notebooks/Perceptron Learning in Python.ipynb

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   "source": [
    "# Perceptron Learning in Python"
   ]
  },
  {
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   "metadata": {},
   "source": [
    "**(C) 2017-2019 by [Damir Cavar](http://damir.cavar.me/)**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Download:** This and various other Jupyter notebooks are available from my [GitHub repo](https://github.com/dcavar/python-tutorial-for-ipython)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**License:** [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/) ([CA BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0/))"
   ]
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is a tutorial related to the discussion of machine learning and NLP in the class *Machine Learning for NLP: Deep Learning* (Topics in Artificial Intelligence) taught at Indiana University in Spring 2018."
   ]
  },
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   "source": [
    "## What is a Perceptron?"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "There are many online examples and tutorials on perceptrons and learning. Here is a list of some articles:\n",
    "- [Wikipedia on Perceptrons](https://en.wikipedia.org/wiki/Perceptron)\n",
    "- Jurafsky and Martin (ed. 3), Chapter 8"
   ]
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   "source": [
    "## Example"
   ]
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   "source": [
    "This is an example that I have taken from a draft of the 3rd edition of Jurafsky and Martin, with slight modifications:"
   ]
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   "source": [
    "We import *numpy* and use its *exp* function. We could use the same function from the *math* module, or some other module like *scipy*. The *sigmoid* function is defined as in the textbook:"
   ]
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   "source": [
    "import numpy as np\n",
    "\n",
    "def sigmoid(z):\n",
    "    return 1 / (1 + np.exp(-z))"
   ]
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   "cell_type": "markdown",
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   "source": [
    "Our example data, **weights** $w$, **bias** $b$, and **input** $x$ are defined as:"
   ]
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   "cell_type": "code",
   "execution_count": 2,
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   "source": [
    "w = np.array([0.2, 0.3, 0.8])\n",
    "b = 0.5\n",
    "x = np.array([0.5, 0.6, 0.1])"
   ]
  },
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   "source": [
    "Our neural unit would compute $z$ as the **dot-product** $w \\cdot x$ and add the **bias** $b$ to it. The sigmoid function defined above will convert this $z$ value to the **activation value** $a$ of the unit:"
   ]
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   "cell_type": "code",
   "execution_count": 3,
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     "text": [
      "z: 0.8600000000000001\n",
      "a: 0.7026606543447316\n"
     ]
    }
   ],
   "source": [
    "z = w.dot(x) + b\n",
    "print(\"z:\", z)\n",
    "print(\"a:\", sigmoid(z))"
   ]
  },
  {
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   "metadata": {},
   "source": [
    "### The XOR Problem"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The power of neural units comes from combining them into larger networks. Minsky and Papert (1969): A single neural unit cannot compute the simple logical function XOR."
   ]
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   "cell_type": "markdown",
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   "source": [
    "The task is to implement a simple **perceptron** to compute logical operations like AND, OR, and XOR.\n",
    "\n",
    "- Input: $x_1$ and $x_2$\n",
    "- Bias: $b = -1$ for AND; $b = 0$ for OR\n",
    "- Weights: $w = [1, 1]$\n",
    "\n",
    "with the following activation function:\n",
    "\n",
    "$$\n",
    "y = \\begin{cases}\n",
    "    \\ 0 & \\quad \\text{if } w \\cdot x + b \\leq 0\\\\\n",
    "    \\ 1 & \\quad \\text{if } w \\cdot x + b > 0\n",
    "  \\end{cases}\n",
    "$$"
   ]
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   "source": [
    "We can define this activation function in Python as:"
   ]
  },
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   "cell_type": "code",
   "execution_count": 4,
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   "outputs": [],
   "source": [
    "def activation(z):\n",
    "    if z > 0:\n",
    "        return 1\n",
    "    return 0"
   ]
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For AND we could implement a perceptron as:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
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   "outputs": [
    {
     "name": "stdout",
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     "text": [
      "0 AND 0: 0\n",
      "1 AND 0: 0\n",
      "0 AND 1: 0\n",
      "1 AND 1: 1\n"
     ]
    }
   ],
   "source": [
    "w = np.array([1, 1])\n",
    "b = -1\n",
    "x = np.array([0, 0])\n",
    "print(\"0 AND 0:\", activation(w.dot(x) + b))\n",
    "x = np.array([1, 0])\n",
    "print(\"1 AND 0:\", activation(w.dot(x) + b))\n",
    "x = np.array([0, 1])\n",
    "print(\"0 AND 1:\", activation(w.dot(x) + b))\n",
    "x = np.array([1, 1])\n",
    "print(\"1 AND 1:\", activation(w.dot(x) + b))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For OR we could implement a perceptron as:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 OR 0: 0\n",
      "1 OR 0: 1\n",
      "0 OR 1: 1\n",
      "1 OR 1: 1\n"
     ]
    }
   ],
   "source": [
    "w = np.array([1, 1])\n",
    "b = 0\n",
    "x = np.array([0, 0])\n",
    "print(\"0 OR 0:\", activation(w.dot(x) + b))\n",
    "x = np.array([1, 0])\n",
    "print(\"1 OR 0:\", activation(w.dot(x) + b))\n",
    "x = np.array([0, 1])\n",
    "print(\"0 OR 1:\", activation(w.dot(x) + b))\n",
    "x = np.array([1, 1])\n",
    "print(\"1 OR 1:\", activation(w.dot(x) + b))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is no way to implement a perceptron for XOR this way."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(C) 2017-2019 by [Damir Cavar](http://damir.cavar.me/)**"
   ]
  }
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