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mlpractical/notebooks/02_Single_layer_models.ipynb

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Initial updates to lab 2 notebook.
2016-09-28 06:07:40 +02:00
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$\\newcommand{\\vct}[1]{\\boldsymbol{#1}}\n",
"\\newcommand{\\mtx}[1]{\\mathbf{#1}}\n",
"\\newcommand{\\tr}{^\\mathrm{T}}\n",
"\\newcommand{\\reals}{\\mathbb{R}}\n",
"\\newcommand{\\lpa}{\\left(}\n",
"\\newcommand{\\rpa}{\\right)}\n",
"\\newcommand{\\lsb}{\\left[}\n",
"\\newcommand{\\rsb}{\\right]}\n",
"\\newcommand{\\lbr}{\\left\\lbrace}\n",
"\\newcommand{\\rbr}{\\right\\rbrace}\n",
"\\newcommand{\\fset}[1]{\\lbr #1 \\rbr}\n",
"\\newcommand{\\pd}[2]{\\frac{\\partial #1}{\\partial #2}}$\n",
"\n",
"# Single layer models\n",
"\n",
"In this lab we will implement a single-layer network model consisting of solely of an affine transformation of the inputs. The relevant material for this was covered in [the slides of the first lecture](http://www.inf.ed.ac.uk/teaching/courses/mlp/2016/mlp01-intro.pdf). \n",
"\n",
"We will first implement the forward propagation of inputs to the network to produce predicted outputs. We will then move on to considering how to use gradients of a cost function evaluated on the outputs to compute the gradients with respect to the model parameters to allow us to perform an iterative gradient-descent training procedure.\n",
"\n",
"## 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 each experiment. This makes it easier to reproduce results as the same random draws will produced each time the experiment is run (e.g. the same random initialisations used for parameters). \n",
"\n",
"Therefore generally when we need to generate random values during this course, we will create a seeded random number generator object as follows:"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"import numpy as np\n",
"\n",
"# Seed a random number generator to be used later\n",
"seed = 27092016 \n",
"rng = np.random.RandomState(seed)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exercise 1: linear and affine transforms\n",
"\n",
"Any *linear transform* (also called a linear map) of a finite-dimensional vector space can be parametrised by a matrix. So for example if we consider $\\vct{x} \\in \\reals^{M}$ as the input space of a model with $M$ dimensional real-valued inputs, then a matrix $\\mtx{W} \\in \\reals^{N\\times M}$ can be used to define a prediction model consisting solely of a linear transform of the inputs\n",
"\n",
"\\begin{equation}\n",
" \\vct{y} = \\mtx{W} \\vct{x}\n",
" \\qquad\n",
" \\Leftrightarrow\n",
" \\qquad\n",
" y_i = \\sum_{j=1}^M \\lpa W_{ij} x_j \\rpa \\qquad \\forall i \\in \\fset{1 \\dots N}\n",
"\\end{equation}\n",
"\n",
"with here $\\vct{y} \\in \\reals^N$ the $N$-dimensional real-valued output of the model. Geometrically we can think of a linear transform doing some combination of rotation, scaling, reflection and shearing of the input.\n",
"\n",
"An *affine transform* consists of a linear transform plus an additional translation parameterised by a vector $\\vct{b} \\in \\reals^N$. A model consisting of an affine transformation of the inputs can then be defined as\n",
"\n",
"\\begin{equation}\n",
" \\vct{y} = \\mtx{W}\\vct{x} + \\vct{b}\n",
" \\qquad\n",
" \\Leftrightarrow\n",
" \\qquad\n",
" y_i = \\sum_{j=1}^M \\lpa W_{ij} x_j \\rpa + b_i \\qquad \\forall i \\in \\fset{1 \\dots N}\n",
"\\end{equation}\n",
"\n",
"In machine learning we will usually refer to the matrix $\\mtx{W}$ as a *weight matrix* and the vector $\\vct{b}$ as a *bias vector*. \n",
"\n",
"Implement *forward propagation* for a single-layer model consisting of an affine transformation of the inputs. This should work for a batch of inputs of shape `(batch_size, input_dim)` producing a batch of outputs of shape `(batch_size, output_dim)`. \n",
"\n",
"You may find the NumPy [`dot`](http://docs.scipy.org/doc/numpy/reference/generated/numpy.dot.html) function and [broadcasting features](http://docs.scipy.org/doc/numpy/user/basics.broadcasting.html) useful."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def fprop(inputs, weights, biases):\n",
" \"\"\"Forward propagates activations through the layer transformation.\n",
"\n",
" For inputs `x`, outputs `y`, weights `W` and biases `b` the layer\n",
" corresponds to `y = W x + b`.\n",
"\n",
" Args:\n",
" inputs: Array of layer inputs of shape (batch_size, input_dim).\n",
" weights: Array of weight parameters of shape \n",
" (output_dim, input_dim).\n",
" biases: Array of bias parameters of shape (output_dim, ).\n",
"\n",
" Returns:\n",
" outputs: Array of layer outputs of shape (batch_size, output_dim).\n",
" \"\"\"\n",
" return weights.dot(inputs.T).T + biases"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Check your implementation by running the test cell below"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"All outputs correct!\n"
]
}
],
"source": [
"inputs = np.array([[0., -1., 2.], [-6., 3., 1.]])\n",
"weights = np.array([[2., -3., -1.], [-5., 7., 2.]])\n",
"biases = np.array([5., -3.])\n",
"true_outputs = np.array([[6., -6.], [-17., 50.]])\n",
"\n",
"if not np.allclose(fprop(inputs, weights, biases), true_outputs):\n",
" print('Wrong outputs computed.')\n",
"else:\n",
" print('All outputs correct!')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exercise 2: visualising random models"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In this exercise you will use your `fprop` implementation to visualise the outputs of a single-layer affine transform model with two-dimensional inputs and a one-dimensional output. In this simple case we can visualise the joint input-output space on a 3D axis.\n",
"\n",
"For this task and the learning experiments later in the notebook we will use a regression dataset from the [UCI machine learning repository](http://archive.ics.uci.edu/ml/index.html). In particular we will use a version of the [Combined Cycle Power Plant dataset](http://archive.ics.uci.edu/ml/datasets/Combined+Cycle+Power+Plant), where the task is to predict the energy output of a power plant given observations of the local ambient conditions (e.g. temperature, pressure and humidity).\n",
"\n",
"The original dataset has four input dimensions and a single target output dimension. We have preprocessed the dataset by [whitening](https://en.wikipedia.org/wiki/Whitening_transformation) it, a common preprocessing step. We will only use the first two dimensions of the whitened inputs (corresponding to the first two principal components of the inputs) so we can easily visualise the joint input-output space.\n",
"\n",
"The dataset has been wrapped in the `CCPPDataProvider` class in the `mlp.data_providers` module. Running the cell below will initialise an instance of this class, get a single batch of inputs and outputs and import the necessary `matplotlib` objects."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"import matplotlib.pyplot as plt\n",
"from mpl_toolkits.mplot3d import Axes3D\n",
"from mlp.data_providers import CCPPDataProvider\n",
"%matplotlib notebook\n",
"\n",
"data_provider = CCPPDataProvider(\n",
" which_set='train',\n",
" input_dims=[0, 1],\n",
" batch_size=5000, \n",
" max_num_batches=1, \n",
" shuffle_order=False\n",
")\n",
"\n",
"input_dim, output_dim = 2, 1\n",
"\n",
"inputs, targets = data_provider.next()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now run the cell below to plot the predicted outputs of a randomly initialised model across the two dimensional input space as well as the true target outputs. This sort of visualisation can be a useful method (in low dimensions) to assess how well the model is likely to be able to fit the data and to judge appropriate initialisation scales for the parameters. Each time you re-run the cell a new set of random parameters will be sampled\n",
"\n",
"Some questions to consider:\n",
"\n",
" * How do the weights and bias initialisation scale affect the sort of predicted input-output relationships?\n",
" * Does the linear form of the model seem a good fit for the data here?"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
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" canvas.addClass('mpl-canvas');\n",
" canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n",
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"\n",
" var rubberband = $('<canvas/>');\n",
" rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n",
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"\n",
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"\n",
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"\n",
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" if (evt.data instanceof Blob) {\n",
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" * Chrome. But how to set the MIME type? It doesn't seem\n",
" * to be part of the websocket stream */\n",
" evt.data.type = \"image/png\";\n",
"\n",
" /* Free the memory for the previous frames */\n",
" if (fig.imageObj.src) {\n",
" (window.URL || window.webkitURL).revokeObjectURL(\n",
" fig.imageObj.src);\n",
" }\n",
"\n",
" fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
" evt.data);\n",
" fig.updated_canvas_event();\n",
" fig.waiting = false;\n",
" return;\n",
" }\n",
" else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n",
" fig.imageObj.src = evt.data;\n",
" fig.updated_canvas_event();\n",
" fig.waiting = false;\n",
" return;\n",
" }\n",
"\n",
" var msg = JSON.parse(evt.data);\n",
" var msg_type = msg['type'];\n",
"\n",
" // Call the \"handle_{type}\" callback, which takes\n",
" // the figure and JSON message as its only arguments.\n",
" try {\n",
" var callback = fig[\"handle_\" + msg_type];\n",
" } catch (e) {\n",
" console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n",
" return;\n",
" }\n",
"\n",
" if (callback) {\n",
" try {\n",
" // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
" callback(fig, msg);\n",
" } catch (e) {\n",
" console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n",
" }\n",
" }\n",
" };\n",
"}\n",
"\n",
"// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
"mpl.findpos = function(e) {\n",
" //this section is from http://www.quirksmode.org/js/events_properties.html\n",
" var targ;\n",
" if (!e)\n",
" e = window.event;\n",
" if (e.target)\n",
" targ = e.target;\n",
" else if (e.srcElement)\n",
" targ = e.srcElement;\n",
" if (targ.nodeType == 3) // defeat Safari bug\n",
" targ = targ.parentNode;\n",
"\n",
" // jQuery normalizes the pageX and pageY\n",
" // pageX,Y are the mouse positions relative to the document\n",
" // offset() returns the position of the element relative to the document\n",
" var x = e.pageX - $(targ).offset().left;\n",
" var y = e.pageY - $(targ).offset().top;\n",
"\n",
" return {\"x\": x, \"y\": y};\n",
"};\n",
"\n",
"/*\n",
" * return a copy of an object with only non-object keys\n",
" * we need this to avoid circular references\n",
" * http://stackoverflow.com/a/24161582/3208463\n",
" */\n",
"function simpleKeys (original) {\n",
" return Object.keys(original).reduce(function (obj, key) {\n",
" if (typeof original[key] !== 'object')\n",
" obj[key] = original[key]\n",
" return obj;\n",
" }, {});\n",
"}\n",
"\n",
"mpl.figure.prototype.mouse_event = function(event, name) {\n",
" var canvas_pos = mpl.findpos(event)\n",
"\n",
" if (name === 'button_press')\n",
" {\n",
" this.canvas.focus();\n",
" this.canvas_div.focus();\n",
" }\n",
"\n",
" var x = canvas_pos.x;\n",
" var y = canvas_pos.y;\n",
"\n",
" this.send_message(name, {x: x, y: y, button: event.button,\n",
" step: event.step,\n",
" guiEvent: simpleKeys(event)});\n",
"\n",
" /* This prevents the web browser from automatically changing to\n",
" * the text insertion cursor when the button is pressed. We want\n",
" * to control all of the cursor setting manually through the\n",
" * 'cursor' event from matplotlib */\n",
" event.preventDefault();\n",
" return false;\n",
"}\n",
"\n",
"mpl.figure.prototype._key_event_extra = function(event, name) {\n",
" // Handle any extra behaviour associated with a key event\n",
"}\n",
"\n",
"mpl.figure.prototype.key_event = function(event, name) {\n",
"\n",
" // Prevent repeat events\n",
" if (name == 'key_press')\n",
" {\n",
" if (event.which === this._key)\n",
" return;\n",
" else\n",
" this._key = event.which;\n",
" }\n",
" if (name == 'key_release')\n",
" this._key = null;\n",
"\n",
" var value = '';\n",
" if (event.ctrlKey && event.which != 17)\n",
" value += \"ctrl+\";\n",
" if (event.altKey && event.which != 18)\n",
" value += \"alt+\";\n",
" if (event.shiftKey && event.which != 16)\n",
" value += \"shift+\";\n",
"\n",
" value += 'k';\n",
" value += event.which.toString();\n",
"\n",
" this._key_event_extra(event, name);\n",
"\n",
" this.send_message(name, {key: value,\n",
" guiEvent: simpleKeys(event)});\n",
" return false;\n",
"}\n",
"\n",
"mpl.figure.prototype.toolbar_button_onclick = function(name) {\n",
" if (name == 'download') {\n",
" this.handle_save(this, null);\n",
" } else {\n",
" this.send_message(\"toolbar_button\", {name: name});\n",
" }\n",
"};\n",
"\n",
"mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n",
" this.message.textContent = tooltip;\n",
"};\n",
"mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
"\n",
"mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n",
"\n",
"mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n",
" // Create a \"websocket\"-like object which calls the given IPython comm\n",
" // object with the appropriate methods. Currently this is a non binary\n",
" // socket, so there is still some room for performance tuning.\n",
" var ws = {};\n",
"\n",
" ws.close = function() {\n",
" comm.close()\n",
" };\n",
" ws.send = function(m) {\n",
" //console.log('sending', m);\n",
" comm.send(m);\n",
" };\n",
" // Register the callback with on_msg.\n",
" comm.on_msg(function(msg) {\n",
" //console.log('receiving', msg['content']['data'], msg);\n",
" // Pass the mpl event to the overriden (by mpl) onmessage function.\n",
" ws.onmessage(msg['content']['data'])\n",
" });\n",
" return ws;\n",
"}\n",
"\n",
"mpl.mpl_figure_comm = function(comm, msg) {\n",
" // This is the function which gets called when the mpl process\n",
" // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
"\n",
" var id = msg.content.data.id;\n",
" // Get hold of the div created by the display call when the Comm\n",
" // socket was opened in Python.\n",
" var element = $(\"#\" + id);\n",
" var ws_proxy = comm_websocket_adapter(comm)\n",
"\n",
" function ondownload(figure, format) {\n",
" window.open(figure.imageObj.src);\n",
" }\n",
"\n",
" var fig = new mpl.figure(id, ws_proxy,\n",
" ondownload,\n",
" element.get(0));\n",
"\n",
" // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
" // web socket which is closed, not our websocket->open comm proxy.\n",
" ws_proxy.onopen();\n",
"\n",
" fig.parent_element = element.get(0);\n",
" fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n",
" if (!fig.cell_info) {\n",
" console.error(\"Failed to find cell for figure\", id, fig);\n",
" return;\n",
" }\n",
"\n",
" var output_index = fig.cell_info[2]\n",
" var cell = fig.cell_info[0];\n",
"\n",
"};\n",
"\n",
"mpl.figure.prototype.handle_close = function(fig, msg) {\n",
" fig.root.unbind('remove')\n",
"\n",
" // Update the output cell to use the data from the current canvas.\n",
" fig.push_to_output();\n",
" var dataURL = fig.canvas.toDataURL();\n",
" // Re-enable the keyboard manager in IPython - without this line, in FF,\n",
" // the notebook keyboard shortcuts fail.\n",
" IPython.keyboard_manager.enable()\n",
" $(fig.parent_element).html('<img src=\"' + dataURL + '\">');\n",
" fig.close_ws(fig, msg);\n",
"}\n",
"\n",
"mpl.figure.prototype.close_ws = function(fig, msg){\n",
" fig.send_message('closing', msg);\n",
" // fig.ws.close()\n",
"}\n",
"\n",
"mpl.figure.prototype.push_to_output = function(remove_interactive) {\n",
" // Turn the data on the canvas into data in the output cell.\n",
" var dataURL = this.canvas.toDataURL();\n",
" this.cell_info[1]['text/html'] = '<img src=\"' + dataURL + '\">';\n",
"}\n",
"\n",
"mpl.figure.prototype.updated_canvas_event = function() {\n",
" // Tell IPython that the notebook contents must change.\n",
" IPython.notebook.set_dirty(true);\n",
" this.send_message(\"ack\", {});\n",
" var fig = this;\n",
" // Wait a second, then push the new image to the DOM so\n",
" // that it is saved nicely (might be nice to debounce this).\n",
" setTimeout(function () { fig.push_to_output() }, 1000);\n",
"}\n",
"\n",
"mpl.figure.prototype._init_toolbar = function() {\n",
" var fig = this;\n",
"\n",
" var nav_element = $('<div/>')\n",
" nav_element.attr('style', 'width: 100%');\n",
" this.root.append(nav_element);\n",
"\n",
" // Define a callback function for later on.\n",
" function toolbar_event(event) {\n",
" return fig.toolbar_button_onclick(event['data']);\n",
" }\n",
" function toolbar_mouse_event(event) {\n",
" return fig.toolbar_button_onmouseover(event['data']);\n",
" }\n",
"\n",
" for(var toolbar_ind in mpl.toolbar_items){\n",
" var name = mpl.toolbar_items[toolbar_ind][0];\n",
" var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
" var image = mpl.toolbar_items[toolbar_ind][2];\n",
" var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
"\n",
" if (!name) { continue; };\n",
"\n",
" var button = $('<button class=\"btn btn-default\" href=\"#\" title=\"' + name + '\"><i class=\"fa ' + image + ' fa-lg\"></i></button>');\n",
" button.click(method_name, toolbar_event);\n",
" button.mouseover(tooltip, toolbar_mouse_event);\n",
" nav_element.append(button);\n",
" }\n",
"\n",
" // Add the status bar.\n",
" var status_bar = $('<span class=\"mpl-message\" style=\"text-align:right; float: right;\"/>');\n",
" nav_element.append(status_bar);\n",
" this.message = status_bar[0];\n",
"\n",
" // Add the close button to the window.\n",
" var buttongrp = $('<div class=\"btn-group inline pull-right\"></div>');\n",
" var button = $('<button class=\"btn btn-mini btn-primary\" href=\"#\" title=\"Stop Interaction\"><i class=\"fa fa-power-off icon-remove icon-large\"></i></button>');\n",
" button.click(function (evt) { fig.handle_close(fig, {}); } );\n",
" button.mouseover('Stop Interaction', toolbar_mouse_event);\n",
" buttongrp.append(button);\n",
" var titlebar = this.root.find($('.ui-dialog-titlebar'));\n",
" titlebar.prepend(buttongrp);\n",
"}\n",
"\n",
"mpl.figure.prototype._root_extra_style = function(el){\n",
" var fig = this\n",
" el.on(\"remove\", function(){\n",
"\tfig.close_ws(fig, {});\n",
" });\n",
"}\n",
"\n",
"mpl.figure.prototype._canvas_extra_style = function(el){\n",
" // this is important to make the div 'focusable\n",
" el.attr('tabindex', 0)\n",
" // reach out to IPython and tell the keyboard manager to turn it's self\n",
" // off when our div gets focus\n",
"\n",
" // location in version 3\n",
" if (IPython.notebook.keyboard_manager) {\n",
" IPython.notebook.keyboard_manager.register_events(el);\n",
" }\n",
" else {\n",
" // location in version 2\n",
" IPython.keyboard_manager.register_events(el);\n",
" }\n",
"\n",
"}\n",
"\n",
"mpl.figure.prototype._key_event_extra = function(event, name) {\n",
" var manager = IPython.notebook.keyboard_manager;\n",
" if (!manager)\n",
" manager = IPython.keyboard_manager;\n",
"\n",
" // Check for shift+enter\n",
" if (event.shiftKey && event.which == 13) {\n",
" this.canvas_div.blur();\n",
" // select the cell after this one\n",
" var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n",
" IPython.notebook.select(index + 1);\n",
" }\n",
"}\n",
"\n",
"mpl.figure.prototype.handle_save = function(fig, msg) {\n",
" fig.ondownload(fig, null);\n",
"}\n",
"\n",
"\n",
"mpl.find_output_cell = function(html_output) {\n",
" // Return the cell and output element which can be found *uniquely* in the notebook.\n",
" // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n",
" // IPython event is triggered only after the cells have been serialised, which for\n",
" // our purposes (turning an active figure into a static one), is too late.\n",
" var cells = IPython.notebook.get_cells();\n",
" var ncells = cells.length;\n",
" for (var i=0; i<ncells; i++) {\n",
" var cell = cells[i];\n",
" if (cell.cell_type === 'code'){\n",
" for (var j=0; j<cell.output_area.outputs.length; j++) {\n",
" var data = cell.output_area.outputs[j];\n",
" if (data.data) {\n",
" // IPython >= 3 moved mimebundle to data attribute of output\n",
" data = data.data;\n",
" }\n",
" if (data['text/html'] == html_output) {\n",
" return [cell, data, j];\n",
" }\n",
" }\n",
" }\n",
" }\n",
"}\n",
"\n",
"// Register the function which deals with the matplotlib target/channel.\n",
"// The kernel may be null if the page has been refreshed.\n",
"if (IPython.notebook.kernel != null) {\n",
" IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n",
"}\n"
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"weights_init_range = 0.5\n",
"biases_init_range = 0.1\n",
"\n",
"# Randomly initialise weights matrix\n",
"weights = rng.uniform(\n",
" low=-weights_init_range, \n",
" high=weights_init_range, \n",
" size=(output_dim, input_dim)\n",
")\n",
"\n",
"# Randomly initialise biases vector\n",
"biases = rng.uniform(\n",
" low=-biases_init_range, \n",
" high=biases_init_range, \n",
" size=output_dim\n",
")\n",
"# Calculate predicted model outputs\n",
"outputs = fprop(inputs, weights, biases)\n",
"\n",
"# Plot target and predicted outputs against inputs on same axis\n",
"fig = plt.figure(figsize=(8, 8))\n",
"ax = fig.add_subplot(111, projection='3d')\n",
"ax.plot(inputs[:, 0], inputs[:, 1], targets[:, 0], 'r.', ms=2)\n",
"ax.plot(inputs[:, 0], inputs[:, 1], outputs[:, 0], 'b.', ms=2)\n",
"ax.set_xlabel('Input dim 1')\n",
"ax.set_ylabel('Input dim 2')\n",
"ax.set_zlabel('Output')\n",
"ax.legend(['Targets', 'Predictions'], frameon=False)\n",
"fig.tight_layout()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exercise 3: computing the SSE cost function and its gradient"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\\begin{equation}\n",
" C = \\frac{1}{2} \\sum_{i=1}^N \\lbr (y_i - t_i)^2 \\rbr\n",
"\\end{equation}\n",
"\n",
"\\begin{equation}\n",
" \\pd{C}{y_k} = y_k - t_k\n",
"\\end{equation}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
" * Implement sum squared error cost function and gradient with respect to outputs"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def cost(outputs, targets):\n",
" \"\"\"Calculates cost function given a batch of outputs and targets.\n",
"\n",
" Args:\n",
" outputs: Array of model outputs of shape (batch_size, output_dim).\n",
" targets: Array of target outputs of shape (batch_size, output_dim).\n",
"\n",
" Returns:\n",
" Scalar cost function value.\n",
" \"\"\"\n",
" return 0.5 * np.mean(np.sum((outputs - targets)**2, axis=1))\n",
" \n",
"def cost_grad(outputs, targets):\n",
" \"\"\"Calculates gradient of cost function with respect to outputs.\n",
"\n",
" Args:\n",
" outputs: Array of model outputs of shape (batch_size, output_dim).\n",
" targets: Array of target outputs of shape (batch_size, output_dim).\n",
"\n",
" Returns:\n",
" Gradient of cost function with respect to outputs.\n",
" \"\"\"\n",
" return outputs - targets"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Check your implementation by running the test cell below"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Cost function and gradient computed correctly!\n"
]
}
],
"source": [
"outputs = np.array([[1., 2.], [-1., 0.], [6., -5.], [-1., 1.]])\n",
"targets = np.array([[0., 1.], [3., -2.], [7., -3.], [1., -2.]])\n",
"true_cost = 5.\n",
"true_cost_grad = np.array([[1., 1.], [-4., 2.], [-1., -2.], [-2., 3.]])\n",
"\n",
"if not cost(outputs, targets) == true_cost:\n",
" print('Cost calculated incorrectly.')\n",
"elif not np.allclose(cost_grad(outputs, targets), true_cost_grad):\n",
" print('Cost gradient calculated incorrectly.')\n",
"else:\n",
" print('Cost function and gradient computed correctly!')"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": true
},
"source": [
"## Exercise 4: computing gradients with respect to the parameters\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\\begin{equation}\n",
" \\pd{C}{W_{ij}} = \\sum_{k=1}^N \\lbr \\pd{C}{y_k} \\pd{y_k}{W_{ij}} \\rbr\n",
" \\qquad\n",
" \\pd{y_k}{W_{ij}} = \\delta_{ik} x_j\n",
"\\end{equation}\n",
"\n",
"\\begin{equation}\n",
" \\pd{C}{b_{i}} = \\sum_{k=1}^N \\lbr \\pd{C}{y_k} \\pd{y_k}{b_{i}} \\rbr\n",
" \\qquad\n",
" \\pd{y_k}{b_i} = \\delta_{ik}\n",
"\\end{equation}\n",
"\n",
" * Implement function to calculate gradient with respect to weight and bias parameters given gradient with respect to outputs "
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def grads_wrt_params(inputs, grads_wrt_outputs):\n",
" \"\"\"Calculates gradients with respect to model parameters.\n",
"\n",
" Args:\n",
" inputs: array of inputs to model of shape (batch_size, input_dim)\n",
" grads_wrt_to_outputs: array of gradients with respect to the model\n",
" outputs of shape (batch_size, output_dim)\n",
"\n",
" Returns:\n",
" list of arrays of gradients with respect to the model parameters\n",
" `[grads_wrt_weights, grads_wrt_biases]`.\n",
" \"\"\"\n",
" grads_wrt_weights = np.dot(grads_wrt_outputs.T, inputs)\n",
" grads_wrt_biases = np.sum(grads_wrt_outputs, axis=0)\n",
" return [grads_wrt_weights, grads_wrt_biases]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Check your implementation by running the test cell below"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"All parameter gradients calculated correctly!\n"
]
}
],
"source": [
"inputs = np.array([[1., 2., 3.], [-1., 4., -9.]])\n",
"grads_wrt_outputs = np.array([[-1., 1.], [2., -3.]])\n",
"true_grads_wrt_weights = np.array([[-3., 6., -21.], [4., -10., 30.]])\n",
"true_grads_wrt_biases = np.array([1., -2.])\n",
"\n",
"grads_wrt_weights, grads_wrt_biases = grads_wrt_params(\n",
" inputs, grads_wrt_outputs)\n",
"\n",
"if not np.allclose(true_grads_wrt_weights, grads_wrt_weights):\n",
" print('Gradients with respect to weights incorrect.')\n",
"elif not np.allclose(true_grads_wrt_biases, grads_wrt_biases):\n",
" print('Gradients with respect to biases incorrect.')\n",
"else:\n",
" print('All parameter gradients calculated correctly!')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exercise 5: wrapping the functions into reusable components\n",
"\n",
"In the previous exercises you implemented methods to compute the predicted outputs of our model, evaluate the cost function and its gradient on the outputs and finally to calculate the gradients of the cost with respect to the model parameters. Together they constitute all the basic ingredients we need to implement a gradient-descent based iterative learning procedure for the model.\n",
"\n",
"Although you could implement training code which directly uses the functions you defined, this would only be usable for this particular model architecture. In subsequent labs we will want to use the affine transform functions as the basis for more interesting multi-layer models. We will therefore wrap the implementations you just wrote in to reusable components that we can build more complex models with later in the course.\n",
"\n",
" * In the `mlp.layers` module, use your implementations of `fprop` and `grad_wrt_params` above to implement the corresponding methods in the skeleton `AffineLayer` class provided.\n",
" * In the `mlp.costs` module use your implementation of `cost` and `cost_grad` to implement the `__call__` and `grad` methods respectively of the skeleton `MeanSquaredErrorCost` class provided. Note `__call__` is a special Python method that allows an object to be used with a function call syntax."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Run the cell below to use your completed `AffineLayer` and `MeanSquaredErrorCost` implementations to train a single-layer model using gradient descent on the CCCP dataset."
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"Epoch 1: 0.0s to complete\n",
" cost(train)=7.66e-02, cost(valid)=7.66e-02, cost(param)=0.00e+00\n",
"Epoch 2: 0.0s to complete\n",
" cost(train)=7.67e-02, cost(valid)=7.66e-02, cost(param)=0.00e+00\n",
"Epoch 3: 0.0s to complete\n",
" cost(train)=7.66e-02, cost(valid)=7.66e-02, cost(param)=0.00e+00\n",
"Epoch 4: 0.0s to complete\n",
" cost(train)=7.65e-02, cost(valid)=7.67e-02, cost(param)=0.00e+00\n",
"Epoch 5: 0.0s to complete\n",
" cost(train)=7.67e-02, cost(valid)=7.68e-02, cost(param)=0.00e+00\n"
]
}
],
"source": [
"from mlp.layers import AffineLayer\n",
"from mlp.costs import MeanSquaredErrorCost\n",
"from mlp.models import SingleLayerModel\n",
"from mlp.initialisers import UniformInit, ConstantInit\n",
"from mlp.learning_rules import GradientDescentLearningRule\n",
"from mlp.optimisers import Optimiser\n",
"import logging\n",
"\n",
"logger = logging.getLogger()\n",
"logger.setLevel(logging.INFO)\n",
"logger.addHandler(logging.StreamHandler())\n",
"\n",
"train_data = CCPPDataProvider('train', [0, 1], batch_size=100)\n",
"valid_data = CCPPDataProvider('train', [0, 1], batch_size=100)\n",
"input_dim, output_dim = 2, 1\n",
"\n",
"layer = AffineLayer(input_dim, output_dim, UniformInit(-0.1, 0.1, rng=rng), ConstantInit(0))\n",
"model = SingleLayerModel(layer)\n",
"cost = MeanSquaredErrorCost()\n",
"learning_rule = GradientDescentLearningRule(learning_rate=1e-3)\n",
"optimiser = Optimiser(model, cost, learning_rule, train_data, valid_data)\n",
"stats = optimiser.train(5, 1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Using similar code to previously we can now visualise the joint input-output space for the trained model. If you implemented the required methods correctly you should now see a much improved fit between predicted and target outputs when running the cell below."
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"collapsed": false
},
"outputs": [
{
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"\n",
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"\n",
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"}\n",
"\n",
"mpl.figure.prototype.handle_cursor = function(fig, msg) {\n",
" var cursor = msg['cursor'];\n",
" switch(cursor)\n",
" {\n",
" case 0:\n",
" cursor = 'pointer';\n",
" break;\n",
" case 1:\n",
" cursor = 'default';\n",
" break;\n",
" case 2:\n",
" cursor = 'crosshair';\n",
" break;\n",
" case 3:\n",
" cursor = 'move';\n",
" break;\n",
" }\n",
" fig.rubberband_canvas.style.cursor = cursor;\n",
"}\n",
"\n",
"mpl.figure.prototype.handle_message = function(fig, msg) {\n",
" fig.message.textContent = msg['message'];\n",
"}\n",
"\n",
"mpl.figure.prototype.handle_draw = function(fig, msg) {\n",
" // Request the server to send over a new figure.\n",
" fig.send_draw_message();\n",
"}\n",
"\n",
"mpl.figure.prototype.handle_image_mode = function(fig, msg) {\n",
" fig.image_mode = msg['mode'];\n",
"}\n",
"\n",
"mpl.figure.prototype.updated_canvas_event = function() {\n",
" // Called whenever the canvas gets updated.\n",
" this.send_message(\"ack\", {});\n",
"}\n",
"\n",
"// A function to construct a web socket function for onmessage handling.\n",
"// Called in the figure constructor.\n",
"mpl.figure.prototype._make_on_message_function = function(fig) {\n",
" return function socket_on_message(evt) {\n",
" if (evt.data instanceof Blob) {\n",
" /* FIXME: We get \"Resource interpreted as Image but\n",
" * transferred with MIME type text/plain:\" errors on\n",
" * Chrome. But how to set the MIME type? It doesn't seem\n",
" * to be part of the websocket stream */\n",
" evt.data.type = \"image/png\";\n",
"\n",
" /* Free the memory for the previous frames */\n",
" if (fig.imageObj.src) {\n",
" (window.URL || window.webkitURL).revokeObjectURL(\n",
" fig.imageObj.src);\n",
" }\n",
"\n",
" fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
" evt.data);\n",
" fig.updated_canvas_event();\n",
" fig.waiting = false;\n",
" return;\n",
" }\n",
" else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n",
" fig.imageObj.src = evt.data;\n",
" fig.updated_canvas_event();\n",
" fig.waiting = false;\n",
" return;\n",
" }\n",
"\n",
" var msg = JSON.parse(evt.data);\n",
" var msg_type = msg['type'];\n",
"\n",
" // Call the \"handle_{type}\" callback, which takes\n",
" // the figure and JSON message as its only arguments.\n",
" try {\n",
" var callback = fig[\"handle_\" + msg_type];\n",
" } catch (e) {\n",
" console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n",
" return;\n",
" }\n",
"\n",
" if (callback) {\n",
" try {\n",
" // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
" callback(fig, msg);\n",
" } catch (e) {\n",
" console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n",
" }\n",
" }\n",
" };\n",
"}\n",
"\n",
"// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
"mpl.findpos = function(e) {\n",
" //this section is from http://www.quirksmode.org/js/events_properties.html\n",
" var targ;\n",
" if (!e)\n",
" e = window.event;\n",
" if (e.target)\n",
" targ = e.target;\n",
" else if (e.srcElement)\n",
" targ = e.srcElement;\n",
" if (targ.nodeType == 3) // defeat Safari bug\n",
" targ = targ.parentNode;\n",
"\n",
" // jQuery normalizes the pageX and pageY\n",
" // pageX,Y are the mouse positions relative to the document\n",
" // offset() returns the position of the element relative to the document\n",
" var x = e.pageX - $(targ).offset().left;\n",
" var y = e.pageY - $(targ).offset().top;\n",
"\n",
" return {\"x\": x, \"y\": y};\n",
"};\n",
"\n",
"/*\n",
" * return a copy of an object with only non-object keys\n",
" * we need this to avoid circular references\n",
" * http://stackoverflow.com/a/24161582/3208463\n",
" */\n",
"function simpleKeys (original) {\n",
" return Object.keys(original).reduce(function (obj, key) {\n",
" if (typeof original[key] !== 'object')\n",
" obj[key] = original[key]\n",
" return obj;\n",
" }, {});\n",
"}\n",
"\n",
"mpl.figure.prototype.mouse_event = function(event, name) {\n",
" var canvas_pos = mpl.findpos(event)\n",
"\n",
" if (name === 'button_press')\n",
" {\n",
" this.canvas.focus();\n",
" this.canvas_div.focus();\n",
" }\n",
"\n",
" var x = canvas_pos.x;\n",
" var y = canvas_pos.y;\n",
"\n",
" this.send_message(name, {x: x, y: y, button: event.button,\n",
" step: event.step,\n",
" guiEvent: simpleKeys(event)});\n",
"\n",
" /* This prevents the web browser from automatically changing to\n",
" * the text insertion cursor when the button is pressed. We want\n",
" * to control all of the cursor setting manually through the\n",
" * 'cursor' event from matplotlib */\n",
" event.preventDefault();\n",
" return false;\n",
"}\n",
"\n",
"mpl.figure.prototype._key_event_extra = function(event, name) {\n",
" // Handle any extra behaviour associated with a key event\n",
"}\n",
"\n",
"mpl.figure.prototype.key_event = function(event, name) {\n",
"\n",
" // Prevent repeat events\n",
" if (name == 'key_press')\n",
" {\n",
" if (event.which === this._key)\n",
" return;\n",
" else\n",
" this._key = event.which;\n",
" }\n",
" if (name == 'key_release')\n",
" this._key = null;\n",
"\n",
" var value = '';\n",
" if (event.ctrlKey && event.which != 17)\n",
" value += \"ctrl+\";\n",
" if (event.altKey && event.which != 18)\n",
" value += \"alt+\";\n",
" if (event.shiftKey && event.which != 16)\n",
" value += \"shift+\";\n",
"\n",
" value += 'k';\n",
" value += event.which.toString();\n",
"\n",
" this._key_event_extra(event, name);\n",
"\n",
" this.send_message(name, {key: value,\n",
" guiEvent: simpleKeys(event)});\n",
" return false;\n",
"}\n",
"\n",
"mpl.figure.prototype.toolbar_button_onclick = function(name) {\n",
" if (name == 'download') {\n",
" this.handle_save(this, null);\n",
" } else {\n",
" this.send_message(\"toolbar_button\", {name: name});\n",
" }\n",
"};\n",
"\n",
"mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n",
" this.message.textContent = tooltip;\n",
"};\n",
"mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
"\n",
"mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n",
"\n",
"mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n",
" // Create a \"websocket\"-like object which calls the given IPython comm\n",
" // object with the appropriate methods. Currently this is a non binary\n",
" // socket, so there is still some room for performance tuning.\n",
" var ws = {};\n",
"\n",
" ws.close = function() {\n",
" comm.close()\n",
" };\n",
" ws.send = function(m) {\n",
" //console.log('sending', m);\n",
" comm.send(m);\n",
" };\n",
" // Register the callback with on_msg.\n",
" comm.on_msg(function(msg) {\n",
" //console.log('receiving', msg['content']['data'], msg);\n",
" // Pass the mpl event to the overriden (by mpl) onmessage function.\n",
" ws.onmessage(msg['content']['data'])\n",
" });\n",
" return ws;\n",
"}\n",
"\n",
"mpl.mpl_figure_comm = function(comm, msg) {\n",
" // This is the function which gets called when the mpl process\n",
" // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
"\n",
" var id = msg.content.data.id;\n",
" // Get hold of the div created by the display call when the Comm\n",
" // socket was opened in Python.\n",
" var element = $(\"#\" + id);\n",
" var ws_proxy = comm_websocket_adapter(comm)\n",
"\n",
" function ondownload(figure, format) {\n",
" window.open(figure.imageObj.src);\n",
" }\n",
"\n",
" var fig = new mpl.figure(id, ws_proxy,\n",
" ondownload,\n",
" element.get(0));\n",
"\n",
" // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
" // web socket which is closed, not our websocket->open comm proxy.\n",
" ws_proxy.onopen();\n",
"\n",
" fig.parent_element = element.get(0);\n",
" fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n",
" if (!fig.cell_info) {\n",
" console.error(\"Failed to find cell for figure\", id, fig);\n",
" return;\n",
" }\n",
"\n",
" var output_index = fig.cell_info[2]\n",
" var cell = fig.cell_info[0];\n",
"\n",
"};\n",
"\n",
"mpl.figure.prototype.handle_close = function(fig, msg) {\n",
" fig.root.unbind('remove')\n",
"\n",
" // Update the output cell to use the data from the current canvas.\n",
" fig.push_to_output();\n",
" var dataURL = fig.canvas.toDataURL();\n",
" // Re-enable the keyboard manager in IPython - without this line, in FF,\n",
" // the notebook keyboard shortcuts fail.\n",
" IPython.keyboard_manager.enable()\n",
" $(fig.parent_element).html('<img src=\"' + dataURL + '\">');\n",
" fig.close_ws(fig, msg);\n",
"}\n",
"\n",
"mpl.figure.prototype.close_ws = function(fig, msg){\n",
" fig.send_message('closing', msg);\n",
" // fig.ws.close()\n",
"}\n",
"\n",
"mpl.figure.prototype.push_to_output = function(remove_interactive) {\n",
" // Turn the data on the canvas into data in the output cell.\n",
" var dataURL = this.canvas.toDataURL();\n",
" this.cell_info[1]['text/html'] = '<img src=\"' + dataURL + '\">';\n",
"}\n",
"\n",
"mpl.figure.prototype.updated_canvas_event = function() {\n",
" // Tell IPython that the notebook contents must change.\n",
" IPython.notebook.set_dirty(true);\n",
" this.send_message(\"ack\", {});\n",
" var fig = this;\n",
" // Wait a second, then push the new image to the DOM so\n",
" // that it is saved nicely (might be nice to debounce this).\n",
" setTimeout(function () { fig.push_to_output() }, 1000);\n",
"}\n",
"\n",
"mpl.figure.prototype._init_toolbar = function() {\n",
" var fig = this;\n",
"\n",
" var nav_element = $('<div/>')\n",
" nav_element.attr('style', 'width: 100%');\n",
" this.root.append(nav_element);\n",
"\n",
" // Define a callback function for later on.\n",
" function toolbar_event(event) {\n",
" return fig.toolbar_button_onclick(event['data']);\n",
" }\n",
" function toolbar_mouse_event(event) {\n",
" return fig.toolbar_button_onmouseover(event['data']);\n",
" }\n",
"\n",
" for(var toolbar_ind in mpl.toolbar_items){\n",
" var name = mpl.toolbar_items[toolbar_ind][0];\n",
" var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
" var image = mpl.toolbar_items[toolbar_ind][2];\n",
" var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
"\n",
" if (!name) { continue; };\n",
"\n",
" var button = $('<button class=\"btn btn-default\" href=\"#\" title=\"' + name + '\"><i class=\"fa ' + image + ' fa-lg\"></i></button>');\n",
" button.click(method_name, toolbar_event);\n",
" button.mouseover(tooltip, toolbar_mouse_event);\n",
" nav_element.append(button);\n",
" }\n",
"\n",
" // Add the status bar.\n",
" var status_bar = $('<span class=\"mpl-message\" style=\"text-align:right; float: right;\"/>');\n",
" nav_element.append(status_bar);\n",
" this.message = status_bar[0];\n",
"\n",
" // Add the close button to the window.\n",
" var buttongrp = $('<div class=\"btn-group inline pull-right\"></div>');\n",
" var button = $('<button class=\"btn btn-mini btn-primary\" href=\"#\" title=\"Stop Interaction\"><i class=\"fa fa-power-off icon-remove icon-large\"></i></button>');\n",
" button.click(function (evt) { fig.handle_close(fig, {}); } );\n",
" button.mouseover('Stop Interaction', toolbar_mouse_event);\n",
" buttongrp.append(button);\n",
" var titlebar = this.root.find($('.ui-dialog-titlebar'));\n",
" titlebar.prepend(buttongrp);\n",
"}\n",
"\n",
"mpl.figure.prototype._root_extra_style = function(el){\n",
" var fig = this\n",
" el.on(\"remove\", function(){\n",
"\tfig.close_ws(fig, {});\n",
" });\n",
"}\n",
"\n",
"mpl.figure.prototype._canvas_extra_style = function(el){\n",
" // this is important to make the div 'focusable\n",
" el.attr('tabindex', 0)\n",
" // reach out to IPython and tell the keyboard manager to turn it's self\n",
" // off when our div gets focus\n",
"\n",
" // location in version 3\n",
" if (IPython.notebook.keyboard_manager) {\n",
" IPython.notebook.keyboard_manager.register_events(el);\n",
" }\n",
" else {\n",
" // location in version 2\n",
" IPython.keyboard_manager.register_events(el);\n",
" }\n",
"\n",
"}\n",
"\n",
"mpl.figure.prototype._key_event_extra = function(event, name) {\n",
" var manager = IPython.notebook.keyboard_manager;\n",
" if (!manager)\n",
" manager = IPython.keyboard_manager;\n",
"\n",
" // Check for shift+enter\n",
" if (event.shiftKey && event.which == 13) {\n",
" this.canvas_div.blur();\n",
" // select the cell after this one\n",
" var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n",
" IPython.notebook.select(index + 1);\n",
" }\n",
"}\n",
"\n",
"mpl.figure.prototype.handle_save = function(fig, msg) {\n",
" fig.ondownload(fig, null);\n",
"}\n",
"\n",
"\n",
"mpl.find_output_cell = function(html_output) {\n",
" // Return the cell and output element which can be found *uniquely* in the notebook.\n",
" // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n",
" // IPython event is triggered only after the cells have been serialised, which for\n",
" // our purposes (turning an active figure into a static one), is too late.\n",
" var cells = IPython.notebook.get_cells();\n",
" var ncells = cells.length;\n",
" for (var i=0; i<ncells; i++) {\n",
" var cell = cells[i];\n",
" if (cell.cell_type === 'code'){\n",
" for (var j=0; j<cell.output_area.outputs.length; j++) {\n",
" var data = cell.output_area.outputs[j];\n",
" if (data.data) {\n",
" // IPython >= 3 moved mimebundle to data attribute of output\n",
" data = data.data;\n",
" }\n",
" if (data['text/html'] == html_output) {\n",
" return [cell, data, j];\n",
" }\n",
" }\n",
" }\n",
" }\n",
"}\n",
"\n",
"// Register the function which deals with the matplotlib target/channel.\n",
"// The kernel may be null if the page has been refreshed.\n",
"if (IPython.notebook.kernel != null) {\n",
" IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n",
"}\n"
],
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"<IPython.core.display.Javascript object>"
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],
"text/plain": [
"<IPython.core.display.HTML object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"data_provider = CCPPDataProvider(\n",
" which_set='train',\n",
" input_dims=[0, 1],\n",
" batch_size=5000, \n",
" max_num_batches=1, \n",
" shuffle_order=False\n",
")\n",
"\n",
"inputs, targets = data_provider.next()\n",
"\n",
"# Calculate predicted model outputs\n",
"outputs = model.fprop(inputs)[-1]\n",
"\n",
"# Plot target and predicted outputs against inputs on same axis\n",
"fig = plt.figure(figsize=(8, 8))\n",
"ax = fig.add_subplot(111, projection='3d')\n",
"ax.plot(inputs[:, 0], inputs[:, 1], targets[:, 0], 'r.', ms=2)\n",
"ax.plot(inputs[:, 0], inputs[:, 1], outputs[:, 0], 'b.', ms=2)\n",
"ax.set_xlabel('Input dim 1')\n",
"ax.set_ylabel('Input dim 2')\n",
"ax.set_zlabel('Output')\n",
"ax.legend(['Targets', 'Predictions'], frameon=False)\n",
"fig.tight_layout()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Optional exercise: visualising training trajectories in parameter space"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Running the cell below will display an interactive widget which plots the trajectories of gradient-based training of the single-layer affine model on the CCPP dataset in the three dimensional parameter space (two weights plus bias) from random initialisations. Also shown on the right is a plot of the evolution of the cost function (evaluated on the current batch) over training. By moving the sliders you can alter the training hyperparameters to investigate the effect they have on how training procedes.\n",
"\n",
"Some questions to explore:\n",
"\n",
" * Are there multiple local minima in parameter space here? Why?\n",
" * What happens to learning for very small learning rates? And very large learning rates?\n",
" * How does the batch size affect learning?"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": false,
"scrolled": false
},
"outputs": [
{
"data": {
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"text/plain": [
"<matplotlib.figure.Figure at 0x7f85000cc6d0>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"from ipywidgets import interact\n",
"%matplotlib inline\n",
"\n",
"def setup_figure():\n",
" # create figure and axes\n",
" fig = plt.figure(figsize=(12, 6))\n",
" ax1 = fig.add_axes([0., 0., 0.5, 1.], projection='3d')\n",
" ax2 = fig.add_axes([0.6, 0.1, 0.4, 0.8])\n",
" # set axes properties\n",
" ax2.spines['right'].set_visible(False)\n",
" ax2.spines['top'].set_visible(False)\n",
" ax2.yaxis.set_ticks_position('left')\n",
" ax2.xaxis.set_ticks_position('bottom')\n",
" ax2.set_yscale('log')\n",
" ax1.set_xlim((-2, 2))\n",
" ax1.set_ylim((-2, 2))\n",
" ax1.set_zlim((-2, 2))\n",
" #set axes labels and title\n",
" ax1.set_title('Parameter trajectories over training')\n",
" ax1.set_xlabel('Weight 1')\n",
" ax1.set_ylabel('Weight 2')\n",
" ax1.set_zlabel('Bias')\n",
" ax2.set_title('Batch costs over training')\n",
" ax2.set_xlabel('Batch update number')\n",
" ax2.set_ylabel('Batch cost')\n",
" return fig, ax1, ax2\n",
"\n",
"def visualise_training(n_epochs=1, batch_size=200, log_lr=-3.5, n_inits=5,\n",
" w_scale=1., b_scale=1., elev=30., azim=0.):\n",
" fig, ax1, ax2 = setup_figure()\n",
" # create seeded random number generator\n",
" rng = np.random.RandomState(1234)\n",
" # create data provider\n",
" data_provider = CCPPDataProvider(\n",
" input_dims=[0, 1],\n",
" batch_size=batch_size, \n",
" shuffle_order=False,\n",
" )\n",
" learning_rate = 10 ** log_lr\n",
" n_batches = data_provider.num_batches\n",
" weights_traj = np.empty((n_inits, n_epochs * n_batches + 1, 1, 2))\n",
" biases_traj = np.empty((n_inits, n_epochs * n_batches + 1, 1))\n",
" costs_traj = np.empty((n_inits, n_epochs * n_batches))\n",
" # randomly initialise parameters\n",
" weights = rng.uniform(-w_scale, w_scale, (n_inits, 1, 2))\n",
" biases = rng.uniform(-b_scale, b_scale, (n_inits, 1))\n",
" # store initial parameters\n",
" weights_traj[:, 0] = weights\n",
" biases_traj[:, 0] = biases\n",
" # iterate across different initialisations\n",
" for i in range(n_inits):\n",
" # iterate across epochs\n",
" for e in range(n_epochs):\n",
" # iterate across batches\n",
" for b, (inputs, targets) in enumerate(data_provider):\n",
" outputs = fprop(inputs, weights[i], biases[i])\n",
" costs_traj[i, e * n_batches + b] = cost(outputs, targets)\n",
" grad_wrt_outputs = cost_grad(outputs, targets)\n",
" weights_grad, biases_grad = grads_wrt_params(inputs, grad_wrt_outputs)\n",
" weights[i] -= learning_rate * weights_grad\n",
" biases[i] -= learning_rate * biases_grad\n",
" weights_traj[i, e * n_batches + b + 1] = weights[i]\n",
" biases_traj[i, e * n_batches + b + 1] = biases[i]\n",
" # choose a different color for each trajectory\n",
" colors = plt.cm.jet(np.linspace(0, 1, n_inits))\n",
" # plot all trajectories\n",
" for i in range(n_inits):\n",
" lines_1 = ax1.plot(\n",
" weights_traj[i, :, 0, 0], \n",
" weights_traj[i, :, 0, 1], \n",
" biases_traj[i, :, 0], \n",
" '-', c=colors[i], lw=2)\n",
" lines_2 = ax2.plot(\n",
" np.arange(n_batches * n_epochs),\n",
" costs_traj[i],\n",
" c=colors[i]\n",
" )\n",
" ax1.view_init(elev, azim)\n",
" plt.show()\n",
"\n",
"w = interact(\n",
" visualise_training,\n",
" elev=(-90, 90, 2),\n",
" azim=(-180, 180, 2), \n",
" n_epochs=(1, 5), \n",
" batch_size=(100, 1000, 100),\n",
" log_lr=(-5., -2.),\n",
" w_scale=(0., 2.),\n",
" b_scale=(0., 2.),\n",
" n_inits=(1, 10)\n",
")\n",
"\n",
"for child in w.widget.children:\n",
" child.layout.width = '100%'"
]
}
],
"metadata": {
"anaconda-cloud": {},
"kernelspec": {
"display_name": "Python [conda env:mlp]",
"language": "python",
"name": "conda-env-mlp-py"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 2
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython2",
"version": "2.7.12"
}
},
"nbformat": 4,
"nbformat_minor": 0
}
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