"This tutorial is about linear transforms - a basic building block of neural networks, including deep learning models.\n",
"\n",
"# Virtual environments and syncing repositories\n",
"\n",
"Before you proceed onwards, remember to activate you virtual environments so you can use the software you installed last week as well as run the notebooks in interactive mode, not through the github.com website.\n",
"\n",
"## Virtual environments\n",
"\n",
"To activate the virtual environment:\n",
" * If you were in last week's Tuesday or Wednesday group type `activate_mlp` or `source ~/mlpractical/venv/bin/activate`\n",
" * If you were in the Monday group:\n",
" + and if you have chosen the **comfy** way type: `workon mlpractical`\n",
" + and if you have chosen the **generic** way, `source` your virutal environment using `source` and specyfing the path to the activate script (you need to localise it yourself, there were not any general recommendations w.r.t dir structure and people have installed it in different places, usually somewhere in the home directories. If you cannot easily find it by yourself, use something like: `find . -iname activate` ):\n",
"\n",
"## On Synchronising repositories\n",
"\n",
"Enter the git mlp repository you set up last week (i.e. `~/mlpractical/repo-mlp`) and once you sync the repository (in one of the two below ways, or look at our short Git FAQ <a href=\"https://github.com/CSTR-Edinburgh/mlpractical/blob/master/gitFAQ.md\">here</a>), start the notebook session by typing:\n",
"\n",
"```\n",
"ipython notebook\n",
"```\n",
"\n",
"### Default way\n",
"\n",
"To avoid potential conflicts between the changes you have made since last week and our additions, we recommend `stash` your changes and `pull` the new code from the mlpractical repository by typing:\n",
"\n",
"1. `git stash save \"Lab1 work\"`\n",
"2. `git pull`\n",
"\n",
"Then, if you need to, you can always (temporaily) restore a desired state of the repository (look <a href=\"https://github.com/CSTR-Edinburgh/mlpractical/blob/master/gitFAQ.md\">here</a>).\n",
"\n",
"**Otherwise** you may also create a branch for each lab separately (again, look <a href=\"https://github.com/CSTR-Edinburgh/mlpractical/blob/master/gitFAQ.md\">here</a> and git tutorials we linked there), this will allow you to keep `master` branch clean, and pull changes into it every week from the central repository. At the same time branching gives you much more flexibility with changes you introduce to the code as potential conflicts will not occur until you try to make an explicit merge.\n",
"\n",
"### For advanced github users\n",
"\n",
"It is OK if you want to keep your changes and merge the new code with whatever you already have, but you need to know what you are doing and how to resolve conflicts.\n",
" \n",
" "
]
},
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"cell_type": "markdown",
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"source": [
"# Single Layer Models\n",
"\n",
"***\n",
"### Note on storing matrices in computer memory\n",
"Although `numpy` can easily handle both formats (possibly with some computational overhead), in our code we will stick with modern (and default) `c`-like approach and use row-wise format (contrary to Fortran that used column-wise approach). \n",
"* vectors are kept row-wise $\\mathbf{x} = (x_1, x_1, \\ldots, x_D) $ (rather than $\\mathbf{x} = (x_1, x_1, \\ldots, x_D)^T$)\n",
"* similarly, in case of matrices we will stick to: $\\left[ \\begin{array}{cccc}\n",
"x_{11} & x_{12} & \\ldots & x_{1D} \\\\\n",
"x_{21} & x_{22} & \\ldots & x_{2D} \\\\\n",
"x_{31} & x_{32} & \\ldots & x_{3D} \\\\ \\end{array} \\right]$ and each row (i.e. $\\left[ \\begin{array}{cccc} x_{11} & x_{12} & \\ldots & x_{1D} \\end{array} \\right]$) represents a single data-point (like one MNIST image or one window of observations)\n",
"In lecture slides you will find the equations following the conventional mathematical column-wise approach, but you can easily map them one way or the other using using matrix transpose.\n",
"The basis of all linear models is so called affine transform, that is a transform that implements some linear transformation and translation of input features. The transforms we are going to use are parameterised by:\n",
"Note, the bias is simply some additve term, and can be easily incorporated into an additional row in weight matrix and an additinal input in the inputs which is set to $1.0$ (as in the below picture taken from the lecture slides). However, here (and in the code) we will keep them separate.\n",
"For instance, for the above example of 5-dimensional input vector by $\\mathbf{x} = (x_1, x_2, x_3, x_4, x_5)$, weight matrix $\\mathbf{W}=\\left[ \\begin{array}{ccc}\n",
"w_{11} & w_{12} & w_{13} \\\\\n",
"w_{21} & w_{22} & w_{23} \\\\\n",
"w_{31} & w_{32} & w_{33} \\\\\n",
"w_{41} & w_{42} & w_{43} \\\\\n",
"w_{51} & w_{52} & w_{53} \\\\ \\end{array} \\right]$, bias vector $\\mathbf{b} = (b_1, b_2, b_3)$ and outputs $\\mathbf{y} = (y_1, y_2, y_3)$, one can write the transformation as follows:\n",
"\n",
"(for the $i$-th output)\n",
"\n",
"(1) $\n",
"\\begin{equation}\n",
" y_i = b_i + \\sum_j x_jw_{ji}\n",
"\\end{equation}\n",
"$\n",
"\n",
"or the equivalent vector form (where $\\mathbf w_i$ is the $i$-th column of $\\mathbf W$, but note, when we **slice** the $i$th column we will get a **vector** $\\mathbf w_i = (w_{1i}, w_{2i}, w_{3i}, w_{4i}, w_{5i})$, hence the transpose for $\\mathbf w_i$ in the below equation):\n",
"\n",
"(2) $\n",
"\\begin{equation}\n",
" y_i = b_i + \\mathbf x \\mathbf w_i^T\n",
"\\end{equation}\n",
"$\n",
"\n",
"The same operation can be also written in matrix form, to compute all the outputs $\\mathbf{y}$ at the same time:\n",
"\n",
"(3) $\n",
"\\begin{equation}\n",
" \\mathbf y=\\mathbf x\\mathbf W + \\mathbf b\n",
"\\end{equation}\n",
"$\n",
"\n",
"This is equivalent to slides 12/13 in lecture 1, except we are using row vectors.\n",
"\n",
"When $\\mathbf{x}$ is a mini-batch (contains $B$ data-points of dimension $D$ each), i.e. $\\left[ \\begin{array}{cccc}\n",
"where $\\mathbf{W} \\in \\mathbb{R}^{D\\times K}$ and both $\\mathbf{X}\\in\\mathbb{R}^{B\\times D}$ and $\\mathbf{Y}\\in\\mathbb{R}^{B\\times K}$ are matrices, and $\\mathbf{b}\\in\\mathbb{R}^{1\\times K}$ needs to be <a href=\"http://docs.scipy.org/doc/numpy/user/basics.broadcasting.html\">broadcasted</a> $B$ times (numpy will do this by default). However, we will not make an explicit distinction between a special case for $B=1$ and $B>1$ and simply use equation (3) instead, although $\\mathbf{x}$ and hence $\\mathbf{y}$ could be matrices. From an implementation point of view, it does not matter.\n",
"The desired functionality for matrix multiplication in numpy is provided by <a href=\"http://docs.scipy.org/doc/numpy/reference/generated/numpy.dot.html\">numpy.dot</a> function. If you haven't use it so far, get familiar with it as we will use it extensively."
]
},
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"metadata": {},
"source": [
"### A general note on random number generators\n",
"\n",
"It is generally a good practice (for machine learning applications **not** for cryptography!) to seed a pseudo-random number generator once at the beginning of the experiment, and use it later through the code where necesarry. This makes it easier to reproduce results since random initialisations can be replicated. As such, within this course we are going use a single random generator object, similar to the below:"
"Using `numpy.dot`, implement **forward** propagation through the linear transform defined by equations (3) and (4) for $B=1$ and $B>1$ i.e. use parameters $\\mathbf{W}$ and $\\mathbf{b}$ with data $\\mathbf{X}$ to determine $\\mathbf{Y}$. Use `MNISTDataProvider` (introduced last week) to generate $\\mathbf{X}$. We are going to write a function for each equation:\n",
"1. `y1_equation_1`: Return the value of the $1^{st}$ dimension of $\\mathbf{y}$ (the output of the first output node) given a single training data point $\\mathbf{x}$ using a sum\n",
"1. `y1_equation_2`: Repeat above using vector multiplication (use `numpy.dot()`)\n",
"1. `y_equation_3`: Return the value of $\\mathbf{y}$ (the whole output layer) given a single training data point $\\mathbf{x}$\n",
"1. `Y_equation_4`: Return the value of $\\mathbf{Y}$ given $\\mathbf{X}$\n",
"Modify (if necessary) examples from Exercise 1 to perform **backward** propagation, that is, given $\\mathbf{y}$ (obtained in previous step) and weight matrix $\\mathbf{W}$, project $\\mathbf{y}$ onto the input space $\\mathbf{x}$ (ignore or set to zero the biases towards $\\mathbf{x}$ in backward pass). Mathematically, we are interested in the following transformation: $\\mathbf{z}=\\mathbf{y}\\mathbf{W}^T$"
"In case you do not fully understand how matrix-vector and/or matrix-matrix products work, consider implementing `my_dot_mat_mat` function (you have been given `my_dot_vec_mat` code to look at as an example) which takes as the input the following arguments:\n",
"Your job is to write a variant that works in a mini-batch mode where both $\\mathbf{x}\\in\\mathbb{R}^{B\\times D}$ and $\\mathbf{y}\\in\\mathbb{R}^{B\\times K}$ are matrices in which each rows contain one of $B$ data-points from mini-batch (rather than $\\mathbf{x}\\in\\mathbb{R}^{D}$ and $\\mathbf{y}\\in\\mathbb{R}^{K}$)."
"We will learn the model with stochastic gradient descent on N data-points using mean square error (MSE) loss function, which is defined as follows:\n",
"Similarly, using the above $\\delta^n_r$ one can express the gradient of the weight $w_{sr}$ (from the s-th input to the r-th output) for linear model and MSE cost as follows:\n",
" * Weight matrix $\\mathbf{W}$: $w_{ik}$ is the weight from input $x_i$ to output $y_k$. Note, here this is really a vector since a single scalar output, y_1.\n",
" * Scalar bias $b$ for the only output in our model \n",
" * Scalar target $t$ for the only output in out model\n",
" \n",
"First, ensure you can make use of data provider (note, for training data has been normalised to zero mean and unit variance, hence different effective range than one can find in file):"
"The below code implements a very simple variant of stochastic gradient descent for the weather regression example. Your task is to implement 5 functions in the next cell and then run two next cells that 1) build sgd functions and 2) run the actual training."
"Modify the above regression problem so the model makes binary classification whether the the weather is going to be one of those \\{rainy, sunny} (look at slide 12 of the 2nd lecture)\n",
"\n",
"Tip: You need to introduce the following changes:\n",
"1. Modify `MetOfficeDataProvider` (for example, inherit from MetOfficeDataProvider to create a new class MetOfficeDataProviderBin) and modify `next()` function so it returns as `targets` either 0 (sunny - if the the amount of rain [before mean/variance normalisation] is equal to 0 or 1 (rainy -- otherwise).\n",
"2. Modify the functions from previous exercise so the fprop implements `sigmoid` on top of affine transform.\n",
"3. Modify cost function to binary cross-entropy\n",
"4. Make sure you compute the gradients correctly (as you have changed both the output and the cost)\n"
