micrograd-example.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Micrograd Example Notebook "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "from micrograd.core.engine import Value\n",
    "from micrograd.nn import Neuron, Layer, MLP\n",
    "from micrograd.utils.draw import draw_dot\n",
    "import numpy as np\n",
    "import torch"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Important Concepts"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Derivative\n",
    "\n",
    "In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).\n",
    "\n",
    "![image.png](derivative_formula.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# define a math function\n",
    "def f(x):\n",
    "    return 3*x**2 - 4*x + 5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "14.00300000000243\n"
     ]
    }
   ],
   "source": [
    "# Derivative of a function with one input\n",
    "# Input\n",
    "x = 3.0\n",
    "# Change the input slightly\n",
    "h = 0.001\n",
    "# Find the slope of the change\n",
    "print((f(x + h) - f(x)) / h)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have a positive slope. Let's consider a negative slope"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-21.996999999998934\n"
     ]
    }
   ],
   "source": [
    "# Input\n",
    "x = -3.0\n",
    "# Change the input slightly \n",
    "h = 0.001\n",
    "# Find the slope of the change \n",
    "print((f(x + h) - f(x))  / h)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, let's consider a derivative of a function with multiple inputs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4.0\n"
     ]
    }
   ],
   "source": [
    "a = 2.0\n",
    "b = -3.0\n",
    "c = 10.0\n",
    "d = a * b + c\n",
    "print(d)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Find the derivative of d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "d1 4.0\n",
      "d2 3.999699999999999\n",
      "slope -3.000000000010772\n"
     ]
    }
   ],
   "source": [
    "h = 0.0001\n",
    "\n",
    "# inputs\n",
    "a = 2.0\n",
    "b = -3.0\n",
    "c = 10.0\n",
    "# expression\n",
    "d1 = a * b + c\n",
    "\n",
    "# Change the input a slightly\n",
    "a += h\n",
    "d2 = a * b + c\n",
    "print('d1', d1)\n",
    "print('d2', d2)\n",
    "\n",
    "print('slope', (d2-d1)/h) # this is how much the function increased when we changed a specific input with respect to a. \n",
    "# Then we normalize it by h to get slope"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Chain Rule\n",
    "\n",
    "The chain rule may also be expressed in Leibniz's notation. If a variable z depends on the variable y, which itself depends on the variable x (that is, y and z are dependent variables), then z depends on x as well, via the intermediate variable y.\n",
    "\n",
    "![chain_rule_image](chain_rule.png)\n",
    "\n",
    "Intuitively, the chain rule states that knowing the instantaneous rate of change of z relative to y and that of y relative to x allows one to calculate the instantaneous rate of change of z relative to x as the product of the two rates of change.\n",
    "\n",
    "As put by George F. Simmons: \"if a car travels twice as fast as a bicycle and the bicycle is four times as fast as a walking man, then the car travels 2 × 4 = 8 times as fast as the man.\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```python\n",
    "dd / dc = 1.0\n",
    "dd / de = 1.0\n",
    "```\n",
    "WANT:\n",
    "```python\n",
    "dL / dc\n",
    "```\n",
    "\n",
    "The chain rule tells us: \n",
    "```python\n",
    "dL / dc = (dL / dd) * (dd / dc)\n",
    "```\n",
    "\n",
    "KNOW: \n",
    "```python\n",
    "dL / dd\n",
    "dd / dc \n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Backpropagation\n",
    "In machine learning, backpropagation is a widely used algorithm for training feedforward neural networks. Backpropagation efficiently computes the gradient of the loss function with respect to the weights of the network.\n",
    "\n",
    "### Gradient Descent\n",
    "In mathematics, gradient descent is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. Gradient descent is an optimization algorithm which is commonly-used to train machine learning models and neural networks.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Math operations with micrograd core data structure - Value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Value(data=2.0)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = Value(2.0)\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Value(data=12.0)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = Value(10.0)\n",
    "a + b # Addition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Value(data=-8.0)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a - b # Subtraction"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Value(data=20.0)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a * b # Multiplication"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Value(data=0.2)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a / b # Division"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Value(data=7.38905609893065)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.exp() # Exponents"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Value(data=4.0)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a ** 2 # Power"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Visualizing math expressions & backpropagation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = Value(2.0, label='a')\n",
    "b = Value(-3.0, label='b')\n",
    "c = Value(10.0, label='c')\n",
    "e = a * b; e.label = 'e'\n",
    "d = e + c; d.label = 'd'\n",
    "f = Value(-2.0, label='f')\n",
    "L = d * f; L.label = 'L'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
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   "source": [
    "draw_dot(L)"
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  {
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   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Do backpropagation - compute the gradient of weights with respect to L output\n",
    "L.backward()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
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   "source": [
    "draw_dot(L)"
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Backpropagation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "# inputs x1, x2\n",
    "x1 = Value(2.0, label='x1')\n",
    "x2 = Value(0.0, label='x2')\n",
    "\n",
    "# weights w1,w2\n",
    "w1 = Value(-3.0, label='w1')\n",
    "w2 = Value(1.0, label='w2')\n",
    "b = Value(6.8813735870195432, label='b')\n",
    "\n",
    "# x1*w1 + x2*w2 + b\n",
    "x1w1 = x1*w1; x1w1.label = 'x1*w1'\n",
    "x2w2 = x2*w2; x2w2.label = 'x2*w2'\n",
    "x1w1x2w2 = x1w1 + x2w2; x1w1x2w2.label = 'x1*w1 + x2*w2'\n",
    "n = x1w1x2w2 + b; n.label = 'n'\n",
    "o = n.tanh(); o.label = 'o'\n",
    "o.backward()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
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   "source": [
    "draw_dot(o)"
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Breaking tanh\n",
    "\n",
    "![tanh_formula](tanh_formula.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
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     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# inputs x1, x2\n",
    "x1 = Value(2.0, label='x1')\n",
    "x2 = Value(0.0, label='x2')\n",
    "\n",
    "# weights w1,w2\n",
    "w1 = Value(-3.0, label='w1')\n",
    "w2 = Value(1.0, label='w2')\n",
    "b = Value(6.8813735870195432, label='b')\n",
    "\n",
    "# x1*w1 + x2*w2 + b\n",
    "x1w1 = x1*w1; x1w1.label = 'x1*w1'\n",
    "x2w2 = x2*w2; x2w2.label = 'x2*w2'\n",
    "x1w1x2w2 = x1w1 + x2w2; x1w1x2w2.label = 'x1*w1 + x2*w2'\n",
    "n = x1w1x2w2 + b; n.label = 'n'\n",
    "# ---- Breaking tanh using exponents\n",
    "e = (2*n).exp()\n",
    "o = (e - 1) / (e + 1)\n",
    "# ----\n",
    "o.label = 'o'\n",
    "o.backward()\n",
    "draw_dot(o)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## PyTorch Implementation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.7071066904050358\n",
      "----\n",
      "x2 0.5000001283844369\n",
      "w2 0.0\n",
      "x1 -1.5000003851533106\n",
      "w1 1.0000002567688737\n"
     ]
    }
   ],
   "source": [
    "x1 = torch.Tensor([2.0]).double(); x1.requires_grad = True\n",
    "x2 = torch.Tensor([0.0]).double(); x2.requires_grad = True\n",
    "w1 = torch.Tensor([-3.0]).double(); w1.requires_grad = True\n",
    "w2 = torch.Tensor([1.0]).double(); w2.requires_grad = True\n",
    "b = torch.Tensor([6.8813735870195432]).double(); b.requires_grad= True\n",
    "n = x1*w1 + x2*w2 + b\n",
    "o = torch.tanh(n)\n",
    "\n",
    "print(o.data.item())\n",
    "o.backward()\n",
    "\n",
    "print('----')\n",
    "print('x2', x2.grad.item())\n",
    "print('w2', w2.grad.item())\n",
    "print('x1', x1.grad.item())\n",
    "print('w1', w1.grad.item())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Neural Network - MLP\n",
    "\n",
    "A multilayer perceptron (MLP) is a fully connected class of feedforward artificial neural network (ANN).\n",
    "\n",
    "![multilayer_perceptron](multilayer_perceptron.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dataset"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Input data\n",
    "xs = [\n",
    "    [2.0, 3.0, -1.0],\n",
    "    [3.0, -1.0, 0.5],\n",
    "    [0.5, 1.0, 1.0],\n",
    "    [1.0, 1.0, -1.0]\n",
    "]\n",
    "\n",
    "ys = [1.0, -1.0, -1.0, 1.0] # desired targets"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Multi-Layer Perceptron"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Define the MLP model\n",
    "model = MLP(3, [4,4,1]) # 3 inputs, 2 hidden layers with 4 neurons, 1 output neuron"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Value(data=7.802903398155113)"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Generate predictions\n",
    "ypred = [model(x) for x in xs] # forward pass\n",
    "# Compute loss function - Mean Squared Error\n",
    "loss = sum((yout - ygt)**2 for ygt, yout in zip(ys, ypred))\n",
    "loss"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Backpropagation - calculate the derivative of the weights with respect to the loss function\n",
    "loss.backward()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Gradient Descent"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Value(data=0.5197949387101655),\n",
       " Value(data=-0.4291460083822245),\n",
       " Value(data=0.48347715831706295),\n",
       " Value(data=-0.7512784795445704),\n",
       " Value(data=0.2189254681192465),\n",
       " Value(data=0.29281518158756303),\n",
       " Value(data=-0.9416613112529886),\n",
       " Value(data=0.5713493568356285),\n",
       " Value(data=-0.5789861579300231),\n",
       " Value(data=-0.10492220587754897),\n",
       " Value(data=-0.6505671522220438),\n",
       " Value(data=0.6346615245637692),\n",
       " Value(data=-0.15735395918016826),\n",
       " Value(data=-0.6578579452925957),\n",
       " Value(data=0.9386433057993429),\n",
       " Value(data=0.050189747543482044),\n",
       " Value(data=-0.5508884564502055),\n",
       " Value(data=0.1914749689750317),\n",
       " Value(data=-0.7291385548141456),\n",
       " Value(data=-0.6384595441216914),\n",
       " Value(data=0.04769368827730136),\n",
       " Value(data=-0.6764877796831275),\n",
       " Value(data=0.7442781618288787),\n",
       " Value(data=0.09411393670019907),\n",
       " Value(data=0.6998889978653702),\n",
       " Value(data=0.6165967103863859),\n",
       " Value(data=0.6358718006149522),\n",
       " Value(data=-0.8702808347573929),\n",
       " Value(data=0.7091036791461329),\n",
       " Value(data=0.393375284131235),\n",
       " Value(data=-0.6596138485643472),\n",
       " Value(data=0.7089102014088249),\n",
       " Value(data=-0.23545868349602306),\n",
       " Value(data=0.15110295399432472),\n",
       " Value(data=0.304995142200877),\n",
       " Value(data=0.021626346472704228),\n",
       " Value(data=0.22519513607411468),\n",
       " Value(data=0.8031124243488104),\n",
       " Value(data=-0.7252866679642207),\n",
       " Value(data=0.6083982083243515),\n",
       " Value(data=0.9764139968399135)]"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Get the parameters of the model - weights and biases\n",
    "model.parameters()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Gradient descent optimization\n",
    "for p in model.parameters():\n",
    "    p.data += -0.01 * p.grad"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.520263721596618"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model.layers[0].neurons[0].w[0].data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.04687828864525957"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model.layers[0].neurons[0].w[0].grad"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Value(data=0.9689930491668445),\n",
       " Value(data=0.9780460095095732),\n",
       " Value(data=0.9718695703068375),\n",
       " Value(data=0.968276993895206)]"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ypred"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Reinitialize the dataset\n",
    "# Input data\n",
    "xs = [\n",
    "    [2.0, 3.0, -1.0],\n",
    "    [3.0, -1.0, 0.5],\n",
    "    [0.5, 1.0, 1.0],\n",
    "    [1.0, 1.0, -1.0]\n",
    "]\n",
    "\n",
    "ys = [1.0, -1.0, -1.0, 1.0] # desired targets"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 7.79664149901795\n",
      "1 7.716244497138515\n",
      "2 7.511684976174127\n",
      "3 6.421747048109483\n",
      "4 2.946795319174857\n",
      "5 0.20897695214744097\n",
      "6 0.11123741959655434\n",
      "7 0.07808457342555372\n",
      "8 0.06038654948533271\n",
      "9 0.04924824150241945\n",
      "10 0.04156366724706306\n",
      "11 0.03593437382115132\n",
      "12 0.03163113840632937\n",
      "13 0.028234551295622492\n",
      "14 0.02548566822603158\n",
      "15 0.02321574336614988\n",
      "16 0.021310013779419698\n",
      "17 0.019687675314695063\n",
      "18 0.018290172432913136\n",
      "19 0.017074028199008367\n",
      "20 0.016006282349765887\n",
      "21 0.015061492516750204\n",
      "22 0.01421970682882314\n",
      "23 0.013465059293279938\n",
      "24 0.012784775487331769\n"
     ]
    }
   ],
   "source": [
    "for k in range(25):\n",
    "    # forward pass\n",
    "    ypred = [model(x) for x in xs]\n",
    "    loss = sum((yout - ygt)**2 for ygt, yout in zip(ys, ypred))\n",
    "    \n",
    "    # zero grad\n",
    "    for p in model.parameters():\n",
    "        p.grad = 0.0\n",
    "    \n",
    "    # backward pass\n",
    "    loss.backward()\n",
    "    \n",
    "    # update\n",
    "    for p in model.parameters():\n",
    "        p.data += -0.1 * p.grad\n",
    "    \n",
    "    print(k, loss.data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Value(data=0.9446501105332974),\n",
       " Value(data=-0.9586892699191314),\n",
       " Value(data=-0.9250671357829152),\n",
       " Value(data=0.9510137298443214)]"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ypred"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1.0, -1.0, -1.0, 1.0]"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ys"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Summary\n",
    "- Neural nets are \"simple\" mathematical expressions \n",
    "- They take input as data. They take input the weights and parameters (weights and biases) of the neural net. \n",
    "- Mathematical expression forward pass followed by a loss function\n",
    "- Loss function tries to measure the accuracy of the predictions. Usually, the loss will be low when your predictions are matching your targets. \n",
    "- Then, we backward the loss. We use backpropagation to get the gradient. \n",
    "- We tune all the parameters to decrease the loss locally. \n",
    "- We have to iterate this process many times - that's what's called the gradient descent. We simply follow the gradient information and that minimizes the loss and the loss is arranged when the loss is minimized and the neural network is doing what you want it to do."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
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