notebooks/hard_concrete.ipynb

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    "In the paper, the hard concrete distribution is used to generate the *gating parameter* $z$.  This notebook is for explaination of it. \n",
    "\n",
    "Now, $\\ln\\alpha$ and $\\beta$ are hyper parameters (the location parameter and temparature parameter). Suppose\n",
    "\n",
    "$$u\\sim \\mathscr{U}(0,1), ~ s=\\sigma(\\frac{\\ln u-\\ln(1-u)+\\ln\\alpha}{\\beta}),~\\bar{s}=s(\\zeta-\\gamma)+\\gamma$$\n",
    "\n",
    "Here $s\\in(0, 1)$ and it's \"stretched\" to $(\\gamma, \\zeta)~~(\\ni\\bar{s})$. Plus, $\\sigma$ is the sigmoid function. \n",
    "\n",
    "Then now you see  $z$ generated from the hard concrete distribution as below.\n",
    "\n",
    "$$z=\\min(1, \\max(0, \\bar{s}))$$\n",
    "\n",
    "This $\\min(1, \\max(0, \\cdot))$ is called hard-sigmoid function. Let's see how $z$ behaves."
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    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from ipywidgets import interact\n",
    "plt.style.use(\"ggplot\")"
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    "def sigmoid(x):\n",
    "    return 1 / (1 + np.exp(-x))\n",
    "\n",
    "def hard_sigmoid(x):\n",
    "    return min(1, max(0, x))\n",
    "\n",
    "def hard_concrete(loc, temp, gamma, zeta):\n",
    "    u = np.random.random()\n",
    "    s = sigmoid((np.log(u) - np.log(1 - u) + loc) / temp)\n",
    "    s = s * (zeta - gamma) + gamma\n",
    "    return hard_sigmoid(s)\n"
   ]
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    "def plot_hard_concreate(loc, temp, gamma, zeta, num=10_000, bins=100, **kwargs):\n",
    "    plt.hist([hard_concrete(loc, temp, gamma, zeta) for _ in range(num)], bins=bins, density=True, **kwargs)\n"
   ]
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   "source": [
    "Now the histograms of $z\\sim\\mathrm{hard~conc.}(\\ln\\alpha, \\beta, \\gamma, \\zeta)$ is shown below. It's an interactive graph so you can change the parameters $\\ln\\alpha, \\beta$."
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    "def interactive_hard_concrete(loc, temp):\n",
    "    plot_hard_concreate(loc, temp, gamma=-0.1, zeta=1.1)\n",
    "\n",
    "interact(interactive_hard_concrete, loc=(-0.5, 0.5), temp=(0.01, 1));\n"
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    "You see that the location parameter $\\ln\\alpha$ determins the ratio of $0$ and $1$ like the hyper parameter of Bernoulli distribution and the temparature $\\beta$ controls the degree of approximation to a Bernoulli distribution; if it is high then the number of $x\\in(0,1)$ increases (less Bernoulli), while $\\beta$ is low then the number of $x$ decreases. When $\\beta\\to 0$, this hard concrete ditiribution is exactly the Bernoulli distribution.\n"
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qldN3GLg2Ij4H/DTwemYeH9fKd9QW+manE4iI34+IU/9l0V3A2yPiBeDjwA2np9rxaNnzJ4EzgS9ExLMRMf3/TnyMWvZcRst+HwFeiYh/Br4C/HZmvnJ6Kh5dy56vBw5GxDeA++kfwjfLG2dExP30NzZ/JCKORsSBiPhYRHxsMORh+m/ULwB3Ar82zvn9pqgkFbGjttAlSd0Z6JJUhIEuSUUY6JJUhIEuSUUY6JJUhIEuSUUY6JJUxP8DuW50TcoVFCYAAAAASUVORK5CYII=\n",
          "text/plain": "<matplotlib.figure.Figure at 0x118f66518>"
         },
         "metadata": {},
         "output_type": "display_data"
        }
       ]
      }
     },
     "1a49807950f842be8f500e8b82a3fc4d": {
      "model_module": "@jupyter-widgets/base",
      "model_module_version": "1.0.0",
      "model_name": "LayoutModel",
      "state": {}
     },
     "1ca6e221d623444d882725afda2d59bc": {
      "model_module": "@jupyter-widgets/controls",
      "model_module_version": "1.0.0",
      "model_name": "SliderStyleModel",
      "state": {
       "description_width": ""
      }
     },
     "1ddbfd1a05f646199f72011640a83c2b": {
      "model_module": "@jupyter-widgets/output",
      "model_module_version": "1.0.0",
      "model_name": "OutputModel",
      "state": {
       "layout": "IPY_MODEL_43f59818d30f4abcab342d05161e5161",
       "outputs": [
        {
         "data": {
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       "layout": "IPY_MODEL_de43c155251248c2bfa2809c5dbf8e59",
       "max": 30000,
       "min": -10000,
       "style": "IPY_MODEL_76560e0c81fa4dbeb5f3e4d2ca097b02",
       "value": 10000
      }
     },
     "5d1b50ac3a424494b6164ad379ced733": {
      "model_module": "@jupyter-widgets/controls",
      "model_module_version": "1.0.0",
      "model_name": "IntSliderModel",
      "state": {
       "description": "bins",
       "layout": "IPY_MODEL_96968120833e4191859087011a91e91b",
       "max": 300,
       "min": -100,
       "style": "IPY_MODEL_561f0587f937432abf667e75dbb42de7",
       "value": 100
      }
     },
     "5e4b55cca4634416bb36c1c587e07428": {
      "model_module": "@jupyter-widgets/output",
      "model_module_version": "1.0.0",
      "model_name": "OutputModel",
      "state": {
       "layout": "IPY_MODEL_b081a6ff7f0f4a25aef154fbe04b5540",
       "outputs": [
        {
         "data": {
          "image/png": "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\n",
          "text/plain": "<matplotlib.figure.Figure at 0x1188affd0>"
         },
         "metadata": {},
         "output_type": "display_data"
        }
       ]
      }
     },
     "60b9ca70164b40f0bd302169ebab3b4b": {
      "model_module": "@jupyter-widgets/controls",
      "model_module_version": "1.0.0",
      "model_name": "FloatSliderModel",
      "state": {
       "description": "temp",
       "layout": "IPY_MODEL_1320ff9b07744f8c8900a532fce6c7df",
       "max": 1.5,
       "min": -0.5,
       "step": 0.1,
       "style": "IPY_MODEL_8177c75f77a2409daeb4a5f67fab29f7",
       "value": 0.5
      }
     },
     "62d89b6cd0554f4da71b3c29c9090b7b": {
      "model_module": "@jupyter-widgets/base",
      "model_module_version": "1.0.0",
      "model_name": "LayoutModel",
      "state": {}
     },
     "6495768150254200b028cf826d7e9b73": {
      "model_module": "@jupyter-widgets/base",
      "model_module_version": "1.0.0",
      "model_name": "LayoutModel",
      "state": {}
     },
     "65f9e163774142b98f84fb203e592550": {
      "model_module": "@jupyter-widgets/output",
      "model_module_version": "1.0.0",
      "model_name": "OutputModel",
      "state": {
       "layout": "IPY_MODEL_70813af1f4b74bc9958fc62458bc34b0",
       "outputs": [
        {
         "data": {
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      "state": {
       "layout": "IPY_MODEL_8511e918c1714c5a80333ad7b467f7af",
       "outputs": [
        {
         "data": {
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