"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, no through github.com website.\n",
"\n",
"## Virtual environments\n",
"\n",
"To activate virtual environment:\n",
" * If you were on Tuesday/Wednesday group type `activate_mlp` or `source ~/mlpractical/venv/bin/activate`\n",
" * If you were on Monday group:\n",
" + and if you have chosen **comfy** way type: workon mlpractival\n",
" + and if you have chosen **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",
"Enter your git mlp repository you set up last week (i.e. ~/mlpractical/repo-mlp) and once you synced the repository (in one of the two below ways), start the notebook session by typing:\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 \"my 1st lab work\"`\n",
"2. `git pull`\n",
"\n",
"Then, once you need you can always (temporaily) restore a desired state of the repository.\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",
"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",
"\n",
"This means, that in this tutorial:\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",
"\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",
"\n",
"***\n",
"\n",
"## Linear and Affine Transforms\n",
"\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",
"\n",
" * Weight matrix $\\mathbf{W} \\in \\mathbb{R}^{D\\times K}$: where element $w_{ik}$ is the weight from input $x_i$ to output $y_k$\n",
" * Bias vector $\\mathbf{b}\\in R^{K}$ : where element $b_{k}$ is the bias for output $k$\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 $\\mathbf{W}=\\left[ \\begin{array}{ccc}\n",
"w_{11} & w_{12} & w_{13} \\\\\n",
"w_{21} & w_{22} & w_{23} \\\\\n",
"w_{31} & w_{32} & w_{33} \\\\\n",
"w_{41} & w_{42} & w_{43} \\\\\n",
"w_{51} & x_{52} & 2_{53} \\\\ \\end{array} \\right]$, 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",
"where both $\\mathbf{X}\\in\\mathbb{R}^{B\\times D}$ and $\\mathbf{Y}\\in\\mathbb{R}^{B\\times K}$ are matrices, and $\\mathbf{b}$ needs to be <a href=\"http://docs.scipy.org/doc/numpy/user/basics.broadcasting.html\">broadcased</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 implementation point of view, it does not matter.\n",
"\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."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Note on random number generators (could move it somewhere else)\n",
"\n",
"It is generally a good practice (for machine learning applications **not** 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 allows to avoid hard to reproduce scenariors where a particular action happens only for a particular sequence of numbers (which you cannot reproduce easily due to unknown random seeds sequence on the way!). As such, within this course we are going use a single random generator object. For instance, the one similar to the below:"
"Using numpy.dot, implement **forward** propagation through the linear transform defined by equations (3) and (4) for $B=1$ and $B>1$. As data ($\\mathbf{x}$) use `MNISTDataProvider` from previous laboratories. For case when $B=1$ write a function to compute the 1st output ($y_1$) using equations (1) and (2). Check if the output is the same as the corresponding one obtained with numpy. \n",
"\n",
"Tip: To generate random data you can use `random_generator.uniform(-0.1, 0.1, (D, 10))` from the preceeding cell."
"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."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"\n",
"#When implementing those, take into account the mini-batch case, for which one is\n",
"#expected to sum the errors for each example\n",
"\n",
"def fprop(x, W, b):\n",
" #code implementing eq. (3)\n",
" raise NotImplementedError('Write me!')\n",
"\n",
"def cost(y, t):\n",
" #Mean Square Error cost, equation (5)\n",
" raise NotImplementedError('Write me!')\n",
"\n",
"def cost_grad(y, t):\n",
" #Gradient of the cost w.r.t y equation (8)\n",
" raise NotImplementedError('Write me!')\n",
"\n",
"def cost_wrt_W(cost_grad, x):\n",
" #Gradient of the cost w.r.t W, equation (9)\n",
" raise NotImplementedError('Write me!')\n",
" \n",
"def cost_wrt_b(cost_grad):\n",
" #Gradient of the cost w.r.t to b, equation (10)\n",
"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",
"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"
