releasing lab04

labs/ex04/template/helpers.py

@@ -0,0 +1,13 @@
+import networkx
+import numpy as np
+
+n_nodes = 16
+def generate_torus_adj_matrix(n_nodes):
+    G = networkx.generators.lattice.grid_2d_graph(int(np.sqrt(n_nodes)), int(np.sqrt(n_nodes)), periodic=True)
+    # Adjacency matrix
+    A = networkx.adjacency_matrix(G).toarray()
+
+    # Add self-loops
+    for i in range(0, A.shape[0]):
+        A[i][i] = 1
+    return A

labs/ex04/template/notebook_lab04.ipynb

@@ -0,0 +1,198 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Useful starting lines\n",
+    "%matplotlib inline\n",
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "try:\n",
+    "    import google.colab\n",
+    "    IN_COLAB = True\n",
+    "except:\n",
+    "    IN_COLAB = False\n",
+    "if IN_COLAB:\n",
+    "    # Clone the entire repo to access the files.\n",
+    "    !git clone -l -s https://github.com/epfml/OptML_course.git cloned-repo\n",
+    "    %cd cloned-repo/labs/ex04/template/\n",
+    "\n",
+    "from helpers import *\n",
+    "%load_ext autoreload\n",
+    "%autoreload 2"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Random Walks"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "In this exercise you will implement a simple random walk on a torus graph and will check its convergence to uniform distribution.\n",
+    "\n",
+    "Torus is a 2D-grid graph and looks like a 'doughnout', as shown in the picture below. \n",
+    "<img src=\"https://github.com/epfml/OptML_course/blob/2ff8711feb70637d0d0f9ac75ec6164c7659c1f5/labs/ex04/template/torus_topology.png?raw=true\" alt=\"Drawing\" style=\"width: 200px;\"/>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**Note:** We will use the networkx library to generate our graph. You can install this using\n",
+    "\n",
+    "```bash\n",
+    "    pip3 install --upgrade --user networkx\n",
+    "```\n",
+    "\n",
+    "Let's generate the probability matrix $\\mathbf{G}$ of a torus graph of size $4\\times 4$, note that we include self-loops too. You can play around with the code in the helpers.py to generate different graphs."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "n_nodes = 16\n",
+    "A = generate_torus_adj_matrix(n_nodes)\n",
+    "degree = # fill in here the degree of a node in the graph\n",
+    "G = A/degree"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Lets generate initial probabitily distribution. Recall that our walk always starts from the node 1."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "x_init = # fill in here"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "As you will prove in Q2, probability distribution at each step evolves as $x_{t + 1} = G x_{t}$. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def random_walk(G, x_init, num_iter):\n",
+    "    ''' Computes probability distribution of random walk after\n",
+    "        num_iter steps.\n",
+    "        Output: \n",
+    "        x: final estimate of probability distribution after\n",
+    "            num_iter steps\n",
+    "        errors: array of differences ||x_{t} - mu||_2^2, where\n",
+    "            mu is uniform distribution\n",
+    "    '''\n",
+    "    x = np.copy(x_init)\n",
+    "    errors = np.zeros(num_iter)\n",
+    "    mu = # fill in here\n",
+    "    for t in range(0, num_iter):\n",
+    "        # ***************************************************\n",
+    "        # INSERT YOUR CODE HERE\n",
+    "        # TODO: simulate probability distribution in random walk\n",
+    "        # ***************************************************\n",
+    "    return x, errors"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Lets run our algorithm for 50 iterations and see at the final probability distribution."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "x, errors = random_walk(G, x_init, num_iter=50)\n",
+    "plt.bar(np.arange(len(x)), x)\n",
+    "plt.xlabel(\"node\")\n",
+    "plt.ylabel(\"probability\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We see that the final disctribution is indeed uniform. Lets now plot how fast did the algorithm converge. We will use logarithmic scale on y-axis to be able to distinguish between sublinear and linear rates."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.semilogy(errors)\n",
+    "plt.xlabel(\"iteration\")\n",
+    "plt.ylabel(\"$||x_{t} - mu||_2^2$\")"
+   ]
+  }
+ ],
+ "metadata": {
+  "anaconda-cloud": {},
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.7.4"
+  },
+  "widgets": {
+   "state": {
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+       "cell_index": 22
+      }
+     ]
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+    "e4a6a7a70ccd42ddb112989c04f2ed3f": {
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+ "nbformat": 4,
+ "nbformat_minor": 1
+}