update week 43
This commit is contained in:
Morten Hjorth-Jensen
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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": "77f844d9",
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"editable": true
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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": "d5b983e3",
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"metadata": {
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"editable": true
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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": "7f6cc12f",
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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": "c0b07e46",
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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": "e0a9840a",
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"metadata": {
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"editable": true
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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": "ff95d379",
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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": "36fa3466",
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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": "e2e5e808",
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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": "419e3904",
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"metadata": {
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"editable": true
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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": "88a9712a",
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"metadata": {
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"editable": true
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"metadata": {},
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"source": [
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"$$\n",
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"\\boldsymbol{X}{W}_h=\\begin{bmatrix} 0 & 0 \\\\\n",
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@@ -215,9 +193,7 @@
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{
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"cell_type": "markdown",
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"id": "036eee14",
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"metadata": {
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"editable": true
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"metadata": {},
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"source": [
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"Assume also that the bias vector for the hidden layer is"
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]
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@@ -225,9 +201,7 @@
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{
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"cell_type": "markdown",
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"id": "3c771918",
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"metadata": {
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"editable": true
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},
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"metadata": {},
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"source": [
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"$$\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": "f5a4150d",
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"metadata": {
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"editable": true
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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": "e4eb7753",
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"metadata": {
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"editable": true
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"source": [
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"$$\n",
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"\\boldsymbol{z}_h=\\boldsymbol{X}\\boldsymbol{W}_h+\\boldsymbol{b}_h=\\begin{bmatrix} 0 & -1 \\\\\n",
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@@ -263,9 +233,7 @@
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{
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"cell_type": "markdown",
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"id": "b5442b32",
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"metadata": {
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"editable": true
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"source": [
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"Let us then assume that our activation function is the RELU function, which simply means that we take the max of $0$ and the elements of the input argument $\\boldsymbol{z}_h$, that is we have"
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]
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@@ -273,9 +241,7 @@
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{
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"cell_type": "markdown",
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"id": "64d99f85",
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"metadata": {
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"editable": true
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"source": [
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"$$\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": "2aee4b6b",
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"metadata": {
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"editable": true
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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": "884541eb",
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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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"$$\n",
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"\\boldsymbol{w}_o=\\begin{bmatrix} 1 \\\\\n",
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@@ -311,9 +273,7 @@
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{
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"cell_type": "markdown",
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"id": "3999a4d5",
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"metadata": {
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"editable": true
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},
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"metadata": {},
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"source": [
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"and multiplying with $\\boldsymbol{a}_h$ gives the output"
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]
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@@ -321,9 +281,7 @@
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{
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"cell_type": "markdown",
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"id": "a4896b4d",
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"metadata": {
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"editable": true
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},
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"metadata": {},
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"source": [
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"$$\n",
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"\\boldsymbol{a}_o=\\boldsymbol{w}_h^T\\begin{bmatrix} 0 & 0 \\\\\n",
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@@ -336,9 +294,7 @@
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{
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"cell_type": "markdown",
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"id": "2c74e7c0",
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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."
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]
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@@ -346,9 +302,7 @@
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{
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"cell_type": "markdown",
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"id": "249b9804",
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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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@@ -359,10 +313,7 @@
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"cell_type": "code",
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"execution_count": 1,
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"id": "afdd3100",
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"metadata": {
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"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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"%matplotlib inline\n",
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@@ -429,9 +380,7 @@
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{
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"cell_type": "markdown",
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"id": "fb383164",
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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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"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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@@ -439,9 +388,7 @@
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{
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"cell_type": "markdown",
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"id": "0d26394c",
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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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@@ -450,10 +397,7 @@
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"cell_type": "code",
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"execution_count": 2,
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"id": "befc368c",
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"metadata": {
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"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 necessary packages\n",
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@@ -515,9 +459,7 @@
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{
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"cell_type": "markdown",
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"id": "03895a9f",
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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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"## Building a neural network code\n",
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"\n",
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@@ -534,9 +476,7 @@
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{
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"cell_type": "markdown",
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"id": "6226915b",
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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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@@ -556,10 +496,7 @@
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"cell_type": "code",
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"execution_count": 3,
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"id": "b6d5ea73",
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"metadata": {
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"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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@@ -697,9 +634,7 @@
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{
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"cell_type": "markdown",
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"id": "edb42bef",
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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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"cell_type": "code",
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"execution_count": 4,
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"id": "108c1209",
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"metadata": {
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"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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"momentum_scheduler = Momentum(eta=1e-3, momentum=0.9)\n",
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@@ -726,9 +658,7 @@
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{
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"cell_type": "markdown",
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"id": "d1c423f0",
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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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"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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@@ -738,10 +668,7 @@
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"cell_type": "code",
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"execution_count": 5,
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"id": "63c8c749",
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"metadata": {
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"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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"weights = np.ones((3,3))\n",
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@@ -760,9 +687,7 @@
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{
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"cell_type": "markdown",
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"id": "7cb80992",
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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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"### Cost functions\n",
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"\n",
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@@ -776,10 +701,7 @@
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"cell_type": "code",
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"execution_count": 6,
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"id": "a181dacb",
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"metadata": {
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"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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@@ -814,9 +736,7 @@
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{
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"cell_type": "markdown",
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"id": "26a7fe37",
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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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"Below we give a short example of how these cost function may be used\n",
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"to obtain results if you wish to test them out on your own using\n",
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"cell_type": "code",
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"execution_count": 7,
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"id": "5894bd31",
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"metadata": {
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"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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"from autograd import grad\n",
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@@ -848,9 +765,7 @@
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{
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"cell_type": "markdown",
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"id": "71141fab",
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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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"### Activation functions\n",
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"\n",
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@@ -864,10 +779,7 @@
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"cell_type": "code",
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"execution_count": 8,
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"id": "433e0886",
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"metadata": {
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"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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@@ -922,9 +834,7 @@
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{
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"cell_type": "markdown",
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"id": "14c9d538",
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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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"Below follows a short demonstration of how to use an activation\n",
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"function. The derivative of the activation function will be important\n",
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@@ -937,10 +847,7 @@
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"cell_type": "code",
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"execution_count": 9,
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"id": "4c3a44ef",
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"metadata": {
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"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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"z = np.array([[4, 5, 6]]).T\n",
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@@ -958,9 +865,7 @@
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{
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"cell_type": "markdown",
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"id": "aa7f604e",
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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 Neural Network\n",
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"\n",
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"cell_type": "code",
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"execution_count": 10,
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"id": "585a227b",
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"metadata": {
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"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 math\n",
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@@ -1454,9 +1356,7 @@
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{
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"cell_type": "markdown",
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"id": "7de704af",
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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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"Before we make a model, we will quickly generate a dataset we can use\n",
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"for our linear regression problem as shown below"
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"cell_type": "code",
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"execution_count": 11,
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"id": "212b0a35",
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"metadata": {
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"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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@@ -1510,9 +1407,7 @@
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{
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"cell_type": "markdown",
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"id": "1ff4ec7b",
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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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"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",
|
||||
@@ -1526,10 +1421,7 @@
|
||||
"cell_type": "code",
|
||||
"execution_count": 12,
|
||||
"id": "57d8eca3",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"input_nodes = X_train.shape[1]\n",
|
||||
@@ -1541,9 +1433,7 @@
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "f1977bae",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"We then fit our model with our training data using the scheduler of our choice."
|
||||
]
|
||||
@@ -1552,10 +1442,7 @@
|
||||
"cell_type": "code",
|
||||
"execution_count": 13,
|
||||
"id": "31ae2a1e",
|
||||
"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",
|
||||
@@ -1567,9 +1454,7 @@
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "96a64da4",
|
||||
"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",
|
||||
@@ -1583,10 +1468,7 @@
|
||||
"cell_type": "code",
|
||||
"execution_count": 14,
|
||||
"id": "246499fc",
|
||||
"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",
|
||||
@@ -1597,9 +1479,7 @@
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "61d11e17",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"We see that given more epochs to train on, the regressor reaches a lower MSE.\n",
|
||||
"\n",
|
||||
@@ -1612,10 +1492,7 @@
|
||||
"cell_type": "code",
|
||||
"execution_count": 15,
|
||||
"id": "b5b91899",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"from sklearn.datasets import load_breast_cancer\n",
|
||||
@@ -1638,10 +1515,7 @@
|
||||
"cell_type": "code",
|
||||
"execution_count": 16,
|
||||
"id": "4bd01b3d",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"input_nodes = X_train.shape[1]\n",
|
||||
@@ -1653,9 +1527,7 @@
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "3a138d66",
|
||||
"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"
|
||||
]
|
||||
@@ -1664,10 +1536,7 @@
|
||||
"cell_type": "code",
|
||||
"execution_count": 17,
|
||||
"id": "faf1534e",
|
||||
"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",
|
||||
@@ -1679,9 +1548,7 @@
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "33d2d346",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Finally, we will create a neural network with 2 hidden layers with activation functions."
|
||||
]
|
||||
@@ -1690,10 +1557,7 @@
|
||||
"cell_type": "code",
|
||||
"execution_count": 18,
|
||||
"id": "a8f0a61c",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"input_nodes = X_train.shape[1]\n",
|
||||
@@ -1710,10 +1574,7 @@
|
||||
"cell_type": "code",
|
||||
"execution_count": 19,
|
||||
"id": "d651702d",
|
||||
"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",
|
||||
@@ -1725,9 +1586,7 @@
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "200b64c0",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"### Multiclass classification\n",
|
||||
"\n",
|
||||
@@ -1740,10 +1599,7 @@
|
||||
"cell_type": "code",
|
||||
"execution_count": 20,
|
||||
"id": "98b8d59b",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"from sklearn.datasets import load_digits\n",
|
||||
@@ -1777,9 +1633,7 @@
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "cadd7e47",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Testing the XOR gate and other gates\n",
|
||||
"\n",
|
||||
@@ -1790,10 +1644,7 @@
|
||||
"cell_type": "code",
|
||||
"execution_count": 21,
|
||||
"id": "a40e5cfe",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"X = np.array([ [0, 0], [0, 1], [1, 0],[1, 1]],dtype=np.float64)\n",
|
||||
@@ -1813,15 +1664,31 @@
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "b635db73",
|
||||
"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
|
||||
}
|
||||
|
||||
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Reference in New Issue
Block a user