some computational stuff
Will King
5 years ago
parent
cdd75d05ea
commit
ea4fec2813
@ -0,0 +1,252 @@
|
||||
{
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 1,
|
||||
"id": "11cc7082-6fb7-49fd-992e-6678f1570ed9",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"import torch\n",
|
||||
"import numpy\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 2,
|
||||
"id": "9cff4845-9f88-4f49-8ad9-d94386ccc7dc",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"a = torch.tensor([2., 3.], requires_grad=True)\n",
|
||||
"b = torch.tensor([6., 4.], requires_grad=True)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 12,
|
||||
"id": "f49549e1-dbcb-4aa7-8036-8bef936b3b98",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def f(a,b):\n",
|
||||
" return a**3+a**4 + a*b**2 -b**2 -b**5"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 14,
|
||||
"id": "7ee977aa-d71e-4cfd-8743-a06216fa2209",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"f(a,b).backward(q)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 15,
|
||||
"id": "bccc63c6-ce7c-402e-99fc-d5b0a9b0deb1",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"tensor([ 92., 169.])"
|
||||
]
|
||||
},
|
||||
"execution_count": 15,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"a.grad"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 16,
|
||||
"id": "fe0575a3-bdc9-4b2f-9564-b01763c09c2b",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"tensor([-6252., -1168.])"
|
||||
]
|
||||
},
|
||||
"execution_count": 16,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"b.grad"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 10,
|
||||
"id": "e653ec77-4034-4797-b5db-bb06778090da",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"\n",
|
||||
"Begin demo \n",
|
||||
"\n",
|
||||
"x = tensor([4.], requires_grad=True)\n",
|
||||
"y = tensor([29.], grad_fn=<AddBackward0>)\n",
|
||||
"\n",
|
||||
"df = <AddBackward0 object at 0x7f388c0cd640>\n",
|
||||
"\n",
|
||||
"gradient of func(x) = \n",
|
||||
"tensor([11.])\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"# gradient_demo.py\n",
|
||||
"\n",
|
||||
"import torch as T\n",
|
||||
"device = T.device(\"cpu\")\n",
|
||||
"\n",
|
||||
"def some_func(x):\n",
|
||||
" result = (x * x) + (3 * x) + 1\n",
|
||||
" return result\n",
|
||||
"\n",
|
||||
"def main():\n",
|
||||
" print(\"\\nBegin demo \\n\")\n",
|
||||
"\n",
|
||||
" x = T.tensor([4.0], dtype=T.float32,\n",
|
||||
" requires_grad=True).to(device)\n",
|
||||
" y = some_func(x)\n",
|
||||
"\n",
|
||||
" print(\"x = \" + str(x))\n",
|
||||
" print(\"y = \" + str(y))\n",
|
||||
" print(\"\")\n",
|
||||
"\n",
|
||||
" df = y.grad_fn\n",
|
||||
" print(\"df = \" + str(df))\n",
|
||||
" print(\"\")\n",
|
||||
"\n",
|
||||
" y.backward() # compute grad of some_func(4)\n",
|
||||
" print(\"gradient of func(x) = \")\n",
|
||||
" print(x.grad) # 2(4) + 3 = 11\n",
|
||||
"\n",
|
||||
"if __name__ == \"__main__\":\n",
|
||||
" main()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "b3470450-834c-43db-8f89-d6f51e2754e0",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Try this\n",
|
||||
" - define a bellman like function \n",
|
||||
" - use `def` \n",
|
||||
" - Find the derivatives with respect to X\n",
|
||||
" - Find the derivatives with respect to $\\theta$\n",
|
||||
" - Invert this to find the time-transition function?\n",
|
||||
" - Inversion is probably impossible. Instead, solve both the policy and value of derivatives together?\n",
|
||||
" \n",
|
||||
" \n",
|
||||
"## Math for conditions\n",
|
||||
"### Optimality\n",
|
||||
"Stationarity and Complementary Slackness conditions\n",
|
||||
"$$\n",
|
||||
"0 = \\frac{\\partial F}{\\partial x} + \\beta \\frac{\\partial G}{\\partial x} \\frac{\\partial V}{\\partial G} + \\lambda \\\\\n",
|
||||
"0 = \\lambda \\cdot g(x,\\theta) \\\n",
|
||||
"$$\n",
|
||||
"### Envelope\n",
|
||||
"$$\n",
|
||||
"0 = \\frac{\\partial F}{\\partial \\theta} + \\frac{\\partial G}{\\partial \\theta} \\frac{\\partial V}{\\partial G} - \\frac{\\partial V}{\\partial \\theta}\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "91e8528c-a2c4-4dfd-87c6-3d367c9d0f2e",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"So, how do you incorporate the situation where you have to iterate multiple times?\n",
|
||||
" - Just add conditions as rows?\n",
|
||||
" - Solve and substitute using some theorem on the inverse of derivatives in multivariate systems?"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 18,
|
||||
"id": "9b9df983-d2b2-4515-a98b-2469d8d392f1",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def utility(c,a=2):\n",
|
||||
" return (c**(1-a))/(1-a)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 19,
|
||||
"id": "ea3a604f-48bb-424b-bb76-649d640a55db",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"-1.0"
|
||||
]
|
||||
},
|
||||
"execution_count": 19,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": []
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 17,
|
||||
"id": "8cc8b408-c0b9-4344-bed7-f6fbb6514a2c",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def bellman(theta, x, V, b=0.95):\n",
|
||||
" pass\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "d7730732-de9e-4d8c-b85f-112bee09762a",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Python 3",
|
||||
"language": "python",
|
||||
"name": "python3"
|
||||
},
|
||||
"language_info": {
|
||||
"codemirror_mode": {
|
||||
"name": "ipython",
|
||||
"version": 3
|
||||
},
|
||||
"file_extension": ".py",
|
||||
"mimetype": "text/x-python",
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.9.5"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 5
|
||||
}
|
||||
Loading…
Reference in New Issue