update
This commit is contained in:
Morten Hjorth-Jensen
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@@ -3,9 +3,7 @@
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{
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"cell_type": "markdown",
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"id": "d5ebb4c0",
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"metadata": {
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"editable": true
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},
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"metadata": {},
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"source": [
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"<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)\n",
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"doconce format html exercisesweek43.do.txt -->\n",
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@@ -15,9 +13,7 @@
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||||
{
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"cell_type": "markdown",
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"id": "812b4e46",
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||||
"metadata": {
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"editable": true
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},
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"metadata": {},
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"source": [
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"# Exercises weeks 43 and 44 \n",
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"**October 23-27, 2023**\n",
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@@ -30,9 +26,7 @@
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{
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"cell_type": "markdown",
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"id": "3230cd2f",
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||||
"metadata": {
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"editable": true
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},
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"metadata": {},
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"source": [
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"# Overarching aims of the exercises weeks 43 and 44\n",
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"\n",
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@@ -70,9 +64,7 @@
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||||
{
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"cell_type": "markdown",
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"id": "e3617d4e",
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"metadata": {
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"editable": true
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},
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"metadata": {},
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"source": [
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"## The AND and XOR Gates\n",
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"\n",
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@@ -108,9 +100,7 @@
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{
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"cell_type": "markdown",
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"id": "54e1e7fc",
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"metadata": {
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||||
"editable": true
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||||
},
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||||
"metadata": {},
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"source": [
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"## Representing the Data Sets\n",
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"\n",
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@@ -120,9 +110,7 @@
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{
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"cell_type": "markdown",
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"id": "6b3c15cb",
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||||
"metadata": {
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||||
"editable": true
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||||
},
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"metadata": {},
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"source": [
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"$$\n",
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"\\boldsymbol{X}=\\begin{bmatrix} 0 & 0 \\\\\n",
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@@ -135,9 +123,7 @@
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||||
{
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"cell_type": "markdown",
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"id": "acc25271",
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||||
"metadata": {
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"editable": true
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},
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"metadata": {},
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"source": [
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"while the vector of outputs is $\\boldsymbol{y}^T=[0,1,1,0]$ for the XOR gate, $\\boldsymbol{y}^T=[0,0,0,1]$ for the AND gate and $\\boldsymbol{y}^T=[0,1,1,1]$ for the OR gate.\n",
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"\n",
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@@ -165,9 +151,7 @@
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{
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"cell_type": "markdown",
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"id": "ffe0a840",
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||||
"metadata": {
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||||
"editable": true
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||||
},
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"metadata": {},
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"source": [
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"## Setting up dimensionalities by hand\n",
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"\n",
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@@ -177,9 +161,7 @@
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{
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"cell_type": "markdown",
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"id": "46abf545",
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||||
"metadata": {
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"editable": true
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||||
},
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"metadata": {},
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"source": [
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"$$\n",
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"\\boldsymbol{W_h}=\\begin{bmatrix} 1 & 1 \\\\\n",
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@@ -190,9 +172,7 @@
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{
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"cell_type": "markdown",
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"id": "86e105cc",
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"metadata": {
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"editable": true
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},
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"metadata": {},
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"source": [
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"Multiplying $\\boldsymbol{X}$ and $\\boldsymbol{W}$ gives"
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]
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@@ -200,9 +180,7 @@
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{
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"cell_type": "markdown",
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"id": "5e1d21f1",
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"metadata": {
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"editable": true
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},
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"metadata": {},
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"source": [
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"$$\n",
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"\\boldsymbol{X}{W}_h=\\begin{bmatrix} 0 & 0 \\\\\n",
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@@ -215,9 +193,7 @@
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{
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"cell_type": "markdown",
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"id": "692c6cbf",
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"metadata": {
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"editable": true
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},
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"metadata": {},
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"source": [
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"Assume also that the bias vector for the hidden layer is"
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]
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@@ -225,9 +201,7 @@
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{
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"cell_type": "markdown",
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"id": "5d81e641",
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"metadata": {
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"editable": true
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},
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"metadata": {},
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"source": [
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"$$\n",
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"\\boldsymbol{b}_h=\\begin{bmatrix} 0 \\\\\n",
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@@ -238,9 +212,7 @@
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{
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"cell_type": "markdown",
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"id": "a42bdee4",
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"metadata": {
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"editable": true
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},
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"metadata": {},
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"source": [
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"Adding it gives us the input to the activation function of the hidden layer"
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]
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@@ -248,9 +220,7 @@
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{
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"cell_type": "markdown",
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"id": "5116b854",
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"metadata": {
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"editable": true
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},
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"metadata": {},
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"source": [
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"$$\n",
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"\\boldsymbol{z}_h=\\boldsymbol{X}\\boldsymbol{W}_h+\\boldsymbol{b}_h=\\begin{bmatrix} 0 & -1 \\\\\n",
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@@ -263,9 +233,7 @@
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{
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"cell_type": "markdown",
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"id": "8dd26d09",
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"metadata": {
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"editable": true
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},
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"metadata": {},
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"source": [
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"Let us then assume that our activation function is the RELU function, which simply means that we take the max of $0$ and the elements of the input argument $\\boldsymbol{z}_h$, that is we have"
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]
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@@ -273,9 +241,7 @@
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{
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"cell_type": "markdown",
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"id": "ebf97c03",
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"metadata": {
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"editable": true
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},
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"metadata": {},
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"source": [
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"$$\n",
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"\\boldsymbol{a}_h=\\mathrm{RELU}(\\boldsymbol{z}_h=\\boldsymbol{X}\\boldsymbol{W}_h+\\boldsymbol{b}_h)=\\begin{bmatrix} 0 & 0 \\\\\n",
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@@ -288,9 +254,7 @@
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{
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"cell_type": "markdown",
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"id": "262a4a3a",
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||||
"metadata": {
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||||
"editable": true
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||||
},
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||||
"metadata": {},
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"source": [
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"Assume also that the bias of the output layer is zero and that the weights of the output layer are"
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]
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@@ -298,9 +262,7 @@
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{
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"cell_type": "markdown",
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"id": "f03ad8e7",
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||||
"metadata": {
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"editable": true
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||||
},
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"metadata": {},
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"source": [
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"$$\n",
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"\\boldsymbol{w}_o=\\begin{bmatrix} 1 \\\\\n",
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@@ -311,9 +273,7 @@
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||||
{
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||||
"cell_type": "markdown",
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"id": "36fe00c0",
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||||
"metadata": {
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||||
"editable": true
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||||
},
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||||
"metadata": {},
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||||
"source": [
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||||
"and multiplying with $\\boldsymbol{a}_h$ gives the output"
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]
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@@ -321,9 +281,7 @@
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{
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||||
"cell_type": "markdown",
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"id": "80ab43ac",
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||||
"metadata": {
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||||
"editable": true
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||||
},
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||||
"metadata": {},
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||||
"source": [
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"$$\n",
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||||
"\\boldsymbol{a}_o=\\begin{bmatrix} 0 & 0 \\\\\n",
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@@ -337,9 +295,7 @@
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||||
{
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||||
"cell_type": "markdown",
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||||
"id": "86bcfe49",
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||||
"metadata": {
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||||
"editable": true
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||||
},
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||||
"metadata": {},
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||||
"source": [
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||||
"the wanted result. Pay attention to the dimensionalities as well."
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||||
]
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||||
@@ -347,9 +303,7 @@
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||||
{
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||||
"cell_type": "markdown",
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||||
"id": "f19a899e",
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||||
"metadata": {
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||||
"editable": true
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||||
},
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||||
"metadata": {},
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||||
"source": [
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||||
"## Setting up the Neural Network\n",
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||||
"\n",
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||||
@@ -360,10 +314,7 @@
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||||
"cell_type": "code",
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||||
"execution_count": 1,
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||||
"id": "901ddca7",
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||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
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||||
"metadata": {},
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||||
"outputs": [],
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||||
"source": [
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||||
"%matplotlib inline\n",
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@@ -430,9 +381,7 @@
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||||
{
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||||
"cell_type": "markdown",
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||||
"id": "53f22266",
|
||||
"metadata": {
|
||||
"editable": true
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||||
},
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||||
"metadata": {},
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||||
"source": [
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||||
"Not an impressive result, but this was our first forward pass with randomly assigned weights. Let us now add the full network with the back-propagation algorithm discussed above."
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||||
]
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||||
@@ -440,9 +389,7 @@
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||||
{
|
||||
"cell_type": "markdown",
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||||
"id": "5f665ac6",
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||||
"metadata": {
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||||
"editable": true
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||||
},
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||||
"metadata": {},
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||||
"source": [
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||||
"## The Code using Scikit-Learn"
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||||
]
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||||
@@ -451,10 +398,7 @@
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||||
"cell_type": "code",
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||||
"execution_count": 2,
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||||
"id": "cb396cda",
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||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
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||||
},
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||||
"metadata": {},
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||||
"outputs": [],
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||||
"source": [
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||||
"# import necessary packages\n",
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||||
@@ -516,9 +460,7 @@
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||||
{
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||||
"cell_type": "markdown",
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||||
"id": "c5978471",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
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||||
"metadata": {},
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||||
"source": [
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||||
"## Building a neural network code\n",
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||||
"\n",
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||||
@@ -535,9 +477,7 @@
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||||
{
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||||
"cell_type": "markdown",
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||||
"id": "7b82dc46",
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||||
"metadata": {
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||||
"editable": true
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||||
},
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||||
"metadata": {},
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||||
"source": [
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||||
"### Learning rate methods\n",
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||||
"\n",
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@@ -557,10 +497,7 @@
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||||
"cell_type": "code",
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"execution_count": 3,
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||||
"id": "de4dedfb",
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||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
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||||
"metadata": {},
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||||
"outputs": [],
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||||
"source": [
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||||
"import autograd.numpy as np\n",
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@@ -698,9 +635,7 @@
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||||
{
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||||
"cell_type": "markdown",
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||||
"id": "bb621fce",
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||||
"metadata": {
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||||
"editable": true
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||||
},
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||||
"metadata": {},
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||||
"source": [
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||||
"### Usage of the above learning rate schedulers\n",
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||||
"\n",
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||||
@@ -714,10 +649,7 @@
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||||
"cell_type": "code",
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||||
"execution_count": 4,
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||||
"id": "34ddb829",
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||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
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||||
},
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||||
"metadata": {},
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||||
"outputs": [],
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||||
"source": [
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||||
"momentum_scheduler = Momentum(eta=1e-3, momentum=0.9)\n",
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@@ -727,9 +659,7 @@
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||||
{
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||||
"cell_type": "markdown",
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||||
"id": "e43498d4",
|
||||
"metadata": {
|
||||
"editable": true
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||||
},
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||||
"metadata": {},
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||||
"source": [
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||||
"Here is a small example for how a segment of code using schedulers\n",
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||||
"could look. Switching out the schedulers is simple."
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||||
@@ -739,10 +669,7 @@
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||||
"cell_type": "code",
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||||
"execution_count": 5,
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||||
"id": "f05b9625",
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||||
"metadata": {
|
||||
"collapsed": false,
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||||
"editable": true
|
||||
},
|
||||
"metadata": {},
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||||
"outputs": [],
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||||
"source": [
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||||
"weights = np.ones((3,3))\n",
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@@ -761,9 +688,7 @@
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||||
{
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||||
"cell_type": "markdown",
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||||
"id": "19cf9841",
|
||||
"metadata": {
|
||||
"editable": true
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||||
},
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||||
"metadata": {},
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||||
"source": [
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||||
"### Cost functions\n",
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"\n",
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@@ -777,10 +702,7 @@
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||||
"cell_type": "code",
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"execution_count": 6,
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||||
"id": "993efa8d",
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||||
"metadata": {
|
||||
"collapsed": false,
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||||
"editable": true
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||||
},
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||||
"metadata": {},
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||||
"outputs": [],
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||||
"source": [
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||||
"import autograd.numpy as np\n",
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@@ -815,9 +737,7 @@
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||||
{
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||||
"cell_type": "markdown",
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||||
"id": "011c734c",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Below we give a short example of how these cost function may be used\n",
|
||||
"to obtain results if you wish to test them out on your own using\n",
|
||||
@@ -828,10 +748,7 @@
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||||
"cell_type": "code",
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||||
"execution_count": 7,
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||||
"id": "9e1b97f5",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
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||||
"outputs": [],
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||||
"source": [
|
||||
"from autograd import grad\n",
|
||||
@@ -849,9 +766,7 @@
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||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "4acd87b2",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"### Activation functions\n",
|
||||
"\n",
|
||||
@@ -865,10 +780,7 @@
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||||
"cell_type": "code",
|
||||
"execution_count": 8,
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||||
"id": "befa86ae",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"import autograd.numpy as np\n",
|
||||
@@ -923,9 +835,7 @@
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||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "c20aa75e",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Below follows a short demonstration of how to use an activation\n",
|
||||
"function. The derivative of the activation function will be important\n",
|
||||
@@ -938,10 +848,7 @@
|
||||
"cell_type": "code",
|
||||
"execution_count": 9,
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||||
"id": "e209f9e5",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"outputs": [],
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||||
"source": [
|
||||
"z = np.array([[4, 5, 6]]).T\n",
|
||||
@@ -959,9 +866,7 @@
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||||
{
|
||||
"cell_type": "markdown",
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||||
"id": "524b409b",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"### The Neural Network\n",
|
||||
"\n",
|
||||
@@ -983,10 +888,7 @@
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||||
"cell_type": "code",
|
||||
"execution_count": 10,
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||||
"id": "083116d3",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"outputs": [],
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||||
"source": [
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||||
"import math\n",
|
||||
@@ -1455,9 +1357,7 @@
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||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "e4ba55d0",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Before we make a model, we will quickly generate a dataset we can use\n",
|
||||
"for our linear regression problem as shown below"
|
||||
@@ -1467,10 +1367,7 @@
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||||
"cell_type": "code",
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||||
"execution_count": 11,
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||||
"id": "c72a22c6",
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||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"import autograd.numpy as np\n",
|
||||
@@ -1511,9 +1408,7 @@
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "25b63b47",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Now that we have our dataset ready for the regression, we can create\n",
|
||||
"our regressor. Note that with the seed parameter, we can make sure our\n",
|
||||
@@ -1527,10 +1422,7 @@
|
||||
"cell_type": "code",
|
||||
"execution_count": 12,
|
||||
"id": "b6b4e461",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"input_nodes = X_train.shape[1]\n",
|
||||
@@ -1542,9 +1434,7 @@
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "d7860b74",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"We then fit our model with our training data using the scheduler of our choice."
|
||||
]
|
||||
@@ -1553,10 +1443,7 @@
|
||||
"cell_type": "code",
|
||||
"execution_count": 13,
|
||||
"id": "00522c73",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"linear_regression.reset_weights() # reset weights such that previous runs or reruns don't affect the weights\n",
|
||||
@@ -1568,9 +1455,7 @@
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "a57bfb12",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Due to the progress bar we can see the MSE (train_error) throughout\n",
|
||||
"the FFNN's training. Note that the fit() function has some optional\n",
|
||||
@@ -1584,10 +1469,7 @@
|
||||
"cell_type": "code",
|
||||
"execution_count": 14,
|
||||
"id": "35259e41",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"linear_regression.reset_weights() # reset weights such that previous runs or reruns don't affect the weights\n",
|
||||
@@ -1598,9 +1480,7 @@
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "4a403370",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"We see that given more epochs to train on, the regressor reaches a lower MSE.\n",
|
||||
"\n",
|
||||
@@ -1613,10 +1493,7 @@
|
||||
"cell_type": "code",
|
||||
"execution_count": 15,
|
||||
"id": "6c791130",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"from sklearn.datasets import load_breast_cancer\n",
|
||||
@@ -1639,10 +1516,7 @@
|
||||
"cell_type": "code",
|
||||
"execution_count": 16,
|
||||
"id": "0ac0258d",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"input_nodes = X_train.shape[1]\n",
|
||||
@@ -1654,9 +1528,7 @@
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "a9c74baa",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"We will now make use of our validation data by passing it into our fit function as a keyword argument"
|
||||
]
|
||||
@@ -1665,10 +1537,7 @@
|
||||
"cell_type": "code",
|
||||
"execution_count": 17,
|
||||
"id": "e5021bbf",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"logistic_regression.reset_weights() # reset weights such that previous runs or reruns don't affect the weights\n",
|
||||
@@ -1680,9 +1549,7 @@
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "ae0148bb",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Finally, we will create a neural network with 2 hidden layers with activation functions."
|
||||
]
|
||||
@@ -1691,10 +1558,7 @@
|
||||
"cell_type": "code",
|
||||
"execution_count": 18,
|
||||
"id": "bb7e4340",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"input_nodes = X_train.shape[1]\n",
|
||||
@@ -1711,10 +1575,7 @@
|
||||
"cell_type": "code",
|
||||
"execution_count": 19,
|
||||
"id": "d0655afb",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"neural_network.reset_weights() # reset weights such that previous runs or reruns don't affect the weights\n",
|
||||
@@ -1726,9 +1587,7 @@
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "dc8a90af",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"### Multiclass classification\n",
|
||||
"\n",
|
||||
@@ -1741,10 +1600,7 @@
|
||||
"cell_type": "code",
|
||||
"execution_count": 20,
|
||||
"id": "9305c08a",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"from sklearn.datasets import load_digits\n",
|
||||
@@ -1778,9 +1634,7 @@
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "8e208051",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Testing the XOR gate and other gates\n",
|
||||
"\n",
|
||||
@@ -1791,10 +1645,7 @@
|
||||
"cell_type": "code",
|
||||
"execution_count": 21,
|
||||
"id": "db016d41",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"X = np.array([ [0, 0], [0, 1], [1, 0],[1, 1]],dtype=np.float64)\n",
|
||||
@@ -1814,15 +1665,31 @@
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "d1384449",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Not bad, but the results depend strongly on the learning reate. Try different learning rates."
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {},
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Python 3 (ipykernel)",
|
||||
"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": "3.9.10"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 5
|
||||
}
|
||||
|
||||
+231
-654
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Load Diff
+141
-390
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Load Diff
+249
-730
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Load Diff
Reference in New Issue
Block a user