f'Did not converge in {max_iterations} iterations') + print(f'Runtime: {time.time() - start_time} seconds') + + return cluster_labels, centroids + diff --git a/doc/LectureNotes/_build/jupyter_execute/clustering_17_0.png b/doc/LectureNotes/_build/jupyter_execute/clustering_17_0.png new file mode 100644 index 000000000..1e6a8c796 Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/clustering_17_0.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/clustering_20_0.png b/doc/LectureNotes/_build/jupyter_execute/clustering_20_0.png new file mode 100644 index 000000000..be592e440 Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/clustering_20_0.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/clustering_24_0.png b/doc/LectureNotes/_build/jupyter_execute/clustering_24_0.png new file mode 100644 index 000000000..e2dbae3dc Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/clustering_24_0.png differ