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yangju2011/build_backpropagation_from_scratch.ipynb

Created February 24, 2023 14:54
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build_backpropagation_from_scratch.ipynb
{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"id": "94fff70c",
"metadata": {},
"outputs": [],
"source": [
"import math\n",
"import numpy as np\n",
"import random\n",
"import tensorflow as tf\n",
"import matplotlib.pyplot as plt\n",
"%matplotlib inline"
]
},
{
"cell_type": "markdown",
"id": "47d2b1bf",
"metadata": {},
"source": [
"# 1. derivative of a scalar\n",
"1-D data, int, float"
]
},
{
"cell_type": "markdown",
"id": "05ea3f52",
"metadata": {},
"source": [
"https://en.wikipedia.org/wiki/Derivative\n",
"\n",
"at x: \n",
"\n",
"$$f'(x) = \\lim_{h \\to 0} \\frac{f(x+h) - f(x)}{h} $$"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "44f6a109",
"metadata": {},
"outputs": [],
"source": [
"def f(x):\n",
" return 3*x**2 - 4*x + 5"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "41df2686",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"20"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f(3)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "c474b946",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([100. , 91.6875, 83.75 , 76.1875, 69. , 62.1875,\n",
" 55.75 , 49.6875, 44. , 38.6875, 33.75 , 29.1875,\n",
" 25. , 21.1875, 17.75 , 14.6875, 12. , 9.6875,\n",
" 7.75 , 6.1875, 5. , 4.1875, 3.75 , 3.6875,\n",
" 4. , 4.6875, 5.75 , 7.1875, 9. , 11.1875,\n",
" 13.75 , 16.6875, 20. , 23.6875, 27.75 , 32.1875,\n",
" 37. , 42.1875, 47.75 , 53.6875])"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"xs = np.arange(-5, 5, 0.25)\n",
"ys = f(xs)\n",
"ys"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "796b6516",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[<matplotlib.lines.Line2D at 0x12e18f9b0>]"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"plt.plot(xs, ys)"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "f3634130",
"metadata": {},
"outputs": [],
"source": [
"def grad(x, h=1e-4):\n",
" return (f(x+h)-f(x))/h"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "9c042ae9",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"14.000300000063248"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"grad(3)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "a4d52e49",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-21.999700000066014"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"grad(-3)"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "2128b766",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.0002999999981767587"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# we can solve this function and 2/3 is the root \n",
"grad(2./3)"
]
},
{
"cell_type": "markdown",
"id": "b93c17b3",
"metadata": {},
"source": [
"# 2. derivative of multiple scalars"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "41c87049",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"4.0\n"
]
}
],
"source": [
"a = 2.0\n",
"b = -3.0\n",
"c = 10.0\n",
"def f3(a,b,c):\n",
" return a*b + c\n",
"print (f3(a,b,c))"
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "9a6b2a4d",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-3.000000000010772"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"def grad_a(a, b, c, h=1e-4):\n",
" return (f3(a+h, b, c)-f3(a, b, c))/h\n",
"grad_wrt_a = grad_a(a, b, c)\n",
"grad_wrt_a "
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "76488224",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2.0000000000042206"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"def grad_b(a, b, c, h=1e-4):\n",
" return (f3(a, b+h, c)-f3(a, b, c))/h\n",
"grad_wrt_b = grad_b(a, b, c)\n",
"grad_wrt_b "
]
},
{
"cell_type": "markdown",
"id": "540dd9f2",
"metadata": {},
"source": [
"## Exercises"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "96ec076f",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"6.336362190988558\n"
]
}
],
"source": [
"# here is a mathematical expression that takes 3 inputs and produces one output\n",
"from math import sin, cos\n",
"\n",
"def f(a, b, c):\n",
" return -a**3 + sin(3*b) - 1.0/c + b**2.5 - a**0.5\n",
"\n",
"print(f(2, 3, 4))"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "905f4f06",
"metadata": {},
"outputs": [],
"source": [
"# write the function df that returns the analytical gradient of f\n",
"# i.e. use your skills from calculus to take the derivative, then implement the formula\n",
"# if you do not calculus then feel free to ask wolframalpha, e.g.:\n",
"# https://www.wolframalpha.com/input?i=d%2Fda%28sin%283*a%29%29%29\n",
"\n",
"def gradf(a, b, c):\n",
" dfa = -3*a**2 - 0.5 * a**-0.5\n",
" dfb = 3 * cos(3*b) + 2.5 * b**1.5\n",
" dfc = c**-2\n",
" return [dfa, dfb, dfc] "
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "0eb2e72b",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"OK for dim 0: expected -12.353553390593273, yours returns -12.353553390593273\n",
"OK for dim 1: expected 10.25699027111255, yours returns 10.25699027111255\n",
"OK for dim 2: expected 0.0625, yours returns 0.0625\n"
]
}
],
"source": [
"# expected answer is the list of \n",
"ans = [-12.353553390593273, 10.25699027111255, 0.0625]\n",
"yours = gradf(2, 3, 4)\n",
"for dim in range(3):\n",
" ok = 'OK' if abs(yours[dim] - ans[dim]) < 1e-5 else 'WRONG!'\n",
" print(f\"{ok} for dim {dim}: expected {ans[dim]}, yours returns {yours[dim]}\")"
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "32e99e3e",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"OK for dim 0: expected -12.353553390593273, yours returns -12.353553380251014\n",
"OK for dim 1: expected 10.25699027111255, yours returns 10.256990368162633\n",
"OK for dim 2: expected 0.0625, yours returns 0.0624999607623522\n"
]
}
],
"source": [
"# now estimate the gradient numerically without any calculus, using\n",
"# the approximation we used in the video.\n",
"# you should not call the function df from the last cell\n",
"\n",
"h = 1e-8\n",
"a = 2\n",
"b = 3\n",
"c = 4\n",
"def grad_wrt_a(a):\n",
" return (f(a+h, b, c)-f(a, b, c))/h\n",
"def grad_wrt_b(b):\n",
" return (f(a, b+h, c)-f(a, b, c))/h\n",
"def grad_wrt_c(c):\n",
" return (f(a, b, c+h)-f(a, b, c))/h\n",
"\n",
"# -----------\n",
"numerical_grad = [grad_wrt_a(a), grad_wrt_b(b), grad_wrt_c(c)]\n",
"# -----------\n",
"\n",
"for dim in range(3):\n",
" ok = 'OK' if abs(numerical_grad[dim] - ans[dim]) < 1e-5 else 'WRONG!'\n",
" print(f\"{ok} for dim {dim}: expected {ans[dim]}, yours returns {numerical_grad[dim]}\")\n"
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "ad733c61",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"OK for dim 0: expected -12.353553390593273, yours returns -12.353553291433172\n",
"OK for dim 1: expected 10.25699027111255, yours returns 10.256990368162633\n",
"OK for dim 2: expected 0.0625, yours returns 0.0624999607623522\n"
]
}
],
"source": [
"# there is an alternative formula that provides a much better numerical \n",
"# approximation to the derivative of a function.\n",
"# learn about it here: https://en.wikipedia.org/wiki/Symmetric_derivative\n",
"# implement it. confirm that for the same step size h this version gives a\n",
"# better approximation.\n",
"\n",
"h = 1e-8\n",
"a = 2\n",
"b = 3\n",
"c = 4\n",
"def grad_wrt_a2(a):\n",
" return (f(a+h, b, c)-f(a-h, b, c))/(2*h)\n",
"def grad_wrt_b2(b):\n",
" return (f(a, b+h, c)-f(a, b-h, c))/(2*h)\n",
"def grad_wrt_c2(c):\n",
" return (f(a, b, c+h)-f(a, b, c-h))/(2*h)\n",
"\n",
"# -----------\n",
"numerical_grad2 = [grad_wrt_a2(a), grad_wrt_b2(b), grad_wrt_c2(c)]\n",
"# -----------\n",
"\n",
"for dim in range(3):\n",
" ok = 'OK' if abs(numerical_grad2[dim] - ans[dim]) < 1e-5 else 'WRONG!'\n",
" print(f\"{ok} for dim {dim}: expected {ans[dim]}, yours returns {numerical_grad2[dim]}\")"
]
},
{
"cell_type": "markdown",
"id": "ce6083a6",
"metadata": {},
"source": [
"# 3. forward pass"
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "8b9dc457",
"metadata": {},
"outputs": [],
"source": [
"# Value is a node in a graph\n",
"class Value:\n",
" \n",
" # left to right, right is parent (root), left is children (leaf).\n",
" # children: tuple of Value\n",
" # tuples are immutable as opposed to lists which are mutable \n",
" # once the graph is created, the structure is immutable \n",
" def __init__(self, data, _children=(), _op='', label=''):\n",
" # data can only be int or float\n",
" self.data = data\n",
" # init gradient as 0, we should zero_grad each time after we do backward propagation \n",
" self.grad = 0.0\n",
" # chain rule function\n",
" # backward is called from the last node\n",
" self._backward = lambda: None \n",
" # previous node that is connected to the current node, previous nodes are children of the current node\n",
" # child node is Value\n",
" self._prev = set(_children)\n",
" self._op = _op\n",
" self.label = label\n",
" \n",
" # The repr() function returns a printable representation of the given object.\n",
" # without it, Value() will return a pointer <__main__.Value at 0x10d8db3c8>\n",
" def __repr__(self):\n",
" return f\"Value(data={self.data}, _op={self._op}, label={self.label})\"\n",
" \n",
" # a+b => a.__add__(b)\n",
" # other is Value\n",
" def __add__(self, other):\n",
" # if other is an instance of Value, use it; otherwise, wrap it => 1 -> Value(1)\n",
" other = other if isinstance(other, Value) else Value(other)\n",
" \n",
" # return type is Value\n",
" # out is the output (next node) of this and other\n",
" # the children/previous nodes of \"out\" are self and other\n",
" out = Value(self.data + other.data, (self, other), '+')\n",
" \n",
" # set the gradient for self and others when we chain rule it\n",
" def _backward():\n",
" # assume we know out.grad already, because it is backward!\n",
" # a+b gradient wrt to a is 1.0\n",
" # Bug warning: \n",
" # using the gradient chain rule, we could use equal: self.grad = 1.0 * out.grad \n",
" # however, if self = other (i.e. a+a), other.grad = 1.0 * out.grad will essentially overwrite self.grad \n",
" # resulting in incorrect grad of 1 instead of 2 wrt a\n",
" # to fix this bug, change = to +=, contribution from out.grad will be used to update the child node\n",
" # gradients will be accumulated\n",
" self.grad += 1.0 * out.grad \n",
" other.grad += 1.0 * out.grad \n",
" \n",
" # set the out node backward function as defined in __add__\n",
" # _backward is a function, assign function, not value. WRONG out._backward = _backward()\n",
" out._backward = _backward\n",
" return out\n",
" \n",
" # other is Value\n",
" def __mul__(self, other):\n",
" other = other if isinstance(other, Value) else Value(other)\n",
"\n",
" # the children/previous nodes of \"out\" are self and other\n",
" out = Value(self.data * other.data, (self, other), '*')\n",
" \n",
" def _backward():\n",
" # a*b gradient wrt to a is b\n",
" self.grad += other.data * out.grad \n",
" other.grad += self.data * out.grad \n",
"\n",
" out._backward = _backward\n",
" return out\n",
" \n",
" # rmul => a*b = b*a\n",
" # https://stackoverflow.com/questions/5181320/under-what-circumstances-are-rmul-called\n",
" # if cannot do b*a, it will check a*b\n",
" # other is Value\n",
" def __rmul__(self, other):\n",
" return self * other\n",
" \n",
" # activation function, non-linear\n",
" def tanh(self):\n",
" x = self.data\n",
" t = (math.exp(2*x) - 1)/(math.exp(2*x) + 1)\n",
" \n",
" # tanh takes a single input self, so the child of tanh is just self\n",
" out = Value(t, (self, ), 'tanh')\n",
" \n",
" def _backward():\n",
" # tanh derivative = 1 - t**2\n",
" self.grad += (1 - t**2) * out.grad\n",
" \n",
" out._backward = _backward\n",
" return out\n",
"\n",
" # softmax takes in a list of Value\n",
" def softmax(self, xs):\n",
" current = self.exp()\n",
" sum_res = sum([x.exp() for x in xs])\n",
" out = Value(current/sum_res, (self, ), 'softmax')\n",
" def _backward():\n",
" self.grad += out.data*(1-out.data) * out.grad\n",
" out._backward = _backward\n",
" return out\n",
" \n",
" def log(self):\n",
" # self.data is not 0\n",
" assert abs(self.data)>1e-8\n",
" out = Value(math.log(self.data), (self,), 'log')\n",
" def _backward():\n",
" self.grad += (1./self.data) * out.grad\n",
" out._backward = _backward\n",
" return out\n",
" \n",
" \n",
" # we could also further break down tanh into an exp operator \n",
" def exp(self):\n",
" x = self.data\n",
" \n",
" out = Value(math.exp(x), (self, ), 'exp')\n",
" \n",
" def _backward():\n",
" # exp derivative is exp\n",
" self.grad += out.data * out.grad\n",
" \n",
" out._backward = _backward\n",
" return out\n",
" \n",
" # we already have *, which is __mul__\n",
" # a/b = a * b**(-1)\n",
" # other is Value\n",
" def __truediv__(self, other):\n",
" return self * other**-1\n",
" \n",
" def __neg__(self):\n",
" return self * -1 \n",
" \n",
" # other is Value\n",
" def __sub__(self, other):\n",
" return self + (-other)\n",
" \n",
" # power function self ** other\n",
" # other is int, float\n",
" def __pow__(self, other):\n",
" # power: other has to be int or float\n",
" assert isinstance(other, (int, float))\n",
" \n",
" # can directly use the ** built in python, wrap it in Value\n",
" out = Value(self.data**other, (self, ), f'**{other}')\n",
" \n",
" def _backward():\n",
" self.grad += other * (self.data**(other - 1)) * out.grad\n",
" \n",
" out._backward = _backward\n",
" return out\n",
" \n",
" # the real backward function\n",
" def backward(self):\n",
" # Topological Sorting: left to right, children to parent\n",
" topo = []\n",
" visited = set()\n",
" def build_topo(v):\n",
" if v not in visited:\n",
" visited.add(v)\n",
" for child in v._prev:\n",
" build_topo(child)\n",
" # parent node will be added only after all the children (leaf) nodes are processed and added\n",
" topo.append(v)\n",
" \n",
" # build from the root node\n",
" build_topo(self)\n",
" \n",
" # we need to initialize the root.grad first to be multiplied in the chain rule\n",
" self.grad = 1.0\n",
" \n",
" # backward function from the root to children \n",
" for node in reversed(topo):\n",
" # if a node has children, its children grad is calculated by _backward\n",
" # calling the function _backward() to update the gradient \n",
" node._backward()"
]
},
{
"cell_type": "code",
"execution_count": 19,
"id": "2cb44cf3",
"metadata": {},
"outputs": [],
"source": [
"# visualize graph\n",
"from graphviz import Digraph\n",
"\n",
"def trace(root):\n",
" nodes, edges = set(), set()\n",
" def build(v):\n",
" if v not in nodes:\n",
" nodes.add(v)\n",
" for child in v._prev:\n",
" edges.add((child, v))\n",
" build(child)\n",
" build(root)\n",
" return nodes, edges\n",
"\n",
"def draw_dot(root):\n",
" dot = Digraph(format='svg', graph_attr={'rankdir':\"LR\"})\n",
" \n",
" nodes, edges = trace(root)\n",
" for n in nodes:\n",
" uid = str(id(n))\n",
" # for any value in the node, create a rec shape\n",
" dot.node(name = uid, label = \"%s | {data %.4f} | {grad %.4f} \"%(n.label, n.data, n.grad), shape='record')\n",
" if n._op:\n",
" # if this value is of some op, create an op node \n",
" dot.node(name=uid+n._op, label=n._op)\n",
" # connect op node to the value node \n",
" dot.edge(uid+n._op, uid)\n",
" \n",
" for n1, n2 in edges:\n",
" # connect to the op node of n2?\n",
" dot.edge(str(id(n1)), str(id(n2)) + n2._op)\n",
" \n",
" return dot"
]
},
{
"cell_type": "code",
"execution_count": 20,
"id": "a80feeb7",
"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' # L = (a*b + c) * f"
]
},
{
"cell_type": "code",
"execution_count": 21,
"id": "c0da2065",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{Value(data=-6.0, _op=*, label=e), Value(data=10.0, _op=, label=c)}"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"d._prev"
]
},
{
"cell_type": "code",
"execution_count": 22,
"id": "bfd0c008",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"'+'"
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"d._op"
]
},
{
"cell_type": "code",
"execution_count": 23,
"id": "5ad299de",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Value(data=-1.0, _op=+, label=)"
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a + b"
]
},
{
"cell_type": "code",
"execution_count": 24,
"id": "11173dc6",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Value(data=-1.0, _op=+, label=)"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a.__add__(b)"
]
},
{
"cell_type": "code",
"execution_count": 25,
"id": "fb36e73c",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Value(data=4.0, _op=+, label=)"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a*b + c"
]
},
{
"cell_type": "code",
"execution_count": 26,
"id": "114a84ca",
"metadata": {},
"outputs": [
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"# all gradients are initialized as 0\n",
"# L = (a*b + c) * f\n",
"draw_dot(L)"
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{
"cell_type": "code",
"execution_count": 27,
"id": "8076f05e",
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{
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"# manually calculate gradient of each node \n",
"def grad_wrt_a():\n",
" h = 1e-5\n",
" \n",
" 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'\n",
" L1 = L.data\n",
" \n",
" a = Value(2.0+ h, 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'\n",
" L2 = L.data\n",
" \n",
" print ((L2-L1)/h)\n",
" \n",
"grad_wrt_a()"
]
},
{
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"execution_count": 28,
"id": "d1b355b5",
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"# manually calcualte the gradient of each node with the chain rule\n",
"L.grad = 1.0\n",
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"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"# squash to -1 to 1 Hyperbolic function\n",
"plt.plot(np.arange(-5,5,0.2), np.tanh(np.arange(-5,5,0.2)))\n",
"# show with grid\n",
"plt.grid()"
]
},
{
"cell_type": "code",
"execution_count": 30,
"id": "49d958da",
"metadata": {},
"outputs": [],
"source": [
"# input neuron\n",
"x1 = Value(2.0, label='x1')\n",
"x2 = Value(0.0, label='x2')\n",
"# weights\n",
"w1 = Value(-3.0, label='w1')\n",
"w2 = Value(1.0, label='w2')\n",
"# bias\n",
"b = Value(6.881373, 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",
"# cell body agg input\n",
"n = x1w1x2w2 + b; \n",
"n.label = 'n'\n",
"\n",
"o = n.tanh(); o.label='o'"
]
},
{
"cell_type": "code",
"execution_count": 31,
"id": "06c93a6f",
"metadata": {},
"outputs": [
{
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{
"cell_type": "code",
"execution_count": 32,
"id": "31b75b11",
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{
"data": {
"text/plain": [
"'tanh'"
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},
"execution_count": 32,
"metadata": {},
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{
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{
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"execution_count": 33,
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"source": [
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{
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"id": "d9219b77",
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{
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"# o.grad is init as 0 by default\n",
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"source": [
"# b is a leaf node, nothing happens upstream\n",
"b._backward()\n",
"draw_dot(o)"
]
},
{
"cell_type": "code",
"execution_count": 39,
"id": "be3a5b85",
"metadata": {},
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{
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"id": "c63b15c4",
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"# call backward() to update all gradients at once\n",
"\n",
"\n",
"# input neuron\n",
"x1 = Value(2.0, label='x1')\n",
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"\n",
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"n = x1w1x2w2 + b; \n",
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"\n",
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"\n",
"\n",
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{
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"metadata": {},
"source": [
"## Make exp an operator "
]
},
{
"cell_type": "code",
"execution_count": 42,
"id": "bb881cdb",
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"metadata": {},
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],
"source": [
"# input neuron\n",
"# migrograd is only scaler value engine, not for arrays!\n",
"x1 = Value(2.0, label='x1')\n",
"x2 = Value(0.0, label='x2')\n",
"# weights\n",
"w1 = Value(-3.0, label='w1')\n",
"w2 = Value(1.0, label='w2')\n",
"# bias\n",
"b = Value(6.881373, label='b')\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",
"# cell body agg input\n",
"n = x1w1x2w2 + b; \n",
"n.label = 'n'\n",
"\n",
"# not use tanh, but exp -> equivalent to tanh\n",
"# OPERATION with forward and backward PASS\n",
"# level of operation is up to you, +,-,**, tanh, relu, ATOMIC operation\n",
"# input -> output = func(input)\n",
"# backward pass is just backward derivative, chain it up\n",
"e = (2*n).exp(); e.label='e'\n",
"o = (e-1)/(e+1); o.label='o'\n",
"\n",
"# draw dot\n",
"o.grad = 1.0\n",
"o.backward()\n",
"draw_dot(o)"
]
},
{
"cell_type": "markdown",
"id": "ebd24c97",
"metadata": {},
"source": [
"# 5. automatic differentiation in tensorflow"
]
},
{
"cell_type": "code",
"execution_count": 43,
"id": "331c0aa5",
"metadata": {},
"outputs": [],
"source": [
"import tensorflow as tf"
]
},
{
"cell_type": "code",
"execution_count": 44,
"id": "d6ab9674",
"metadata": {},
"outputs": [
{
"data": {
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]
},
"execution_count": 44,
"metadata": {},
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}
],
"source": [
"x1 = tf.constant([2.0])\n",
"x2 = tf.constant([0.0])\n",
"w1 = tf.Variable([-3.0])\n",
"w2 = tf.Variable([1.0])\n",
"b = tf.Variable([6.8813])\n",
"\n",
"n = x1*w1 + x2*w2 + b; \n",
"n"
]
},
{
"cell_type": "code",
"execution_count": 45,
"id": "dbe1d034",
"metadata": {},
"outputs": [
{
"data": {
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]
},
"execution_count": 45,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"o = tf.math.tanh(n)\n",
"o"
]
},
{
"cell_type": "code",
"execution_count": 46,
"id": "1f3e45da",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.7070700526237488"
]
},
"execution_count": 46,
"metadata": {},
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}
],
"source": [
"o.numpy().item()"
]
},
{
"cell_type": "code",
"execution_count": 47,
"id": "a4b0f0c6",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"0.7070700526237488\n",
"----\n",
"x1 -1.5001559257507324\n",
"x2 0.5000519752502441\n",
"w1 1.0001039505004883\n",
"w2 0.0\n",
"b 0.5000519752502441\n"
]
}
],
"source": [
"x1 = tf.constant([2.0])\n",
"x2 = tf.constant([0.0])\n",
"w1 = tf.Variable([-3.0])\n",
"w2 = tf.Variable([1.0])\n",
"b = tf.Variable([6.8813])\n",
"\n",
"with tf.GradientTape(watch_accessed_variables=False, persistent=True) as tape:\n",
" tape.watch([x1, x2, w1, w2, b])\n",
" n = x1*w1 + x2*w2 + b; \n",
" o = tf.math.tanh(n)\n",
" \n",
"do_dx1 = tape.gradient(o, x1)\n",
"do_dx2 = tape.gradient(o, x2)\n",
"do_dw1 = tape.gradient(o, w1)\n",
"do_dw2 = tape.gradient(o, w2)\n",
"do_db = tape.gradient(o, b)\n",
"\n",
"print (o.numpy().item())\n",
"print('----')\n",
"print (\"x1\", do_dx1.numpy().item())\n",
"print (\"x2\", do_dx2.numpy().item())\n",
"print (\"w1\", do_dw1.numpy().item())\n",
"print (\"w2\", do_dw2.numpy().item())\n",
"print (\"b\", do_db.numpy().item())"
]
},
{
"cell_type": "markdown",
"id": "cb11ea11",
"metadata": {},
"source": [
"# 6. MLP"
]
},
{
"cell_type": "code",
"execution_count": 48,
"id": "6a1be963",
"metadata": {},
"outputs": [],
"source": [
"# neural net is specific math abstract\n",
"class Neuron:\n",
" \n",
" # number of input coming to this neuron\n",
" def __init__(self, n_in):\n",
" # weight for each input, same bias\n",
" self.w = [Value(random.uniform(-1,1)) for _ in range(n_in)]\n",
" # trigger happiness\n",
" self.b = Value(random.uniform(-1,1))\n",
" \n",
" # https://www.geeksforgeeks.org/__call__-in-python/\n",
" # enables Python programmers to write classes where the instances behave like functions and can be called like a functio\n",
" # n = Neuron(); n() or n(x) => invoke __call__\n",
" def __call__(self, x):\n",
" # w * x + b -> tanh -> out\n",
" # dot product of w and x\n",
" # feed 2 numbers to the Neuron as input\n",
" # zip takes 2 iterator, iterate the tumple\n",
" # activation\n",
" act = sum((wi * xi for wi, xi in zip(self.w, x)), self.b)\n",
" out = act.tanh()\n",
" return out\n",
" \n",
" def parameters(self):\n",
" # this gives a list of weights + bias \n",
" # parameter is weight AND bias...\n",
" return self.w + [self.b]\n",
" \n",
"class Layer:\n",
" \n",
" # n_in defines the input to each Neuron\n",
" # n_out defines the number of neurons in this layer \n",
" def __init__(self, n_in, n_out):\n",
" self.neurons = [Neuron(n_in) for _ in range(n_out)]\n",
" \n",
" def __call__(self, x):\n",
" # each Neuron takes n_in, and has n_out\n",
" # each layer has a list of Neuron\n",
" outs = [n(x) for n in self.neurons]\n",
" # if a layer has a single neuron, return the Neuron, else, return the list\n",
" return outs[0] if len(outs)==1 else outs\n",
" \n",
" \n",
" def parameters(self):\n",
" return [p for neuron in self.neurons for p in neuron.parameters()]\n",
" \n",
"# multi-layer perception\n",
"class MLP:\n",
" \n",
" # n_in defines the number of neurons in the first layer (input)\n",
" # n_out defines the number of neurons in each later layer, n_outs is a list of layers\n",
" def __init__(self, n_in, n_outs):\n",
" size = [n_in] + n_outs\n",
" # the n_out of current layer is n_in for next layer\n",
" # fully connected NN\n",
" self.layers = [Layer(size[i], size[i+1]) for i in range(len(n_outs))]\n",
" \n",
" # x is a list of \n",
" def __call__(self, x):\n",
" for layer in self.layers:\n",
" # layer is Layer, Layer(x) is to call n(x) in sequence\n",
" x = layer(x)\n",
" return x\n",
" \n",
" def parameters(self):\n",
" return [p for layer in self.layers for p in layer.parameters()]\n",
" "
]
},
{
"cell_type": "code",
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"id": "60897c66",
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{
"data": {
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"[Value(data=0.3881160484826891, _op=, label=),\n",
" Value(data=-0.15693349023222614, _op=, label=),\n",
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]
},
"execution_count": 49,
"metadata": {},
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],
"source": [
"neuron = Neuron(3)\n",
"neuron.w"
]
},
{
"cell_type": "code",
"execution_count": 50,
"id": "f1cbc146",
"metadata": {},
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{
"data": {
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"source": [
"neuron.b"
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{
"cell_type": "code",
"execution_count": 51,
"id": "d307443f",
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{
"data": {
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],
"source": [
"xs = [1.0]\n",
"neuron(xs)"
]
},
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"source": [
"# left we have 1 weight\n",
"draw_dot(neuron(xs))"
]
},
{
"cell_type": "code",
"execution_count": 53,
"id": "afb0af57",
"metadata": {},
"outputs": [
{
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"source": [
"xs = [1.0, 2.0, 3.0]\n",
"neuron(xs)"
]
},
{
"cell_type": "code",
"execution_count": 54,
"id": "ec4563ac",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[Value(data=-0.7982881295742965, _op=, label=), Value(data=0.21181420036863563, _op=, label=)]\n",
"[Value(data=-0.7497782653842755, _op=, label=), Value(data=0.555544388931462, _op=, label=)]\n",
"[Value(data=-0.9464441374064965, _op=, label=), Value(data=0.5089774092959138, _op=, label=)]\n"
]
}
],
"source": [
"layer = Layer(2, 3)\n",
"# this layer has 3 neurons, each neuron has 2 input\n",
"for neuron in layer.neurons:\n",
" print (neuron.w)"
]
},
{
"cell_type": "code",
"execution_count": 55,
"id": "9e947248",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\t <__main__.Layer object at 0x12e40f048>\n",
"\t\t <__main__.Neuron object at 0x12e40f0f0>\n",
"\t\t [Value(data=0.2433628928010092, _op=, label=), Value(data=-0.7551841903930812, _op=, label=)]\n",
"\t\t <__main__.Neuron object at 0x12e40f240>\n",
"\t\t [Value(data=0.3136327527830496, _op=, label=), Value(data=0.1673093452664174, _op=, label=)]\n",
"\t\t <__main__.Neuron object at 0x12e40f320>\n",
"\t\t [Value(data=-0.3963744139076282, _op=, label=), Value(data=0.5531002947326045, _op=, label=)]\n",
"\t <__main__.Layer object at 0x12e40f400>\n",
"\t\t <__main__.Neuron object at 0x12e40f438>\n",
"\t\t [Value(data=-0.8890517129288229, _op=, label=), Value(data=-0.4074469200671813, _op=, label=), Value(data=-0.10151646749100962, _op=, label=)]\n",
"\t\t <__main__.Neuron object at 0x12e40f550>\n",
"\t\t [Value(data=-0.9941308841542051, _op=, label=), Value(data=0.5235207826560115, _op=, label=), Value(data=0.40059567697003273, _op=, label=)]\n",
"\t\t <__main__.Neuron object at 0x12e40f668>\n",
"\t\t [Value(data=-0.3957439769632747, _op=, label=), Value(data=-0.7021758473360142, _op=, label=), Value(data=0.3531180349775369, _op=, label=)]\n",
"\t\t <__main__.Neuron object at 0x12e40f780>\n",
"\t\t [Value(data=0.824463971413687, _op=, label=), Value(data=-0.7559149986155553, _op=, label=), Value(data=-0.06125339835043553, _op=, label=)]\n"
]
}
],
"source": [
"nn = MLP(2, [3, 4])\n",
"# this mlp has 2 layers\n",
"# first layer 2 in, 3 out\n",
"# second layer 3 in, 4 out \n",
"for layer in nn.layers:\n",
" print ('\\t', layer)\n",
" for neuron in layer.neurons:\n",
" print ('\\t\\t', neuron)\n",
" print ('\\t\\t', neuron.w)"
]
},
{
"cell_type": "markdown",
"id": "1c3eda96",
"metadata": {},
"source": [
"# 7. backpropagation to update gradient in MLP manually"
]
},
{
"cell_type": "code",
"execution_count": 56,
"id": "ab4cb3c8",
"metadata": {},
"outputs": [],
"source": [
"# 3 input neuron, 4 layers, 4 neuron, 2 neuron, 1 neuron\n",
"nn = MLP(3, [4, 2, 1])"
]
},
{
"cell_type": "code",
"execution_count": 57,
"id": "69490dc9",
"metadata": {},
"outputs": [
{
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],
"source": [
"# backpropagate it to adjust the weights! \n",
"\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]\n",
"\n",
"ys_preds = [nn(x) for x in xs]\n",
"\n",
"# using random weights\n",
"ys_preds "
]
},
{
"cell_type": "code",
"execution_count": 58,
"id": "835f65eb",
"metadata": {},
"outputs": [
{
"data": {
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}
],
"source": [
"# tune the weight\n",
"# calculate the single number LOSS to tweak the weight\n",
"# MSE loss\n",
"# ypt: ground truth\n",
"# yout: y output \n",
"# we want to minimize the LOSS\n",
"loss = sum(((y_pred - y_gt)**2 for y_gt, y_pred in zip(ys, ys_preds)), Value(0.0))\n",
"loss"
]
},
{
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"id": "c6ce179e",
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