juliaL13.ipynb

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    "# Gradient Descent\n",
    "\n",
    "CPSC 340: Machine Learning and Data Mining\n",
    "\n",
    "The University of British Columbia\n",
    "\n",
    "2018 Winter Term 1\n",
    "\n",
    "By Mike Gelbart (converted to Julia by Jason Chau)"
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    "using Distributions\n",
    "using MultivariateStats\n",
    "using Plots\n",
    "using Plots.PlotMeasures\n",
    "using Printf\n",
    "using LinearAlgebra\n",
    "#= run once\n",
    "    using Pkg\n",
    "    Pkg.add(\"Optim\")\n",
    "=#\n",
    "using Optim"
   ]
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    "Learning goals:\n",
    "\n",
    "- Why do we need gradient descent? Can't we use th normal equations for everything?\n",
    "- What is the step size/learning rate?\n",
    "- How do I implement gradient descent?\n",
    "- What's the difference between the data space and the parameter space?"
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    "## Amin\n",
    "\n",
    "- Assignment 3 released\n",
    "- Midterm: October 18th at 6:30pm in BUCH A102 and A104\n",
    "    - Duration: 80min"
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    "## Motivation 1: large-scale least squares\n",
    "\n",
    "- Normal equations costs $O(nd^2+d^3)$\n",
    "- Can we do better for large $d$?"
   ]
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   "source": [
    "## Motivation 2: least squares with outliers"
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       "Plots.PyPlotBackend()"
      ]
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   "source": [
    "# ENV[\"GRDIR\"]=\"\"\n",
    "# Pkg.build(\"GR\")\n",
    "# using Pkg\n",
    "# Pkg.add(\"LaTeXStrings\")\n",
    "# pyplot()"
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       "  1562.23,721.972 1564.03,722.076 1565.83,722.179 1567.62,722.282 1569.42,722.385 1571.21,722.489 1573.01,722.592 1574.81,722.695 1576.6,722.798 1578.4,722.901 \n",
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       "  \n",
       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function gen_outlier_data(;n=40,Noutliers=3)\n",
    "    # generate random data\n",
    "    x = rand(Normal(),n)\n",
    "    y = 10*x\n",
    "    # add random outliers\n",
    "    y[1:Noutliers] = -100*(x[1:Noutliers]+rand(Normal(),Noutliers))\n",
    "\n",
    "    X = reshape(x,(n,1)) # reshape for linear regression\n",
    "    \n",
    "    return (X,y)\n",
    "end\n",
    "\n",
    "# fit a linear regression model\n",
    "(X,y) = gen_outlier_data()\n",
    "lr = llsq(X,y)\n",
    "function predict(x)\n",
    "    return lr[1] + lr[2]*x\n",
    "end\n",
    "\n",
    "grid = range(minimum(X)-0.3,stop=maximum(X)+.3,length=1000)\n",
    "\n",
    "plot(X,y,\n",
    "    seriestype=:scatter,\n",
    "    markersize=20,\n",
    "    legend=false,\n",
    "    size=(2000,1500),\n",
    "    thickness_scaling=5,\n",
    "    left_margin=-80px,\n",
    "    bottom_margin=-45px\n",
    ")\n",
    "plot!(grid,broadcast(predict,grid))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Can we do better?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Gradient descent\n",
    "\n",
    "#### use of the gradient: optimization\n",
    "\n",
    "The gradient gives us the direction of fastest increase of a function with respect to its parameters.\n",
    "\n",
    "The (negative) gradient gives us a direction of travel (fastest decrease), if we want to minimize a function!\n",
    "\n",
    "If we are trying to minimize $f(w)$, then gradient descent works by starting with some initial guess $w^0$ and then updating with\n",
    "\n",
    "$$w^{t+1} = w^t - \\alpha^t\\, \\nabla f(w^t)$$\n",
    "\n",
    "- $w^t$ refers to the value of $w$ (the weights/parameters) at iteration $t$ of gradient descent.\n",
    "- $\\alpha^t$ is the \"learning rate\" or \"step size\" at iteration $t$.\n",
    "- Gradient descent is an algorithm for minimization. The corresponding method for maximization is called gradient ascent.\n",
    "- The step size should be small to guarantee that the loss goes down. If it's too big the loss could go up.\n",
    "- Converges to a global optimium if $f$ is convex. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- We can use this to minimize losses like the squared error (above) and robust losses (coming soon):\n",
    "\n",
    "\n",
    "$$ f(w) = \\sum_{i=1}^n\\left(w^T x_i-y_i\\right)^2 $$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "#### magnitude vs. direction\n",
    "\n",
    "In the above we are making use of the magnitude of the gradient, not just its direction. This actually makes sense. When the magnitude is small, we are in a flatter area and want to take smaller steps. \n",
    "\n",
    "However, some methods exist that just use the direction. For example given a direction you can do a [line search](https://en.wikipedia.org/wiki/Line_search).\n",
    "\n",
    "#### picking the learning rate\n",
    "\n",
    "In general picking $\\alpha$ is a pain, especially when we get to stochastic gradient descent (later in the course). There is theory on this regarding convergence guarantees and convergence rates. The learning rate is sometimes decreased over time. Fancier methods pick the learning rate adaptively (see a3 code).\n",
    "\n",
    "#### termination conditions\n",
    "\n",
    "We can stop when $||\\nabla f||$ is sufficiently small (because this indicates we're at a local minimum), or when a specified maximum number of iterations are reached (because we want to limit the computational expense)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Least squares via normal equations vs. gradient descent:\n",
    "\n",
    "- Normal equations cost $O(nd^2 + d^3)$.\n",
    "  - forming $X^TX$ costs $O(nd^2)$ and solving a $d\\times d$ linear system costs $O(d^3)$\n",
    "- Gradient descent costs $O(ndt)$ to run for $t$ iterations.\n",
    "  - computing $\\nabla f(w)=X^TXw-X^Ty$ only costs $O(nd)$ \n",
    "- Gradient descent can be faster when $d$ is very large\n",
    "  - well, this is just big-O, we don't know the coeffcients\n",
    "  - matrix operations are fast; iteration is slow, especially in Python\n",
    "  - assuming $t$ doesn't depend on $d$ (see CPSC 540)\n",
    "  - one advantage is that we can control the accuracy by controlling the number of iterations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Experiments: does it work?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1000-element Array{Float64,1}:\n",
       "  2.054647150943276  \n",
       "  1.207473180866666  \n",
       " -0.47792947643876954\n",
       " -1.460928409364932  \n",
       " -0.3006121417811932 \n",
       " -1.9968464747372672 \n",
       " -0.1285219090667731 \n",
       "  0.4450246690343144 \n",
       "  0.5888910589145907 \n",
       " -0.2069410464854636 \n",
       " -1.2310111080923543 \n",
       "  0.05874169316501323\n",
       " -0.36360540520446294\n",
       "  ⋮                  \n",
       " -0.14727865634373283\n",
       " -0.8398518013878338 \n",
       "  0.2496131164060855 \n",
       "  0.3560637238343483 \n",
       " -0.14963192658269955\n",
       "  0.20339407998985495\n",
       " -0.9359934881981689 \n",
       "  0.9320768190558173 \n",
       "  0.8238273126778948 \n",
       " -0.6201891387440464 \n",
       " -0.3585623272458735 \n",
       " -0.6884156417898623 "
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# generate data\n",
    "d = 10\n",
    "n = 1000\n",
    "\n",
    "X = rand(Normal(),n,d)\n",
    "y = rand(Normal(),n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Approach 1: sklearn's `LinearRegression` in Python"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Intercept: 0.053389\n",
      "Weights: [0.0525225, -0.00708656, 0.00274481, 0.00555818, -0.013154, -0.0100317, -0.0103753, 0.0159919, -0.0180807, -0.00160599]"
     ]
    }
   ],
   "source": [
    "# least squares\n",
    "\n",
    "lr = llsq(X,y)\n",
    "Printf.@printf(\"Intercept: %f\\n\",lr[end])\n",
    "Printf.@printf(\"Weights: %s\",lr[1:end-1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Approach 3: gradient descent"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "grad_f (generic function with 1 method)"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "alpha = 1e-5\n",
    "\n",
    "grad_f(w) = X'*(X*w) - X'*y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Intercept: 0.053305\n",
      "Weights: [0.0525097, -0.00706368, 0.00274044, 0.00561045, -0.0130505, -0.0100767, -0.010404, 0.0159321, -0.0180885]"
     ]
    }
   ],
   "source": [
    "w = zeros(size(X)[2]) # initial guess\n",
    "\n",
    "g = grad_f(w)\n",
    "while LinearAlgebra.norm(g) > 0.001\n",
    "    g = grad_f(w)\n",
    "    \n",
    "    w = w - alpha*g\n",
    "end\n",
    "    \n",
    "Printf.@printf(\"Intercept: %f\\n\", w[end])\n",
    "Printf.@printf(\"Weights: %s\", w[1:end-1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From the above, you can see that all three methods generate roughly the same weights. This is comforting."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Robust regression with gradient descent\n",
    "\n",
    "Is this only useful when $d$ is large?\n",
    "\n",
    "Answer: no! Another use: changing the loss function (so we can't use the normal equations).\n",
    "\n",
    "For example let's consider the absolute value objective:\n",
    "\n",
    "$$ f(w) = \\sum_{i=1}^n\\left|w^T x_i-y_i\\right| $$\n",
    "\n",
    "This is the sum of absolute values instead of the sum of squares. This will hopefully give us robust regression because big distances aren't amplified in the objective (let's dwell on this reasoning for a bit).\n",
    "\n",
    "**Note on smoothness:** Non-smooth functions are harder to minimize than smooth functions. Unlike smooth functions, the gradient may not get smaller near a minimizer. \n",
    "\n",
    "Experiments below: this time we add outliers to mess up least squares, and then try switching to the absolute value objective. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "w = [-0.250539]\n",
      "b = -7.304202"
     ]
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       "  \n",
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       "</svg>\n"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(X,y) = gen_outlier_data()\n",
    "x = X[:]\n",
    "\n",
    "# fit a linear regression model\n",
    "lr = llsq(X,y)\n",
    "predict(x) = lr[end].+x*lr[1:end-1]'\n",
    "\n",
    "Printf.@printf(\"w = %s\\n\", lr[1:end-1])\n",
    "Printf.@printf(\"b = %f\", lr[2])\n",
    "\n",
    "plot(X,y,\n",
    "    seriestype=:scatter,\n",
    "    markersize=20,\n",
    "    size=(2000,1500),\n",
    "    xlabel=\"x\",\n",
    "    ylabel=\"y\",\n",
    "    legend=false,\n",
    "    thickness_scaling=5,\n",
    "    left_margin=-100px,\n",
    "    bottom_margin=-45px\n",
    ")\n",
    "grid = range(minimum(X)-0.3,stop=maximum(X)+0.3,length=1000)\n",
    "plot!(grid, predict(grid))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(BONUS \"SLIDE\")\n",
    "\n",
    "For linear regression in one dimension our L1 objective becomes\n",
    "\n",
    "$$f(w,\\beta)=\\displaystyle \\sum_{i=1}^n\\left|w x_i + \\beta - y_i\\right| $$\n",
    "\n",
    "The gradient (where defined) is\n",
    "\n",
    "$$\\frac{\\partial f}{\\partial w} = \\sum_{i=1}^n  x_i \\textrm{sign}\\left(w x_i + \\beta - y_i\\right)$$ \n",
    "\n",
    "$$\\frac{\\partial f}{\\partial \\beta} = \\sum_{i=1}^n  \\textrm{sign}\\left(w x_i + \\beta - y_i\\right)$$ \n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "w = 9.994218\n",
      "β = -0.014016"
     ]
    },
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       "  763.635,345.241 765.096,344.908 766.556,344.575 768.017,344.241 769.478,343.908 770.939,343.575 772.4,343.242 773.861,342.908 775.321,342.575 776.782,342.242 \n",
       "  778.243,341.909 779.704,341.575 781.165,341.242 782.626,340.909 784.086,340.576 785.547,340.242 787.008,339.909 788.469,339.576 789.93,339.243 791.391,338.909 \n",
       "  792.851,338.576 794.312,338.243 795.773,337.91 797.234,337.576 798.695,337.243 800.155,336.91 801.616,336.576 803.077,336.243 804.538,335.91 805.999,335.577 \n",
       "  807.46,335.243 808.92,334.91 810.381,334.577 811.842,334.244 813.303,333.91 814.764,333.577 816.225,333.244 817.685,332.911 819.146,332.577 820.607,332.244 \n",
       "  822.068,331.911 823.529,331.578 824.99,331.244 826.45,330.911 827.911,330.578 829.372,330.245 830.833,329.911 832.294,329.578 833.754,329.245 835.215,328.912 \n",
       "  836.676,328.578 838.137,328.245 839.598,327.912 841.059,327.579 842.519,327.245 843.98,326.912 845.441,326.579 846.902,326.246 848.363,325.912 849.824,325.579 \n",
       "  851.284,325.246 852.745,324.912 854.206,324.579 855.667,324.246 857.128,323.913 858.589,323.579 860.049,323.246 861.51,322.913 862.971,322.58 864.432,322.246 \n",
       "  865.893,321.913 867.354,321.58 868.814,321.247 870.275,320.913 871.736,320.58 873.197,320.247 874.658,319.914 876.118,319.58 877.579,319.247 879.04,318.914 \n",
       "  880.501,318.581 881.962,318.247 883.423,317.914 884.883,317.581 886.344,317.248 887.805,316.914 889.266,316.581 890.727,316.248 892.188,315.915 893.648,315.581 \n",
       "  895.109,315.248 896.57,314.915 898.031,314.582 899.492,314.248 900.953,313.915 902.413,313.582 903.874,313.249 905.335,312.915 906.796,312.582 908.257,312.249 \n",
       "  909.717,311.915 911.178,311.582 912.639,311.249 914.1,310.916 915.561,310.582 917.022,310.249 918.482,309.916 919.943,309.583 921.404,309.249 922.865,308.916 \n",
       "  924.326,308.583 925.787,308.25 927.247,307.916 928.708,307.583 930.169,307.25 931.63,306.917 933.091,306.583 934.552,306.25 936.012,305.917 937.473,305.584 \n",
       "  938.934,305.25 940.395,304.917 941.856,304.584 943.316,304.251 944.777,303.917 946.238,303.584 947.699,303.251 949.16,302.918 950.621,302.584 952.081,302.251 \n",
       "  953.542,301.918 955.003,301.585 956.464,301.251 957.925,300.918 959.386,300.585 960.846,300.251 962.307,299.918 963.768,299.585 965.229,299.252 966.69,298.918 \n",
       "  968.151,298.585 969.611,298.252 971.072,297.919 972.533,297.585 973.994,297.252 975.455,296.919 976.915,296.586 978.376,296.252 979.837,295.919 981.298,295.586 \n",
       "  982.759,295.253 984.22,294.919 985.68,294.586 987.141,294.253 988.602,293.92 990.063,293.586 991.524,293.253 992.985,292.92 994.445,292.587 995.906,292.253 \n",
       "  997.367,291.92 998.828,291.587 1000.29,291.254 1001.75,290.92 1003.21,290.587 1004.67,290.254 1006.13,289.921 1007.59,289.587 1009.05,289.254 1010.51,288.921 \n",
       "  1011.98,288.587 1013.44,288.254 1014.9,287.921 1016.36,287.588 1017.82,287.254 1019.28,286.921 1020.74,286.588 1022.2,286.255 1023.66,285.921 1025.12,285.588 \n",
       "  1026.58,285.255 1028.04,284.922 1029.51,284.588 1030.97,284.255 1032.43,283.922 1033.89,283.589 1035.35,283.255 1036.81,282.922 1038.27,282.589 1039.73,282.256 \n",
       "  1041.19,281.922 1042.65,281.589 1044.11,281.256 1045.57,280.923 1047.04,280.589 1048.5,280.256 1049.96,279.923 1051.42,279.59 1052.88,279.256 1054.34,278.923 \n",
       "  1055.8,278.59 1057.26,278.257 1058.72,277.923 1060.18,277.59 1061.64,277.257 1063.1,276.923 1064.57,276.59 1066.03,276.257 1067.49,275.924 1068.95,275.59 \n",
       "  1070.41,275.257 1071.87,274.924 1073.33,274.591 1074.79,274.257 1076.25,273.924 1077.71,273.591 1079.17,273.258 1080.63,272.924 1082.09,272.591 1083.56,272.258 \n",
       "  1085.02,271.925 1086.48,271.591 1087.94,271.258 1089.4,270.925 1090.86,270.592 1092.32,270.258 1093.78,269.925 1095.24,269.592 1096.7,269.259 1098.16,268.925 \n",
       "  1099.62,268.592 1101.09,268.259 1102.55,267.926 1104.01,267.592 1105.47,267.259 1106.93,266.926 1108.39,266.593 1109.85,266.259 1111.31,265.926 1112.77,265.593 \n",
       "  1114.23,265.259 1115.69,264.926 1117.15,264.593 1118.62,264.26 1120.08,263.926 1121.54,263.593 1123,263.26 1124.46,262.927 1125.92,262.593 1127.38,262.26 \n",
       "  1128.84,261.927 1130.3,261.594 1131.76,261.26 1133.22,260.927 1134.68,260.594 1136.15,260.261 1137.61,259.927 1139.07,259.594 1140.53,259.261 1141.99,258.928 \n",
       "  1143.45,258.594 1144.91,258.261 1146.37,257.928 1147.83,257.595 1149.29,257.261 1150.75,256.928 1152.21,256.595 1153.68,256.262 1155.14,255.928 1156.6,255.595 \n",
       "  1158.06,255.262 1159.52,254.929 1160.98,254.595 1162.44,254.262 1163.9,253.929 1165.36,253.595 1166.82,253.262 1168.28,252.929 1169.74,252.596 1171.21,252.262 \n",
       "  1172.67,251.929 1174.13,251.596 1175.59,251.263 1177.05,250.929 1178.51,250.596 1179.97,250.263 1181.43,249.93 1182.89,249.596 1184.35,249.263 1185.81,248.93 \n",
       "  1187.27,248.597 1188.74,248.263 1190.2,247.93 1191.66,247.597 1193.12,247.264 1194.58,246.93 1196.04,246.597 1197.5,246.264 1198.96,245.931 1200.42,245.597 \n",
       "  1201.88,245.264 1203.34,244.931 1204.8,244.598 1206.27,244.264 1207.73,243.931 1209.19,243.598 1210.65,243.265 1212.11,242.931 1213.57,242.598 1215.03,242.265 \n",
       "  1216.49,241.931 1217.95,241.598 1219.41,241.265 1220.87,240.932 1222.33,240.598 1223.8,240.265 1225.26,239.932 1226.72,239.599 1228.18,239.265 1229.64,238.932 \n",
       "  1231.1,238.599 1232.56,238.266 1234.02,237.932 1235.48,237.599 1236.94,237.266 1238.4,236.933 1239.86,236.599 1241.33,236.266 1242.79,235.933 1244.25,235.6 \n",
       "  1245.71,235.266 1247.17,234.933 1248.63,234.6 1250.09,234.267 1251.55,233.933 1253.01,233.6 1254.47,233.267 1255.93,232.934 1257.39,232.6 1258.85,232.267 \n",
       "  1260.32,231.934 1261.78,231.601 1263.24,231.267 1264.7,230.934 1266.16,230.601 1267.62,230.267 1269.08,229.934 1270.54,229.601 1272,229.268 1273.46,228.934 \n",
       "  1274.92,228.601 1276.38,228.268 1277.85,227.935 1279.31,227.601 1280.77,227.268 1282.23,226.935 1283.69,226.602 1285.15,226.268 1286.61,225.935 1288.07,225.602 \n",
       "  1289.53,225.269 1290.99,224.935 1292.45,224.602 1293.91,224.269 1295.38,223.936 1296.84,223.602 1298.3,223.269 1299.76,222.936 1301.22,222.603 1302.68,222.269 \n",
       "  1304.14,221.936 1305.6,221.603 1307.06,221.27 1308.52,220.936 1309.98,220.603 1311.44,220.27 1312.91,219.937 1314.37,219.603 1315.83,219.27 1317.29,218.937 \n",
       "  1318.75,218.603 1320.21,218.27 1321.67,217.937 1323.13,217.604 1324.59,217.27 1326.05,216.937 1327.51,216.604 1328.97,216.271 1330.44,215.937 1331.9,215.604 \n",
       "  1333.36,215.271 1334.82,214.938 1336.28,214.604 1337.74,214.271 1339.2,213.938 1340.66,213.605 1342.12,213.271 1343.58,212.938 1345.04,212.605 1346.5,212.272 \n",
       "  1347.97,211.938 1349.43,211.605 1350.89,211.272 1352.35,210.939 1353.81,210.605 1355.27,210.272 1356.73,209.939 1358.19,209.606 1359.65,209.272 1361.11,208.939 \n",
       "  1362.57,208.606 1364.03,208.273 1365.5,207.939 1366.96,207.606 1368.42,207.273 1369.88,206.939 1371.34,206.606 1372.8,206.273 1374.26,205.94 1375.72,205.606 \n",
       "  1377.18,205.273 1378.64,204.94 1380.1,204.607 1381.56,204.273 1383.03,203.94 1384.49,203.607 1385.95,203.274 1387.41,202.94 1388.87,202.607 1390.33,202.274 \n",
       "  1391.79,201.941 1393.25,201.607 1394.71,201.274 1396.17,200.941 1397.63,200.608 1399.09,200.274 1400.56,199.941 1402.02,199.608 1403.48,199.275 1404.94,198.941 \n",
       "  1406.4,198.608 1407.86,198.275 1409.32,197.942 1410.78,197.608 1412.24,197.275 1413.7,196.942 1415.16,196.609 1416.62,196.275 1418.09,195.942 1419.55,195.609 \n",
       "  1421.01,195.275 1422.47,194.942 1423.93,194.609 1425.39,194.276 1426.85,193.942 1428.31,193.609 1429.77,193.276 1431.23,192.943 1432.69,192.609 1434.15,192.276 \n",
       "  1435.61,191.943 1437.08,191.61 1438.54,191.276 1440,190.943 1441.46,190.61 1442.92,190.277 1444.38,189.943 1445.84,189.61 1447.3,189.277 1448.76,188.944 \n",
       "  1450.22,188.61 1451.68,188.277 1453.14,187.944 1454.61,187.611 1456.07,187.277 1457.53,186.944 1458.99,186.611 1460.45,186.278 1461.91,185.944 1463.37,185.611 \n",
       "  1464.83,185.278 1466.29,184.945 1467.75,184.611 1469.21,184.278 1470.67,183.945 1472.14,183.611 1473.6,183.278 1475.06,182.945 1476.52,182.612 1477.98,182.278 \n",
       "  1479.44,181.945 1480.9,181.612 1482.36,181.279 1483.82,180.945 1485.28,180.612 1486.74,180.279 1488.2,179.946 1489.67,179.612 1491.13,179.279 1492.59,178.946 \n",
       "  1494.05,178.613 1495.51,178.279 1496.97,177.946 1498.43,177.613 1499.89,177.28 1501.35,176.946 1502.81,176.613 1504.27,176.28 1505.73,175.947 1507.2,175.613 \n",
       "  1508.66,175.28 1510.12,174.947 1511.58,174.614 1513.04,174.28 1514.5,173.947 1515.96,173.614 1517.42,173.281 1518.88,172.947 1520.34,172.614 1521.8,172.281 \n",
       "  1523.26,171.947 1524.73,171.614 1526.19,171.281 1527.65,170.948 1529.11,170.614 1530.57,170.281 1532.03,169.948 1533.49,169.615 1534.95,169.281 1536.41,168.948 \n",
       "  1537.87,168.615 1539.33,168.282 1540.79,167.948 1542.26,167.615 1543.72,167.282 1545.18,166.949 1546.64,166.615 1548.1,166.282 1549.56,165.949 1551.02,165.616 \n",
       "  1552.48,165.282 1553.94,164.949 1555.4,164.616 1556.86,164.283 1558.32,163.949 1559.79,163.616 1561.25,163.283 1562.71,162.95 1564.17,162.616 1565.63,162.283 \n",
       "  1567.09,161.95 1568.55,161.617 1570.01,161.283 1571.47,160.95 1572.93,160.617 1574.39,160.283 1575.85,159.95 1577.32,159.617 1578.78,159.284 1580.24,158.95 \n",
       "  1581.7,158.617 1583.16,158.284 1584.62,157.951 1586.08,157.617 1587.54,157.284 1589,156.951 1590.46,156.618 1591.92,156.284 1593.38,155.951 1594.85,155.618 \n",
       "  1596.31,155.285 1597.77,154.951 1599.23,154.618 1600.69,154.285 1602.15,153.952 1603.61,153.618 1605.07,153.285 1606.53,152.952 1607.99,152.619 1609.45,152.285 \n",
       "  1610.91,151.952 1612.37,151.619 1613.84,151.286 1615.3,150.952 1616.76,150.619 1618.22,150.286 1619.68,149.953 1621.14,149.619 1622.6,149.286 1624.06,148.953 \n",
       "  1625.52,148.619 1626.98,148.286 1628.44,147.953 1629.9,147.62 1631.37,147.286 1632.83,146.953 1634.29,146.62 1635.75,146.287 1637.21,145.953 1638.67,145.62 \n",
       "  1640.13,145.287 1641.59,144.954 1643.05,144.62 1644.51,144.287 1645.97,143.954 1647.43,143.621 1648.9,143.287 1650.36,142.954 1651.82,142.621 1653.28,142.288 \n",
       "  1654.74,141.954 1656.2,141.621 1657.66,141.288 1659.12,140.955 1660.58,140.621 1662.04,140.288 1663.5,139.955 1664.96,139.622 1666.43,139.288 1667.89,138.955 \n",
       "  1669.35,138.622 1670.81,138.289 1672.27,137.955 1673.73,137.622 1675.19,137.289 1676.65,136.955 1678.11,136.622 1679.57,136.289 1681.03,135.956 1682.49,135.622 \n",
       "  1683.96,135.289 1685.42,134.956 1686.88,134.623 1688.34,134.289 1689.8,133.956 1691.26,133.623 1692.72,133.29 1694.18,132.956 1695.64,132.623 1697.1,132.29 \n",
       "  1698.56,131.957 1700.02,131.623 1701.49,131.29 1702.95,130.957 1704.41,130.624 1705.87,130.29 1707.33,129.957 1708.79,129.624 1710.25,129.291 1711.71,128.957 \n",
       "  1713.17,128.624 1714.63,128.291 1716.09,127.958 1717.55,127.624 1719.02,127.291 1720.48,126.958 1721.94,126.625 1723.4,126.291 1724.86,125.958 1726.32,125.625 \n",
       "  1727.78,125.291 1729.24,124.958 1730.7,124.625 1732.16,124.292 1733.62,123.958 1735.08,123.625 1736.55,123.292 1738.01,122.959 1739.47,122.625 1740.93,122.292 \n",
       "  1742.39,121.959 1743.85,121.626 1745.31,121.292 1746.77,120.959 1748.23,120.626 1749.69,120.293 1751.15,119.959 1752.61,119.626 1754.08,119.293 1755.54,118.96 \n",
       "  1757,118.626 1758.46,118.293 1759.92,117.96 1761.38,117.627 1762.84,117.293 1764.3,116.96 1765.76,116.627 1767.22,116.294 1768.68,115.96 1770.14,115.627 \n",
       "  1771.61,115.294 1773.07,114.961 1774.53,114.627 1775.99,114.294 1777.45,113.961 1778.91,113.628 1780.37,113.294 1781.83,112.961 1783.29,112.628 1784.75,112.294 \n",
       "  1786.21,111.961 1787.67,111.628 1789.13,111.295 1790.6,110.961 1792.06,110.628 1793.52,110.295 1794.98,109.962 1796.44,109.628 1797.9,109.295 1799.36,108.962 \n",
       "  1800.82,108.629 1802.28,108.295 1803.74,107.962 1805.2,107.629 1806.66,107.296 1808.13,106.962 1809.59,106.629 1811.05,106.296 1812.51,105.963 1813.97,105.629 \n",
       "  1815.43,105.296 1816.89,104.963 1818.35,104.63 1819.81,104.296 1821.27,103.963 1822.73,103.63 1824.19,103.297 1825.66,102.963 1827.12,102.63 1828.58,102.297 \n",
       "  1830.04,101.964 1831.5,101.63 1832.96,101.297 1834.42,100.964 1835.88,100.63 1837.34,100.297 1838.8,99.964 1840.26,99.6307 1841.72,99.2974 1843.19,98.9642 \n",
       "  1844.65,98.6309 1846.11,98.2977 1847.57,97.9644 1849.03,97.6312 1850.49,97.2979 1851.95,96.9646 1853.41,96.6314 1854.87,96.2981 1856.33,95.9649 1857.79,95.6316 \n",
       "  \n",
       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w = 0\n",
    "β = 0\n",
    "α = 1\n",
    "\n",
    "Nsteps = 1000\n",
    "\n",
    "for t in 1:Nsteps-1\n",
    "    dLdw = sum(x.*broadcast(sign,(w .* x).+(β.-y)))\n",
    "    dLdβ = sum(broadcast(sign,(w .* x).+(β.-y)))\n",
    "    w -= (α/t)*dLdw # we are decreasing the step size over time to deal with the non-smoothness\n",
    "    β -= (α/t)*dLdβ # the details are beyond the scope of the course, see CPSC 540\n",
    "end\n",
    "    \n",
    "Printf.@printf(\"w = %f\\n\",w)\n",
    "Printf.@printf(\"β = %f\",β)\n",
    "\n",
    "plot(x,y,\n",
    "    seriestype=:scatter,\n",
    "    xlabel=\"x\",\n",
    "    ylabel=\"y\",\n",
    "    markersize=20,\n",
    "    size=(2000,1500),\n",
    "    thickness_scaling=5,\n",
    "    left_margin=-100px,\n",
    "    bottom_margin=-45px,\n",
    "    legend=false\n",
    ")\n",
    "grid = range(minimum(x)-0.3,stop=maximum(x)+0.3,length=1000)\n",
    "predict(x) = x*w'.+β\n",
    "plot!(grid,predict(grid))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Gradient descent is not just for robust regression!\n",
    "\n",
    "We can use it for a large class of models. With some caveats, **you are now empowered to minimize things!** \n",
    "\n",
    "## Why gradient descent?\n",
    "\n",
    "A common misconception from previous terms of CPSC 340:\n",
    "\n",
    "> We use gradient descent because of the non-smooth loss.\n",
    "\n",
    "Not true!! We use it for large $d$. We also use it because the normal equations only apply to linear least squares. We can use gradient descent in all sorts of cases. The non-smoothness just complicates things a bit."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## (bonus \"slide\") Gradient descent is not the only game in town \n",
    "\n",
    "There are many optimization algorithms out there. Gradient descent is not applicable in all contexts and it is certainly not the best method in all contexts! We focus on it because:\n",
    "\n",
    "- it is relatively simple to understand and implement.\n",
    "- the time complexity is linear in $d$ per iteration.\n",
    "- we can generalize it to _stochastic gradient descent_, which is coming later in the course.\n",
    "\n",
    "See CPSC 406 for much more. (Through Python's scientific libraries like [scipy.optimize](https://docs.scipy.org/doc/scipy/reference/optimize.html) you can access a variety of methods as black-box optimizers. For example see scipy's [implementation of the popular L-BFGS method](https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.optimize.fmin_l_bfgs_b.html), which is more sophisticated but still linear in $d$ per iteration.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The different decisions we make\n",
    "\n",
    "We should try to keep separate (in our minds) the following choices:\n",
    "\n",
    "- the model (e.g. linear)\n",
    "- the loss (e.g. squared error)\n",
    "- the optimization method (e.g. gradient descent)\n",
    "\n",
    "We can (roughly) pick these independently of each other. Not quite because the choice of optimization method may depend on the choice of model/loss: e.g. we can use the normal equations as our optimization method if we're doing least squares."
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## What space am I in?\n",
    "\n",
    "A key skill is being able to move around (conceptually) between $x$ (data) space and $w$ (parameter) space. The above plots are in $y$ vs. $x$ space. Here's what the loss looks like in parameter space:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
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     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function plot_loss(loss_fun, X, y) # reading/understanding this plotting code is optional\n",
    "    m = 100\n",
    "    w_lin = range(-20.0, stop=20.0, length=m)\n",
    "    b_lin = range(-20.0, stop=20.0, length=m)\n",
    "    \n",
    "    pred = w_lin*X' .+ b_lin\n",
    "    \n",
    "    contour(w_lin, b_lin, (w,b) -> sum((w.*X.+b-y).^2),\n",
    "        nlevels=10,\n",
    "        w=4,\n",
    "        xlabel=\"w\",\n",
    "        ylabel=\"beta\",\n",
    "        title=\"Loss in w-beta space\",\n",
    "        size=(2000,1500),\n",
    "        thickness_scaling=5,\n",
    "        left_margin=-95px,\n",
    "        bottom_margin=-40px\n",
    "    )\n",
    "    \n",
    "    loss = loss_fun(pred, y)\n",
    "    imin = argmin(loss)\n",
    "    plot!([w_lin[imin]], [b_lin[imin]], markersize=20, seriestype=:scatter, legend=false)\n",
    "end\n",
    "squared_loss_fun(pred,y) = [sum((pred[i,:]-y).^2) for i in 1:size(pred)[1]]\n",
    "plot_loss(squared_loss_fun, X, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Key intuition: every point in weight space corressponds to a _model_ (in this case a line) in input-output space.\n",
    "\n",
    "- In general, doing supervised learning in $d$ dimensions, the data space will be mapping from $d$ dimensions to $1$ dimensional. \n",
    "- For linear models, the number of parameters is $d+1$ ($d$ weights plus an intercept), but this isn't necessarily the case for other models."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 157,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
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       "  2147.46,217.6 2155.41,218.817 2163.54,220.143 2171.68,221.55 2179.81,223.039 2182.87,223.63 2187.95,224.62 2196.08,226.291 2204.21,228.045 2211.37,229.66 \n",
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       "  2058.37,507.041 2057.8,506.962 2049.67,505.993 2041.54,505.201 2033.4,504.583 2025.27,504.14 2017.14,503.872 2009,503.776 2000.87,503.854 1992.73,504.103 \n",
       "  1984.6,504.525 1976.47,505.117 1968.33,505.88 1960.2,506.812 1958.52,507.041 1952.06,507.937 1943.93,509.24 1935.8,510.716 1927.66,512.363 1924.5,513.071 \n",
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       "  2058.94,183.339 2052,183.443 2045.06,183.646 2038.12,183.947 2032.55,184.269 2031.19,184.349 2024.25,184.854 2017.31,185.459 2010.37,186.163 2003.43,186.966 \n",
       "  1996.49,187.869 1989.56,188.87 1982.62,189.97 1975.68,191.169 1975.31,191.237 1968.74,192.481 1961.8,193.894 1954.86,195.406 1947.93,197.017 1943.11,198.205 \n",
       "  1940.99,198.734 1934.05,200.567 1927.11,202.499 1920.17,204.53 1918.08,205.173 1913.23,206.681 1906.3,208.941 1899.36,211.301 1896.99,212.141 1892.42,213.782 \n",
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       "  1836.91,237.819 1832.61,240.013 1829.97,241.375 1823.03,245.066 1819.54,246.981 1816.1,248.889 1809.16,252.844 1807.27,253.948 1802.22,256.948 1795.7,260.916 \n",
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       "  2094.11,462.987 2093.63,462.895 2086.69,461.776 2079.75,460.86 2072.82,460.147 2065.88,459.635 2058.94,459.325 2052,459.215 2045.06,459.304 2038.12,459.593 \n",
       "  2031.19,460.079 2024.25,460.764 2017.31,461.645 2010.37,462.723 2008.93,462.987 2003.43,464.022 1996.49,465.528 1989.56,467.233 1982.62,469.137 1979.92,469.955 \n",
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       "  1885.48,521.788 1880.96,525.698 1878.54,527.861 1873.37,532.666 1871.6,534.356 1866.3,539.634 1864.66,541.314 1859.7,546.602 1857.73,548.783 1853.54,553.57 \n",
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       "  1823.03,596.428 1819.68,602.346 1816.1,608.983 1815.92,609.314 1812.44,616.281 1809.16,623.206 1809.14,623.249 1806.1,630.217 1803.23,637.185 1802.22,639.799 \n",
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       "  1864.66,401.525 1866.66,400.275 1871.6,397.249 1878.27,393.307 1878.54,393.152 1885.48,389.282 1890.97,386.339 1892.42,385.576 1899.36,382.073 1904.94,379.371 \n",
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     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "w = 3.000952\n",
      "β = -8.996891"
     ]
    }
   ],
   "source": [
    "# a few iterations of GD showing the loss space and line space\n",
    "# you don't need to read/understand this code (it's optional)\n",
    "\n",
    "x = rand(Normal(),40)\n",
    "y = 3 .* x .- 9\n",
    "X = reshape(x,(40,1))\n",
    "\n",
    "w = 0\n",
    "β = 0\n",
    "α = 0.02\n",
    "\n",
    "Nsteps = 5\n",
    "\n",
    "function make_plot_pairs(w,β,t;titles=false)\n",
    "    scatter = plot(x,y,markersize=20,\n",
    "        seriestype=:scatter,\n",
    "        size=(1000,750),\n",
    "        ylabel=\"iteration $t\",\n",
    "        title = titles ? \"Data space\" : \"\",\n",
    "        thickness_scaling = 5,\n",
    "        left_margin = -80px,\n",
    "        legend=false\n",
    "    )\n",
    "    model = plot!(x,w .* x .+ β)\n",
    "    \n",
    "    m = 100\n",
    "    w_lin = range(-10.0, stop=10.0, length=m)\n",
    "    β_lin = range(-10.0, stop=10.0, length=m)\n",
    "    pred = w_lin*X'.+β_lin\n",
    "    loss = squared_loss_fun(pred, y) \n",
    "    imin = argmin(loss)\n",
    "    CS = contour(w_lin, β_lin, (w,b)->sum((w.*X.+b - y).^2),\n",
    "        nlevels=10,\n",
    "        w=4,\n",
    "        title=titles ? \"Parameter space\" : \"\",\n",
    "        size=(1000,750),\n",
    "        thickness_scaling=5,\n",
    "        left_margin=-80px,\n",
    "        legend=false\n",
    "    )\n",
    "    plot!([w_lin[imin]], [β_lin[imin]], seriestype=:scatter, markersize=20)\n",
    "    plot!([w], [β], seriestype=:scatter, markersize=20)\n",
    "    display(plot(scatter,CS,\n",
    "        size=(2400,1000)\n",
    "    ))\n",
    "end\n",
    "\n",
    "# plt.figure(figsize=(8, 20))\n",
    "make_plot_pairs(w,β,0,titles=true)    \n",
    "for t in 0:Nsteps-1\n",
    "    dLdw = sum(x .* (w .* x .+ β-y))\n",
    "    dLdβ = sum(w .* x .+ β-y)\n",
    "    w -= α*dLdw\n",
    "    β -= α*dLdβ\n",
    "    \n",
    "    make_plot_pairs(w,β,t)\n",
    "end\n",
    "Printf.@printf(\"w = %f\\n\",w)\n",
    "Printf.@printf(\"β = %f\",β)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Robust regression intuition (in one dimension)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 199,
   "metadata": {},
   "outputs": [
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      ]
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     "metadata": {},
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    }
   ],
   "source": [
    "n = 40\n",
    "Noutliers = 3\n",
    "x = rand(Normal(),n)\n",
    "y = 2 .* x\n",
    "y[1:Noutliers] = y[1:Noutliers] -20*rand(Normal(),Noutliers) # outliers are all negative\n",
    "\n",
    "# plt.figure(figsize=(10,3))\n",
    "# plt.subplot(1,2,1)\n",
    "plot1 = plot(x,y,\n",
    "    seriestype=:scatter,\n",
    "    markersize=20,\n",
    "    xlabel=\"x\",\n",
    "    ylabel=\"y\",\n",
    "    thickness_scaling=5,\n",
    "    size=(1000,750),\n",
    "    left_margin=-80px,\n",
    "    bottom_margin=-45px,\n",
    "    titlefontsize=5,\n",
    "    legend=false\n",
    ")\n",
    "pred1 = 2 .* x\n",
    "L2_loss = sum((pred1-y).^2)\n",
    "L1_loss = sum(broadcast(abs,pred1-y))\n",
    "plot!(x, pred1,\n",
    "    title=\"L2 loss = $L2_loss, L1 loss = $L1_loss\"\n",
    ")\n",
    "# plt.subplot(1,2,2)\n",
    "plot2 = plot(x,y,markersize=20,\n",
    "    seriestype=:scatter,\n",
    "    xlabel=\"x\",\n",
    "    ylabel=\"y\",\n",
    "    thickness_scaling=5,\n",
    "    size=(1000,750),\n",
    "    left_margin=-80px,\n",
    "    bottom_margin=-45px,\n",
    "    legend=false,\n",
    "    titlefontsize=5\n",
    ")\n",
    "pred2 = 2 .* x .- 1\n",
    "L2_loss = sum((pred2-y).^2)\n",
    "L1_loss = sum(broadcast(abs,pred2-y))\n",
    "plot!(x, pred2,\n",
    "    title=\"L2 loss = $L2_loss, L1 loss = $L1_loss\"\n",
    ")\n",
    "display(plot(plot1,plot2,size=(2400,1500)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider what happens to the **L1** loss if we move the line down by 1 unit towards the outliers:\n",
    "\n",
    "- The loss goes down by 1 for 3 points, and up by 1 for 37 points, so overall it goes up by 34 (see above).\n",
    "- So, even the slightest perturbation towards the outliers is unfavorable: it really is robust!\n",
    "- Every point gets an equal vote.\n",
    "\n",
    "Consider what happens to the **L2** loss if we move the line down by 1 unit towards the outliers:\n",
    "\n",
    "- The loss goes up very slightly for the 37 points, but goes down massively for 3 points, so overall it goes down (see above).\n",
    "- Thus, moving the line is favorable. \n",
    "- 3 points have overruled 37 points.\n",
    "\n",
    "In fact, this is an extension of the idea that the median of a bunch of numbers minimizes the L1 error and the mean minimizes the L2 error. The robustness of the median that we discussed early on in the course is this same effect. This is also related to $k$-means taking the mean and thus minimizing L2."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Digression: mean and median"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- The **mean** is the value that minimizes the **L2** error between itself and a set of numbers.\n",
    "- The **median** is the value that minimizes the **L1** error between itself and a set of numbers."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 184,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mean of the numbers: 151.771429\n",
      "Median of the numbers: 4.000000\n"
     ]
    }
   ],
   "source": [
    "nums = [1,7,4,-3,3.4,1000,50]\n",
    "\n",
    "Printf.@printf(\"Mean of the numbers: %f\\n\", mean(nums))\n",
    "Printf.@printf(\"Median of the numbers: %f\\n\", median(nums))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 217,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Minimizer of L2: 10065.197737618111\n",
      "Minimizer of L1: 758.9674393903247"
     ]
    }
   ],
   "source": [
    "l2_loss(x) = sum((x-nums).^2)\n",
    "l1_loss(x) = sum(broadcast(abs,x-nums))\n",
    "\n",
    "x0 = zeros(7)\n",
    "\n",
    "Printf.@printf(\"Minimizer of L2: %s\\n\", Optim.minimum(optimize(l2_loss, x0)))\n",
    "Printf.@printf(\"Minimizer of L1: %s\", Optim.minimum(optimize(l1_loss, x0)))"
   ]
  }
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