1976 lines
158 KiB
Plaintext
1976 lines
158 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "672d74cf-2fad-4493-9e3d-7ac2286dbfee",
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"metadata": {},
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"outputs": [],
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"source": [
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"import math\n",
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"import numpy as np\n",
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"import matplotlib.pyplot as plt\n",
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"%matplotlib inline"
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]
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},
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{
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"cell_type": "markdown",
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"id": "e7295cd8-1f33-40d2-aca5-e226e5b945ba",
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"metadata": {},
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"source": [
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"Let's define a function f"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "07a0fdd9-e9c4-4a28-9201-99429a638573",
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"metadata": {},
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"outputs": [],
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"source": [
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"def f(x):\n",
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" return 3*x**2 - 4*x + 5"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"id": "67093e12-40d4-4abd-a81d-2f7f63881e19",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"20.0"
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]
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},
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"execution_count": 3,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"f(3.0)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "ec1297be-8d68-4d59-af98-39e45eb9857b",
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"metadata": {},
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"source": [
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"We can also plot it for a range of values"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"id": "0a3f756f-886b-455d-b1f2-53060088dce9",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"[<matplotlib.lines.Line2D at 0x229ffcad010>]"
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]
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},
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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},
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{
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"data": {
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"image/png": 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",
|
|
"text/plain": [
|
|
"<Figure size 640x480 with 1 Axes>"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"xs = np.arange(-5, 5, 0.25)\n",
|
|
"ys = f(xs)\n",
|
|
"plt.plot(xs, ys)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "5c9cb820-159e-420d-b0d0-a8706e19f9d5",
|
|
"metadata": {},
|
|
"source": [
|
|
"Now, what's a derivate?\n",
|
|
"> It is sensitivity of the function to the change of the output with respect to the input.\n",
|
|
"\n",
|
|
"In simpler terms, (f(x+h) - f(x))/h, where h tends to zero. This gives us the slope."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 5,
|
|
"id": "7866b622-e10c-4003-844f-9044e7bdc080",
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"3.0000002482211127e-05"
|
|
]
|
|
},
|
|
"execution_count": 5,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"h = 0.00001\n",
|
|
"x = 2/3\n",
|
|
"(f(x+h) - f(x))/h"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "4f027e6d-d889-4827-867e-335a96c20045",
|
|
"metadata": {},
|
|
"source": [
|
|
"Let's define a function with mupltiple inputs"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 6,
|
|
"id": "a3bcd24a-4a48-4d85-8d49-3fb751589b40",
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"4.0\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"a = 2.0\n",
|
|
"b = -3.0\n",
|
|
"c = 10.0\n",
|
|
"\n",
|
|
"d = a*b + c\n",
|
|
"print(d)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 7,
|
|
"id": "78dcb378-190f-4421-8d95-af79cdf05b18",
|
|
"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",
|
|
"a = 2.0\n",
|
|
"b = -3.0\n",
|
|
"c = 10.0\n",
|
|
"\n",
|
|
"d1 = a*b + c\n",
|
|
"a += h\n",
|
|
"d2 = a*b + c\n",
|
|
"\n",
|
|
"print('d1', d1)\n",
|
|
"print('d2', d2)\n",
|
|
"print('slope', (d2 - d1)/h) # By the good old derivation (wrt to a), we know that this will be b"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 8,
|
|
"id": "9fb1208f-1a1e-4fdb-a290-a7d0e8a5050a",
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"d1 4.0\n",
|
|
"d2 4.0002\n",
|
|
"slope 2.0000000000042206\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"h = 0.0001\n",
|
|
"a = 2.0\n",
|
|
"b = -3.0\n",
|
|
"c = 10.0\n",
|
|
"\n",
|
|
"d1 = a*b + c\n",
|
|
"b += h\n",
|
|
"d2 = a*b + c\n",
|
|
"\n",
|
|
"print('d1', d1)\n",
|
|
"print('d2', d2)\n",
|
|
"print('slope', (d2 - d1)/h) # By the good old derivation (wrt to b), we know that this will be a"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 9,
|
|
"id": "e7bfb7ef-4c16-44c6-afce-ac20ab858b4f",
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
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"text": [
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"d1 4.0\n",
|
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"d2 4.0001\n",
|
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"slope 0.9999999999976694\n"
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]
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}
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],
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"source": [
|
|
"h = 0.0001\n",
|
|
"a = 2.0\n",
|
|
"b = -3.0\n",
|
|
"c = 10.0\n",
|
|
"\n",
|
|
"d1 = a*b + c\n",
|
|
"c += h\n",
|
|
"d2 = a*b + c\n",
|
|
"\n",
|
|
"print('d1', d1)\n",
|
|
"print('d2', d2)\n",
|
|
"print('slope', (d2 - d1)/h) # By the good old derivation (wrt to c), we know that this will be 1"
|
|
]
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|
},
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{
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"cell_type": "markdown",
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"id": "ad332637-3dbe-4630-bb5a-d5a76e935a9b",
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"metadata": {},
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"source": [
|
|
"The NN will be mathematically very large expressions. We now start by building the data structures for this. Let's start by making the `Value` object from mircograd"
|
|
]
|
|
},
|
|
{
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"cell_type": "code",
|
|
"execution_count": 131,
|
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"id": "e11f7575-00be-44a4-9bde-ae9288b1922a",
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"metadata": {
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"scrolled": true
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},
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"outputs": [
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{
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|
"data": {
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"text/plain": [
|
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"Value(data=4.0)"
|
|
]
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|
},
|
|
"execution_count": 131,
|
|
"metadata": {},
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|
"output_type": "execute_result"
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|
}
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|
],
|
|
"source": [
|
|
"class Value:\n",
|
|
"\n",
|
|
" def __init__(self, data, _children=(), _op='', label=''): # Define a empty tuple `children` to keep the pointers of other Value objects\n",
|
|
" # Define a empty set `op` to keep track of what created that Value object\n",
|
|
" self.data = data\n",
|
|
" self.grad = 0.0 # this is the derivative of Value wrt to its nodes; initalized as 0 (we assume at the beginning that every Value doesn't impact the output)\n",
|
|
" self._backward = lambda: None # for a leaf node, this should be nothing.\n",
|
|
" self._prev = set(_children) # This will be a empty set when we define a new Value object (a, b, c)\n",
|
|
" self._op = _op\n",
|
|
" self.label = label\n",
|
|
"\n",
|
|
" def __repr__(self):\n",
|
|
" return f\"Value(data={self.data})\"\n",
|
|
"\n",
|
|
" def __add__(self, other): # a.__add__(b)\n",
|
|
" out = Value(self.data + other.data, (self, other), '+') # Since self.data and other.data is python floating point number,\n",
|
|
" # the addition is according to whatever is defined in the python kernel\n",
|
|
" def _backward():\n",
|
|
" self.grad += 1.0 * out.grad\n",
|
|
" other.grad += 1.0 * out.grad\n",
|
|
" out._backward = _backward\n",
|
|
" \n",
|
|
" return out\n",
|
|
" \n",
|
|
" def __mul__(self, other): # a.__mul__(b) # You can't name it mult or multi, because we are defining magic methods for the Value object\n",
|
|
" out = Value(self.data * other.data, (self, other), '*')\n",
|
|
"\n",
|
|
" def _backward():\n",
|
|
" self.grad += other.data * out.grad\n",
|
|
" other.grad += self.data * out.grad\n",
|
|
" out._backward = _backward\n",
|
|
" \n",
|
|
" return out\n",
|
|
"\n",
|
|
" def tanh(self):\n",
|
|
" x = self.data\n",
|
|
" t = (math.exp(2*x) - 1)/(math.exp(2*x) + 1)\n",
|
|
" out = Value(t, (self, ), 'tanh')\n",
|
|
"\n",
|
|
" def _backward():\n",
|
|
" self.grad += (1 - t**2) * out.grad\n",
|
|
" out._backward = _backward\n",
|
|
" \n",
|
|
" return out\n",
|
|
"\n",
|
|
" def backward(self):\n",
|
|
" \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",
|
|
" topo.append(v)\n",
|
|
" build_topo(self)\n",
|
|
" \n",
|
|
" self.grad = 1.0\n",
|
|
" \n",
|
|
" for node in reversed(topo):\n",
|
|
" node._backward()\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' #(a.__mul__(b)).__add__(c); you can also call this manually.\n",
|
|
"f = Value(-2.0, label='f')\n",
|
|
"L = d * f; L.label = 'L'\n",
|
|
"d"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 11,
|
|
"id": "aa1c0bd9-5c01-4ff6-b397-bf8b37ca0a70",
|
|
"metadata": {},
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"outputs": [
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|
{
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"data": {
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"text/plain": [
|
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"{Value(data=-6.0), Value(data=10.0)}"
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]
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},
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|
"execution_count": 11,
|
|
"metadata": {},
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|
"output_type": "execute_result"
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|
}
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|
],
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|
"source": [
|
|
"d._prev # This will gives us the Value objects that made d (the children of d, ik sounds wrong but it is what it is)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 12,
|
|
"id": "7d88f630-07a0-4e12-a0e6-177094ed4999",
|
|
"metadata": {},
|
|
"outputs": [
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{
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|
"data": {
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|
"text/plain": [
|
|
"set()"
|
|
]
|
|
},
|
|
"execution_count": 12,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
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|
}
|
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],
|
|
"source": [
|
|
"c._prev # Has no children"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 13,
|
|
"id": "a24046f3-53a4-4bea-ae0b-756bb9a80dd3",
|
|
"metadata": {},
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|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"'+'"
|
|
]
|
|
},
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|
"execution_count": 13,
|
|
"metadata": {},
|
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"output_type": "execute_result"
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}
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],
|
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"source": [
|
|
"d._op"
|
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]
|
|
},
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|
{
|
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"cell_type": "code",
|
|
"execution_count": 27,
|
|
"id": "cdf24235-ee48-4675-bb65-bd78d1ae201e",
|
|
"metadata": {},
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"outputs": [],
|
|
"source": [
|
|
"# imported from mircograd's codebase\n",
|
|
"\n",
|
|
"from graphviz import Digraph\n",
|
|
"\n",
|
|
"def trace(root):\n",
|
|
" # builds a set of all nodes and edges in a graph\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'}) # LR = left to right\n",
|
|
" \n",
|
|
" nodes, edges = trace(root)\n",
|
|
" for n in nodes:\n",
|
|
" uid = str(id(n))\n",
|
|
" # for any value in the graph, create a rectangular ('record') node for it\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 a result of some operation, create an op node for it\n",
|
|
" dot.node(name = uid + n._op, label = n._op)\n",
|
|
" # and connect this node to it\n",
|
|
" dot.edge(uid + n._op, uid)\n",
|
|
"\n",
|
|
" for n1, n2 in edges:\n",
|
|
" # connect n1 to the op node of n2\n",
|
|
" dot.edge(str(id(n1)), str(id(n2)) + n2._op)\n",
|
|
"\n",
|
|
" return dot"
|
|
]
|
|
},
|
|
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"The value of the forward pass is 8. We would now like to run backpropagation. We will start at the end and reverse and calculate the gradient along all these intermediate values. For every single value here, we are going to calculate the derivative of value (L) wrt every node. \n",
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"a.grad 6.000000000000227\n",
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"b.grad -3.9999999999995595\n",
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"def local_staging(): # just for our local testing\n",
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" h = 0.001\n",
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" \n",
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" a = Value(2.0, label='a')\n",
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" b = Value(-3.0, label='b')\n",
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" c = Value(10.0, label='c')\n",
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" e = a*b; e.label = 'e'\n",
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" d = e + c; d.label = 'd'\n",
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" f = Value(-2.0, label='f')\n",
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" L = d * f; L.label = 'L'\n",
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" L1 = L.data\n",
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" \n",
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" a = Value(2.0 + h, label='a') # changing a by h\n",
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" b = Value(-3.0, label='b')\n",
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" c = Value(10.0, label='c')\n",
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" e = a*b; e.label = 'e'\n",
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" d = e + c; d.label = 'd'\n",
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" f = Value(-2.0, label='f')\n",
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" L = d * f; L.label = 'L'\n",
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" L2 = L.data\n",
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"\n",
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" print('a.grad',(L2 - L1)/h)\n",
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"\n",
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" a = Value(2.0, label='a') \n",
|
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" b = Value(-3.0 + h, label='b') # changing b by h\n",
|
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" c = Value(10.0, label='c')\n",
|
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" e = a*b; e.label = 'e'\n",
|
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" d = e + c; d.label = 'd'\n",
|
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" f = Value(-2.0, label='f')\n",
|
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" L = d * f; L.label = 'L'\n",
|
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" L2 = L.data\n",
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"\n",
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" print('b.grad',(L2 - L1)/h) \n",
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|
"\n",
|
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" a = Value(2.0, label='a') \n",
|
|
" b = Value(-3.0, label='b')\n",
|
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" c = Value(10.0 + h, label='c') # changing c by h\n",
|
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" e = a*b; e.label = 'e'\n",
|
|
" d = e + c; d.label = 'd'\n",
|
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" f = Value(-2.0, label='f')\n",
|
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" L = d * f; L.label = 'L'\n",
|
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" L2 = L.data\n",
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"\n",
|
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" print('c.grad',(L2 - L1)/h) \n",
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|
"\n",
|
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" a = Value(2.0, label='a') \n",
|
|
" b = Value(-3.0, label='b')\n",
|
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" c = Value(10.0, label='c')\n",
|
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" e = a*b; e.label = 'e'\n",
|
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" e.data += h # changing e by h\n",
|
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" d = e + c; d.label = 'd'\n",
|
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" f = Value(-2.0, label='f')\n",
|
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" L = d * f; L.label = 'L'\n",
|
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" L2 = L.data\n",
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"\n",
|
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" print('e.grad',(L2 - L1)/h) \n",
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" \n",
|
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" a = Value(2.0, label='a')\n",
|
|
" b = Value(-3.0, label='b')\n",
|
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" c = Value(10.0, label='c')\n",
|
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" e = a*b; e.label = 'e'\n",
|
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" d = e + c; d.label = 'd'\n",
|
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" f = Value(-2.0 + h, label='f') # changing f by h\n",
|
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" L = d * f; L.label = 'L'\n",
|
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" L2 = L.data\n",
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"\n",
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" print('f.grad',(L2 - L1)/h) # should be d\n",
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"\n",
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" a = Value(2.0, label='a')\n",
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" b = Value(-3.0, label='b')\n",
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" c = Value(10.0, label='c')\n",
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" e = a*b; e.label = 'e'\n",
|
|
" d = e + c; d.label = 'd'\n",
|
|
" d.data += h # changing d by h\n",
|
|
" f = Value(-2.0, label='f') \n",
|
|
" L = d * f; L.label = 'L'\n",
|
|
" L2 = L.data \n",
|
|
"\n",
|
|
" print('d.grad',(L2 - L1)/h) # should be f\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",
|
|
" L2 = L.data + h # changing L by h\n",
|
|
"\n",
|
|
" print('L.grad',(L2 - L1)/h) # should be 1\n",
|
|
"\n",
|
|
"local_staging()"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 34,
|
|
"id": "78e46a2f-7ba5-458f-beb3-133ebca6a839",
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"L.grad = 1.0"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "90580c47-bf74-4599-81c9-8bccb8f3c69d",
|
|
"metadata": {},
|
|
"source": [
|
|
"L = d * f\n",
|
|
"\n",
|
|
"dL/dd = ?\n",
|
|
"By calculus, we know that this will be f"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 38,
|
|
"id": "bbd2d2c2-f136-4244-a901-ff7c0165efe2",
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"f.grad = 4.0\n",
|
|
"d.grad = -2.0"
|
|
]
|
|
},
|
|
{
|
|
"attachments": {
|
|
"8211c8be-62e8-4102-aaca-a5bd17838677.png": {
|
|
"image/png": 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"
|
|
}
|
|
},
|
|
"cell_type": "markdown",
|
|
"id": "57e6a7b9-30af-4273-95f1-906bc3da4756",
|
|
"metadata": {},
|
|
"source": [
|
|
"to find dL/dc, let's start with dd/dc \n",
|
|
"\n",
|
|
"d = c + e;\n",
|
|
"so, dd/dc = 1 (and dd/de = 1)\n",
|
|
"\n",
|
|
"This is a local derivative to study how c & e affects d. But we have to do this for L, and in a NN, there will be 1000s of such local derivatives.\n",
|
|
"\n",
|
|
"So, let's use chain rule\n",
|
|
"\n",
|
|
"\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",
|
|
"so,\n",
|
|
"\n",
|
|
"dL/dc = dL/dd * dd/dc = -2.0 * 1 = -2.0"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 45,
|
|
"id": "416bb5f7-8e8a-49d6-b6b8-784e0bf26447",
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"c.grad = -2.0\n",
|
|
"e.grad = -2.0"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "f5989699-809e-4b6b-bba6-7af1d836fe95",
|
|
"metadata": {},
|
|
"source": [
|
|
"Now, to find dL/da and dL/db,\n",
|
|
"\n",
|
|
"dL/da = (dL/dd * dd/de) * de/da = (dL/de) * de/da\n",
|
|
"\n",
|
|
"dL/db = (dL/dd * dd/de) * de/db = (dL/de) * de/db\n",
|
|
"\n",
|
|
"de/da = d(a*b)/da = b\n",
|
|
"\n",
|
|
"de/db = d(a*b)/db = a\n",
|
|
"\n",
|
|
"dL/da = -2.0 * -3.0 = 6.0\n",
|
|
"\n",
|
|
"dL/db = -2.0 * 2.0 = -4.0"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 50,
|
|
"id": "9c66472a-bcd8-42c7-93c0-2c44c8941469",
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"a.grad = 6.0\n",
|
|
"b.grad = -4.0"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "bbab9a31-8408-475a-98e3-58796dbe78cd",
|
|
"metadata": {},
|
|
"source": [
|
|
"This was manual backpropagation! Let's nudge our inputs towards their gradients to increase L (single, manual optimization step) "
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 62,
|
|
"id": "15926195-7075-48e8-bf02-d8b06ad904a4",
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"-7.713030143999999\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"a.data += 0.001 * a.grad # 0.001 is the step size\n",
|
|
"b.data += 0.001 * b.grad\n",
|
|
"c.data += 0.001 * c.grad\n",
|
|
"f.data += 0.001 * f.grad\n",
|
|
"\n",
|
|
"e = a * b\n",
|
|
"d = e + c\n",
|
|
"L = d * f\n",
|
|
"\n",
|
|
"print(L.data)"
|
|
]
|
|
},
|
|
{
|
|
"attachments": {
|
|
"34f0ebc5-cb10-4db9-b7a9-9f501a3732da.webp": {
|
|
"image/webp": 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"
|
|
}
|
|
},
|
|
"cell_type": "markdown",
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"id": "4eff7563-3a93-493d-8788-bb055a23d07a",
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"metadata": {},
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"source": [
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"Let's look at another backpropagation using a neuron\n",
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"\n",
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"\n",
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"\n",
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"activation function is usually sigmoid or tanh [[more here]](https://en.wikipedia.org/wiki/Activation_function)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 123,
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"id": "29f8a9d0-16a6-49e3-b1db-52084db2ce24",
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"metadata": {},
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"outputs": [],
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"source": [
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"# inputs x1,x2\n",
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"x1 = Value(2.0, label='x1')\n",
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"x2 = Value(0.0, label='x2')\n",
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"# weights w1,w2\n",
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"w1 = Value(-3.0, label='w1')\n",
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"w2 = Value(1.0, label='w2')\n",
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"# bias of the neuron\n",
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"b = Value(6.8813735870195432, label='b')\n",
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"# x1*w1 + x2*w2 + b\n",
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"x1w1 = x1*w1; x1w1.label = 'x1*w1'\n",
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"x2w2 = x2*w2; x2w2.label = 'x2*w2'\n",
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"x1w1x2w2 = x1w1 + x2w2; x1w1x2w2.label = 'x1*w1 + x2*w2'\n",
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"n = x1w1x2w2 + b; n.label = 'n'\n",
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"o = n.tanh(); o.label = 'output'"
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]
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},
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"id": "1a1bdc0d-23e1-44bc-8b8e-1e88cdb64b2c",
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"metadata": {
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"scrolled": true
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}
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"draw_dot(o)"
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"o.grad = 1.0"
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]
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{
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"cell_type": "code",
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"execution_count": 80,
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"metadata": {},
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"0.4999999999999999\n"
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]
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}
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],
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"source": [
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"# o = tanh(n)\n",
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"# do/dn = 1 - tanh(n)**2 = 1 - o**2\n",
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"print(1 - o.data**2)\n",
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"n.grad = 0.5"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 78,
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"id": "aa8737b2-6e53-4aa5-bb2c-2221e1dce011",
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"metadata": {},
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"outputs": [],
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"source": [
|
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"# a plus is just a distributor of gradient, so n.grad will flow to b and x1*w1 + x2*w2, and x1w1x2w2 will flow to x1w1 & x2w2\n",
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"x1w1x2w2.grad = 0.5\n",
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"b.grad = 0.5\n",
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"x1w1.grad = 0.5\n",
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"x2w2.grad = 0.5"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 82,
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"id": "ff460f77-a807-4fce-93b7-079662202a70",
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"metadata": {},
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"outputs": [],
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"source": [
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"# a multi is front grad * the other child\n",
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"x1.grad = w1.data * x1w1.grad\n",
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"w1.grad = x1.data * x1w1.grad\n",
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"x2.grad = w2.data * x2w2.grad\n",
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"w2.grad = x2.data * x2w2.grad"
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]
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},
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{
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"cell_type": "markdown",
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"id": "d6965ba7-fefa-4c10-aae7-49ee2e9da76d",
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"metadata": {},
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"source": [
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"Doing this manually is ridiculous. Let's implement this backprop in our Value object."
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]
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},
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{
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"cell_type": "code",
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|
"execution_count": 101,
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"id": "32f79032-f121-4b86-b77d-dacdbdd01ee4",
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"metadata": {},
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"outputs": [],
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"source": [
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"o.grad = 1.0 # we need to set the base case for o"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 102,
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"id": "66038692-d288-4052-9861-4aeeb8a84836",
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"metadata": {},
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"outputs": [],
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"source": [
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"o._backward()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 108,
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"id": "ac1b36fa-1eae-473a-9182-2e18c0d18c1b",
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"metadata": {},
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"outputs": [],
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"source": [
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"n._backward()\n",
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|
"x1w1x2w2._backward()\n",
|
|
"x1w1._backward()\n",
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"x2w2._backward()"
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]
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},
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{
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"cell_type": "markdown",
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"id": "aeb2bc8d-76cb-4a0a-8fec-848a5ea6600e",
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"metadata": {},
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"source": [
|
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"Now, we get rid of us doing _backward manually. For this, we use topological sort [[more here](https://en.wikipedia.org/wiki/Topological_sorting)]"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 110,
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"id": "4adbf2b0-5130-4908-8536-6da47081037a",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"[Value(data=6.881373587019543),\n",
|
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" Value(data=-3.0),\n",
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" Value(data=2.0),\n",
|
|
" Value(data=-6.0),\n",
|
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" Value(data=0.0),\n",
|
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" Value(data=1.0),\n",
|
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" Value(data=0.0),\n",
|
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" Value(data=-6.0),\n",
|
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" Value(data=0.8813735870195432),\n",
|
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" Value(data=0.7071067811865476)]"
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]
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},
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|
"execution_count": 110,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
|
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"# imported from mircograd's codebase\n",
|
|
"\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",
|
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" build_topo(child)\n",
|
|
" topo.append(v)\n",
|
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"build_topo(o)\n",
|
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"topo"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 120,
|
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"id": "d9626671-9a47-41ce-a7d3-3517a3f61888",
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"metadata": {},
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"outputs": [],
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"source": [
|
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"o.grad = 1.0\n",
|
|
"\n",
|
|
"topo = []\n",
|
|
"visited = set()\n",
|
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"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",
|
|
" topo.append(v)\n",
|
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"build_topo(o)\n",
|
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"\n",
|
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"for node in reversed(topo):\n",
|
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" node._backward()"
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]
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},
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{
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"cell_type": "markdown",
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"id": "174e74ee-c432-41e6-acd4-a1e8c3ccbf7b",
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"metadata": {},
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"source": [
|
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"Now let's put this inside our Value object"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 125,
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"id": "ebcd762e-0b83-4303-a6aa-c60f2b4e24c7",
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"metadata": {},
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"outputs": [],
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"source": [
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"o.backward()"
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]
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},
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{
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"cell_type": "markdown",
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"id": "3507e421-b28b-486b-8b6b-4126bafe1964",
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"metadata": {},
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"source": [
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"We have a bug"
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]
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},
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{
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"id": "82af7380-6803-4d7d-8088-8ee42edd0cdd",
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"metadata": {},
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"outputs": [
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"metadata": {},
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}
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],
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"source": [
|
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"a = Value(3.0, label='a')\n",
|
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"b = a + a ; b.label = 'b'\n",
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"b.backward()\n",
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"draw_dot(b)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "88b4e727-0ab8-448e-8076-45fcb46cbe0a",
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"metadata": {},
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"source": [
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"The grad of a should be 2, NOT 1!"
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]
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},
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{
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"cell_type": "code",
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"id": "8d38e9af-126f-4574-9fd7-5ea1e9285d23",
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],
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"text/plain": [
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]
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},
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"execution_count": 133,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"a = Value(-2.0, label='a')\n",
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"b = Value(3.0, label='b')\n",
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"d = a * b ; d.label = 'd'\n",
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"e = a + b ; e.label = 'e'\n",
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"f = d * e ; f.label = 'f'\n",
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"\n",
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"f.backward() # Fixed\n",
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"\n",
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"draw_dot(f)"
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]
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}
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],
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