ESL-LASSO.ipynb

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  {
   "cell_type": "markdown",
   "id": "ongoing-hunger",
   "metadata": {},
   "source": [
    "# ESL-LASSO implementation "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "excessive-there",
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots, LinearAlgebra, Random, Distributions, Suppressor, ForwardDiff"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "loved-place",
   "metadata": {},
   "source": [
    "## ESL loss and penalization term\n",
    "\n",
    "- Exponentiasl squared loss with tuning parameter $\\gamma$, $\\phi_\\gamma(t)= 1-  \\exp (-t^2/\\gamma)$, which leads to the objective function\n",
    "$$ 1- \\frac{1}{n}\\sum_{i = 1}^ n \\exp \\left\\{ -(Y_i- x_i^T\\beta)^2/\\gamma\\right\\}  . $$\n",
    "In contrast to the traditional OLS loss, the flatness of this loss function on the tail enables the robust estimates;\n",
    "\n",
    "- Adaptive LASSO regularizer with tuning parameters $\\tau_{nj}$, $$\\mathcal{p}_{\\lambda_{nj}} = \\tau_{nj}|\\beta_j|/|\\tilde{\\beta_j}|,$$ where $\\tilde{\\beta_j}$ is an pilot robust regression estimator, e.g., MM-estimator, S-estimator, OLS estimator, which should at least be $\\sqrt{n}$-consistent to $\\beta^*$. The goal of adding such penalty term is to guarantee a **consistent** vairable selection procedure with **orcal property**. In the paper, the initial estimator $\\tilde{\\beta_j}$ is selected as the MM-estimator."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "blessed-officer",
   "metadata": {},
   "source": [
    "Therefore, the ESL-LASSO estimator is obtained by minimizing the following ojectibve function \n",
    "$$\\ell_n(\\beta)i= - \\frac{1}{n} \\sum_{i = 1}^ n \\exp \\left\\{ -(Y_i- x_i^T\\beta)^2/\\gamma\\right\\} + \\sum_{j = 1}^d \\frac{\\tau_{nj}}{|\\tilde{\\beta_j}|} |\\beta_j|.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "molecular-bulletin",
   "metadata": {},
   "source": [
    "The tricky part for the optimization is that the loss function $\\phi_\\gamma(t)$ is convex but quasi-convex."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "metric-stockholm",
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function ϕ(t::Float64, gamma::Float64)\n",
    "    return 1- exp(- t^2/gamma)\n",
    "end\n",
    "\n",
    "# comparison of reg loss function \n",
    "x1 = -3: .1:3\n",
    "y1 = hcat(ϕ.(x1,1.),x1.^2)\n",
    "p1 = plot(x1, y1, title = \"Regression Loss\", lw = 3, label = [\"ϕ\" \"L2 loss\"])\n",
    "\n",
    "# effect of gamma in ESL\n",
    "x = -10: .1:10\n",
    "y = hcat(ϕ.(x,.1), ϕ.(x,1.),ϕ.(x,10.))\n",
    "p2 = plot(x, y, title = \"ϕ with different γ\", lw = 3, label = [\"γ = 0.1\" \"γ = 1\"  \"γ = 10\"], legend = :bottomright)\n",
    "\n",
    "plot(p1, p2, layout = (1,2))\n",
    "#savefig(\"tex/figures/esl.png\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "continental-history",
   "metadata": {},
   "source": [
    "In the above plot(right), we have illustrated the influence of the $\\gamma$ to the shape of the ELS loss $\\phi_\\gamma$. Statistically, the choice of $\\gamma$ controls the level of robustness. As shown in the right figure, when $\\gamma = 0.1$, the loss is almost invariant to the increase of $|t|$ when $|t| > 1$; while $\\gamma = 10$, the loss is sensitive to the change of $|t|$ when $|t|\\leq 5$. The design of this loss aligns with the intuition that outliers tend to produce abnormally large residual value and a robust model should not be stable these large residual, meaning that the loss function should be relatively flat in the range of large value. And the value of $\\gamma$ corresponds to width of this ra|nge."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "piano-needle",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ℓ (generic function with 1 method)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# esl empirical loss(l*)\n",
    "function ESL(beta::Vector,dat::Array{Float64, 2}, gamma::Float64)\n",
    "    n =size(dat)[1]\n",
    "    X = dat[:,2:end]\n",
    "    Y = dat[:,1]\n",
    "    res = Y - X*beta\n",
    "    Loss = sum( ϕ.(res, gamma))/n\n",
    "    return Loss\n",
    "end \n",
    "\n",
    "# adaptive lasso penalty\n",
    "function adaLASSO(beta::Vector, lambda::Vector)\n",
    "    return lambda'* abs.(beta)\n",
    "end\n",
    "\n",
    "# the objective function \n",
    "function ℓ(beta::Vector,dat::Array{Float64, 2}, gamma::Float64, lambda::Vector)\n",
    "    return ESL(beta, dat, gamma) + adaLASSO(beta, lambda)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "functional-savings",
   "metadata": {},
   "source": [
    "## Computation\n",
    "\n",
    "### 1. Dealing with the non-convex loss\n",
    "\n",
    "#### 1.1. Quadratic approximation (paper)\n",
    "In the paper, due to the non-convexity of the ESL loss function, for the sake of the computation, the authors consider a quadratic approximation to the loss function at the intial estimator $\\tilde{\\beta}$. Let \n",
    "$$\\ell^*(\\beta) =  -\\frac{1}{n}\\sum_{i = 1}^ n \\exp \\left\\{ -(Y_i- x_i^T\\beta)^2/\\gamma\\right\\},$$ \n",
    "then the actual objevtive function for computation becomes \n",
    "$$\\tilde{\\ell}(\\beta) = \\frac{1}{2} (\\beta - \\tilde{\\beta})^T \\nabla^2 \\ell^*(\\tilde{\\beta}) (\\beta - \\tilde{\\beta}) + \\sum_{j = 1}^d \\frac{\\tau_{nj}}{|\\tilde{\\beta_j}|} |\\beta_j|.$$\n",
    "Here $\\ell^*, \\tilde{\\ell}$ actually depends on $n$, but we abuse the notation for convenience by dropping the subscript $n$.   Then there are plenty of efficient algorithms---such as ISTA, LARS, corrdinate descent, ADMM---can be used to solve this type of problem.  \n",
    "\n",
    "However, there is an obvious flaw of this global convex relaxation strategy---**L2 loss is NOT robust**.  To overcome the influence of outliers, the loss function of a robust regression is usually bounded above (meaning that it has to be non-convex), in which case the the outliers cannot leverage the estimates arbitrarily. Therefore, the main concern of this global quadratic appriximation is that the squared loss can potentially ruin the robustness of the estimates. Such quadrtic approximation is  acceptable **only when the initial estimator $\\tilde{\\beta}$ is close to the optimum**. As displayed in the left figure above, the ESL can be approximated well by quadratic function around $0$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "rapid-cuisine",
   "metadata": {},
   "source": [
    "#### 1.2. Proximal gradient algorithm (We proposed)\n",
    "\n",
    "We propose an simple iterative algorithm of solving this $L_1$ penalized optimization problem, \n",
    "$$\\beta_{k+1} \\gets \\text{Prox}_{\\alpha_k, \\mathcal{P}}(\\beta_k - \\alpha_k \\nabla \\ell_n^*(\\beta_k)), \\quad \\alpha_k \\leq \\left(\\frac{2}{\\gamma} \\sigma_1\\left(\\frac{1}{n} \\sum_{i=1}^n x_i x_i^T \\right)\\right)^{-1}, $$\n",
    "where $ \\text{Prox}_{\\alpha_k, \\mathcal{P}}(\\cdot)$ is the proximal operator and $\\mathcal{P}(\\beta) = \\sum_{j = 1}^d \\frac{\\tau_{nj}}{|\\tilde{\\beta_j}|} |\\beta_j|$. This simple algorithm is designed by using the idea of the majorization-maximization method---by optimizing a sequence of quadratic upper bounds of the target function and decreasing the objective value at each step (the detailed interpretation of this algorithm will be included in the qualify report). \n",
    "\n",
    "- The prox can be written as the **soft-thresholding operator**: Let $\\beta^+_k := \\beta_k - \\alpha_k  \\nabla \\ell_n^*(\\beta_k) $, $\\lambda_j = \\frac{\\tau_{nj}}{|\\tilde{\\beta_j}|}$,\n",
    "$$\\left[\\text{Prox}_{\\alpha_k, \\mathcal{P}} (\\beta^+_k) \\right]_j= \\left[S_{\\lambda_j\\alpha_k }(\\beta^+_k)\\right]_{j}=\\left\\{\\begin{array}{ll}\n",
    "\\left[ \\beta^+_k\\right]_j-\\lambda_j\\alpha_k & \\text { if } \\left[ \\beta^+_k\\right]_j>\\lambda \\\\\n",
    "0 & \\text { if }-\\lambda_j\\alpha_k  \\leq \\left[ \\beta^+_k\\right]_j \\leq \\lambda_j\\alpha_k , \\quad i=1, \\ldots, d \\\\\n",
    "\\left[ \\beta^+_k\\right]_j+\\lambda_j\\alpha_k  & \\text { if } \\left[ \\beta^+_k\\right]_j<-\\lambda\n",
    "\\end{array}\\right.$$\n",
    "The implementation of this is simple and can be very efficient with broadcasting.\n",
    "\n",
    "Write $ \\ell_i^*(\\beta) = 1- \\exp \\left\\{ -(Y_i- x_i^T\\beta)^2/\\gamma\\right\\}$ and hence $\\ell^* = \\frac{1}{n}\\sum_{i=1}^n \\ell_i^*$. We can examine the gradient and Hessian as follows.\n",
    "- The grad: \n",
    "$$\\nabla \\ell_i^*(\\beta) = \\frac{2 r_i(\\beta)}{\\gamma}\\exp\\left(-\\frac{2 r_i(\\beta)^2}{\\gamma} \\right) x_i,$$\n",
    "where $r_i(\\beta) = x_i^T\\beta - y_i$.\n",
    "- The Hessian:\n",
    "$$\\begin{aligned}\n",
    "\\nabla^2 \\ell_i^*(\\beta) & = \\frac{2}{\\gamma} \\left[ \\exp\\left(-\\frac{2 r_i(\\beta)^2}{\\gamma} \\right) \\left(1- \\frac{2 r_i(\\beta)^2}{\\gamma} \\right)\\right] (x_i x_i^T) \\\\\n",
    "& \\preceq \\frac{2}{\\gamma} (x_i x_i^T)\n",
    "\\end{aligned}.$$\n",
    "This upper bound determines the Lipshitz constant for the grad of $\\ell^*$ and we can use the L-smoothness of $\\ell^*$ to construct a tight quadratic upper bounds for it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "atomic-copying",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "∇ESL (generic function with 1 method)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# soft-thresholding operator\n",
    "function sfth(beta::Array{Float64,1}, lambda::Array{Float64,1})\n",
    "    a = max.(abs.(beta) - lambda, 0 )\n",
    "    return a.* sign.(beta)\n",
    "end\n",
    "\n",
    "# proximalGD algorithm of solving l1 penalized obj\n",
    "function proxGD(x0::Array{Float64,1}, obj::Function, grd::Function, lrt::Function, sfth::Function, lambda::Array{Float64,1}, num_iters::Int = 100000)\n",
    "    x = x0\n",
    "    for i = 1:num_iters\n",
    "        xx = x - lrt(i).* grd(x)\n",
    "        x = sfth(xx, lrt(i)*lambda)\n",
    "        if i%10000 == 0\n",
    "            println(x,\"\\r\")\n",
    "        end\n",
    "    end\n",
    "    return x, obj(x)\n",
    "end\n",
    "\n",
    "# true grad of l*\n",
    "function ∇ESL( beta::Vector,dat::Array{Float64, 2}, gamma::Float64)\n",
    "    X = dat[:, 2:end]\n",
    "    Y = dat[:,1]\n",
    "    n = size(dat)[1]\n",
    "    res = X*beta - Y\n",
    "    D = (res/gamma) .* exp.(-res.^2/gamma) .* X\n",
    "    return  2 * mean(D, dims  = 1)[1,:]\n",
    "end    \n",
    "# we can also use ForwardDiff (autograd), which is equivalent"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "standard-examination",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "#109 (generic function with 1 method)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#1d synthetic dataset\n",
    "Random.seed!(2021)\n",
    "n1 = 200\n",
    "X = randn(n1,1)\n",
    "Y = 30*X + randn(n1,1) .+ 3\n",
    "dat = hcat(Y,X)\n",
    "λ =[log(20.)/20] #logn/n  \n",
    "γ = .3\n",
    "#grd and obj\n",
    "esl = x -> ESL(x, dat, γ)\n",
    "esl_grd = x-> ∇ESL(x, dat, γ)\n",
    "esl_autogrd = x-> ForwardDiff.gradient(esl, x)\n",
    "obj = x-> ℓ(x, dat, γ, λ)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "conscious-composition",
   "metadata": {},
   "source": [
    "Note that because of the non-convexity nature of this optimization problem, there is no global convergence guarantee on the proximal GD algorithm. In fact, we illustrate below to show that this algorithm will stuck at some place."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "valid-uncertainty",
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       "  184.045,598.037 184.89,598.892 185.736,599.746 186.581,600.601 187.427,601.456 188.272,602.311 189.118,603.165 189.963,604.02 190.809,604.875 191.654,605.73 \n",
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       "  209.41,623.681 210.255,624.535 211.101,625.39 211.946,626.245 212.792,627.1 213.637,627.955 214.483,628.81 215.328,629.665 216.174,630.519 217.019,631.374 \n",
       "  217.865,632.229 218.71,633.084 219.556,633.939 220.401,634.794 221.247,635.649 222.092,636.504 222.938,637.358 223.783,638.213 224.629,639.068 225.474,639.923 \n",
       "  226.32,640.778 227.165,641.633 228.011,642.488 228.856,643.343 229.702,644.198 230.547,645.053 231.393,645.908 232.238,646.762 233.084,647.617 233.929,648.472 \n",
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       "  260.14,674.976 260.985,675.831 261.831,676.686 262.676,677.541 263.522,678.395 264.367,679.25 265.213,680.105 266.058,680.96 266.904,681.815 267.749,682.67 \n",
       "  268.595,683.525 269.44,684.38 270.286,685.235 271.131,686.09 271.977,686.945 272.822,687.8 273.668,688.655 274.513,689.51 275.359,690.365 276.204,691.22 \n",
       "  277.05,692.075 277.895,692.93 278.741,693.785 279.586,694.64 280.432,695.495 281.277,696.35 282.123,697.205 282.968,698.06 283.814,698.915 284.659,699.77 \n",
       "  285.505,700.625 286.35,701.48 287.196,702.335 288.041,703.19 288.887,704.045 289.732,704.9 290.578,705.755 291.423,706.61 292.269,707.465 293.114,708.32 \n",
       "  293.96,709.175 294.805,710.03 295.651,710.885 296.496,711.74 297.342,712.595 298.187,713.45 299.033,714.305 299.878,715.16 300.724,716.015 301.569,716.87 \n",
       "  302.415,717.725 303.26,718.58 304.106,719.435 304.951,720.29 305.797,721.145 306.642,722 307.488,722.855 308.334,723.71 309.179,724.565 310.025,725.42 \n",
       "  310.87,726.275 311.716,727.13 312.561,727.985 313.407,728.841 314.252,729.696 315.098,730.551 315.943,731.406 316.789,732.261 317.634,733.116 318.48,733.971 \n",
       "  319.325,734.826 320.171,735.681 321.016,736.536 321.862,737.391 322.707,738.246 323.553,739.101 324.398,739.956 325.244,740.811 326.089,741.666 326.935,742.521 \n",
       "  327.78,743.376 328.626,744.231 329.471,745.086 330.317,745.941 331.162,746.796 332.008,747.651 332.853,748.506 333.699,749.361 334.544,750.216 335.39,751.071 \n",
       "  336.235,751.926 337.081,752.781 337.926,753.636 338.772,754.491 339.617,755.346 340.463,756.201 341.308,757.056 342.154,757.911 342.999,758.766 343.845,759.621 \n",
       "  344.69,760.477 345.536,761.332 346.381,762.187 347.227,763.042 348.072,763.897 348.918,764.752 349.763,765.607 350.609,766.462 351.454,767.317 352.3,768.172 \n",
       "  353.145,769.027 353.991,769.882 354.836,770.737 355.682,771.592 356.527,772.447 357.373,773.302 358.218,774.157 359.064,775.012 359.909,775.867 360.755,776.722 \n",
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       "  378.51,794.678 379.356,795.533 380.201,796.389 381.047,797.244 381.892,798.099 382.738,798.954 383.583,799.809 384.429,800.664 385.274,801.519 386.12,802.374 \n",
       "  386.965,803.229 387.811,804.084 388.656,804.939 389.502,805.795 390.347,806.65 391.193,807.505 392.038,808.36 392.884,809.215 393.729,810.07 394.575,810.925 \n",
       "  395.42,811.78 396.266,812.636 397.111,813.491 397.957,814.346 398.802,815.201 399.648,816.056 400.493,816.911 401.339,817.767 402.184,818.622 403.03,819.477 \n",
       "  403.875,820.332 404.721,821.187 405.566,822.043 406.412,822.898 407.257,823.753 408.103,824.608 408.948,825.464 409.794,826.319 410.639,827.174 411.485,828.029 \n",
       "  412.33,828.885 413.176,829.74 414.021,830.595 414.867,831.451 415.712,832.306 416.558,833.161 417.403,834.017 418.249,834.872 419.094,835.727 419.94,836.583 \n",
       "  420.785,837.438 421.631,838.294 422.476,839.149 423.322,840.005 424.167,840.86 425.013,841.716 425.858,842.571 426.704,843.427 427.549,844.282 428.395,845.138 \n",
       "  429.24,845.994 430.086,846.849 430.931,847.705 431.777,848.561 432.622,849.416 433.468,850.272 434.313,851.128 435.159,851.984 436.004,852.84 436.85,853.696 \n",
       "  437.695,854.552 438.541,855.407 439.386,856.263 440.232,857.119 441.077,857.976 441.923,858.832 442.768,859.688 443.614,860.544 444.459,861.4 445.305,862.256 \n",
       "  446.15,863.113 446.996,863.969 447.841,864.825 448.687,865.682 449.532,866.538 450.378,867.395 451.223,868.251 452.069,869.108 452.914,869.965 453.76,870.821 \n",
       "  454.605,871.678 455.451,872.535 456.296,873.392 457.142,874.249 457.987,875.106 458.833,875.963 459.678,876.82 460.524,877.677 461.369,878.535 462.215,879.392 \n",
       "  463.06,880.249 463.906,881.107 464.751,881.964 465.597,882.822 466.442,883.68 467.288,884.537 468.133,885.395 468.979,886.253 469.824,887.111 470.67,887.969 \n",
       "  471.515,888.827 472.361,889.685 473.206,890.544 474.052,891.402 474.897,892.261 475.743,893.119 476.588,893.978 477.434,894.836 478.279,895.695 479.125,896.554 \n",
       "  479.97,897.413 480.816,898.272 481.661,899.132 482.507,899.991 483.352,900.85 484.198,901.71 485.043,902.569 485.889,903.429 486.734,904.289 487.58,905.149 \n",
       "  488.425,906.008 489.271,906.869 490.116,907.729 490.962,908.589 491.807,909.449 492.653,910.31 493.498,911.17 494.344,912.031 495.189,912.892 496.035,913.753 \n",
       "  496.88,914.614 497.726,915.475 498.571,916.336 499.417,917.197 500.262,918.059 501.108,918.92 501.953,919.782 502.799,920.643 503.644,921.505 504.49,922.367 \n",
       "  505.335,923.229 506.181,924.091 507.026,924.953 507.872,925.815 508.717,926.678 509.563,927.54 510.408,928.403 511.254,929.265 512.099,930.128 512.945,930.991 \n",
       "  513.79,931.854 514.636,932.717 515.481,933.58 516.327,934.443 517.172,935.306 518.018,936.169 518.863,937.033 519.709,937.896 520.554,938.76 521.4,939.623 \n",
       "  522.245,940.487 523.091,941.351 523.936,942.214 524.782,943.078 525.627,943.942 526.473,944.806 527.318,945.67 528.164,946.534 529.009,947.398 529.855,948.262 \n",
       "  530.7,949.126 531.546,949.99 532.391,950.854 533.237,951.718 534.082,952.582 534.928,953.447 535.773,954.311 536.619,955.175 537.464,956.039 538.31,956.903 \n",
       "  539.155,957.768 540.001,958.632 540.846,959.496 541.692,960.36 542.537,961.224 543.383,962.088 544.228,962.952 545.074,963.816 545.919,964.68 546.765,965.544 \n",
       "  547.61,966.408 548.456,967.272 549.301,968.136 550.147,969 550.992,969.863 551.838,970.727 552.683,971.59 553.529,972.454 554.374,973.317 555.22,974.181 \n",
       "  556.065,975.044 556.911,975.907 557.756,976.77 558.602,977.633 559.447,978.496 560.293,979.358 561.138,980.221 561.984,981.083 562.829,981.946 563.675,982.808 \n",
       "  564.52,983.67 565.366,984.532 566.211,985.394 567.057,986.256 567.902,987.117 568.748,987.979 569.593,988.84 570.439,989.701 571.284,990.562 572.13,991.423 \n",
       "  572.975,992.283 573.821,993.144 574.666,994.004 575.512,994.865 576.357,995.725 577.203,996.585 578.048,997.444 578.894,998.304 579.739,999.163 580.585,1000.02 \n",
       "  581.43,1000.88 582.276,1001.74 583.121,1002.6 583.967,1003.46 584.812,1004.32 585.658,1005.17 586.503,1006.03 587.349,1006.89 588.194,1007.75 589.04,1008.61 \n",
       "  589.885,1009.46 590.731,1010.32 591.576,1011.18 592.422,1012.03 593.267,1012.89 594.113,1013.75 594.958,1014.6 595.804,1015.46 596.649,1016.32 597.495,1017.17 \n",
       "  598.34,1018.03 599.186,1018.88 600.031,1019.74 600.877,1020.59 601.722,1021.45 602.568,1022.3 603.413,1023.16 604.259,1024.01 605.104,1024.87 605.95,1025.72 \n",
       "  606.795,1026.58 607.641,1027.43 608.487,1028.29 609.332,1029.14 610.178,1029.99 611.023,1030.85 611.869,1031.7 612.714,1032.56 613.56,1033.41 614.405,1034.26 \n",
       "  615.251,1035.12 616.096,1035.97 616.942,1036.82 617.787,1037.68 618.633,1038.53 619.478,1039.38 620.324,1040.24 621.169,1041.09 622.015,1041.94 622.86,1042.79 \n",
       "  623.706,1043.65 624.551,1044.5 625.397,1045.35 626.242,1046.21 627.088,1047.06 627.933,1047.91 628.779,1048.76 629.624,1049.62 630.47,1050.47 631.315,1051.32 \n",
       "  632.161,1052.17 633.006,1053.03 633.852,1053.88 634.697,1054.73 635.543,1055.59 636.388,1056.44 637.234,1057.29 638.079,1058.14 638.925,1059 639.77,1059.85 \n",
       "  640.616,1060.7 641.461,1061.56 642.307,1062.41 643.152,1063.26 643.998,1064.12 644.843,1064.97 645.689,1065.82 646.534,1066.68 647.38,1067.53 648.225,1068.38 \n",
       "  649.071,1069.24 649.916,1070.09 650.762,1070.94 651.607,1071.8 652.453,1072.65 653.298,1073.51 654.144,1074.36 654.989,1075.21 655.835,1076.07 656.68,1076.92 \n",
       "  657.526,1077.78 658.371,1078.63 659.217,1079.49 660.062,1080.34 660.908,1081.2 661.753,1082.05 662.599,1082.91 663.444,1083.76 664.29,1084.62 665.135,1085.47 \n",
       "  665.981,1086.33 666.826,1087.19 667.672,1088.04 668.517,1088.9 669.363,1089.75 670.208,1090.61 671.054,1091.47 671.899,1092.32 672.745,1093.18 673.59,1094.04 \n",
       "  674.436,1094.89 675.281,1095.75 676.127,1096.61 676.972,1097.46 677.818,1098.32 678.663,1099.18 679.509,1100.03 680.354,1100.89 681.2,1101.75 682.045,1102.61 \n",
       "  682.891,1103.47 683.736,1104.32 684.582,1105.18 685.427,1106.04 686.273,1106.9 687.118,1107.76 687.964,1108.61 688.809,1109.47 689.655,1110.33 690.5,1111.19 \n",
       "  691.346,1112.05 692.191,1112.91 693.037,1113.77 693.882,1114.63 694.728,1115.48 695.573,1116.34 696.419,1117.2 697.264,1118.06 698.11,1118.92 698.955,1119.78 \n",
       "  699.801,1120.64 700.646,1121.5 701.492,1122.36 702.337,1123.22 703.183,1124.08 704.028,1124.94 704.874,1125.8 705.719,1126.66 706.565,1127.52 707.41,1128.38 \n",
       "  708.256,1129.24 709.101,1130.1 709.947,1130.96 710.792,1131.83 711.638,1132.69 712.483,1133.55 713.329,1134.41 714.174,1135.27 715.02,1136.13 715.865,1136.99 \n",
       "  716.711,1137.85 717.556,1138.72 718.402,1139.58 719.247,1140.44 720.093,1141.3 720.938,1142.16 721.784,1143.03 722.629,1143.89 723.475,1144.75 724.32,1145.61 \n",
       "  725.166,1146.48 726.011,1147.34 726.857,1148.2 727.702,1149.06 728.548,1149.93 729.393,1150.79 730.239,1151.65 731.084,1152.51 731.93,1153.38 732.775,1154.24 \n",
       "  733.621,1155.1 734.466,1155.97 735.312,1156.83 736.157,1157.69 737.003,1158.56 737.848,1159.42 738.694,1160.28 739.539,1161.15 740.385,1162.01 741.23,1162.87 \n",
       "  742.076,1163.74 742.921,1164.6 743.767,1165.46 744.612,1166.33 745.458,1167.19 746.303,1168.05 747.149,1168.92 747.994,1169.78 748.84,1170.64 749.685,1171.51 \n",
       "  750.531,1172.37 751.376,1173.23 752.222,1174.1 753.067,1174.96 753.913,1175.82 754.758,1176.69 755.604,1177.55 756.449,1178.41 757.295,1179.27 758.14,1180.14 \n",
       "  758.986,1181 759.831,1181.86 760.677,1182.72 761.522,1183.58 762.368,1184.44 763.213,1185.31 764.059,1186.17 764.904,1187.03 765.75,1187.89 766.595,1188.75 \n",
       "  767.441,1189.61 768.286,1190.47 769.132,1191.33 769.977,1192.19 770.823,1193.05 771.668,1193.9 772.514,1194.76 773.359,1195.62 774.205,1196.48 775.05,1197.34 \n",
       "  775.896,1198.19 776.741,1199.05 777.587,1199.91 778.432,1200.76 779.278,1201.62 780.123,1202.48 780.969,1203.33 781.814,1204.19 782.66,1205.04 783.505,1205.9 \n",
       "  784.351,1206.75 785.196,1207.61 786.042,1208.46 786.887,1209.31 787.733,1210.17 788.578,1211.02 789.424,1211.87 790.269,1212.72 791.115,1213.58 791.96,1214.43 \n",
       "  792.806,1215.28 793.651,1216.13 794.497,1216.98 795.342,1217.83 796.188,1218.68 797.033,1219.53 797.879,1220.38 798.724,1221.23 799.57,1222.08 800.415,1222.93 \n",
       "  801.261,1223.78 802.106,1224.63 802.952,1225.48 803.797,1226.33 804.643,1227.18 805.488,1228.02 806.334,1228.87 807.179,1229.72 808.025,1230.57 808.87,1231.42 \n",
       "  809.716,1232.27 810.561,1233.12 811.407,1233.96 812.252,1234.81 813.098,1235.66 813.943,1236.51 814.789,1237.36 815.634,1238.21 816.48,1239.06 817.325,1239.91 \n",
       "  818.171,1240.76 819.016,1241.61 819.862,1242.46 820.707,1243.31 821.553,1244.16 822.398,1245.01 823.244,1245.86 824.089,1246.71 824.935,1247.56 825.78,1248.42 \n",
       "  826.626,1249.27 827.471,1250.12 828.317,1250.98 829.162,1251.83 830.008,1252.68 830.853,1253.54 831.699,1254.39 832.544,1255.25 833.39,1256.11 834.235,1256.96 \n",
       "  835.081,1257.82 835.926,1258.68 836.772,1259.54 837.617,1260.4 838.463,1261.26 839.308,1262.12 840.154,1262.98 840.999,1263.84 841.845,1264.7 842.69,1265.56 \n",
       "  843.536,1266.43 844.381,1267.29 845.227,1268.16 846.072,1269.02 846.918,1269.89 847.763,1270.76 848.609,1271.62 849.454,1272.49 850.3,1273.36 851.145,1274.23 \n",
       "  851.991,1275.1 852.836,1275.97 853.682,1276.84 854.527,1277.72 855.373,1278.59 856.218,1279.47 857.064,1280.34 857.909,1281.22 858.755,1282.09 859.6,1282.97 \n",
       "  860.446,1283.85 861.291,1284.73 862.137,1285.61 862.982,1286.49 863.828,1287.37 864.673,1288.25 865.519,1289.13 866.364,1290.02 867.21,1290.9 868.055,1291.79 \n",
       "  868.901,1292.67 869.746,1293.56 870.592,1294.45 871.437,1295.33 872.283,1296.22 873.128,1297.11 873.974,1298 874.819,1298.89 875.665,1299.78 876.51,1300.67 \n",
       "  877.356,1301.57 878.201,1302.46 879.047,1303.35 879.892,1304.25 880.738,1305.14 881.583,1306.04 882.429,1306.93 883.274,1307.83 884.12,1308.72 884.965,1309.62 \n",
       "  885.811,1310.52 886.656,1311.42 887.502,1312.31 888.347,1313.21 889.193,1314.11 890.038,1315.01 890.884,1315.91 891.729,1316.8 892.575,1317.7 893.42,1318.6 \n",
       "  894.266,1319.5 895.111,1320.4 895.957,1321.3 896.802,1322.19 897.648,1323.09 898.493,1323.99 899.339,1324.89 900.184,1325.78 901.03,1326.68 901.875,1327.58 \n",
       "  902.721,1328.47 903.566,1329.37 904.412,1330.26 905.257,1331.15 906.103,1332.05 906.948,1332.94 907.794,1333.83 908.64,1334.72 909.485,1335.61 910.331,1336.5 \n",
       "  911.176,1337.38 912.022,1338.27 912.867,1339.15 913.713,1340.04 914.558,1340.92 915.404,1341.8 916.249,1342.68 917.095,1343.56 917.94,1344.44 918.786,1345.31 \n",
       "  919.631,1346.19 920.477,1347.06 921.322,1347.93 922.168,1348.8 923.013,1349.67 923.859,1350.53 924.704,1351.4 925.55,1352.26 926.395,1353.12 927.241,1353.98 \n",
       "  928.086,1354.84 928.932,1355.7 929.777,1356.55 930.623,1357.41 931.468,1358.26 932.314,1359.11 933.159,1359.96 934.005,1360.81 934.85,1361.65 935.696,1362.5 \n",
       "  936.541,1363.34 937.387,1364.18 938.232,1365.02 939.078,1365.86 939.923,1366.7 940.769,1367.53 941.614,1368.37 942.46,1369.2 943.305,1370.04 944.151,1370.87 \n",
       "  944.996,1371.7 945.842,1372.53 946.687,1373.35 947.533,1374.18 948.378,1375.01 949.224,1375.83 950.069,1376.66 950.915,1377.48 951.76,1378.31 952.606,1379.13 \n",
       "  953.451,1379.95 954.297,1380.78 955.142,1381.6 955.988,1382.42 956.833,1383.24 957.679,1384.06 958.524,1384.88 959.37,1385.7 960.215,1386.53 961.061,1387.35 \n",
       "  961.906,1388.17 962.752,1388.99 963.597,1389.81 964.443,1390.63 965.288,1391.46 966.134,1392.28 966.979,1393.1 967.825,1393.92 968.67,1394.75 969.516,1395.57 \n",
       "  970.361,1396.4 971.207,1397.22 972.052,1398.05 972.898,1398.88 973.743,1399.71 974.589,1400.54 975.434,1401.37 976.28,1402.2 977.125,1403.03 977.971,1403.86 \n",
       "  978.816,1404.7 979.662,1405.53 980.507,1406.37 981.353,1407.2 982.198,1408.04 983.044,1408.88 983.889,1409.72 984.735,1410.56 985.58,1411.41 986.426,1412.25 \n",
       "  987.271,1413.1 988.117,1413.94 988.962,1414.79 989.808,1415.64 990.653,1416.49 991.499,1417.34 992.344,1418.2 993.19,1419.05 994.035,1419.91 994.881,1420.76 \n",
       "  995.726,1421.62 996.572,1422.48 997.417,1423.34 998.263,1424.21 999.108,1425.07 999.954,1425.93 1000.8,1426.8 1001.64,1427.67 1002.49,1428.54 1003.34,1429.41 \n",
       "  1004.18,1430.28 1005.03,1431.15 1005.87,1432.02 1006.72,1432.9 1007.56,1433.77 1008.41,1434.65 1009.25,1435.53 1010.1,1436.4 1010.95,1437.28 1011.79,1438.16 \n",
       "  1012.64,1439.04 1013.48,1439.92 1014.33,1440.81 1015.17,1441.69 1016.02,1442.57 1016.86,1443.45 1017.71,1444.34 1018.55,1445.22 1019.4,1446.11 1020.25,1446.99 \n",
       "  1021.09,1447.87 1021.94,1447.05 1022.78,1446.22 1023.63,1445.39 1024.47,1444.57 1025.32,1443.74 1026.16,1442.91 1027.01,1442.08 1027.86,1441.26 1028.7,1440.43 \n",
       "  1029.55,1439.59 1030.39,1438.76 1031.24,1437.93 1032.08,1437.1 1032.93,1436.26 1033.77,1435.43 1034.62,1434.59 1035.46,1433.75 1036.31,1432.91 1037.16,1432.07 \n",
       "  1038,1431.23 1038.85,1430.39 1039.69,1429.54 1040.54,1428.7 1041.38,1427.85 1042.23,1427 1043.07,1426.15 1043.92,1425.29 1044.77,1424.44 1045.61,1423.58 \n",
       "  1046.46,1422.73 1047.3,1421.87 1048.15,1421.01 1048.99,1420.15 1049.84,1419.28 1050.68,1418.42 1051.53,1417.55 1052.37,1416.69 1053.22,1415.82 1054.07,1414.95 \n",
       "  1054.91,1414.08 1055.76,1413.21 1056.6,1412.34 1057.45,1411.47 1058.29,1410.6 1059.14,1409.73 1059.98,1408.86 1060.83,1407.98 1061.68,1407.11 1062.52,1406.24 \n",
       "  1063.37,1405.37 1064.21,1404.5 1065.06,1403.63 1065.9,1402.76 1066.75,1401.9 1067.59,1401.03 1068.44,1400.16 1069.28,1399.3 1070.13,1398.44 1070.98,1397.57 \n",
       "  1071.82,1396.71 1072.67,1395.85 1073.51,1395 1074.36,1394.14 1075.2,1393.28 1076.05,1392.42 1076.89,1391.57 1077.74,1390.71 1078.59,1389.86 1079.43,1389 \n",
       "  1080.28,1388.15 1081.12,1387.29 1081.97,1386.44 1082.81,1385.58 1083.66,1384.72 1084.5,1383.86 1085.35,1383 1086.19,1382.14 1087.04,1381.27 1087.89,1380.4 \n",
       "  1088.73,1379.53 1089.58,1378.66 1090.42,1377.79 1091.27,1376.91 1092.11,1376.03 1092.96,1375.15 1093.8,1374.26 1094.65,1373.38 1095.5,1372.49 1096.34,1371.59 \n",
       "  1097.19,1370.7 1098.03,1369.8 1098.88,1368.91 1099.72,1368.01 1100.57,1367.11 1101.41,1366.2 1102.26,1365.3 1103.1,1364.4 1103.95,1363.5 1104.8,1362.6 \n",
       "  1105.64,1361.7 1106.49,1360.81 1107.33,1359.91 1108.18,1359.02 1109.02,1358.13 1109.87,1357.25 1110.71,1356.37 1111.56,1355.49 1112.41,1354.62 1113.25,1353.75 \n",
       "  1114.1,1352.89 1114.94,1352.03 1115.79,1351.18 1116.63,1350.34 1117.48,1349.5 1118.32,1348.67 1119.17,1347.85 1120.01,1347.04 1120.86,1346.23 1121.71,1345.44 \n",
       "  1122.55,1344.65 1123.4,1343.87 1124.24,1343.1 1125.09,1342.35 1125.93,1341.6 1126.78,1340.86 1127.62,1340.14 1128.47,1339.42 1129.32,1338.72 1130.16,1338.02 \n",
       "  1131.01,1337.34 1131.85,1336.67 1132.7,1336 1133.54,1335.35 1134.39,1334.71 1135.23,1334.07 1136.08,1333.45 1136.92,1332.83 1137.77,1332.21 1138.62,1331.6 \n",
       "  1139.46,1331 1140.31,1330.39 1141.15,1329.79 1142,1329.18 1142.84,1328.57 1143.69,1327.96 1144.53,1327.33 1145.38,1326.7 1146.23,1326.06 1147.07,1325.4 \n",
       "  1147.92,1324.73 1148.76,1324.05 1149.61,1323.34 1150.45,1322.63 1151.3,1321.89 1152.14,1321.13 1152.99,1320.35 1153.83,1319.56 1154.68,1318.74 1155.53,1317.91 \n",
       "  1156.37,1317.06 1157.22,1316.19 1158.06,1315.3 1158.91,1314.4 1159.75,1313.48 1160.6,1312.56 1161.44,1311.62 1162.29,1310.68 1163.14,1309.73 1163.98,1308.78 \n",
       "  1164.83,1307.83 1165.67,1306.89 1166.52,1305.95 1167.36,1305.02 1168.21,1304.1 1169.05,1303.19 1169.9,1302.29 1170.74,1301.41 1171.59,1300.54 1172.44,1299.68 \n",
       "  1173.28,1298.84 1174.13,1298 1174.97,1297.16 1175.82,1296.33 1176.66,1295.49 1177.51,1294.65 1178.35,1293.79 1179.2,1292.92 1180.05,1292.03 1180.89,1291.12 \n",
       "  1181.74,1290.2 1182.58,1289.25 1183.43,1288.3 1184.27,1287.32 1185.12,1286.33 1185.96,1285.34 1186.81,1284.33 1187.65,1283.33 1188.5,1282.32 1189.35,1281.31 \n",
       "  1190.19,1280.31 1191.04,1279.32 1191.88,1278.34 1192.73,1277.37 1193.57,1276.42 1194.42,1275.49 1195.26,1274.57 1196.11,1273.68 1196.96,1272.8 1197.8,1271.95 \n",
       "  1198.65,1271.13 1199.49,1270.32 1200.34,1269.54 1201.18,1268.77 1202.03,1268.03 1202.87,1267.3 1203.72,1266.59 1204.56,1265.89 1205.41,1265.2 1206.26,1264.52 \n",
       "  1207.1,1263.84 1207.95,1263.16 1208.79,1262.48 1209.64,1261.8 1210.48,1261.1 1211.33,1260.4 1212.17,1259.69 1213.02,1258.96 1213.87,1258.22 1214.71,1257.46 \n",
       "  1215.56,1256.68 1216.4,1255.89 1217.25,1255.08 1218.09,1254.27 1218.94,1253.44 1219.78,1252.62 1220.63,1251.81 1221.48,1251.01 1222.32,1250.25 1223.17,1249.52 \n",
       "  1224.01,1248.83 1224.86,1248.18 1225.7,1247.57 1226.55,1246.98 1227.39,1246.39 1228.24,1245.78 1229.08,1245.12 1229.93,1244.41 1230.78,1243.65 1231.62,1242.86 \n",
       "  1232.47,1242.07 1233.31,1241.33 1234.16,1240.68 1235,1240.16 1235.85,1239.78 1236.69,1239.51 1237.54,1239.32 1238.39,1239.12 1239.23,1238.8 1240.08,1238.3 \n",
       "  1240.92,1237.54 1241.77,1236.53 1242.61,1235.33 1243.46,1234.02 1244.3,1232.7 1245.15,1231.45 1245.99,1230.28 1246.84,1229.19 1247.69,1228.18 1248.53,1227.25 \n",
       "  1249.38,1226.44 1250.22,1225.81 1251.07,1225.37 1251.91,1225.16 1252.76,1225.24 1253.6,1225.62 1254.45,1226.16 1255.3,1226.53 1256.14,1226.36 1256.99,1225.46 \n",
       "  1257.83,1223.87 1258.68,1221.67 1259.52,1218.96 1260.37,1215.9 1261.21,1212.72 1262.06,1209.58 1262.9,1206.69 1263.75,1204.39 1264.6,1202.76 1265.44,1201.47 \n",
       "  1266.29,1200.22 1267.13,1198.92 1267.98,1197.53 1268.82,1196.01 1269.67,1194.58 1270.51,1193.53 1271.36,1192.79 1272.21,1191.99 1273.05,1190.98 1273.9,1189.96 \n",
       "  1274.74,1189.06 1275.59,1188.26 1276.43,1187.55 1277.28,1186.98 1278.12,1186.38 1278.97,1185.5 1279.81,1184.64 1280.66,1184.29 1281.51,1184.52 1282.35,1184.83 \n",
       "  1283.2,1184.95 1284.04,1184.96 1284.89,1184.85 1285.73,1184.47 1286.58,1183.92 1287.42,1183.45 1288.27,1183.13 1289.12,1182.86 1289.96,1182.56 1290.81,1182.23 \n",
       "  1291.65,1181.84 1292.5,1181.26 1293.34,1180.42 1294.19,1179.48 1295.03,1178.7 1295.88,1178.16 1296.72,1177.75 1297.57,1177.32 1298.42,1176.83 1299.26,1176.26 \n",
       "  1300.11,1175.63 1300.95,1174.95 1301.8,1174.22 1302.64,1173.48 1303.49,1172.74 1304.33,1171.98 1305.18,1171.19 1306.03,1170.34 1306.87,1169.44 1307.72,1168.5 \n",
       "  1308.56,1167.55 1309.41,1166.57 1310.25,1165.56 1311.1,1164.5 1311.94,1163.38 1312.79,1162.2 1313.63,1160.98 1314.48,1159.75 1315.33,1158.52 1316.17,1157.32 \n",
       "  1317.02,1156.13 1317.86,1154.97 1318.71,1153.81 1319.55,1152.65 1320.4,1151.5 1321.24,1150.35 1322.09,1149.22 1322.94,1148.11 1323.78,1147.03 1324.63,1145.98 \n",
       "  1325.47,1144.96 1326.32,1143.97 1327.16,1142.99 1328.01,1142 1328.85,1141.01 1329.7,1139.99 1330.54,1138.94 1331.39,1137.86 1332.24,1136.76 1333.08,1135.65 \n",
       "  1333.93,1134.54 1334.77,1133.44 1335.62,1132.37 1336.46,1131.32 1337.31,1130.3 1338.15,1129.32 1339,1128.38 1339.85,1127.47 1340.69,1126.59 1341.54,1125.74 \n",
       "  1342.38,1124.92 1343.23,1124.12 1344.07,1123.33 1344.92,1122.57 1345.76,1121.82 1346.61,1121.09 1347.45,1120.37 1348.3,1119.65 1349.15,1118.95 1349.99,1118.24 \n",
       "  1350.84,1117.54 1351.68,1116.83 1352.53,1116.11 1353.37,1115.38 1354.22,1114.63 1355.06,1113.85 1355.91,1113.05 1356.76,1112.21 1357.6,1111.35 1358.45,1110.45 \n",
       "  1359.29,1109.52 1360.14,1108.57 1360.98,1107.58 1361.83,1106.57 1362.67,1105.54 1363.52,1104.5 1364.36,1103.45 1365.21,1102.39 1366.06,1101.33 1366.9,1100.28 \n",
       "  1367.75,1099.23 1368.59,1098.2 1369.44,1097.18 1370.28,1096.18 1371.13,1095.19 1371.97,1094.23 1372.82,1093.28 1373.67,1092.35 1374.51,1091.45 1375.36,1090.56 \n",
       "  1376.2,1089.69 1377.05,1088.84 1377.89,1088 1378.74,1087.19 1379.58,1086.39 1380.43,1085.6 1381.27,1084.83 1382.12,1084.07 1382.97,1083.33 1383.81,1082.59 \n",
       "  1384.66,1081.86 1385.5,1081.13 1386.35,1080.41 1387.19,1079.69 1388.04,1078.97 1388.88,1078.24 1389.73,1077.51 1390.58,1076.78 1391.42,1076.03 1392.27,1075.27 \n",
       "  1393.11,1074.51 1393.96,1073.73 1394.8,1072.94 1395.65,1072.13 1396.49,1071.31 1397.34,1070.48 1398.18,1069.64 1399.03,1068.78 1399.88,1067.92 1400.72,1067.04 \n",
       "  1401.57,1066.16 1402.41,1065.27 1403.26,1064.37 1404.1,1063.47 1404.95,1062.57 1405.79,1061.67 1406.64,1060.76 1407.49,1059.85 1408.33,1058.94 1409.18,1058.04 \n",
       "  1410.02,1057.13 1410.87,1056.22 1411.71,1055.31 1412.56,1054.41 1413.4,1053.5 1414.25,1052.6 1415.09,1051.69 1415.94,1050.79 1416.79,1049.88 1417.63,1048.98 \n",
       "  1418.48,1048.07 1419.32,1047.16 1420.17,1046.25 1421.01,1045.34 1421.86,1044.42 1422.7,1043.51 1423.55,1042.59 1424.4,1041.67 1425.24,1040.75 1426.09,1039.83 \n",
       "  1426.93,1038.9 1427.78,1037.98 1428.62,1037.05 1429.47,1036.12 1430.31,1035.19 1431.16,1034.26 1432,1033.33 1432.85,1032.41 1433.7,1031.48 1434.54,1030.55 \n",
       "  1435.39,1029.63 1436.23,1028.7 1437.08,1027.78 1437.92,1026.86 1438.77,1025.94 1439.61,1025.03 1440.46,1024.11 1441.31,1023.2 1442.15,1022.3 1443,1021.39 \n",
       "  1443.84,1020.49 1444.69,1019.59 1445.53,1018.7 1446.38,1017.81 1447.22,1016.92 1448.07,1016.04 1448.91,1015.16 1449.76,1014.28 1450.61,1013.4 1451.45,1012.53 \n",
       "  1452.3,1011.66 1453.14,1010.8 1453.99,1009.93 1454.83,1009.07 1455.68,1008.21 1456.52,1007.35 1457.37,1006.5 1458.22,1005.64 1459.06,1004.79 1459.91,1003.94 \n",
       "  1460.75,1003.09 1461.6,1002.24 1462.44,1001.39 1463.29,1000.54 1464.13,999.687 1464.98,998.837 1465.82,997.987 1466.67,997.136 1467.52,996.284 1468.36,995.431 \n",
       "  1469.21,994.577 1470.05,993.721 1470.9,992.863 1471.74,992.004 1472.59,991.142 1473.43,990.278 1474.28,989.411 1475.13,988.542 1475.97,987.669 1476.82,986.795 \n",
       "  1477.66,985.917 1478.51,985.036 1479.35,984.153 1480.2,983.266 1481.04,982.376 1481.89,981.484 1482.73,980.589 1483.58,979.691 1484.43,978.791 1485.27,977.888 \n",
       "  1486.12,976.984 1486.96,976.077 1487.81,975.168 1488.65,974.258 1489.5,973.346 1490.34,972.433 1491.19,971.519 1492.04,970.604 1492.88,969.689 1493.73,968.773 \n",
       "  1494.57,967.857 1495.42,966.942 1496.26,966.027 1497.11,965.112 1497.95,964.198 1498.8,963.284 1499.64,962.372 1500.49,961.461 1501.34,960.551 1502.18,959.642 \n",
       "  1503.03,958.735 1503.87,957.83 1504.72,956.925 1505.56,956.023 1506.41,955.122 1507.25,954.223 1508.1,953.326 1508.95,952.43 1509.79,951.536 1510.64,950.644 \n",
       "  1511.48,949.753 1512.33,948.865 1513.17,947.978 1514.02,947.092 1514.86,946.208 1515.71,945.326 1516.56,944.446 1517.4,943.567 1518.25,942.689 1519.09,941.813 \n",
       "  1519.94,940.939 1520.78,940.066 1521.63,939.194 1522.47,938.324 1523.32,937.455 1524.16,936.588 1525.01,935.721 1525.86,934.856 1526.7,933.993 1527.55,933.13 \n",
       "  1528.39,932.269 1529.24,931.409 1530.08,930.55 1530.93,929.692 1531.77,928.835 1532.62,927.979 1533.47,927.124 1534.31,926.27 1535.16,925.417 1536,924.565 \n",
       "  1536.85,923.713 1537.69,922.862 1538.54,922.012 1539.38,921.163 1540.23,920.314 1541.07,919.466 1541.92,918.618 1542.77,917.771 1543.61,916.924 1544.46,916.078 \n",
       "  1545.3,915.231 1546.15,914.385 1546.99,913.539 1547.84,912.693 1548.68,911.847 1549.53,911.001 1550.38,910.155 1551.22,909.309 1552.07,908.462 1552.91,907.616 \n",
       "  1553.76,906.768 1554.6,905.921 1555.45,905.073 1556.29,904.224 1557.14,903.375 1557.98,902.526 1558.83,901.675 1559.68,900.824 1560.52,899.972 1561.37,899.12 \n",
       "  1562.21,898.267 1563.06,897.413 1563.9,896.558 1564.75,895.702 1565.59,894.845 1566.44,893.988 1567.29,893.129 1568.13,892.27 1568.98,891.41 1569.82,890.549 \n",
       "  1570.67,889.687 1571.51,888.824 1572.36,887.96 1573.2,887.096 1574.05,886.23 1574.89,885.364 1575.74,884.498 1576.59,883.63 1577.43,882.762 1578.28,881.893 \n",
       "  1579.12,881.023 1579.97,880.153 1580.81,879.283 1581.66,878.411 1582.5,877.54 1583.35,876.668 1584.2,875.795 1585.04,874.923 1585.89,874.05 1586.73,873.176 \n",
       "  1587.58,872.303 1588.42,871.429 1589.27,870.556 1590.11,869.682 1590.96,868.808 1591.8,867.934 1592.65,867.061 1593.5,866.187 1594.34,865.314 1595.19,864.441 \n",
       "  1596.03,863.567 1596.88,862.695 1597.72,861.822 1598.57,860.95 1599.41,860.078 1600.26,859.207 1601.11,858.336 1601.95,857.465 1602.8,856.594 1603.64,855.725 \n",
       "  1604.49,854.855 1605.33,853.986 1606.18,853.118 1607.02,852.25 1607.87,851.382 1608.71,850.515 1609.56,849.649 1610.41,848.782 1611.25,847.917 1612.1,847.052 \n",
       "  1612.94,846.187 1613.79,845.323 1614.63,844.46 1615.48,843.596 1616.32,842.734 1617.17,841.872 1618.02,841.01 1618.86,840.148 1619.71,839.288 1620.55,838.427 \n",
       "  1621.4,837.567 1622.24,836.707 1623.09,835.848 1623.93,834.989 1624.78,834.131 1625.62,833.273 1626.47,832.415 1627.32,831.557 1628.16,830.7 1629.01,829.843 \n",
       "  1629.85,828.987 1630.7,828.13 1631.54,827.274 1632.39,826.418 1633.23,825.563 1634.08,824.708 1634.93,823.852 1635.77,822.998 1636.62,822.143 1637.46,821.289 \n",
       "  1638.31,820.434 1639.15,819.58 1640,818.726 1640.84,817.873 1641.69,817.019 1642.53,816.166 1643.38,815.313 1644.23,814.46 1645.07,813.607 1645.92,812.754 \n",
       "  1646.76,811.901 1647.61,811.049 1648.45,810.197 1649.3,809.344 1650.14,808.492 1650.99,807.64 1651.84,806.788 1652.68,805.937 1653.53,805.085 1654.37,804.234 \n",
       "  1655.22,803.382 1656.06,802.531 1656.91,801.68 1657.75,800.829 1658.6,799.978 1659.44,799.127 1660.29,798.277 1661.14,797.426 1661.98,796.576 1662.83,795.726 \n",
       "  1663.67,794.875 1664.52,794.025 1665.36,793.176 1666.21,792.326 1667.05,791.476 1667.9,790.627 1668.75,789.778 1669.59,788.928 1670.44,788.079 1671.28,787.231 \n",
       "  1672.13,786.382 1672.97,785.533 1673.82,784.685 1674.66,783.837 1675.51,782.989 1676.35,782.141 1677.2,781.293 1678.05,780.446 1678.89,779.598 1679.74,778.751 \n",
       "  1680.58,777.904 1681.43,777.058 1682.27,776.211 1683.12,775.365 1683.96,774.519 1684.81,773.673 1685.66,772.827 1686.5,771.981 1687.35,771.136 1688.19,770.291 \n",
       "  1689.04,769.446 1689.88,768.602 1690.73,767.757 1691.57,766.913 1692.42,766.07 1693.26,765.226 1694.11,764.383 1694.96,763.54 1695.8,762.697 1696.65,761.855 \n",
       "  1697.49,761.012 1698.34,760.17 1699.18,759.329 1700.03,758.488 1700.87,757.646 1701.72,756.806 1702.57,755.965 1703.41,755.125 1704.26,754.285 1705.1,753.446 \n",
       "  1705.95,752.607 1706.79,751.768 1707.64,750.929 1708.48,750.091 1709.33,749.253 1710.17,748.416 1711.02,747.578 1711.87,746.741 1712.71,745.905 1713.56,745.069 \n",
       "  1714.4,744.233 1715.25,743.397 1716.09,742.562 1716.94,741.727 1717.78,740.892 1718.63,740.058 1719.48,739.224 1720.32,738.39 1721.17,737.557 1722.01,736.724 \n",
       "  1722.86,735.891 1723.7,735.059 1724.55,734.227 1725.39,733.395 1726.24,732.564 1727.08,731.732 1727.93,730.902 1728.78,730.071 1729.62,729.241 1730.47,728.411 \n",
       "  1731.31,727.581 1732.16,726.751 1733,725.922 1733.85,725.093 1734.69,724.264 1735.54,723.435 1736.39,722.606 1737.23,721.778 1738.08,720.95 1738.92,720.122 \n",
       "  1739.77,719.294 1740.61,718.466 1741.46,717.639 1742.3,716.811 1743.15,715.984 1743.99,715.156 1744.84,714.329 1745.69,713.502 1746.53,712.674 1747.38,711.847 \n",
       "  1748.22,711.02 1749.07,710.192 1749.91,709.365 1750.76,708.537 1751.6,707.71 1752.45,706.882 1753.3,706.054 1754.14,705.227 1754.99,704.398 1755.83,703.57 \n",
       "  1756.68,702.742 1757.52,701.913 1758.37,701.084 1759.21,700.255 1760.06,699.425 1760.9,698.595 1761.75,697.765 1762.6,696.935 1763.44,696.104 1764.29,695.273 \n",
       "  1765.13,694.441 1765.98,693.609 1766.82,692.776 1767.67,691.943 1768.51,691.11 1769.36,690.276 1770.21,689.441 1771.05,688.606 1771.9,687.771 1772.74,686.935 \n",
       "  1773.59,686.098 1774.43,685.261 1775.28,684.423 1776.12,683.584 1776.97,682.745 1777.81,681.905 1778.66,681.065 1779.51,680.223 1780.35,679.381 1781.2,678.539 \n",
       "  1782.04,677.696 1782.89,676.852 1783.73,676.007 1784.58,675.161 1785.42,674.315 1786.27,673.468 1787.12,672.62 1787.96,671.772 1788.81,670.923 1789.65,670.073 \n",
       "  1790.5,669.222 1791.34,668.37 1792.19,667.518 1793.03,666.665 1793.88,665.811 1794.72,664.956 1795.57,664.101 1796.42,663.245 1797.26,662.388 1798.11,661.53 \n",
       "  1798.95,660.672 1799.8,659.813 1800.64,658.953 1801.49,658.092 1802.33,657.231 1803.18,656.369 1804.03,655.506 1804.87,654.643 1805.72,653.779 1806.56,652.914 \n",
       "  1807.41,652.049 1808.25,651.183 1809.1,650.316 1809.94,649.449 1810.79,648.581 1811.64,647.712 1812.48,646.843 1813.33,645.974 1814.17,645.104 1815.02,644.233 \n",
       "  1815.86,643.362 1816.71,642.49 1817.55,641.618 1818.4,640.745 1819.24,639.872 1820.09,638.999 1820.94,638.125 1821.78,637.25 1822.63,636.376 1823.47,635.501 \n",
       "  1824.32,634.625 1825.16,633.749 1826.01,632.873 1826.85,631.997 1827.7,631.12 1828.55,630.243 1829.39,629.366 1830.24,628.489 1831.08,627.611 1831.93,626.734 \n",
       "  1832.77,625.856 1833.62,624.978 1834.46,624.1 1835.31,623.221 1836.15,622.343 1837,621.464 1837.85,620.586 1838.69,619.707 1839.54,618.828 1840.38,617.949 \n",
       "  1841.23,617.071 1842.07,616.192 1842.92,615.313 1843.76,614.434 1844.61,613.555 1845.46,612.677 1846.3,611.798 1847.15,610.92 1847.99,610.041 1848.84,609.163 \n",
       "  1849.68,608.284 1850.53,607.406 1851.37,606.528 1852.22,605.65 1853.06,604.772 1853.91,603.895 1854.76,603.017 1855.6,602.14 1856.45,601.263 1857.29,600.386 \n",
       "  1858.14,599.509 1858.98,598.632 1859.83,597.756 1860.67,596.88 1861.52,596.004 1862.37,595.128 1863.21,594.252 1864.06,593.377 1864.9,592.502 1865.75,591.627 \n",
       "  1866.59,590.752 1867.44,589.878 1868.28,589.004 1869.13,588.13 1869.97,587.257 1870.82,586.383 1871.67,585.51 1872.51,584.637 1873.36,583.765 1874.2,582.893 \n",
       "  1875.05,582.021 1875.89,581.149 1876.74,580.277 1877.58,579.406 1878.43,578.535 1879.28,577.665 1880.12,576.794 1880.97,575.924 1881.81,575.054 1882.66,574.185 \n",
       "  1883.5,573.316 1884.35,572.447 1885.19,571.578 1886.04,570.71 1886.88,569.842 1887.73,568.974 1888.58,568.106 1889.42,567.239 1890.27,566.372 1891.11,565.505 \n",
       "  1891.96,564.639 1892.8,563.772 1893.65,562.906 1894.49,562.041 1895.34,561.175 1896.19,560.31 1897.03,559.445 1897.88,558.581 1898.72,557.716 1899.57,556.852 \n",
       "  1900.41,555.988 1901.26,555.125 1902.1,554.262 1902.95,553.398 1903.79,552.536 1904.64,551.673 1905.49,550.811 1906.33,549.949 1907.18,549.087 1908.02,548.225 \n",
       "  1908.87,547.364 1909.71,546.503 1910.56,545.642 1911.4,544.781 1912.25,543.92 1913.1,543.06 1913.94,542.2 1914.79,541.34 1915.63,540.481 1916.48,539.621 \n",
       "  1917.32,538.762 1918.17,537.903 1919.01,537.044 1919.86,536.186 1920.7,535.327 1921.55,534.469 1922.4,533.611 1923.24,532.753 1924.09,531.896 1924.93,531.038 \n",
       "  1925.78,530.181 1926.62,529.324 1927.47,528.467 1928.31,527.61 1929.16,526.754 1930.01,525.897 1930.85,525.041 1931.7,524.185 1932.54,523.329 1933.39,522.473 \n",
       "  1934.23,521.618 1935.08,520.762 1935.92,519.907 1936.77,519.052 1937.61,518.197 1938.46,517.342 1939.31,516.487 1940.15,515.632 1941,514.778 1941.84,513.924 \n",
       "  1942.69,513.069 1943.53,512.215 1944.38,511.361 1945.22,510.508 1946.07,509.654 1946.92,508.8 1947.76,507.947 1948.61,507.093 1949.45,506.24 1950.3,505.387 \n",
       "  1951.14,504.534 1951.99,503.681 1952.83,502.828 1953.68,501.975 1954.52,501.123 1955.37,500.27 1956.22,499.417 1957.06,498.565 1957.91,497.713 1958.75,496.86 \n",
       "  1959.6,496.008 1960.44,495.156 1961.29,494.304 1962.13,493.452 1962.98,492.6 1963.83,491.749 1964.67,490.897 1965.52,490.045 1966.36,489.193 1967.21,488.342 \n",
       "  1968.05,487.49 1968.9,486.639 1969.74,485.787 1970.59,484.936 1971.43,484.085 1972.28,483.234 1973.13,482.382 1973.97,481.531 1974.82,480.68 1975.66,479.829 \n",
       "  1976.51,478.978 1977.35,478.127 1978.2,477.276 1979.04,476.425 1979.89,475.574 1980.74,474.723 1981.58,473.872 1982.43,473.021 1983.27,472.17 1984.12,471.319 \n",
       "  1984.96,470.468 1985.81,469.618 1986.65,468.767 1987.5,467.916 1988.34,467.065 1989.19,466.214 1990.04,465.364 1990.88,464.513 1991.73,463.662 1992.57,462.811 \n",
       "  1993.42,461.961 1994.26,461.11 1995.11,460.259 1995.95,459.408 1996.8,458.557 1997.65,457.706 1998.49,456.856 1999.34,456.005 2000.18,455.154 2001.03,454.303 \n",
       "  2001.87,453.452 2002.72,452.601 2003.56,451.75 2004.41,450.899 2005.25,450.048 2006.1,449.197 2006.95,448.346 2007.79,447.495 2008.64,446.644 2009.48,445.793 \n",
       "  2010.33,444.941 2011.17,444.09 2012.02,443.239 2012.86,442.388 2013.71,441.536 2014.56,440.685 2015.4,439.833 2016.25,438.982 2017.09,438.13 2017.94,437.279 \n",
       "  2018.78,436.427 2019.63,435.575 2020.47,434.724 2021.32,433.872 2022.16,433.02 2023.01,432.168 2023.86,431.316 2024.7,430.464 2025.55,429.612 2026.39,428.76 \n",
       "  2027.24,427.908 2028.08,427.056 2028.93,426.203 2029.77,425.351 2030.62,424.498 2031.47,423.646 2032.31,422.793 2033.16,421.941 2034,421.088 2034.85,420.235 \n",
       "  2035.69,419.382 2036.54,418.529 2037.38,417.676 2038.23,416.823 2039.07,415.97 2039.92,415.117 2040.77,414.264 2041.61,413.41 2042.46,412.557 2043.3,411.704 \n",
       "  2044.15,410.85 2044.99,409.996 2045.84,409.143 2046.68,408.289 2047.53,407.435 2048.38,406.581 2049.22,405.727 2050.07,404.873 2050.91,404.019 2051.76,403.164 \n",
       "  2052.6,402.31 2053.45,401.455 2054.29,400.601 2055.14,399.746 2055.98,398.892 2056.83,398.037 2057.68,397.182 2058.52,396.327 2059.37,395.472 2060.21,394.617 \n",
       "  2061.06,393.762 2061.9,392.906 2062.75,392.051 2063.59,391.196 2064.44,390.34 2065.29,389.485 2066.13,388.629 2066.98,387.773 2067.82,386.917 2068.67,386.061 \n",
       "  2069.51,385.205 2070.36,384.349 2071.2,383.493 2072.05,382.637 2072.89,381.781 2073.74,380.924 2074.59,380.068 2075.43,379.211 2076.28,378.354 2077.12,377.498 \n",
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      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#plot the objective function\n",
    "xx= -100:.1:150\n",
    "yy= [[i] for i in xx]\n",
    "plot(xx, obj.(yy),lw = 2, title = \"Prox GD visualization\", label = \"obj function\")\n",
    "# check the conv of proxGD\n",
    "lrt = i-> .1 # step length\n",
    "a,b = proxGD([100.], obj, esl_grd, lrt, sfth, λ)\n",
    "plot!(a, [b], mark = 3, label = \"Last iteration\",  legend = :bottomright)\n",
    "# plot!(xx, 100 .*(esl.(yy).-0.8), lw  = 2,label = \"esl loss\", title = \"ESL empirical loss\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "korean-google",
   "metadata": {},
   "source": [
    "### 2. Tuning parameter selection\n",
    "\n",
    "#### 2.1. $\\tau_{nj}$ in the penalty term\n",
    "\n",
    "The choice of $\\tau_{nj}$ is simply proposed as $\\tau_{nj} = \\log(n)/n$, which ensures the oracal property for the variable selection. For notational convenience, we drop the index $j$ by writing $\\tau_n = \\log(n)$ for the rest of document. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "caroline-equity",
   "metadata": {},
   "source": [
    "#### 2.2. $\\gamma$ in the loss function\n",
    "\n",
    "**The influence of $\\gamma$ to the optimization**\n",
    "\n",
    "The selection of $\\gamma$ is crucial in terms of robustness. In previous section, we find that the change of $\\gamma$ leads to a drastic variation of the optimization land scape f the ESL--- larger $\\gamma$ leads to a smoother $\\ell^*$ , where $\\ell^*$ is the empirical ESL loss without regularizer. The following figures displayes the shape of $\\ell^*$ under different $\\gamma$ with the previous simulated data. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "cloudy-boston",
   "metadata": {},
   "outputs": [],
   "source": [
    "plt = plot( xlim = (-100, 150), title = \"ℓ* under different γ\")\n",
    "xx= -100:.1:150\n",
    "yy= [[i] for i in xx]\n",
    "\n",
    "for γ = [1. 5. 20.]\n",
    "    esl = x -> ESL(x, dat, γ)\n",
    "    plot!( xx, esl.(yy),lw = 3, label = \"γ = $γ\", legend = :bottomleft)\n",
    "end \n",
    "\n",
    "#savefig(\"tex/figures/lstar.png\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "structured-french",
   "metadata": {},
   "source": [
    "**How to select $\\gamma$**\n",
    "\n",
    "The author propose a data-dependent procedure to **learn** the valule of $\\gamma$, which includes three steps:\n",
    "1. Find pseudo outliers $D_m$;\n",
    "\n",
    "2. Learn $\\gamma$ by optimizing the **determinant of the asymptotic covariance matrix** of $\\hat{\\beta}_n$---$\\hat{V}(\\gamma)=\\left\\{\\hat{I}_{1}\\left(\\hat{\\boldsymbol{\\beta}}_{n}\\right)\\right\\}^{-1} \\tilde{\\Sigma}_{2}\\left\\{\\hat{I}_{1}\\left(\\hat{\\boldsymbol{\\beta}}_{n}\\right)\\right\\}^{-1}$---within the range \n",
    "$$G = \\left\\{\\gamma: \\frac{2m}{n} + \\frac{2}{n}\\sum_{ i =m+ 1}^n \\phi_\\gamma(r_i(\\beta_n))\\leq 1\\right\\}, $$ where\n",
    "$$ \\begin{aligned}\n",
    "\\hat{I}_{1}\\left(\\hat{\\boldsymbol{\\beta}}_{n}\\right) &=\\frac{2}{\\gamma}\\left\\{\\frac{1}{n} \\sum_{i=1}^{n} \\exp \\left(-r_{i}^{2}\\left(\\hat{\\boldsymbol{\\beta}}_{n}\\right) / \\gamma\\right)\\left(\\frac{2 r_{i}^{2}\\left(\\hat{\\boldsymbol{\\beta}}_{n}\\right)}{\\gamma}-1\\right)\\right\\}\\left(\\frac{1}{n} \\sum_{i=1}^{n} \\mathbf{x}_{i} \\mathbf{x}_{i}^{T}\\right) \\\\\n",
    "\\tilde{\\Sigma}_{2} &=\\operatorname{cov}\\left\\{\\exp \\left(-r_{1}^{2}\\left(\\hat{\\boldsymbol{\\beta}}_{n}\\right) / \\gamma\\right) \\frac{2 r_{1}\\left(\\hat{\\boldsymbol{\\beta}}_{n}\\right)}{\\gamma} \\mathbf{x}_{1}, \\cdots, \\exp \\left(-r_{n}^{2}\\left(\\hat{\\boldsymbol{\\beta}}_{n}\\right) / \\gamma\\right) \\frac{2 r_{n}\\left(\\hat{\\boldsymbol{\\beta}}_{n}\\right)}{\\gamma} \\mathbf{x}_{n}\\right\\}\n",
    "\\end{aligned}.$$\n",
    " The key notice here is that in step 2, the target under minimization **depends on the value of** $\\beta_n$. \n",
    "\n",
    "3. Update $\\beta_n$ by optimizing the objective function (using proximal GD algorithm presented above) with $\\tau_j = \\log(n)/n$ and $\\gamma$ obtained in step 2.\n",
    "\n",
    "Repeat Steps 1-3 until $\\beta_n$ and $\\gamma$ converge. The intial value of $\\beta_n$ is chosen as the MM-estimator $\\tilde{\\beta_n}$, which is used in the adaptive LASSO penelty. The key design of this procedure is the Step 2, where the choice of $\\gamma$ aims to endow $\\beta_n$  with the orcal property (by minimizing the asymptotic cov) and high breakdown point (by searching $\\gamma$ within **some range**).\n",
    "  "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cardiac-guinea",
   "metadata": {},
   "source": [
    "However, there is no smart algorithm that efficiently solves the constrained optimization problem in Step2.  In fact, as described in the paper, the way of finding the $\\gamma$ in Step 2 is literally done by grid search (see their Figure 1). It is acceptable in practice since it is a 1-d optimization problem and the complexity of evaluating the function is $O(d^3)$ (involving a matrix inverse)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "amber-strength",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "gamma_plot (generic function with 1 method)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "######################################\n",
    "#####functions for step 2#############\n",
    "######################################\n",
    "\n",
    "\n",
    "# find pseudo outlier\n",
    "function find_pseudo_out(beta::Vector,dat::Array{Float64, 2})\n",
    "    X = dat[:, 2:end]\n",
    "    Y = dat[:,1]\n",
    "    n = size(dat)[1]\n",
    "    res = X*beta - Y\n",
    "    Sn = 1.4826 * median( abs.(res .- median(res)))\n",
    "    ind = findall(x-> abs(x)≥ 2.5Sn, res)\n",
    "    D_out = dat[ind, 1:end]\n",
    "    D_good = dat[1:end .∉ [ind], 1:end]\n",
    "    return D_out, D_good \n",
    "end\n",
    "\n",
    "# use logdet for numerical stability\n",
    "function logdet_V( beta::Vector,dat::Array{Float64, 2}, gamma::Float64)\n",
    "    X = dat[:, 2:end]\n",
    "    Y = dat[:,1]\n",
    "    n = size(dat)[1]\n",
    "    res = X*beta - Y\n",
    "    #I^{-1}\n",
    "    a = 2/gamma * mean(exp.(-res.^2/gamma).*(2*res.^2/gamma.-1))\n",
    "    cov_x = X'*X/n\n",
    "    inv_I = inv(cov_x)/a\n",
    "    #Σ\n",
    "    D = (2res/gamma) .* exp.(-res.^2/gamma) .* X\n",
    "    st_D = D .- mean(D, dims = 1)\n",
    "    Σ = st_D'*st_D/n\n",
    "    return logdet(inv_I*Σ*inv_I)\n",
    "end\n",
    "\n",
    "#find range G\n",
    "function ξ( beta::Vector,dat_good::Array{Float64, 2}, gamma::Float64, n)\n",
    "    m = n - size(dat_good)[1]\n",
    "    return 2m/n + 2/n*(n-m)*ESL(beta, dat_good, gamma)\n",
    "end\n",
    "\n",
    "function get_G(beta::Vector,dat_good::Array{Float64, 2}, gm::Vector, n)\n",
    "    xxi = gm -> ξ(beta,dat_good, gm, n) \n",
    "    ξ_val = xxi.(gm)\n",
    "    ind = findall(x -> x ≤ 1, ξ_val)\n",
    "    G = gm[ind]\n",
    "    return G, ξ_val[ind]\n",
    "end\n",
    "\n",
    "# find the optimal gamma\n",
    "function search_gamma(beta::Vector, dat::Array{Float64, 2}, γ_seq::Vector) \n",
    "    n = size(dat)[1]\n",
    "    _ ,dat_good = find_pseudo_out(beta, dat)  #find pseudo outliers\n",
    "    xxi = gm -> ξ(beta,dat_good, gm, n)\n",
    "    G, _ = get_G(beta, dat_good, γ_seq, n) # find G \n",
    "    dv = gm -> logdet_V(beta,dat, gm)\n",
    "    gamma_opt = G[argmin(dv.(G))]  #find optimal γ\n",
    "    return gamma_opt\n",
    "end\n",
    "\n",
    "\n",
    "# making plots for the G and optimal gamma\n",
    "function gamma_plot(beta::Vector, dat::Array{Float64, 2}, gm::Vector)\n",
    "     n = size(dat)[1]\n",
    "    _ ,dat_good = find_pseudo_out(beta, dat)  #find pseudo outliers\n",
    "    \n",
    "    #p1 plot G and ξ val\n",
    "    xxi = gm -> ξ(beta,dat_good, gm, n)\n",
    "    p1 = plot(gm, xxi.(gm), ylim =(0,1.2) , lw = 3, xlabel = \"γ\", ylabel= \"ξ(γ)\" , title  = \"Find G\", label = \"\")\n",
    "    G, xi = get_G(beta, dat_good, Array(gm), n) # find G and corresponding ξ values\n",
    "    plot!(G, xi, lw=3, label = \" G\")\n",
    "    \n",
    "    #p2 plot logdev and opt γ    \n",
    "    dv = gm -> logdet_V(beta,dat, gm ) \n",
    "    p2 = plot(gm, dv.(gm), lw = 3, xlabel= \"γ\", ylabel= \"logdet(V(γ))\", title = \"Find γ\",label = \"\")\n",
    "    gamma_opt = search_gamma(beta, dat, gm)  #find optimal γ\n",
    "    plot!([gamma_opt], [minimum(dv.(gm))], mark = 5, label = \"optimal γ\")\n",
    "    \n",
    "    # show the plot\n",
    "    plot(p1, p2, layout = (1,2))\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "czech-justice",
   "metadata": {},
   "source": [
    "The way of finding G and $\\gamma$ is just by looking at those plots. The key limitation of Step2 is the automation of the inference procedure---to perform a grid search, we must prespecify a reasonable range, which is obviously problem specific and requires some tuning.   We replicate the Figure 1. of the original paper as follows to illustrate how to find to determine set G and find $\\gamma$ in Step 2. we generate a synthetic dataset with $n = 800$, $d= 8$, $\\beta  = [1.,1.5, 2.,1.,0.,0.,0.,0. ]$. We generate $(X_i)_{i =1}^n$ i.i.d from a multinormal distribution $\\mathcal{N}(0, \\Omega_2) $, where $(i,j)$-th element of $\\Omega_2$ is $0.5^{|i-j|}$. And $Y_i  = X_i^T \\beta + \\epsilon$, where $\\epsilon\\sim \\text{Cauchy}(0,1)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "concrete-panama",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "800×9 Array{Float64,2}:\n",
       "  6.66663    2.26654    1.76371    …   0.732585    0.0533183  -0.628258\n",
       "  3.17446   -0.568045  -0.290432       0.0294616   0.674915   -0.747509\n",
       "  0.134793   0.923698  -0.0678524     -0.329027    0.138088    0.0111555\n",
       "  5.41037   -0.824903   0.565764       1.157       1.69248     0.372737\n",
       " -1.85427    0.620967  -0.318664       0.827126   -0.901563   -1.35518\n",
       "  5.16145    0.746819   0.829235   …  -0.0549827  -1.28709    -0.214125\n",
       " -4.52562   -0.537082  -0.671303      -0.350171   -0.460876    0.627194\n",
       "  5.85796    0.86393    0.723557       1.11474     0.448124    1.36133\n",
       "  7.86234    0.338392   1.59968        0.532641   -0.337809    1.46504\n",
       "  4.44315    0.212861   1.69755        1.01033    -0.427638   -0.188114\n",
       " -3.01251    0.132596  -1.4708     …  -0.969108   -0.382803    0.630117\n",
       " -4.57044   -1.6881    -2.29064        0.199812    1.43047     0.559259\n",
       " -6.9991    -1.29659   -1.01802        0.380041   -0.360525   -0.571099\n",
       "  ⋮                                ⋱                          \n",
       " 13.7966     1.5711     1.8907         1.79122     0.887667   -0.451644\n",
       "  1.11221    0.718644  -0.121195       0.302175    0.833809    1.16917\n",
       " -0.722021  -2.05619   -0.30243    …   1.09662     1.76296     0.179305\n",
       "  6.9648     1.81636    0.706522       0.731417    1.23115     0.0772814\n",
       "  6.37124    1.41525    1.12144        1.26496     0.191028   -0.948206\n",
       "  3.31693    1.50176    0.496673       0.255184    0.776699    0.810546\n",
       " -5.23919   -1.9817    -1.34673        0.669125   -0.795261    0.588071\n",
       " -0.520372   0.558671  -0.0480614  …   0.821946   -0.102941   -1.06214\n",
       " -9.80401   -0.739697   0.0532407      1.32492     3.66052     3.86674\n",
       "  5.34331    1.05355    1.62287       -0.1465     -0.765123   -0.0226485\n",
       " -2.62941   -0.437845   0.15889        0.531542    0.82632    -0.360996\n",
       " -1.83953    0.26488   -0.355048       1.09868     2.0824     -0.112896"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Random.seed!(202102)\n",
    "# simulate dataset as in the paper\n",
    "Omega2 = 0.5.^ (abs.(Array(1:8) .- Array(1:8)'))\n",
    "# isposdef!(Omega2)\n",
    "n = 800\n",
    "L  = cholesky(Omega2)\n",
    "X =( L.L * randn(8,n))'\n",
    "beta_true  = [1.,1.5, 2.,1.,0.,0.,0.,0. ]\n",
    "Y  = X*beta_true + rand(Cauchy(),n) \n",
    "Dat = hcat(Y, X)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "functional-margin",
   "metadata": {},
   "outputs": [
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       "  1504.62,1295.88 1510.53,1312.18 1516.44,1325.35 1522.35,1336.09 1528.27,1344.9 1534.18,1352.16 1540.09,1358.18 1546,1363.18 1551.91,1367.35 1557.82,1370.83 \n",
       "  1563.74,1373.75 1569.65,1376.18 1575.56,1378.22 1581.47,1379.92 1587.38,1381.34 1593.29,1382.51 1599.21,1383.47 1605.12,1384.26 1611.03,1384.89 1616.94,1385.39 \n",
       "  1622.85,1385.77 1628.76,1386.05 1634.67,1386.24 1640.59,1386.35 1646.5,1386.4 1652.41,1386.38 1658.32,1386.3 1664.23,1386.17 1670.14,1386 1676.06,1385.79 \n",
       "  1681.97,1385.54 1687.88,1385.26 1693.79,1384.95 1699.7,1384.62 1705.61,1384.25 1711.53,1383.86 1717.44,1383.46 1723.35,1383.03 1729.26,1382.58 1735.17,1382.12 \n",
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       "  1800.2,1376.23 1806.11,1375.64 1812.02,1375.04 1817.93,1374.44 1823.85,1373.83 1829.76,1373.22 1835.67,1372.6 1841.58,1371.98 1847.49,1371.35 1853.4,1370.72 \n",
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       "  2273.13,1325.14 2279.04,1324.54 2284.95,1323.93 2290.86,1323.33 2296.77,1322.73 2302.68,1322.12 2308.6,1321.53 2314.51,1320.93 2320.42,1320.33 2326.33,1319.73 \n",
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      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gm = 0.1:0.1:15\n",
    "gamma_plot(beta_true, Dat,Array(gm))\n",
    "# savefig(\"tex/figures/gamma_plot.png\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "theoretical-coast",
   "metadata": {},
   "source": [
    "### Complete inference algorithm \n",
    "\n",
    "Now we provide the code for the complete procedure of ESL-LASSO inference, which uses the complicated adaptive optimization techiques to find the crucial parameter $\\gamma$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "presidential-utility",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "esl_lasso (generic function with 1 method)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "############################################################\n",
    "###The complete procedure (proposed by paper)###############\n",
    "############################################################\n",
    "\n",
    "function esl_lasso( beta0::Array{Float64,1}, dat::Array{Float64, 2}; γ_seq::Array{Float64,1} = Array(.1:.1:50), opt_lrt::Function = i-> .1, opt_iters::Int = 50000, reps::Int = 2)\n",
    "    ### x0: init_estimator (e.g. MM_estimator)\n",
    "    ### dat: dataset s.t. Y = dat[:,1], X = dat[:, 2:end] \n",
    "    ### γ_seq: γ values to perform grid search (e.g. .1:.1:20)\n",
    "    ### opt_alg: proxGD\n",
    "    ### opt_lrt: smaller than Lipschitz\n",
    "    ### opt_iters: niter for proxGD\n",
    "    ### reps: number of reptitions of step1-3\n",
    "    n =  size(dat)[1]\n",
    "    X = dat[:,2:end]\n",
    "    beta = beta0\n",
    "    for k = 1:reps \n",
    "        # find the optimal gamma\n",
    "        gamma_opt = search_gamma(beta, dat, γ_seq)\n",
    "        println(\"γ = $gamma_opt\")\n",
    "        #adaLASSO parameter\n",
    "        lambda = (log(n)/n)./ (abs.(beta) .+ 1e-10)  \n",
    "        \n",
    "        # update beta using proxGD\n",
    "        obj = b-> ℓ(b, dat, gamma_opt, lambda)\n",
    "        esl_grd = b-> ∇ESL(b, dat, gamma_opt)\n",
    "        lip_cont = abs(eigmax(X'*X/n)*2/gamma_opt)\n",
    "        lrt = i-> min(opt_lrt(i), lip_cont)\n",
    "        beta, _ = proxGD(beta, obj, esl_grd, lrt, sfth, lambda, opt_iters)\n",
    "    end\n",
    "        \n",
    "    return beta\n",
    "end\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "sensitive-basis",
   "metadata": {},
   "source": [
    "Below we perform a complete ESL-LASSO inference procedure for the previous synthetic dataset with initial estimates chosen as the MM-estimator. As suggested in the paper, we only repeat the Step1-3 once and the estimates produced is quite close to the true one. In general, with large sample size (around 1000), the ESL-LASSO inference procedure provides prretty accurate estimates. However,  as we try different number of repetitions of Step1-3, we find some issue. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "ecological-riverside",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "γ = 3.8\n",
      "[0.9977869453659487, 1.6104736028691897, 1.9765504278147588, 0.8509747847377364, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9977869453659487, 1.6104736028691897, 1.9765504278147588, 0.8509747847377364, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9977869453659487, 1.6104736028691897, 1.9765504278147588, 0.8509747847377364, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9977869453659487, 1.6104736028691897, 1.9765504278147588, 0.8509747847377364, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9977869453659487, 1.6104736028691897, 1.9765504278147588, 0.8509747847377364, -0.0, -0.0, -0.0, 0.0]\n",
      "γ = 3.3\n",
      "[0.9988964837673584, 1.6151806221044158, 1.9769381285595293, 0.8418929498041625, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9988964837673584, 1.6151806221044158, 1.9769381285595293, 0.8418929498041625, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9988964837673584, 1.6151806221044158, 1.9769381285595293, 0.8418929498041625, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9988964837673584, 1.6151806221044158, 1.9769381285595293, 0.8418929498041625, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9988964837673584, 1.6151806221044158, 1.9769381285595293, 0.8418929498041625, -0.0, -0.0, -0.0, 0.0]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "8-element Array{Float64,1}:\n",
       "  0.9988964837673584\n",
       "  1.6151806221044158\n",
       "  1.9769381285595293\n",
       "  0.8418929498041625\n",
       " -0.0\n",
       " -0.0\n",
       " -0.0\n",
       "  0.0"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# performing esl_lasso inference with above simulated dataset \n",
    "# beta_true  = [1.,1.5, 2.,1.,0.,0.,0.,0. ]\n",
    "opt_lrt = i-> .1\n",
    "#MM estimator (computed from R)\n",
    "MM = [1.013508 ,1.483255 ,2.052222, 1.009798, -0.07373979 ,-0.09211071 ,-0.00702801 ,-0.01657055]\n",
    "esl_lasso(MM, Dat)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "vanilla-treasurer",
   "metadata": {},
   "source": [
    "With the same datasets and initial value, we perform ESL-LASSO inferece again by repeating Step1-3 6 times. As described in the paper, by keep iterating Step1-3, eventually $\\gamma$ and $\\hat{\\beta}_n$ will converge together. However, as shown in the follwoing outputs, the choice of $\\gamma$ and the estiamtes $\\hat{\\beta}_n$ diverge to some wild value, yielding to nonsense estimates. This issue can be reproduced on different datasets and initial values---increasing the repetition of Step1-3 tends to produce worse estimates, especially when the sample size is small.  However, we do find out that with very large sample size (n = 1999), repeating Step1-3 can still maintain accurate estimates, but repeating Step1-3 seems to be useless in this case since the estimates obtianed in the first round is already accurate. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "unable-dragon",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "γ = 3.8\n",
      "[0.9977869453659487, 1.6104736028691897, 1.9765504278147588, 0.8509747847377364, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9977869453659487, 1.6104736028691897, 1.9765504278147588, 0.8509747847377364, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9977869453659487, 1.6104736028691897, 1.9765504278147588, 0.8509747847377364, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9977869453659487, 1.6104736028691897, 1.9765504278147588, 0.8509747847377364, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9977869453659487, 1.6104736028691897, 1.9765504278147588, 0.8509747847377364, -0.0, -0.0, -0.0, 0.0]\n",
      "γ = 3.3\n",
      "[0.9988964837673584, 1.6151806221044158, 1.9769381285595293, 0.8418929498041625, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9988964837673584, 1.6151806221044158, 1.9769381285595293, 0.8418929498041625, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9988964837673584, 1.6151806221044158, 1.9769381285595293, 0.8418929498041625, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9988964837673584, 1.6151806221044158, 1.9769381285595293, 0.8418929498041625, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9988964837673584, 1.6151806221044158, 1.9769381285595293, 0.8418929498041625, -0.0, -0.0, -0.0, 0.0]\n",
      "γ = 3.3\n",
      "[0.9988440334496892, 1.615339001211779, 1.9772862968149372, 0.8411045846123926, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9988440334496892, 1.615339001211779, 1.9772862968149372, 0.8411045846123926, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9988440334496892, 1.615339001211779, 1.9772862968149372, 0.8411045846123926, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9988440334496892, 1.615339001211779, 1.9772862968149372, 0.8411045846123926, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9988440334496892, 1.615339001211779, 1.9772862968149372, 0.8411045846123926, -0.0, -0.0, -0.0, 0.0]\n",
      "γ = 3.3\n",
      "[0.9988343500594301, 1.6153479450088561, 1.9773240427158325, 0.8410326399759992, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9988343500594301, 1.6153479450088561, 1.9773240427158325, 0.8410326399759992, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9988343500594301, 1.6153479450088561, 1.9773240427158325, 0.8410326399759992, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9988343500594301, 1.6153479450088561, 1.9773240427158325, 0.8410326399759992, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9988343500594301, 1.6153479450088561, 1.9773240427158325, 0.8410326399759992, -0.0, -0.0, -0.0, 0.0]\n",
      "γ = 3.3\n",
      "[0.9988332705891819, 1.6153486999009412, 1.977327626158716, 0.8410260072156049, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9988332705891819, 1.6153486999009412, 1.977327626158716, 0.8410260072156049, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9988332705891819, 1.6153486999009412, 1.977327626158716, 0.8410260072156049, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9988332705891819, 1.6153486999009412, 1.977327626158716, 0.8410260072156049, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9988332705891819, 1.6153486999009412, 1.977327626158716, 0.8410260072156049, -0.0, -0.0, -0.0, 0.0]\n",
      "γ = 3.3\n",
      "[0.9988331622354776, 1.61534877154021, 1.9773279583507406, 0.8410253940309074, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9988331622354776, 1.61534877154021, 1.9773279583507406, 0.8410253940309074, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9988331622354776, 1.61534877154021, 1.9773279583507406, 0.8410253940309074, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9988331622354776, 1.61534877154021, 1.9773279583507406, 0.8410253940309074, -0.0, -0.0, -0.0, 0.0]\n",
      "[0.9988331622354776, 1.61534877154021, 1.9773279583507406, 0.8410253940309074, -0.0, -0.0, -0.0, 0.0]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "8-element Array{Float64,1}:\n",
       "  0.9988331622354776\n",
       "  1.61534877154021\n",
       "  1.9773279583507406\n",
       "  0.8410253940309074\n",
       " -0.0\n",
       " -0.0\n",
       " -0.0\n",
       "  0.0"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# repeat Step1-3 5 times\n",
    "esl_lasso(MM, Dat;reps =  6) "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "composed-laundry",
   "metadata": {},
   "source": [
    "Remarks:  After I conduced the full simulation study, the performance of this ESL_LASSO is **shockingly bad**. In fact, there is no hope to reproduce their simulation results and the proposed methods cannot handle the outlier at all. Given that I did not follow the steps described in the paper explicitly (e.g., global quadratic approximation, coordinate descent optimization) though, I could not speculate how they obtain the experiment, the proposed method as good as what claimed in the paper. To be more specific, the key problem of their method is the selection of the $\\gamma$. For the first two simulation datasets described in the paper, the $\\gamma$ selected was abnormlaly large and the resulting estimator is just all $0$. For the third datasets, the variable selection seems to be consistent but the estimator is accurate, even when the sample size gets large (over $1000$) where PENSE performs really well. \n",
    "\n",
    "To illustrate and diagnose the problme of the $\\gamma$ selection, I will include some plots in the final report."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "protecting-slovakia",
   "metadata": {},
   "source": [
    "To recover their method, I came up with my own idea of selecting the partameter $\\gamma$, which is able to signficantly boost the performance of ESL_LASSO and is also much more reliable in contrast to the original method. The detailed procedure is desribed in the next section."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "greenhouse-preview",
   "metadata": {},
   "source": [
    "### New procedure of choosing $\\gamma$ \n",
    "\n",
    "The intuition of our method is from the expression of the loss function\n",
    "$$ 1- \\frac{1}{n}\\sum_{i = 1}^ n \\exp \\left\\{ -(Y_i- x_i^T\\beta)^2/\\gamma\\right\\}  ,$$\n",
    "where the $\\gamma$ serves as a scale parameter that controls the importance of the residual error. Therefore, a naive idea is to use the estimated variance of the residual. However, considering the presence of outliers, estimating the residual vairance with the raw datset is not appropriate since the outlier can introduce a huge influence to the sample variance, leading to an overestimation. Thus, our goal is find a relatively robust variance estimator. \n",
    "\n",
    "The Step 1 (detecting pseudo outliers) of the original paper inspires us that we can first use this step to discard the potential outliers from the original dataset, and estimate the residual variance use the \" washed\" dataset. The detailed steps are described below:\n",
    "\n",
    "1'. Find pseudo outliers $D_m$ and the \"washed\" dataset $D_{n\\setminus m}$;\n",
    "\n",
    "2'. $\\gamma \\gets \\frac{1}{n-m} \\sum_{i = m+1}^n ( Y_i - X_i \\beta )^2$.\n",
    "\n",
    "3'. Obtain $\\hat \\beta$ by solving ESL-LASSO with proximal GD algorithm.\n",
    "\n",
    "Note that Step2' also depend on the current estimates of $\\beta$. For the initial round, we use the initial value of $\\beta$ (MM-estimator) in Step 2' and then repeat Step1'-3' until convergence. In practice, we find that the value of $\\gamma$ and $\\beta$ do not vary much across different rounds, so we propose to just repeat Step1'-2' once. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "radio-certification",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "our_esl_lasso (generic function with 1 method)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#########################################################\n",
    "###The complete procedure (proposed by us)###############\n",
    "#########################################################\n",
    "\n",
    "function our_esl_lasso(beta0::Array{Float64,1}, dat::Array{Float64, 2}; opt_lrt::Function = i-> .1, opt_iters::Int = 50000, reps::Int = 2)\n",
    "    n =  size(dat)[1]\n",
    "    X = dat[:,2:end]\n",
    "    beta = beta0\n",
    "    \n",
    "    for k  = 1:reps\n",
    "        # find pseudo outlier \n",
    "        _, dat_good = find_pseudo_out(beta::Vector,dat::Array{Float64, 2})\n",
    "        \n",
    "        # use the residual variance of \"washed\" dataset as gamma \n",
    "        gamma =sum((dat_good[:,1] - dat_good[:,2:end]*beta).^2)/size(dat_good)[1]\n",
    "        println(\"γ = $gamma\")\n",
    "        \n",
    "        #adaLASSO parameter\n",
    "        lambda = (log(n)/n)./ (abs.(beta) .+ 1e-10)  \n",
    "        \n",
    "        # update beta using proxGD\n",
    "        obj = b-> ℓ(b, dat, gamma, lambda)\n",
    "        esl_grd = b-> ∇ESL(b, dat, gamma)\n",
    "        lip_cont = abs(eigmax(X'*X/n)*2/gamma)\n",
    "        lrt = i-> min(opt_lrt(i), lip_cont)\n",
    "        beta, _ = proxGD(beta, obj, esl_grd, lrt, sfth, lambda, opt_iters)\n",
    "    end\n",
    "\n",
    "    return beta\n",
    "end\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cloudy-indiana",
   "metadata": {},
   "source": [
    "Finally, we perform our version of ESL_LASSO on the same synthetic dataset, and compare with the paper version of ESL_LASSO. Note that the true coefficients is given by $\\beta  = [1.,1.5, 2.,1.,0.,0.,0.,0. ]$, by simply looking at the following outputs, we conclude that our modification is succsessful."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "latest-morgan",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "γ = 1.9723628353858573\n",
      "[1.0053791844171298, 1.6126674426234946, 1.9626192763295387, 0.8576626841872692, -0.0, -0.0, -0.0, 0.0]\n",
      "[1.0053791844171298, 1.6126674426234946, 1.9626192763295387, 0.8576626841872692, -0.0, -0.0, -0.0, 0.0]\n",
      "[1.0053791844171298, 1.6126674426234946, 1.9626192763295387, 0.8576626841872692, -0.0, -0.0, -0.0, 0.0]\n",
      "[1.0053791844171298, 1.6126674426234946, 1.9626192763295387, 0.8576626841872692, -0.0, -0.0, -0.0, 0.0]\n",
      "[1.0053791844171298, 1.6126674426234946, 1.9626192763295387, 0.8576626841872692, -0.0, -0.0, -0.0, 0.0]\n",
      "γ = 1.954794471286978\n",
      "[1.0030704680253917, 1.617114833872204, 1.9648135917459186, 0.84866166875117, -0.0, -0.0, -0.0, 0.0]\n",
      "[1.0030704680253917, 1.617114833872204, 1.9648135917459186, 0.84866166875117, -0.0, -0.0, -0.0, 0.0]\n",
      "[1.0030704680253917, 1.617114833872204, 1.9648135917459186, 0.84866166875117, -0.0, -0.0, -0.0, 0.0]\n",
      "[1.0030704680253917, 1.617114833872204, 1.9648135917459186, 0.84866166875117, -0.0, -0.0, -0.0, 0.0]\n",
      "[1.0030704680253917, 1.617114833872204, 1.9648135917459186, 0.84866166875117, -0.0, -0.0, -0.0, 0.0]\n",
      "γ = 1.955612868101835\n",
      "[1.00289243446513, 1.617310180387177, 1.965131925541069, 0.8479680235223855, -0.0, -0.0, -0.0, 0.0]\n",
      "[1.00289243446513, 1.617310180387177, 1.965131925541069, 0.8479680235223855, -0.0, -0.0, -0.0, 0.0]\n",
      "[1.00289243446513, 1.617310180387177, 1.965131925541069, 0.8479680235223855, -0.0, -0.0, -0.0, 0.0]\n",
      "[1.00289243446513, 1.617310180387177, 1.965131925541069, 0.8479680235223855, -0.0, -0.0, -0.0, 0.0]\n",
      "[1.00289243446513, 1.617310180387177, 1.965131925541069, 0.8479680235223855, -0.0, -0.0, -0.0, 0.0]\n",
      "γ = 1.9556829787886938\n",
      "[1.0028797650442562, 1.6173218274864902, 1.965159637176434, 0.84791334425513, -0.0, -0.0, -0.0, 0.0]\n",
      "[1.0028797650442562, 1.6173218274864902, 1.965159637176434, 0.84791334425513, -0.0, -0.0, -0.0, 0.0]\n",
      "[1.0028797650442562, 1.6173218274864902, 1.965159637176434, 0.84791334425513, -0.0, -0.0, -0.0, 0.0]\n",
      "[1.0028797650442562, 1.6173218274864902, 1.965159637176434, 0.84791334425513, -0.0, -0.0, -0.0, 0.0]\n",
      "[1.0028797650442562, 1.6173218274864902, 1.965159637176434, 0.84791334425513, -0.0, -0.0, -0.0, 0.0]\n",
      "γ = 1.955688637448828\n",
      "[1.002878844834052, 1.6173226384798582, 1.9651618847124537, 0.8479090291880539, -0.0, -0.0, -0.0, 0.0]\n",
      "[1.002878844834052, 1.6173226384798582, 1.9651618847124537, 0.8479090291880539, -0.0, -0.0, -0.0, 0.0]\n",
      "[1.002878844834052, 1.6173226384798582, 1.9651618847124537, 0.8479090291880539, -0.0, -0.0, -0.0, 0.0]\n",
      "[1.002878844834052, 1.6173226384798582, 1.9651618847124537, 0.8479090291880539, -0.0, -0.0, -0.0, 0.0]\n",
      "[1.002878844834052, 1.6173226384798582, 1.9651618847124537, 0.8479090291880539, -0.0, -0.0, -0.0, 0.0]\n",
      "γ = 1.9556890877351563\n",
      "[1.0028787759898872, 1.6173226987279188, 1.9651620637704577, 0.8479086889134222, -0.0, -0.0, -0.0, 0.0]\n",
      "[1.0028787759898872, 1.6173226987279188, 1.9651620637704577, 0.8479086889134222, -0.0, -0.0, -0.0, 0.0]\n",
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     "execution_count": 15,
     "metadata": {},
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   ],
   "source": [
    "our_esl_lasso(MM, Dat;reps = 6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "civic-robinson",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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