basic parabola notebook

claesenm · Jul 9, 2015 · 62fcced · 62fcced
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+{
+ "metadata": {
+  "signature": "sha256:56f96cf17697048e79058823a57f2bcddffd55678dfbbcba6f2b6ae8a0b99738"
+ },
+ "nbformat": 3,
+ "nbformat_minor": 0,
+ "worksheets": [
+  {
+   "cells": [
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "In this example we will use various Optunity solvers to minimize the following simple 2d parabola:\n",
+      "$$f(x, y) = (x - x_{off})^2 + (y - y_{off})^2,$$\n",
+      "where $x_{off}$ and $y_{off}$ are randomly determined offsets."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "def create_objective_function():\n",
+      "    xoff = random.random()\n",
+      "    yoff = random.random()\n",
+      "    def f(x, y):\n",
+      "        return (x - xoff)**2 + (y - yoff - 5)**2\n",
+      "    return f"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 1
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "We start with the necessary imports."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "%matplotlib inline # comment this line when running the notebook yourself\n",
+      "import math\n",
+      "import optunity\n",
+      "import random\n",
+      "import numpy as np\n",
+      "import matplotlib.pyplot as plt\n",
+      "import matplotlib.ticker as ticker"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 2
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "First we check which solvers are available. This is available via `optunity.available_solvers`."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "solvers = optunity.available_solvers()\n",
+      "print('Available solvers: ' + ', '.join(solvers))"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "Available solvers: particle swarm, tpe, sobol, nelder-mead, random search, cma-es, grid search\n"
+       ]
+      }
+     ],
+     "prompt_number": 3
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "To run an experiment, we start by generating an objective function."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "f = create_objective_function()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 4
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Now we use every available solver to optimize the objective function within the box $x\\in\\ ]-5, 5[$ and $y\\in\\ ]-5, 5[$ and save the results."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "logs = {}\n",
+      "for solver in solvers:\n",
+      "    pars, details, _ = optunity.minimize(f, num_evals=100, x=[-5, 5], y=[-5, 5], solver_name=solver)\n",
+      "    logs[solver] = np.array([details.call_log['args']['x'],\n",
+      "                             details.call_log['args']['y']])"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 6
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Finally, lets look at the results, that is the trace of each solver along with contours of the objective function."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "# make sure different traces are somewhat visually separable\n",
+      "colors =  ['r', 'g', 'b', 'y', 'k', 'y', 'r', 'g']\n",
+      "markers = ['x', '+', 'o', 's', 'p', 'x', '+', 'o']\n",
+      "\n",
+      "# compute contours of the objective function\n",
+      "delta = 0.025\n",
+      "x = np.arange(-5.0, 5.0, delta)\n",
+      "y = np.arange(-5.0, 5.0, delta)\n",
+      "X, Y = np.meshgrid(x, y)\n",
+      "Z = f(X, Y)\n",
+      "\n",
+      "CS = plt.contour(X, Y, Z)\n",
+      "plt.clabel(CS, inline=1, fontsize=10, alpha=0.5)\n",
+      "for i, solver in enumerate(solvers):\n",
+      "    plt.scatter(logs[solver][0,:], logs[solver][1,:], c=colors[i], marker=markers[i], alpha=0.80)\n",
+      "\n",
+      "plt.xlim([-5, 5])\n",
+      "plt.ylim([-5, 5])\n",
+      "plt.axis('equal')\n",
+      "plt.legend(solvers)\n",
+      "plt.show()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 7
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Now lets see the performance of the solvers across in 100 repeated experiments. We will do 100 experiments for each solver and then report the resulting statistics. This may take a while to run."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "optima = dict([(s, []) for s in solvers])\n",
+      "for i in range(100):\n",
+      "    f = create_objective_function()\n",
+      "\n",
+      "    for solver in solvers:\n",
+      "        pars, details, _ = optunity.minimize(f, num_evals=100, x=[-5, 5], y=[-5, 5],\n",
+      "                                             solver_name=solver)\n",
+      "        # the above line can be parallelized by adding `pmap=optunity.pmap`\n",
+      "        # however this is incompatible with IPython\n",
+      "\n",
+      "        optima[solver].append(details.optimum)\n",
+      "        logs[solver] = np.array([details.call_log['args']['x'],\n",
+      "                                 details.call_log['args']['y']])\n",
+      "\n",
+      "from collections import OrderedDict\n",
+      "log_optima = OrderedDict()\n",
+      "means = OrderedDict()\n",
+      "std = OrderedDict()\n",
+      "for k, v in optima.items():\n",
+      "    log_optima[k] = [-math.log10(val) for val in v]\n",
+      "    means[k] = sum(log_optima[k]) / len(v)\n",
+      "    std[k] = np.std(log_optima[k])\n",
+      "\n",
+      "plt.barh(np.arange(len(means)), means.values(), height=0.8, xerr=std.values(), alpha=0.5)\n",
+      "plt.xlabel('number of correct digits')\n",
+      "plt.yticks(np.arange(len(means))+0.4, list(means.keys()))\n",
+      "plt.tight_layout()\n",
+      "plt.show()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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w7GY54VjSwDCvm7eEcALAYsJJBZcQTnxoVf+QSwnHm5YTTmA4ItZzLPC9uPz3\nhBMtWr1Ofv5pwjGyq+J27yAM771OCOHz4jauJ4TKLYQhwVYnSVxICJ1Vcd/vblH73YQTRpYD/04Y\nCnyuRbuLgZ8QTpLYJm53CeGU+y+2aC9J0ls2dJbgxsBdhFvFqwd5DEpSai4m9MKmEHqPSwutRpIk\nSZIkSZIkSZIkSZIkqef8f6+4mZcrLUD7AAAAAElFTkSuQmCC\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x7fd78b35ce90>"
+       ]
+      }
+     ],
+     "prompt_number": 8
+    }
+   ],
+   "metadata": {}
+  }
+ ]
+}