05_Transfer_functions.ipynb

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    "# Introduction\n",
    "\n",
    "This tutorial focuses on implementation of alternatives to sigmoid transfer functions for hidden units.  (*Transfer functions* are also called *activation functions* or *nonlinearities*.) First, we will work with hyperboilc tangent (tanh) and then unbounded (or partially unbounded) piecewise linear functions: Rectifying Linear Units (ReLU) and Maxout.\n",
    "\n",
    "\n",
    "## Virtual environments\n",
    "\n",
    "Before you proceed onwards, remember to activate your virtual environment by typing `activate_mlp` or `source ~/mlpractical/venv/bin/activate` (or if you did the original install the \"comfy way\" type: `workon mlpractical`).\n",
    "\n",
    "\n",
    "## Syncing the git repository\n",
    "\n",
    "Look <a href=\"https://github.com/CSTR-Edinburgh/mlpractical/blob/master/gitFAQ.md\">here</a> for more details. But in short, we recommend to create a separate branch for this lab, as follows:\n",
    "\n",
    "1. Enter the mlpractical directory `cd ~/mlpractical/repo-mlp`\n",
    "2. List the branches and check which are currently active by typing: `git branch`\n",
    "3. If you have followed our recommendations, you should be in the `lab4` branch, please commit your local changed to the repo index by typing:\n",
    "```\n",
    "git commit -am \"finished lab4\"\n",
    "```\n",
    "4. Now you can switch to `master` branch by typing: \n",
    "```\n",
    "git checkout master\n",
    " ```\n",
    "5. To update the repository (note, assuming master does not have any conflicts), if there are some, have a look <a href=\"https://github.com/CSTR-Edinburgh/mlpractical/blob/master/gitFAQ.md\">here</a>\n",
    "```\n",
    "git pull\n",
    "```\n",
    "6. And now, create the new branch & switch to it by typing:\n",
    "```\n",
    "git checkout -b lab5\n",
    "```"
   ]
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    "# Overview of alternative transfer functions\n",
    "\n",
    "Now, we briefly summarise some other possible choices for hidden layer transfer functions.\n",
    "\n",
    "## Tanh\n",
    "\n",
    "Given a linear activation $a_{i}$ tanh implements the following operation:\n",
    "\n",
    "(1) $h_i(a_i) = \\mbox{tanh}(a_i) = \\frac{\\exp(a_i) - \\exp(-a_i)}{\\exp(a_i) + \\exp(-a_i)}$\n",
    "\n",
    "Hence, the derivative of $h_i$ with respect to $a_i$ is:\n",
    "\n",
    "(2) $\\begin{align}\n",
    "\\frac{\\partial h_i}{\\partial a_i} &= 1 - h^2_i\n",
    "\\end{align}\n",
    "$\n",
    "\n",
    "\n",
    "## ReLU\n",
    "\n",
    "Given a linear activation $a_{i}$ relu implements the following operation:\n",
    "\n",
    "(3) $h_i(a_i) = \\max(0, a_i)$\n",
    "\n",
    "Hence, the gradient is :\n",
    "\n",
    "(4) $\\begin{align}\n",
    "\\frac{\\partial h_i}{\\partial a_i} &=\n",
    "\\begin{cases}\n",
    "     1     & \\quad \\text{if } a_i > 0 \\\\\n",
    "     0       & \\quad \\text{if } a_i \\leq 0 \\\\\n",
    "\\end{cases}\n",
    "\\end{align}\n",
    "$\n",
    "\n",
    "ReLU implements a form of data-driven sparsity, that is, on average the activations are sparse (many of them are 0) but the general sparsity pattern will depend on particular data-point. This is different from sparsity obtained in model's parameters one can obtain with $L1$ regularisation as the latter affect all data-points in the same way.\n",
    "\n",
    "## Maxout\n",
    "\n",
    "Maxout is an example of data-driven type of non-linearity in which the transfer function can be learned from data. That is, the model can build a non-linear transfer function from piecewise linear components.  These linear components, depending on the number of linear regions used in the pooling operator (given by parameter $K$), can approximate  arbitrary functions, such as ReLU, abs, etc.\n",
    "\n",
    "Given some subset (group, pool) of $K$ linear activations $a_{j}, a_{j+1}, \\ldots, a_{j+K}$ at the $l$-th layer, maxout implements the following operation:\n",
    "\n",
    "(5) $h_i(a_j, a_{j+1}, \\ldots, a_{j+K}) = \\max(a_j, a_{j+1}, \\ldots, a_{j+K})$\n",
    "\n",
    "Hence, the gradient of $h_i$ w.r.t to the pooling region $a_{j}, a_{j+1}, \\ldots, a_{j+K}$  is :\n",
    "\n",
    "(6) $\\begin{align}\n",
    "\\frac{\\partial h_i}{\\partial (a_j, a_{j+1}, \\ldots, a_{j+K})} &=\n",
    "\\begin{cases}\n",
    "     1     & \\quad \\text{for the max activation}  \\\\\n",
    "     0       & \\quad \\text{otherwise} \\\\\n",
    "\\end{cases}\n",
    "\\end{align}\n",
    "$\n",
    "\n",
    "Implementation tips are given in Exercise 3.\n",
    "\n",
    "# On weight initialisation\n",
    "\n",
    "Activation functions directly affect the \"network dynamics\", that is, the magnitudes of the statistics each layer is producing. For example, *squashing* non-linearities like sigmoid or tanh bring the linear activations to a certain bounded range. ReLU, on the contrary, has an unbounded positive side. This directly affects all statistics collected in forward and backward passes as well as the gradients w.r.t paramters - hence also the pace at which the model learns. That is why learning rate is usually required to be tuned for given the characterictics of the non-linearities used. \n",
    "\n",
    "Another important hyperparameter is the initial range used to initialise the weight matrices.  We have largely ignored it so far (although if you did further experiments in coursework 1, you may have found setting it had an effect on training deeper networks with 4 or 5 hidden layers).  However, for sigmoidal non-linearities (sigmoid, tanh) the initialisation range is an important hyperparameter and a considerable amount of research has been put into determining what is the best strategy for choosing it. In fact, one of the early triggers of the recent resurgence of deep learning was pre-training - techniques for initialising weights in an unsupervised manner so that one can effectively train deeper models in supervised fashion later.  \n",
    "\n",
    "## Sigmoidal transfer functions\n",
    "\n",
    "Y. LeCun in [Efficient Backprop](http://link.springer.com/chapter/10.1007%2F3-540-49430-8_2) recommends the following setting of the initial range $r$ for sigmoidal units (assuming that the data has been normalised to zero mean, unit variance): \n",
    "\n",
    "(7) $ r = \\frac{1}{\\sqrt{N_{IN}}} $\n",
    "\n",
    "where $N_{IN}$ is the number of inputs to the given layer and the weights are then sampled from the (usually uniform) distribution $U(-r,r)$. The motivation is to keep the initial forward-pass signal in the linear region of the sigmoid non-linearity so that the gradients are large enough for training to proceed (note that the sigmoidal non-linearities saturate when activations are either very positive or very negative, leading to very small gradients and hence poor learning dynamics).\n",
    "\n",
    "The initialisation used in (7) however leads to different magnitudes of activations/gradients at different layers (due to multiplicative nature of the computations) and more recently, [Glorot et. al](http://jmlr.org/proceedings/papers/v9/glorot10a/glorot10a.pdf) proposed the so-called *normalised initialisation*, which ensures the variance of the forward signal (activations) is approximately the same in each layer. The same applies to the gradients obtained in backward pass.  \n",
    "\n",
    "The $r$ in the *normalised initialisation* for $\\mbox{tanh}$ non-linearity is then:\n",
    "\n",
    "(8) $ r = \\frac{\\sqrt{6}}{\\sqrt{N_{IN}+N_{OUT}}} $\n",
    "\n",
    "For the sigmoid (logistic) non-linearity, to get similiar characteristics, one should scale $r$ in (8) by 4, that is:\n",
    "\n",
    "(9) $ r = \\frac{4\\sqrt{6}}{\\sqrt{N_{IN}+N_{OUT}}} $\n",
    "\n",
    "## Piece-wise linear transfer functions (ReLU, Maxout)\n",
    "\n",
    "For unbounded transfer functions initialisation is not as crucial as for sigmoidal ones. This is due to the fact that their gradients do not diminish (they are acutally more likely to explode) and they do not saturate (ReLU saturates at 0, but not on the positive slope, where gradient is 1 everywhere).  (In practice ReLU is sometimes \"clipped\" with a maximum value, typically 20).\n"
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    "# Exercise 1:  Implement the tanh transfer function\n",
    "\n",
    "Your implementation should follow the code conventions used to build other layer types (for example, Sigmoid and Softmax). Test your solution by training a one-hidden-layer model with 100 hidden units, similiar to the one used in Task 3a in the coursework. \n",
    "\n",
    "Tune the learning rate and compare the initial ranges in equations (7) and (8). Note that there might not be much difference for one-hidden-layer model, but you can easily notice a substantial gain from using (8) (or (9) for  logistic sigmoid activation) for deeper models, for example, the 5 hidden-layer network from the first coursework.\n",
    "\n",
    "Implementation tip: Use numpy.tanh() to compute the non-linearity.  Use the irange argument when creating the given layer type to provide the initial sampling range."
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    "# Exercise 2: Implement ReLU\n",
    "\n",
    "Again, your implementation should follow the conventions used to build Linear, Sigmoid and Softmax layers. As in exercise 1, test your solution by training a one-hidden-layer model with 100 hidden units, similiar to the one used in Task 3a in the coursework. Tune the learning rate (start with the initial one set to 0.1) with the initial weight range set to 0.05."
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    "# Exercise 3: Implement Maxout\n",
    "\n",
    "As with the previous two exercises, your implementation should follow the conventions used to build the Linear, Sigmoid and Softmax layers. For now implement only non-overlapping pools (i.e. the pool in which all activations $a_{j}, a_{j+1}, \\ldots, a_{j+K}$ belong to only one pool). As before, test your solution by training a one-hidden-layer model with 100 hidden units, similiar to the one used in Task 3a in the coursework. Use the same optimisation hyper-parameters (learning rate, initial weights range) as you used for ReLU models. Tune the pool size $K$ (but keep the number of total parameters fixed).\n",
    "\n",
    "Note: The Max operator reduces dimensionality, hence for example, to get 100 hidden maxout units with pooling size set to $K=2$ the size of linear part needs to be set to $100K$ (assuming non-overlapping pools). This affects how you compute the total number of weights in the model.\n",
    "\n",
    "Implementation tips: To back-propagate through the maxout layer, one needs to keep track of which linear activation $a_{j}, a_{j+1}, \\ldots, a_{j+K}$ was the maximum in each pool. The convenient way to do so is by storing the indices of the maximum units in the fprop function and then in the backprop stage pass the gradient only through those (i.e. for example, one can build an auxiliary matrix where each element is either 1 (if unit was maximum, and passed forward through the max operator for a given data-point) or 0 otherwise. Then in the backward pass it suffices to upsample the maxout *igrads* signal to the linear layer dimension and element-wise multiply by the aforemenioned auxiliary matrix.\n",
    "\n",
    "*Optional:* Implement the generic pooling mechanism by introducing an additional *stride* hyper-parameter $0<S\\leq K$. It specifies how many units you move to build the next pool. For instance, for non-overlapping pooling with $S=K=3$ one would build the first two maxout units as: $h_1=\\max(a_1,a_2,a_3)$ and $h_2=\\max(a_4,a_5,a_6)$. However, after setting $S=1$ the pools should share some subset of linear activations: $h_1=\\max(a_1,a_2,a_3)$ and $h_2=\\max(a_2,a_3,a_4)$."
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    "# Exercise 4: Train all the above models with dropout\n",
    "\n",
    "Try all of the above non-linearities with dropout training. Use the dropout hyper-parameters $\\{p_{inp}, p_{hid}\\}$ that worked best for sigmoid models from the previous lab.\n",
    "\n",
    "Note: the code for dropout you were asked to implement last week has not been given as a solution for this week - as a result you need to move/merge the required dropout parts from your previous *lab4* branch (or implement it if you haven't already done so). \n"
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