{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# DEEP LEARNING\n",
    "\n",
    "Xiaoqi Zhuang 45521225\n",
    "\n",
    "Ziqing Yan 45551781\n",
    "\n",
    "Zhipei Tao 45495184\n",
    "\n",
    "Dabang Sheng 45687514"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Statistics\n",
    "using LinearAlgebra\n",
    "using Flux.Data.MNIST, PyPlot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Construction of Deep Neural Networks and Backpropagation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "331×781 Array{RGBA{N0f8},2} with eltype RGBA{Normed{UInt8,8}}:\n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)  …  RGBA{N0f8}(1.0,1.0,1.0,1.0)\n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)     RGBA{N0f8}(1.0,1.0,1.0,1.0)\n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)     RGBA{N0f8}(1.0,1.0,1.0,1.0)\n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)     RGBA{N0f8}(1.0,1.0,1.0,1.0)\n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)     RGBA{N0f8}(1.0,1.0,1.0,1.0)\n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)  …  RGBA{N0f8}(1.0,1.0,1.0,1.0)\n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)     RGBA{N0f8}(1.0,1.0,1.0,1.0)\n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)     RGBA{N0f8}(1.0,1.0,1.0,1.0)\n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)     RGBA{N0f8}(1.0,1.0,1.0,1.0)\n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)     RGBA{N0f8}(1.0,1.0,1.0,1.0)\n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)  …  RGBA{N0f8}(1.0,1.0,1.0,1.0)\n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)     RGBA{N0f8}(1.0,1.0,1.0,1.0)\n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)     RGBA{N0f8}(1.0,1.0,1.0,1.0)\n",
       " ⋮                            ⋱  ⋮                          \n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)     RGBA{N0f8}(1.0,1.0,1.0,1.0)\n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)  …  RGBA{N0f8}(1.0,1.0,1.0,1.0)\n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)     RGBA{N0f8}(1.0,1.0,1.0,1.0)\n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)     RGBA{N0f8}(1.0,1.0,1.0,1.0)\n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)     RGBA{N0f8}(1.0,1.0,1.0,1.0)\n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)     RGBA{N0f8}(1.0,1.0,1.0,1.0)\n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)  …  RGBA{N0f8}(1.0,1.0,1.0,1.0)\n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)     RGBA{N0f8}(1.0,1.0,1.0,1.0)\n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)     RGBA{N0f8}(1.0,1.0,1.0,1.0)\n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)     RGBA{N0f8}(1.0,1.0,1.0,1.0)\n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)     RGBA{N0f8}(1.0,1.0,1.0,1.0)\n",
       " RGBA{N0f8}(1.0,1.0,1.0,1.0)  …  RGBA{N0f8}(1.0,1.0,1.0,1.0)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "imgs   = MNIST.images()\n",
    "labels = MNIST.labels();## The Construction of Deep Neural Networks and BackpropagationImageMagick.load_(Diagram)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Backpropagation algorithm is one of two ways to compute the derivatives\n",
    "$\\frac{\\partial F}{\\partial x}$ backpropagation is the process taking the error and feeding backward to the \n",
    "error though the net work.\n",
    "mathematics of gradient descent tells us how to take an error to nudge weight then we calculated the error coming out of the hidden layer and keep going back and that is back propagation and how the hidden errors are calculated."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One dimentional example:\n",
    "Input i = 1.5, intial weight y = 0.8, desired output = 0.5, actural output a = i*w = 1.2\n",
    "MSE = C = $(a-y)^{2}$\n",
    "\n",
    "$\\frac{\\partial C}{\\partial a}$ = 2(a-y) = 2*(1.2-0.5), $\\frac{\\partial a}{\\partial w}$ = i = 1.5\n",
    "\n",
    "$\\frac{\\partial C}{\\partial w}$ = $\\frac{\\partial C}{\\partial a}$$\\frac{\\partial a}{\\partial w}$ = 2(a-y)*i = 2(1.5w-0.5)*1.5 = 4.5w-1.5\n",
    "\n",
    "set learning rate r = 0.1\n",
    "\n",
    "then $w_{1}$  = $w_{0}$ -r*$\\frac{\\partial C}{\\partial w}$ = 0.8-0.1(4.5$w_{0}$-1.5) = 0.55$w_{0} +0.15$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "onedim (generic function with 1 method)"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function onedim(w0,DO)\n",
    "    ini = 1.5;\n",
    "    r = 0.1;\n",
    "    AO = ini *w0;\n",
    "    i = 0;\n",
    "    while (AO-DO)^2 > 0.000000000000000001\n",
    "        i += 1\n",
    "        w0 = 0.55*w0 + 0.15\n",
    "        AO = ini *w0\n",
    "    end\n",
    "    return DO,i,AO\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.5, 35, 0.500000000572522)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "onedim(0.8,0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, let's see a complicte example here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As figure shows that, there are three inputs, two hidden layers with two neurons of each, and one output y.\n",
    "\n",
    "In this case, we have 12 weights in total and let's just ignor the biase for now.\n",
    "\n",
    "$H1 = n1 *w1 + n2 *w3 + n3 *w5$ and $outH1 = \\frac{ 1}{ 1+e^{-H1}}$\n",
    "\n",
    "$H2 = n1 *w2 + n2 *w4 + n3 *w6$ and $outH2 = \\frac{ 1}{ 1+e^{-H2}}$\n",
    "\n",
    "$H3 = outH1 *w7 + outH2 *w9$ and $outH3 = \\frac{ 1}{ 1+e^{-H3}}$\n",
    "\n",
    "$H4 = outH1 *w8 + outH2 *w10$ and $outH4 = \\frac{ 1}{ 1+e^{-H4}}$\n",
    "\n",
    "$y = outH3 *w11 + outH4 *w12$ and $outy = \\frac{ 1}{ 1+e^{y}}$\n",
    "\n",
    "$E = (outy - T)^{2}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For example, we want to find $\\frac{\\partial E}{\\partial w11}$ and $\\frac{\\partial E}{\\partial w7}$\n",
    "\n",
    "$\\frac{\\partial E}{\\partial w11}$ = $\\frac{\\partial E}{\\partial outy}$$\\frac{\\partial outy}{\\partial y}$$\\frac{\\partial y}{\\partial w11}$ = $ 2*(outy-T)* \\frac{ e^{-y}}{ (1+e^{-y})^{2}}*outH3$\n",
    "\n",
    "$\\frac{\\partial E}{\\partial w7}$ = $\\frac{\\partial E}{\\partial outy}$$\\frac{\\partial outy}{\\partial y}$$\\frac{\\partial y}{\\partial outH3}$$\\frac{\\partial outH3}{\\partial H3}$$\\frac{\\partial H3}{\\partial w7}$ = $ 2*(outy-T)* \\frac{ e^{-y}}{ (1+e^{-y})^{2}}*w11*\\frac{ e^{-H3}}{ (1+e^{-H3})^{2}}*outH1$\n",
    "\n",
    "and more...\n",
    "\n",
    "Let let's put this on code. we will use function called relu and sigmoid which we will define them first."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sigmoidPrime (generic function with 1 method)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function relu(X)\n",
    "    rel = max.(0,X)\n",
    "    return rel #, X\n",
    "end\n",
    "function Leaky_relu(X)\n",
    "    if X >= 0\n",
    "        return X\n",
    "    else \n",
    "        return 0.01*X\n",
    "    end\n",
    "end\n",
    "function sigmoid(X)\n",
    "    sigma = 1 ./(1 .+ exp.(.-X))\n",
    "    return sigma \n",
    "end\n",
    "function sigmoidPrime(X)\n",
    "    sigmaP = (exp.(.-X)) ./((1 .+ exp.(.-X)).^2)\n",
    "    return sigmaP\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is the main code part."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "dim3221 (generic function with 1 method)"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#input = [1,2,3], T = must be less than 1!!\n",
    "function dim3221(input,T)\n",
    "    w = rand(12,1);\n",
    "    dedw = zeros(12,1);\n",
    "    h = zeros(4,1);\n",
    "    out_h = zeros(4,1);\n",
    "    r = 0.01; #set learning rate r = 0.01\n",
    "    k = 0;\n",
    "    h[1] = input[1]*w[1]+input[2]*w[3]+input[3]*w[5];\n",
    "    h[2] = input[1]*w[2]+input[2]*w[4]+input[3]*w[6];\n",
    "    out_h[1] = sigmoid(relu(h[1]));\n",
    "    out_h[2] = sigmoid(relu(h[2]));\n",
    "    h[3] = out_h[1]*w[7]+out_h[2]*w[9];\n",
    "    h[4] = out_h[1]*w[8]+out_h[2]*w[10];\n",
    "    out_h[3] = sigmoid(relu(h[3]));\n",
    "    out_h[4] = sigmoid(relu(h[4]));\n",
    "    y = out_h[3]*w[11]+out_h[4]*w[12];\n",
    "    out_y = sigmoid(relu(y));\n",
    "    E = out_y-T;\n",
    "    while E^2 > 0.000000001 # we can change the \n",
    "        k = k + 1\n",
    "        dedw[1]  = 2*E*sigmoidPrime(y)*(w[11]*sigmoidPrime(h[3])*w[7]*sigmoidPrime(h[1])*input[1]\n",
    "                            +w[12]*sigmoidPrime(h[4])*w[8]*sigmoidPrime(h[1])*input[1]);\n",
    "        dedw[2]  = 2*E*sigmoidPrime(y)*(w[11]*sigmoidPrime(h[3])*w[9]*sigmoidPrime(h[2])*input[1]\n",
    "                            +w[12]*sigmoidPrime(h[4])*w[10]*sigmoidPrime(h[2])*input[1]);\n",
    "        dedw[3]  = 2*E*sigmoidPrime(y)*(w[11]*sigmoidPrime(h[3])*w[7]*sigmoidPrime(h[1])*input[2]\n",
    "                            +w[12]*sigmoidPrime(h[4])*w[8]*sigmoidPrime(h[1])*input[2]);\n",
    "        dedw[4]  = 2*E*sigmoidPrime(y)*(w[11]*sigmoidPrime(h[3])*w[9]*sigmoidPrime(h[2])*input[2]\n",
    "                            +w[12]*sigmoidPrime(h[4])*w[10]*sigmoidPrime(h[2])*input[2]);\n",
    "        dedw[5]  = 2*E*sigmoidPrime(y)*(w[11]*sigmoidPrime(h[3])*w[7]*sigmoidPrime(h[1])*input[3]\n",
    "                            +w[12]*sigmoidPrime(h[4])*w[8]*sigmoidPrime(h[1])*input[3]);\n",
    "        dedw[6]  = 2*E*sigmoidPrime(y)*(w[11]*sigmoidPrime(h[3])*w[9]*sigmoidPrime(h[2])*input[3]\n",
    "                            +w[12]*sigmoidPrime(h[4])*w[10]*sigmoidPrime(h[2])*input[3]);\n",
    "        dedw[7]  = 2*E*sigmoidPrime(y)*w[11]*sigmoidPrime(h[3])*out_h[1];\n",
    "        dedw[8]  = 2*E*sigmoidPrime(y)*w[12]*sigmoidPrime(h[4])*out_h[1];\n",
    "        dedw[9]  = 2*E*sigmoidPrime(y)*w[11]*sigmoidPrime(h[3])*out_h[2];\n",
    "        dedw[10] = 2*E*sigmoidPrime(y)*w[12]*sigmoidPrime(h[4])*out_h[2];\n",
    "        dedw[11] = 2*E*sigmoidPrime(y)*out_h[3];\n",
    "        dedw[12] = 2*E*sigmoidPrime(y)*out_h[4];\n",
    "        for i = 1: length(w)\n",
    "            w[i] = w[i] - r*dedw[i]\n",
    "        end\n",
    "        h[1] = input[1]*w[1]+input[2]*w[3]+input[3]*w[5];\n",
    "        h[2] = input[1]*w[2]+input[2]*w[4]+input[3]*w[6];\n",
    "        out_h[1] = sigmoid(relu(h[1]));\n",
    "        out_h[2] = sigmoid(relu(h[2]));\n",
    "        h[3] = out_h[1]*w[7]+out_h[2]*w[9];\n",
    "        h[4] = out_h[1]*w[8]+out_h[2]*w[10];\n",
    "        out_h[3] = sigmoid(relu(h[3]));\n",
    "        out_h[4] = sigmoid(relu(h[4]));\n",
    "        y = out_h[3]*w[11]+out_h[4]*w[12];\n",
    "        out_y = sigmoid(relu(y));\n",
    "        E = out_y-T;\n",
    "    end\n",
    "    return k,out_y,T,E \n",
    "    # k means how many time the while loop runs\n",
    "    # out_y means our output\n",
    "    # T is the value we want\n",
    "    # E means the error\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(38130, 0.8999683820338049, 0.9, -3.1617966195107705e-5)"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dim3221([0.4,0.1,0.9],0.9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$V_{L}$ = $b_{L}$ + $A_{L}$$v_{L-1}$\n",
    "or simply\n",
    "w = b +Av\n",
    "\n",
    "Our goal is to find the derivatives $\\frac{\\partial w_{i}}{\\partial b_{i}}$ and $\\frac{\\partial w_{i}}{\\partial A_{jk}} $\n",
    "\n",
    "$\\delta_{ij} = 1,  for  $i = j$, otherwise = 0$\n",
    "\n",
    "$\\frac{\\partial w_{i}}{\\partial b_{i}} = \\delta_{ij} $\n",
    "$\\frac{\\partial w_{i}}{\\partial A_{jk}} = \\delta_{ij}v_{k} $\n",
    "\n",
    "$\\left[\\begin{array}{1}{w_{1}} \\\\ {w_{2}}\\end{array}\\right]$ = $\\left[\\begin{array}{1}{b_{1}} \\\\ {b_{2}}\\end{array}\\right]$ +$\\left[\\begin{array}{1}{a_{11}v_{1}+a_{12}v_{2}} \\\\ {a_{21}v_{1}+a_{22}v_{2}}\\end{array}\\right]$\n",
    "\n",
    "derivatives of \n",
    "$\\frac{\\partial w_{1}}{\\partial b_{1}} = 1,$\n",
    "$\\frac{\\partial w_{1}}{\\partial b_{2}} = 0,$\n",
    "$\\frac{\\partial w_{1}}{\\partial a_{11}} = v_{1},$\n",
    "$\\frac{\\partial w_{1}}{\\partial a_{12}} = v_{2},$\n",
    "$\\frac{\\partial w_{1}}{\\partial a_{21}} = 0,$\n",
    "$\\frac{\\partial w_{1}}{\\partial a_{22}} = 0$\n",
    "\n",
    "$ M  = \\left[\\begin{array}{cc}\n",
    "        {1} & O^{T} \\\\ \n",
    "        {b} & A \\end{array}\\right]$ has\n",
    "$ M\\left[\\begin{array}{c}\n",
    "        {1} \\\\ \n",
    "        {v}\\end{array}\\right]$ = $ \\left[\\begin{array}{c}\n",
    "        {1}\\\\ \n",
    "        {b+ Av} \\end{array}\\right]$\n",
    "        \n",
    "$ M  = \\left[\\begin{array}{cc}\n",
    "        {1} & O^{T} \\\\ \n",
    "        {b} & A \\end{array}\\right],$\n",
    "$\\frac{\\partial w_{i}}{\\partial M_{jk}} = \\delta_{ij}v_{k} $ for i > 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$v_{1} = R(b_{1}+A_{1}v_{0})$ and $w = b_{2}+A_{2}v_{1} = b_{2}+A_{2}R(b_{1}+A_{1}v_{0})$\n",
    "\n",
    "By chain rule, equation 5\n",
    "\n",
    "$\\frac{\\partial w}{\\partial A_{1}} = \\frac{\\partial [A_{2}R(b_{1}+A_{1}v_{0})]}{\\partial A_{1}} $\n",
    "= $A_{2}R^{'}(b_{1}+A_{1}v_{0}) \\frac{\\partial (b_{1}+A_{1}v_{0})}{\\partial A_{1}}$\n",
    "\n",
    "notice that how these formulas go BACKWARDS from w to v.\n",
    "write A and b for the matrix $A_{L-1}$ and the vector $b_{L-1}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$w = A_{L}(R(Av+b))$  and $ = A_{L}R^{'}(b+Av)  \\frac{\\partial (b+Av)}{\\partial A} $\n",
    "\n",
    "$F = x^{2}(x+y)$\n",
    "nodes $ c = x^{2} $and $s = x+y$ and F=cs\n",
    "\n",
    "for example $x = 2$ and $y = 3$, the edges lead to $c = 2$ and $s = 5$ and $F = 20$\n",
    "this agrees with the alfebra that we normally crowd into one line:\n",
    "$ F = x^{2}(x+y) = 2^{2}(2+3) = 4(5) = 20$\n",
    "$c = x^{2}= 4 $ and $s = x+y = 5$ and $F = cs = 20$\n",
    "\n",
    "now er compute the derivative of each step\n",
    "$\\frac{\\partial c}{\\partial x} = 2x,$\n",
    "$\\frac{\\partial s}{\\partial x} = 1,$\n",
    "$\\frac{\\partial F}{\\partial c} = s,$\n",
    "$\\frac{\\partial F}{\\partial s} = c$\n",
    "\n",
    "$\\frac{\\partial F}{\\partial x} =\\frac{\\partial F}{\\partial c} \\frac{\\partial c}{\\partial x} +\\frac{\\partial F}{\\partial s}\\frac{\\partial s}{\\partial x}$\n",
    "= (s)(2x)+(c)(1) = (5)(4)+(4)(1) = $24$\n",
    "\n",
    "Training the network = optimizing the weights\n",
    "$F(v) =A_{L}(RA_{L-1}(...(RA_{2}(RA_{1}v)))) $ \n",
    "is forward through the net.\n",
    "\n",
    "The derivatives of F with respect to the matrices A (and the bias vectors b) \n",
    "are the easiest for the last matrix $A_{L}$ in $A_{L}v_{L-1}.$ the dericative of Av with respect to A contains v's: $\\frac{\\partial F_{i}}{\\partial A_{jk}} = \\delta_{ij}v_{k}$. Next is the derivative of $A_{L}ReLU(A_{L-1}v_{L-1})$ with respect to $A_{L-1}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##  SGD and ADAM"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1. full-batch gradient descent "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The above neural network has one output. In general, we will have multiple outputs and take the highest value as the result. If there are several outputs, the loss function will be more complicated and will take a long time by using the traditional gradient descent method."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following exalmple will use gradient descent to fill the data. We design the data is $ y = 3 + x$, but the computer does not know and it will use gradient descent to minimize the loss function $ (Ax-y)^2 $ to find the coefficient and intercept."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, we assume the function is $ y = \\beta_1 + \\beta_2 x$. To fit the data, we hope minimize the loss function $ L(\\beta)=\\frac{1}{N} \\sum_{j=1}^{N}\\left(y_{j}-\\hat{y}_{j}\\right)^{2}=\\sum_{j=1}^{N} \\frac{1}{N}\\left(\\beta_{0}+\\beta_{1} x_{j}-{y}_{j}\\right)^{2}$. \n",
    "\n",
    "${y}_{j}$ is the real data and $\\hat{y}_j$ is the data according to the predict function by real $x$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The gradient is $\\nabla L=\\left(\\frac{\\partial L}{\\partial \\beta_{0}}, \\frac{\\partial L}{\\partial \\beta_{1}}\\right)=\\left(\\frac{2}{N} \\sum_{j=1}^{N}\\left(\\beta_{0}+\\beta_{1} x_{j}-{y}_{j}\\right), \\frac{2}{N} \\sum_{j=1}^{N} x_{j}\\left(\\beta_{0}+\\beta_{1} x_{j}-{y}_{j}\\right)\\right)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The process is $ \\beta_{n+1} = \\beta_{n} - \\alpha \\nabla L$. $\\alpha$ is the learing rate which should be tested many times."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We want that the algorithm will find that $\\beta_1 = 3$ and $\\beta_2 = 1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "50-element Array{Int64,1}:\n",
       "  4\n",
       "  5\n",
       "  6\n",
       "  7\n",
       "  8\n",
       "  9\n",
       " 10\n",
       " 11\n",
       " 12\n",
       " 13\n",
       " 14\n",
       " 15\n",
       " 16\n",
       "  ⋮\n",
       " 42\n",
       " 43\n",
       " 44\n",
       " 45\n",
       " 46\n",
       " 47\n",
       " 48\n",
       " 49\n",
       " 50\n",
       " 51\n",
       " 52\n",
       " 53"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Create the dataset.\n",
    "x = collect(1:1:50)\n",
    "y = 3 .+ x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Set the original $\\beta = \\left[\\begin{array}{l}{0} \\\\ {0}\\end{array}\\right]$. $\\beta_1$ is the intercept and $\\beta_2$ is the coefficient.\n",
    "\n",
    "The learning rate is $0.001$ and the error is $0.01$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.001"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "beta = [0.0;0.0]\n",
    "alpha = 0.001\n",
    "tol_L = 0.001"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Bgrad (generic function with 1 method)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#calculate gradient\n",
    "function Bgrad(beta,x,y)\n",
    "    grad = [0.0;0.0]\n",
    "    grad[1] = 2 * mean(beta[1].+beta[2]*x-y)\n",
    "    grad[2] = 2 * mean(x.*(beta[1].+beta[2]*x-y))\n",
    "    return grad\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "newBeta (generic function with 1 method)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#update new beta\n",
    "function newBeta(beta,alpha,grad)\n",
    "    beta = beta - alpha * grad \n",
    "    return beta\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "loss_function (generic function with 1 method)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# the loss_function\n",
    "function loss_function(beta,x,y)\n",
    "    error =  (beta[1] .+ beta[2] * x - y) .^ 2\n",
    "    loss = sqrt(mean(error))\n",
    "    return loss\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "times:1 beta[0.057, 1.87] loss:22.97561727135966\n",
      "times:2 beta[0.01851599999999999, 0.526303] loss:16.53951009038752\n",
      "times:3 beta[0.04863751499999999, 1.491696433] loss:11.92712146468593\n",
      "times:4 beta[0.02946372188699999, 0.7979731442739999] loss:8.629609967204736\n",
      "times:5 beta[0.045708164085251995, 1.296350605739305] loss:6.282806372006868\n",
      "times:6 beta[0.03650286686437693, 0.9381854993165704] loss:4.626576869401185\n",
      "times:7 beta[0.04558240066550309, 1.1954593507799358] loss:3.4750724723379323\n",
      "times:8 beta[0.041522808974395364, 1.0105309430568454] loss:2.69422758823238\n",
      "times:9 beta[0.04690268526054746, 1.1433316505705478] loss:2.184236077329377\n",
      "times:10 beta[0.04549896571092843, 1.0478391695926292] loss:1.866934088551124\n",
      "times:11 beta[0.048968170130282485, 1.1163788681508275] loss:1.6795306076853342\n",
      "times:12 beta[0.048934911514329715, 1.0670589748592123] loss:1.573757818976739\n",
      "times:13 beta[0.05141703397348123, 1.102423034538714] loss:1.5159397450519156\n",
      "times:14 beta[0.05209062514405985, 1.0769404155030946] loss:1.4848769790942882\n",
      "times:15 beta[0.0540624827031139, 1.0951771002019342] loss:1.4682471446492504\n",
      "times:16 beta[0.05510032562740903, 1.0820008325373542] loss:1.4592497497223553\n",
      "times:17 beta[0.05680808251674915, 1.0913952864637193] loss:1.4542442504036548\n",
      "times:18 beta[0.05803330674206597, 1.0845723673971592] loss:1.4513142963992176\n",
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     "text": [
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      "times:6537 beta[2.8752708787380534, 1.00370587157669] loss:0.061431380081311784\n",
      "times:6538 beta[2.875331337530166, 1.0037040752638726] loss:0.0614016030172227\n",
      "times:6539 beta[2.8753917670166484, 1.003702279821765] loss:0.06137184038669596\n",
      "times:6540 beta[2.875452167211705, 1.0037004852499454] loss:0.0613420921827315\n",
      "times:6541 beta[2.8755125381295343, 1.0036986915479922] loss:0.06131235839833943\n",
      "times:6542 beta[2.8755728797843276, 1.0036968987154833] loss:0.06128263902652933\n",
      "times:6543 beta[2.875633192190269, 1.0036951067519977] loss:0.06125293406031577\n",
      "times:6544 beta[2.875693475361537, 1.003693315657114] loss:0.0612232434927139\n",
      "times:6545 beta[2.8757537293123008, 1.0036915254304108] loss:0.06119356731674678\n",
      "times:6546 beta[2.875813954056725, 1.003689736071468] loss:0.06116390552543779\n",
      "times:6547 beta[2.8758741496089666, 1.0036879475798646] loss:0.061134258111814506\n",
      "times:6548 beta[2.8759343159831756, 1.0036861599551798] loss:0.061104625068906196\n",
      "times:6549 beta[2.875994453193495, 1.0036843731969942] loss:0.06107500638974967\n",
      "times:6550 beta[2.8760545612540613, 1.003682587304887] loss:0.06104540206738085\n",
      "times:6551 beta[2.876114640179004, 1.0036808022784391] loss:0.061015812094842053\n",
      "times:6552 beta[2.8761746899824456, 1.00367901811723] loss:0.060986236465176535\n",
      "times:6553 beta[2.876234710678502, 1.0036772348208414] loss:0.06095667517143098\n",
      "times:6554 beta[2.876294702281282, 1.0036754523888531] loss:0.06092712820665854\n",
      "times:6555 beta[2.876354664804888, 1.003673670820847] loss:0.0608975955639113\n",
      "times:6556 beta[2.8764145982634153, 1.0036718901164035] loss:0.06086807723624994\n",
      "times:6557 beta[2.876474502670952, 1.0036701102751044] loss:0.06083857321673409\n",
      "times:6558 beta[2.8765343780415797, 1.0036683312965315] loss:0.06080908349842827\n",
      "times:6559 beta[2.8765942243893736, 1.0036665531802664] loss:0.06077960807440023\n",
      "times:6560 beta[2.8766540417284014, 1.003664775925891] loss:0.060750146937722176\n",
      "times:6561 beta[2.8767138300727244, 1.0036629995329878] loss:0.06072070008146782\n",
      "times:6562 beta[2.8767735894363966, 1.0036612240011389] loss:0.0606912674987155\n",
      "times:6563 beta[2.876833319833466, 1.0036594493299271] loss:0.060661849182546426\n",
      "times:6564 beta[2.8768930212779726, 1.0036576755189355] loss:0.06063244512604563\n",
      "times:6565 beta[2.876952693783951, 1.0036559025677467] loss:0.060603055322299954\n",
      "times:6566 beta[2.8770123373654277, 1.003654130475944] loss:0.06057367976440322\n",
      "times:6567 beta[2.877071952036424, 1.0036523592431112] loss:0.06054431844544784\n",
      "times:6568 beta[2.8771315378109525, 1.0036505888688316] loss:0.06051497135853294\n",
      "times:6569 beta[2.8771910947030204, 1.0036488193526891] loss:0.0604856384967599\n",
      "times:6570 beta[2.8772506227266272, 1.0036470506942679] loss:0.0604563198532329\n",
      "times:6571 beta[2.8773101218957664, 1.003645282893152] loss:0.06042701542106085\n",
      "times:6572 beta[2.877369592224424, 1.0036435159489259] loss:0.060397725193355095\n",
      "times:6573 beta[2.87742903372658, 1.0036417498611745] loss:0.06036844916322962\n",
      "times:6574 beta[2.8774884464162067, 1.0036399846294823] loss:0.060339187323803875\n",
      "times:6575 beta[2.877547830307271, 1.0036382202534346] loss:0.060309939668197886\n",
      "times:6576 beta[2.8776071854137313, 1.0036364567326166] loss:0.06028070618953762\n",
      "times:6577 beta[2.8776665117495406, 1.0036346940666137] loss:0.060251486880950174\n",
      "times:6578 beta[2.8777258093286444, 1.0036329322550115] loss:0.060222281735568546\n",
      "times:6579 beta[2.8777850781649814, 1.003631171297396] loss:0.06019309074652672\n",
      "times:6580 beta[2.8778443182724844, 1.003629411193353] loss:0.06016391390696271\n",
      "times:6581 beta[2.8779035296650783, 1.0036276519424692] loss:0.060134751210016414\n",
      "times:6582 beta[2.877962712356682, 1.0036258935443307] loss:0.060105602648836416\n",
      "times:6583 beta[2.878021866361208, 1.003624135998524] loss:0.060076468216567085\n",
      "times:6584 beta[2.8780809916925607, 1.0036223793046366] loss:0.0600473479063627\n",
      "times:6585 beta[2.8781400883646393, 1.003620623462255] loss:0.06001824171137529\n",
      "times:6586 beta[2.878199156391335, 1.0036188684709666] loss:0.059989149624764135\n",
      "times:6587 beta[2.878258195786533, 1.003617114330359] loss:0.05996007163969264\n",
      "times:6588"
     ]
    }
   ],
   "source": [
    "i = 1\n",
    "loss = loss_function(beta,x,y)\n",
    "while loss >tol_L\n",
    "    grad = Bgrad(beta,x,y)\n",
    "    beta = newBeta(beta,alpha,grad)\n",
    "    loss = loss_function(beta,x,y)\n",
    "    println(\"times:\",i,\" \",\"beta\",beta,\" \",\"loss:\",loss)\n",
    "    i = i+1\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It runs 6587 times and the loss is still less than $0.01$. But it works, we can see that it can find the real $\\beta$, which is $\\left[\\begin{array}{l}{3} \\\\ {1}\\end{array}\\right]$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The problem is that I try many times to find the leanring rate $\\alpha = 0.001$ and everytime it will run for very lone time."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "However, now our dataset is just one-dimensional and have 50 elements, which is much smaller than the real dataset."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.Stochastic Gradient Decent, SGD"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hence, we introduce an new method: Stochastic Gradient Decent(also SGD) to help us reduce time."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The difference is that everytime it will just use one random number in $x$ and $y$ and run many times."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Sgrad (generic function with 1 method)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#calculate the stochastic gradient\n",
    "function Sgrad(beta,x,y)\n",
    "    grad = [0.0;0.0]\n",
    "    r = rand(1:length(x))\n",
    "    grad[1] = 2 * mean(beta[1].+beta[2]*x[r]-y[r])\n",
    "    grad[2] = 2 * mean(x[r].*(beta[1].+beta[2]*x[r]-y[r]))\n",
    "    return grad\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.01"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "beta = [1;1]\n",
    "alpha = 0.0001\n",
    "tol_L = 0.01"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "times:100 beta[1.0123040779870913, 1.0703609286969182] loss:1.0336412587647363\n",
      "times:200 beta[1.0227872409210828, 1.0647900060933946] loss:0.9898733859585349\n",
      "times:300 beta[1.0306900082080137, 1.0505152382744385] loss:0.9976996637200308\n",
      "times:400 beta[1.0411705737008619, 1.056776628154044] loss:0.965638984984428\n",
      "times:500 beta[1.0518331193279378, 1.0620371316721133] loss:0.9672586228958588\n",
      "times:600 beta[1.061734123189938, 1.0503753558647473] loss:0.9776437835190781\n",
      "times:700 beta[1.0713067079684067, 1.0545290153028246] loss:0.9533500223711977\n",
      "times:800 beta[1.080427319027703, 1.0455294012593233] loss:1.0035536045349005\n",
      "times:900 beta[1.0907210598033992, 1.0652821245209498] loss:0.9733099962613385\n",
      "times:1000 beta[1.0994437688694092, 1.047401617221089] loss:0.9728965417921616\n",
      "times:1100 beta[1.1089425924942768, 1.0608984692186807] loss:0.9416283085576184\n",
      "times:1200 beta[1.1160777844519714, 1.055958171213927] loss:0.927866131729102\n",
      "times:1300 beta[1.124646667181242, 1.0560069966848251] loss:0.9236885879903142\n",
      "times:1400 beta[1.1332572742702245, 1.0438526481624324] loss:0.9801676885385971\n",
      "times:1500 beta[1.1442407040940064, 1.053712793424037] loss:0.9149269695764359\n",
      "times:1600 beta[1.1517511281930635, 1.0479994678536724] loss:0.9324703519710097\n",
      "times:1700 beta[1.1608323330106154, 1.0484439784867712] loss:0.9237723310251273\n",
      "times:1800 beta[1.1705624000916446, 1.0630531524823381] loss:0.9365031817387454\n",
      "times:1900 beta[1.1793540916256975, 1.0502552232373181] loss:0.9036719901135386\n",
      "times:2000 beta[1.1887618911954725, 1.0445911899724751] loss:0.931973371411588\n",
      "times:2100 beta[1.1977143433530602, 1.052847901374794] loss:0.887885733345808\n",
      "times:2200 beta[1.205566775700082, 1.05542883533197] loss:0.8859895654429037\n",
      "times:2300 beta[1.2149013298684042, 1.0561285557992273] loss:0.8838907293322144\n",
      "times:2400 beta[1.2242289668360833, 1.061235777736612] loss:0.9092892305659843\n",
      "times:2500 beta[1.2322612975808536, 1.0642879876301075] loss:0.9365741564240638\n",
      "times:2600 beta[1.2421343722388423, 1.0566428167259858] loss:0.8754524162196443\n",
      "times:2700 beta[1.2513899978460885, 1.0509229000883258] loss:0.8617367468326678\n",
      "times:2800 beta[1.260073197924862, 1.0459973611972684] loss:0.8729769438394549\n",
      "times:2900 beta[1.2701662041703214, 1.0574006290327584] loss:0.8700387739755079\n",
      "times:3000 beta[1.280245324420506, 1.0578502382026074] loss:0.8699173072905391\n",
      "times:3100 beta[1.2902551744762971, 1.0522597150570432] loss:0.8431892003075775\n",
      "times:3200 beta[1.2989410725321044, 1.0395862444283124] loss:0.8970320244272341\n",
      "times:3300 beta[1.3075328747284671, 1.0519225575206583] loss:0.8349739155585789\n",
      "times:3400 beta[1.316390233221777, 1.0423276601672549] loss:0.859203431373785\n",
      "times:3500 beta[1.3233968477631382, 1.0508318623586996] loss:0.826310896322071\n",
      "times:3600 beta[1.3300862274002565, 1.0473973264654821] loss:0.8249945609543479\n",
      "times:3700 beta[1.3383116613571675, 1.052893925645908] loss:0.8249468050456499\n",
      "times:3800 beta[1.3472279817541095, 1.0600317216338666] loss:0.8748530933767482\n",
      "times:3900 beta[1.3563862067589596, 1.0538558372834195] loss:0.8228460472057513\n",
      "times:4000 beta[1.3646831892560793, 1.0538307311559394] loss:0.8200195858750804\n",
      "times:4100 beta[1.3725144223919596, 1.0575129877242961] loss:0.8454157913812554\n",
      "times:4200 beta[1.3808969911115667, 1.0453613686016212] loss:0.8014419253224866\n",
      "times:4300 beta[1.3893828612775228, 1.0479154649895916] loss:0.7932614574572691\n",
      "times:4400 beta[1.3992309423553722, 1.0592271425411584] loss:0.8594746972411568\n",
      "times:4500 beta[1.4055357610820676, 1.051163457742707] loss:0.7931693778141998\n",
      "times:4600 beta[1.4120053948842979, 1.0414341336638513] loss:0.7999579111663546\n",
      "times:4700 beta[1.4217033979086147, 1.044968361644588] loss:0.77935547367015\n",
      "times:4800 beta[1.429495849216307, 1.0451860080966302] loss:0.7746882789474012\n",
      "times:4900 beta[1.4375557844576499, 1.046784570264398] loss:0.7696109178734952\n",
      "times:5000 beta[1.4448084354388686, 1.052150103250409] loss:0.7855905612820694\n",
      "times:5100 beta[1.4525098331785429, 1.0457458918407316] loss:0.7621942484324443\n",
      "times:5200 beta[1.460054851026043, 1.0487161101627058] loss:0.7634443228005725\n",
      "times:5300 beta[1.4670114550019744, 1.0369703053713104] loss:0.7956298633317717\n",
      "times:5400 beta[1.4740631381344644, 1.0438682459483026] loss:0.752762646963045\n",
      "times:5500 beta[1.4816281839361614, 1.0467816113705355] loss:0.7494469839749418\n",
      "times:5600 beta[1.4885973044608591, 1.0448029948163577] loss:0.7443981448335395\n",
      "times:5700 beta[1.4961789846518498, 1.0399481893868185] loss:0.7534589138232621\n",
      "times:5800 beta[1.5051601422934522, 1.0433080817073288] loss:0.7369322785291471\n",
      "times:5900 beta[1.5119623765815309, 1.0401696802402005] loss:0.7423345294309358\n",
      "times:6000 beta[1.5193444950538704, 1.04567155054464] loss:0.7309324689759336\n",
      "times:6100 beta[1.527349246373768, 1.042976061543033] loss:0.7256546241303684\n",
      "times:6200 beta[1.5343602770935578, 1.0396800766872998] loss:0.7306324697617113\n",
      "times:6300 beta[1.5407858346526262, 1.0436022617307743] loss:0.7187297995259825\n",
      "times:6400 beta[1.5488260084863748, 1.0470349081873058] loss:0.7239495475639915\n",
      "times:6500 beta[1.5549557998121102, 1.040444441752776] loss:0.7154035735521677\n",
      "times:6600 beta[1.562808913821612, 1.0463570593377383] loss:0.7159562518771192\n",
      "times:6700 beta[1.5698431425121253, 1.043811663772279] loss:0.7054583918743689\n",
      "times:6800 beta[1.5764864639057024, 1.0421564089338902] loss:0.7011161866332075\n",
      "times:6900 beta[1.5825771934976893, 1.0365902786149646] loss:0.7165404208010072\n",
      "times:7000 beta[1.5890306726364671, 1.0489781579818789] loss:0.7251310971368659\n",
      "times:7100 beta[1.5972469121967559, 1.0472184655644812] loss:0.7097784053956555\n",
      "times:7200 beta[1.6039520450827975, 1.04328090553262] loss:0.6896306756594259\n",
      "times:7300 beta[1.6114428004682, 1.0410411687051553] loss:0.6839156967988164\n",
      "times:7400 beta[1.6173224984829944, 1.0419948281344424] loss:0.6815332905529274\n",
      "times:7500 beta[1.6231360573409728, 1.0370524933021525] loss:0.6874224949353319\n",
      "times:7600 beta[1.6290914324080061, 1.0404940294003826] loss:0.675230009591443\n",
      "times:7700 beta[1.635524266892446, 1.0391607569357986] loss:0.6732239226204856\n",
      "times:7800 beta[1.6405944000964212, 1.037108176371288] loss:0.6763536554117147\n",
      "times:7900 beta[1.6469374116909212, 1.0374434272133986] loss:0.6712494563348973\n",
      "times:8000 beta[1.6539534285655586, 1.0388508863326071] loss:0.6637798151535657\n",
      "times:8100 beta[1.659403039733422, 1.0339273669248699] loss:0.682467136096381\n",
      "times:8200 beta[1.6662661163383434, 1.0379308905061053] loss:0.6587409547852006\n",
      "times:8300 beta[1.6729549538001784, 1.0313927617842484] loss:0.6945969768701963\n",
      "times:8400 beta[1.6808115257378067, 1.0466919112300352] loss:0.6859568315438026\n",
      "times:8500 beta[1.686443711315743, 1.0368618488503445] loss:0.6500235552644138\n",
      "times:8600 beta[1.693186961699179, 1.039940187876677] loss:0.6444712716465535\n",
      "times:8700 beta[1.7003857405572467, 1.0376511067613672] loss:0.640690378043765\n",
      "times:8800 beta[1.706746441376245, 1.0410899583746862] loss:0.6417604962405438\n",
      "times:8900 beta[1.711774039567294, 1.035451163089753] loss:0.6398058123186288\n",
      "times:9000 beta[1.7191457832107142, 1.0427938879079233] loss:0.6460059697847281\n",
      "times:9100 beta[1.725873337804632, 1.037631886231702] loss:0.6275618410713018\n",
      "times:9200 beta[1.7318039014515665, 1.0369939747048853] loss:0.6249230037772506\n",
      "times:9300 beta[1.737368811201834, 1.0427730318968569] loss:0.6407465898341189\n",
      "times:9400 beta[1.743011781320184, 1.0396802668861316] loss:0.6228875082836046\n",
      "times:9500 beta[1.748673402290266, 1.0303699066353227] loss:0.6476907789874401\n",
      "times:9600 beta[1.7552487688830827, 1.0356114093614874] loss:0.614359073143165\n",
      "times:9700 beta[1.7609180456422326, 1.0330326229713862] loss:0.620196319713462\n",
      "times:9800 beta[1.7666055779318173, 1.0340785947711966] loss:0.6120715033199466\n",
      "times:9900 beta[1.771569296156544, 1.0390628050948223] loss:0.6097098416891364\n",
      "times:10000 beta[1.7775792426694237, 1.0350253670343779] loss:0.603239149776832\n",
      "times:10100 beta[1.783841252456174, 1.0342367058311375] loss:0.6015262922164832\n",
      "times:10200 beta[1.7895209606027047, 1.0350901775391863] loss:0.596721091427257\n",
      "times:10300 beta[1.795426325313823, 1.0350228114140267] loss:0.5936885948187707\n",
      "times:10400 beta[1.8016002909259667, 1.0313235267114418] loss:0.6033632467811922\n",
      "times:10500 beta[1.8076170169715575, 1.0321040428456079] loss:0.5952402610390657\n",
      "times:10600 beta[1.8136482996457228, 1.0425567241806928] loss:0.622405577408994\n",
      "times:10700 beta[1.8180049557267355, 1.0345026273635418] loss:0.5824250071635567\n",
      "times:10800 beta[1.8245806159083897, 1.0382049751080797] loss:0.586893789583794\n",
      "times:10900 beta[1.8313402332733277, 1.0387191301682295] loss:0.5874351180405053\n",
      "times:11000 beta[1.8374172165627038, 1.0390089392794455] loss:0.5874254985864388\n",
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      "times:93700 beta[2.9782106472210774, 1.000674469374536] loss:0.010761338622207003\n",
      "times:93800 beta[2.9783158448979963, 1.0006605130376807] loss:0.010690686042434781\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "times:93900 beta[2.9784102497236584, 1.000607408107527] loss:0.010679560228500196\n",
      "times:94000 beta[2.978521027169251, 1.0005738864671587] loss:0.010744228873772691\n",
      "times:94100 beta[2.978627615325734, 1.0006217478049522] loss:0.010533259769162919\n",
      "times:94200 beta[2.9787271809497375, 1.0005663797277118] loss:0.01065149948835897\n",
      "times:94300 beta[2.978821530263816, 1.000625655265883] loss:0.010431261519157343\n",
      "times:94400 beta[2.978917317722075, 1.0006742814996474] loss:0.01047866785473108\n",
      "times:94500 beta[2.979029870859114, 1.0006119885510376] loss:0.01033309650271269\n",
      "times:94600 beta[2.97915075855794, 1.0006609151048644] loss:0.010340826585483664\n",
      "times:94700 beta[2.979254326735108, 1.0007052193388446] loss:0.010545222625700418\n",
      "times:94800 beta[2.9793597751377945, 1.0005555923312206] loss:0.010304271743825597\n",
      "times:94900 beta[2.97946161099151, 1.000522225027126] loss:0.010437718519255033\n",
      "times:95000 beta[2.9795527682674896, 1.000502867641834] loss:0.010525607081254737\n",
      "times:95100 beta[2.9796540228753203, 1.000659823478121] loss:0.010151795446329917\n",
      "times:95200 beta[2.9797459628782077, 1.0005205017554848] loss:0.01025462176181105\n",
      "times:95300 beta[2.979837271262302, 1.0006404623808962] loss:0.010004927932021682\n",
      "times:95400 beta[2.9799334293416426, 1.0006259291886288] loss:0.009921886434926901\n"
     ]
    }
   ],
   "source": [
    "i = 1\n",
    "loss = loss_function(beta,x,y)\n",
    "while  loss >tol_L\n",
    "    grad = Sgrad(beta,x,y)\n",
    "    beta = newBeta(beta,alpha,grad)\n",
    "    if i % 100 == 0\n",
    "        loss = loss_function(beta,x,y)\n",
    "        println(\"times:\",i,\" \",\"beta\",beta,\" \",\"loss:\",loss)\n",
    "    end\n",
    "    i = i+1\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It runs 95400 times ( 95400/50 = 1908 times) , which is much quicker than previous gradient descent method."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4. Mini-batch Stochastic Gradient Decent"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "SGD just uses one random sample and the Full-batch gradient decent uses all samples. Now we want to have a trade-off between speed and stability. So in the Mini-batch Stochastic Gradient Decent, we use a number designed by us to choose samples."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "minibatch_grad (generic function with 1 method)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function minibatch_grad(beta,batch_size,x,y)\n",
    "    grad = [0.0;0.0]\n",
    "    r_1 = rand(1:(length(x)-10))\n",
    "    r_2 = r_1 + batch_size\n",
    "    grad[1] = 2 * mean(beta[1].+beta[2]*x[r_1:r_2]-y[r_1:r_2])\n",
    "    grad[2] = 2 * mean(x[r_1:r_2].*(beta[1].+beta[2]*x[r_1:r_2]-y[r_1:r_2]))\n",
    "    return grad\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "beta = [1;1]\n",
    "alpha = 0.0001\n",
    "tol_L = 0.01\n",
    "batch_size = 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "times:100 beta[1.0077668378258218, 1.0643812550537828] loss:0.9929970176560121\n",
      "times:200 beta[1.0137381836562147, 1.04977338871153] loss:1.0149204182434075\n",
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      "times:400 beta[1.0281074908074712, 1.0576229045598962] loss:0.9715903735662713\n",
      "times:500 beta[1.0361789339059067, 1.0552323224533264] loss:0.9714705710597099\n",
      "times:600 beta[1.0439115922407536, 1.0509844359572642] loss:0.9857202436000019\n",
      "times:700 beta[1.050899454936243, 1.0788549526785634] loss:1.1396170624850317\n",
      "times:800 beta[1.0568145323192801, 1.0623798041117607] loss:0.966750897687041\n",
      "times:900 beta[1.0640173877612225, 1.053574219482042] loss:0.9604354049177186\n",
      "times:1000 beta[1.071817357301597, 1.0621053967780005] loss:0.9601634217633518\n",
      "times:1100 beta[1.0786893242279703, 1.0714000763219305] loss:1.0352654520144373\n",
      "times:1200 beta[1.0859551640311946, 1.0728643967748395] loss:1.052986911379925\n",
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      "times:1500 beta[1.1040079484041705, 1.0546144342138632] loss:0.9351416408126968\n",
      "times:1600 beta[1.1117962089201485, 1.0606670986943199] loss:0.9396148046995948\n",
      "times:1700 beta[1.1184121435471128, 1.060924345781943] loss:0.9383882516973089\n",
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      "times:2800 beta[1.1927172205173187, 1.0617986866267588] loss:0.9213448945852389\n",
      "times:2900 beta[1.1995565412407205, 1.0495422801167869] loss:0.8942200376942702\n",
      "times:3000 beta[1.2057860767136723, 1.0689302136960488] loss:0.9953921258254934\n",
      "times:3100 beta[1.211712071841534, 1.0681780079816352] loss:0.9851249039805097\n",
      "times:3200 beta[1.2176975376266264, 1.0531037530453993] loss:0.8778297097037226\n",
      "times:3300 beta[1.2244881062113353, 1.0515068784352795] loss:0.8752157109649424\n",
      "times:3400 beta[1.231946086979577, 1.0598777660135588] loss:0.8971130794809784\n",
      "times:3500 beta[1.237747302348732, 1.0553236397792187] loss:0.8723210468960446\n",
      "times:3600 beta[1.2453043368428824, 1.0578655769539682] loss:0.8804654486495591\n",
      "times:3700 beta[1.2515782815818268, 1.0538334531901727] loss:0.8629275263693151\n",
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      "times:3900 beta[1.2639505914037736, 1.0541554145445902] loss:0.8583960424851074\n",
      "times:4000 beta[1.2711272269242284, 1.059585969525077] loss:0.885014255396505\n",
      "times:4100 beta[1.2762218438306654, 1.0612841083458675] loss:0.8989243638333817\n",
      "times:4200 beta[1.2825756686925958, 1.0548317787502024] loss:0.8532327792175755\n",
      "times:4300 beta[1.2884601515208192, 1.0506062125013544] loss:0.8429919787969058\n",
      "times:4400 beta[1.2951577425689378, 1.0531792404930507] loss:0.8429586176379112\n",
      "times:4500 beta[1.3019865862150943, 1.0552484675411284] loss:0.8481064192230781\n",
      "times:4600 beta[1.307746649770788, 1.0540593925853696] loss:0.8408481387990554\n",
      "times:4700 beta[1.3137711904725786, 1.050610256660652] loss:0.8306405566460546\n",
      "times:4800 beta[1.3198691213521243, 1.0547446219097514] loss:0.8395576298417706\n",
      "times:4900 beta[1.3266557184439614, 1.0534064046437819] loss:0.8312642271044101\n",
      "times:5000 beta[1.3335505278055353, 1.057795623469533] loss:0.8560040093567275\n",
      "times:5100 beta[1.3399797360905137, 1.0556180240594484] loss:0.8382370727225548\n",
      "times:5200 beta[1.3454235731320983, 1.0555201744622211] loss:0.8360380708794193\n",
      "times:5300 beta[1.3505303868094607, 1.046180663835381] loss:0.8165656920563431\n",
      "times:5400 beta[1.3548562153375305, 1.0454934772043911] loss:0.8162654443117183\n",
      "times:5500 beta[1.3592549095324153, 1.046221900199476] loss:0.8114448217141225\n",
      "times:5600 beta[1.3654573562564383, 1.0544845664551465] loss:0.8236021821597828\n",
      "times:5700 beta[1.3716089537919758, 1.05151236350628] loss:0.807286540323535\n",
      "times:5800 beta[1.3768554974736422, 1.0468520644700503] loss:0.8004210768399462\n",
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      "times:118600 beta[2.971903973809383, 1.0009029821490714] loss:0.013982378977800903\n",
      "times:118700 beta[2.972000836717734, 1.0008592292590925] loss:0.01381374122455705\n",
      "times:118800 beta[2.972088097533923, 1.0008970104820016] loss:0.013890520018913456\n",
      "times:118900 beta[2.9721824090240894, 1.000914209248849] loss:0.01394088257493305\n",
      "times:119000 beta[2.972292534364598, 1.0009415120434435] loss:0.014081338023814173\n",
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      "times:119200 beta[2.972499405340982, 1.000836148386207] loss:0.013556344153466059\n",
      "times:119300 beta[2.9726065539259037, 1.0008682483681417] loss:0.013586226271135612\n",
      "times:119400 beta[2.972693013919771, 1.0010177174486377] loss:0.01474894002027253\n",
      "times:119500 beta[2.972766503863353, 1.0008425344919492] loss:0.013449116036762602\n",
      "times:119600 beta[2.972855437447863, 1.0008560525838999] loss:0.013448516695966008\n",
      "times:119700 beta[2.972966645922465, 1.000825497543983] loss:0.013330770944155603\n",
      "times:119800 beta[2.9730733800463387, 1.0009838449131792] loss:0.014316288403591452\n",
      "times:119900 beta[2.973164118302391, 1.0007851485910602] loss:0.01322180455607892\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "times:120000 beta[2.973258410330374, 1.0008342084988902] loss:0.013222515532791345\n",
      "times:120100 beta[2.9733497127797137, 1.000821445952881] loss:0.013154866748325347\n",
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      "times:127400 beta[2.979560310889924, 1.0007071109213654] loss:0.010484580114657499\n",
      "times:127500 beta[2.979628955135077, 1.0006066993221296] loss:0.010033224116514871\n",
      "times:127600 beta[2.9796998648263773, 1.0006566883482764] loss:0.010121299285066146\n",
      "times:127700 beta[2.979765970995549, 1.0005501327300772] loss:0.010076510113880425\n",
      "times:127800 beta[2.9798449366877153, 1.0006035767315444] loss:0.00992778019569344\n"
     ]
    }
   ],
   "source": [
    "i = 1\n",
    "loss = loss_function(beta,x,y)\n",
    "while  loss >tol_L\n",
    "    grad = minibatch_grad(beta,batch_size,x,y)\n",
    "    beta = newBeta(beta,alpha,grad)\n",
    "    if i % 100 == 0\n",
    "        loss = loss_function(beta,x,y)\n",
    "        println(\"times:\",i,\" \",\"beta\",beta,\" \",\"loss:\",loss)\n",
    "    end\n",
    "    i = i+1\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In general, we almost use mini-batch stochastic gradient decent in the deep learning."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "However, mini-batch stochastic gradient decent still does not solve the typical problems like finding the local minimum."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3. ADAM"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to suppress the oscillation of SGD, ADAM believes that the gradient descent process can add inertia.\n",
    "\n",
    "Thus we change the gradient to be $ m_{t}=\\beta_{1} \\cdot m_{t-1}+\\left(1-\\beta_{1}\\right) \\cdot g_{t}$, $\\beta_1 = 0.9$\n",
    "\n",
    "We want it to be adapitive, so we want our algorithm can metric history update frequency. The idea is to use gradients from early steps.\n",
    "\n",
    "Thus we introduce a new variable $V_{t}=\\sum_{\\tau=1}^{t} g_{\\tau}^{2}$. However, it is a typical decreasing stepsize, which may make the process end early. Therefore, we use a period instead of the whole process.Then $V_{t}=\\beta_{2} * V_{t-1}+\\left(1-\\beta_{2}\\right) g_{t}^{2}$ , $\\beta_2 = 0.99$\n",
    "\n",
    "Now, our expression is $ \\beta_{n+1} = \\beta_n - \\frac{\\alpha}{\\sqrt{v_t}+\\epsilon} * m_t$\n",
    "\n",
    "$\\epsilon$ is a very samll number to ensure that no division by $0$ and is always set to be $10e-8$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "momentum (generic function with 1 method)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function momentum(grad, b_1,m)\n",
    "    m = b_1*m .+ (1-b_1)*grad\n",
    "    return m\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "v_t (generic function with 1 method)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function v_t(grad,b_2,v)\n",
    "    v = b_2*v .+ (1-b_2)*grad.^2\n",
    "    return v\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "adam_beta (generic function with 1 method)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function adam_beta(beta,alpha,m,v)\n",
    "    beta = beta - alpha .* m / (norm(v)+10e-8)\n",
    "    return beta\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.99"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "beta = [1;1]\n",
    "alpha = 0.01\n",
    "tol_L = 0.01\n",
    "m = 0\n",
    "v = 0\n",
    "b_1 = 0.9\n",
    "b_2 = 0.99"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "At the beginning, $m_t, v_t$ will be close to $0$, so we will transfer them to $\\tilde{m}_{t}=\\frac{m_{t}}{\\left(1-\\beta_{1}^{t}\\right)}$,$\\tilde{v}_{t}=\\frac{v_{t}}{\\left(1-\\beta_{2}^{t}\\right)}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "times:100 beta[1.0000057682225438, 1.0001268656309568] loss:1.9967599974880255\n",
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      "times:3000 beta[1.0000057682225438, 1.0001268656309568] loss:1.9967599974880255\n",
      "times:3100 beta[1.0000057682225438, 1.0001268656309568] loss:1.9967599974880255\n",
      "times:3200 beta[1.0000057682225438, 1.0001268656309568] loss:1.9967599974880255\n",
      "times:3300 beta[1.0000057682225438, 1.0001268656309568] loss:1.9967599974880255\n",
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
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      "times:167800 beta[2.954173989212675, 1.0021650505586002] loss:0.03262202826592005\n",
      "times:167900 beta[2.9579790439027467, 1.0009020523127936] loss:0.02304692330831078\n",
      "times:168000 beta[2.961101519290736, 1.000400040988255] loss:0.029272335639178432\n",
      "times:168100 beta[2.964225770239968, 1.0039352448014547] loss:0.08599336467452792\n",
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      "times:168300 beta[2.968962584573259, 1.001246904150158] loss:0.01800989670607861\n",
      "times:168400 beta[2.9712030971945476, 1.002385328537619] loss:0.04701866082507276\n",
      "times:168500 beta[2.972689694266748, 1.0016611372982573] loss:0.02830377215869164\n",
      "times:168600 beta[2.974426378737469, 1.0012136290644134] loss:0.018319648031524485\n",
      "times:168700 beta[2.975999262693267, 1.001327437791691] loss:0.021539656926863847\n",
      "times:168800 beta[2.977512708148257, 1.001554273687896] loss:0.028232823029312144\n",
      "times:168900 beta[2.97851261100743, 1.0005162951838718] loss:0.011169809898757778\n",
      "times:169000 beta[2.9801444881817374, 1.0010461968363296] loss:0.016567499125868874\n",
      "times:169100 beta[2.980719505390016, 0.999686157095929] loss:0.027656840575962103\n",
      "times:169200 beta[2.9824760685955134, 1.0009571630592993] loss:0.015432959718844041\n",
      "times:169300 beta[2.98377318279571, 1.000338083889529] loss:0.009036009204591655\n"
     ]
    }
   ],
   "source": [
    "i = 1\n",
    "loss = loss_function(beta,x,y)\n",
    "while loss > tol_L\n",
    "    grad = Sgrad(beta,x,y)\n",
    "    m = momentum(grad, b_1,m)\n",
    "    m = m /(1-b_1^i)\n",
    "    v = v_t(grad, b_2,v)\n",
    "    v = v /(1-b_2^i)\n",
    "    beta = adam_beta(beta,alpha,m,v)\n",
    "    if i % 100 == 0\n",
    "        loss = loss_function(beta,x,y)\n",
    "        println(\"times:\",i,\" \",\"beta\",beta,\" \",\"loss:\",loss)\n",
    "    end\n",
    "    i = i+1\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It runs $168600 / 50 = 3372$ times."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## CNN"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Edge detection using convolution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "imgs   = MNIST.images()\n",
    "labels = MNIST.labels();## Edge detection using convolution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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      "text/plain": [
       "28×28 Array{Gray{N0f8},2} with eltype ColorTypes.Gray{FixedPointNumbers.Normed{UInt8,8}}:\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)  …  Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)  …  Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)  …  Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " ⋮                                 ⋱                                  \n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)  …  Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)  …  Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sample_image = imgs[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "image = float.(Sample_image);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Array{Float64,2}:\n",
       " -0.5  0.0  0.5\n",
       " -1.0  0.0  1.0\n",
       " -0.5  0.0  0.5"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x_filter = 1/2*[-1 0 1;\n",
    "                -2 0 2;\n",
    "                -1 0 1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "edge_detecter (generic function with 1 method)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function edge_detecter(img, filter)\n",
    "    N = Int(sqrt(length(img)))\n",
    "    n = Int(sqrt(length(filter)))\n",
    "    result_image = []\n",
    "    for j in 1:N-n+1\n",
    "        for i in 1:N-n+1\n",
    "            window_matrix = img[i:i+n-1,j:j+n-1]\n",
    "            filtered_value = ones(n)'*(window_matrix .* filter)*ones(n)\n",
    "            push!(result_image, filtered_value)\n",
    "        end\n",
    "    end\n",
    "    return reshape(result_image,(N-2,N-2))\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Plots.GRBackend()"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "gr()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
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     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "filtered_image_1 = edge_detecter(image, x_filter);\n",
    "filtered_image_x = abs.(Int.(round.(10 .* filtered_image_1 ));)\n",
    "heatmap(filtered_image_x, yflip = true)"
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  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
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       "  0.0   0.0   0.0\n",
       "  0.5   1.0   0.5"
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     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "y_filter = 1/2*[-1 -2 -1;\n",
    "                0 0 0;\n",
    "                1 2 1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "filtered_image_2 = edge_detecter(image, y_filter);\n",
    "filtered_image_y = abs.(Int.(round.(10 .* filtered_image_2));)\n",
    "heatmap(filtered_image_y, yflip = true)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "edge_image = filtered_image_x + filtered_image_y\n",
    "heatmap(edge_image, yflip = true)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Smoothing images using convolution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "smoothing (generic function with 1 method)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function smoothing(img, smoother)\n",
    "    N = Int(sqrt(length(img)))\n",
    "    n = Int(sqrt(length(smoother)))\n",
    "    weight = ones(n)'*smoother*ones(n)\n",
    "    result_image = []\n",
    "    for j in 1:N-n+1\n",
    "        for i in 1:N-n+1\n",
    "            window_matrix = img[i:i+n-1,j:j+n-1]\n",
    "            smoothed_value = ones(n)'*(window_matrix .* smoother)*ones(n)\n",
    "            push!(result_image, smoothed_value)\n",
    "        end\n",
    "    end\n",
    "    return reshape(result_image,(N-2,N-2))\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Array{Int64,2}:\n",
       " 1  2  1\n",
       " 2  8  2\n",
       " 1  2  1"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smooth_matrix = [1 2 1;\n",
    "                 2 8 2;\n",
    "                 1 2 1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
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       "</svg>\n"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smoothed_image_1 = smoothing(image, smooth_matrix);\n",
    "#smoothed_image = abs.(Int.(round.(10 .* smoothed_image_1)))\n",
    "smoothed_image = Int.(round.(10 .* smoothed_image_1))\n",
    "heatmap(smoothed_image, yflip = true)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Array{Tuple{Int64,Int64},1}:\n",
       " (28, 28)\n",
       " (26, 26)\n",
       " (26, 26)\n",
       " (26, 26)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[size(Sample_image),size(filtered_image_x),size(filtered_image_y),size(smoothed_image)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Edge issue"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$n\\times n$ matrix $\\left[\\begin{matrix} A \\end{matrix}\\right]$, could be extended to $\\left[\\begin{matrix} 0 & 0 & 0\\\\ 0 & A_{n\\times n} & 0 \\\\ 0 & 0 & 0 \\end{matrix}\\right]$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "extend_edge (generic function with 1 method)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function extend_edge(img, value, edge_num)\n",
    "    N = Int(sqrt(length(img)))\n",
    "    head_edge = fill(value, N+2*edge_num, edge_num)\n",
    "    side_edge = fill(value,N, edge_num)\n",
    "    return [head_edge';\n",
    "            side_edge img side_edge;\n",
    "            head_edge']\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(28, 28)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "extended_image = extend_edge(image, 0, 1)\n",
    "filtered_extended_image = edge_detecter(extended_image, x_filter);\n",
    "filtered_extended_image_x = Int.(round.(10 .* filtered_extended_image));\n",
    "size(filtered_extended_image_x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Max Pooling"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "max_pooling (generic function with 1 method)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function max_pooling(img, pooling_size, step)\n",
    "    N = Int(sqrt(length(img)))\n",
    "    output_array = []\n",
    "    if N%pooling_size == 0\n",
    "        for j in 1:step:N-pooling_size\n",
    "            for i in 1:step:N-pooling_size\n",
    "                window = img[i:i+pooling_size,j:j+pooling_size]\n",
    "                result = maximum(window)\n",
    "                push!(output_array, result)\n",
    "                \n",
    "            end\n",
    "        end\n",
    "    end\n",
    "    if N%pooling_size != 0\n",
    "        for j in 1:step:N-pooling_size\n",
    "            for i in 1:step:N-pooling_size\n",
    "                window = img[i:i+pooling_size,j:j+pooling_size]\n",
    "                result = maximum(window)\n",
    "                push!(output_array, result)\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "    size = Int(sqrt(length(output_array)))\n",
    "    output_image = reshape(output_array,size,size)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pooled_image_1 = max_pooling(image, 2, 2);\n",
    "pooled_image = Int.(round.(10 .* pooled_image_1))\n",
    "heatmap(pooled_image, yflip=true)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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