maxkapur/.gitignore

## .gitignore
.ipynb_checkpoints

## admit.csv

          
            admit
            gre
            gpa
            rank

            
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              1
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              1

            
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              1
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              1

            
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              1
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              1

            
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              0
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              1
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              1

            
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              1
              520
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              3

            
              0
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              1
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              4
              3

            
              0
              540
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              1
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              4
              1

            
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              0
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              1
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              1
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              2

            
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              1
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              0
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              1
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              0
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              0
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              0
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              0
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              0
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              0
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              0
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              0
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              4
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              0
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              0
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              4
              3

            
              0
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              0
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              0
              540
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              3.48
              3

            
              1
              800
              3.43
              2

            
              0
              500
              3.53
              4

            
              1
              620
              3.37
              2

            
              0
              520
              2.62
              2

            
              1
              620
              3.23
              3

            
              0
              620
              3.33
              3

            
              0
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              3

            
              0
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              0
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              0
              700
              4
              2

            
              1
              540
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              0
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              0
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              3.54
              3

            
              0
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              3

            
              0
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              1
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              3.47
              2

            
              0
              560
              3.59
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              0
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              3.07
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              1
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              3.23
              4

            
              0
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              3.63
              3

            
              0
              580
              3.77
              4

            
              0
              800
              3.31
              3

            
              1
              580
              3.2
              2

            
              1
              700
              4
              1

            
              0
              420
              3.92
              4

            
              1
              600
              3.89
              1

            
              1
              780
              3.8
              3

            
              0
              740
              3.54
              1

            
              1
              640
              3.63
              1

            
              0
              540
              3.16
              3

            
              0
              580
              3.5
              2

            
              0
              740
              3.34
              4

            
              0
              580
              3.02
              2

            
              0
              460
              2.87
              2

            
              0
              640
              3.38
              3

            
              1
              600
              3.56
              2

            
              1
              660
              2.91
              3

            
              0
              340
              2.9
              1

            
              1
              460
              3.64
              1

            
              0
              460
              2.98
              1

            
              1
              560
              3.59
              2

            
              0
              540
              3.28
              3

            
              0
              680
              3.99
              3

            
              1
              480
              3.02
              1

            
              0
              800
              3.47
              3

            
              0
              800
              2.9
              2

            
              1
              720
              3.5
              3

            
              0
              620
              3.58
              2

            
              0
              540
              3.02
              4

            
              0
              480
              3.43
              2

            
              1
              720
              3.42
              2

            
              0
              580
              3.29
              4

            
              0
              600
              3.28
              3

            
              0
              380
              3.38
              2

            
              0
              420
              2.67
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              1
              800
              3.53
              1

            
              0
              620
              3.05
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              1
              660
              3.49
              2

            
              0
              480
              4
              2

            
              0
              500
              2.86
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              0
              700
              3.45
              3

            
              0
              440
              2.76
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              1
              520
              3.81
              1

            
              1
              680
              2.96
              3

            
              0
              620
              3.22
              2

            
              0
              540
              3.04
              1

            
              0
              800
              3.91
              3

            
              0
              680
              3.34
              2

            
              0
              440
              3.17
              2

            
              0
              680
              3.64
              3

            
              0
              640
              3.73
              3

            
              0
              660
              3.31
              4

            
              0
              620
              3.21
              4

            
              1
              520
              4
              2

            
              1
              540
              3.55
              4

            
              1
              740
              3.52
              4

            
              0
              640
              3.35
              3

            
              1
              520
              3.3
              2

            
              1
              620
              3.95
              3

            
              0
              520
              3.51
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              0
              640
              3.81
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              0
              680
              3.11
              2

            
              0
              440
              3.15
              2

            
              1
              520
              3.19
              3

            
              1
              620
              3.95
              3

            
              1
              520
              3.9
              3

            
              0
              380
              3.34
              3

            
              0
              560
              3.24
              4

            
              1
              600
              3.64
              3

            
              1
              680
              3.46
              2

            
              0
              500
              2.81
              3

            
              1
              640
              3.95
              2

            
              0
              540
              3.33
              3

            
              1
              680
              3.67
              2

            
              0
              660
              3.32
              1

            
              0
              520
              3.12
              2

            
              1
              600
              2.98
              2

            
              0
              460
              3.77
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              1
              580
              3.58
              1

            
              1
              680
              3
              4

            
              1
              660
              3.14
              2

            
              0
              660
              3.94
              2

            
              0
              360
              3.27
              3

            
              0
              660
              3.45
              4

            
              0
              520
              3.1
              4

            
              1
              440
              3.39
              2

            
              0
              600
              3.31
              4

            
              1
              800
              3.22
              1

            
              1
              660
              3.7
              4

            
              0
              800
              3.15
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              0
              420
              2.26
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              1
              620
              3.45
              2

            
              0
              800
              2.78
              2

            
              0
              680
              3.7
              2

            
              0
              800
              3.97
              1

            
              0
              480
              2.55
              1

            
              0
              520
              3.25
              3

            
              0
              560
              3.16
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              0
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              0
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              0
              720
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## ProjectedGradient.ipynb
{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Julia implementations of projected gradient descent and Newton's method\n",
    "\n",
    "Max Kapur | [maxkapur.com](https://www.maxkapur.com/)\n",
    "\n",
    "Consider the equality-constrained convex minimization problem\n",
    "\n",
    "$$\\min f(x) \\text{ s.t. } Ax = b$$\n",
    "\n",
    "There are many ways to minimize a [convex function](https://en.wikipedia.org/wiki/Convex_function) over an [affine set](https://en.wikipedia.org/wiki/Affine_space). Two of them are projected gradient descent and Newton's method. Computationally, the two are very similar. Given a feasible starting point $\\hat x$, compute the objective gradient and (in the case of Newton's) Hessian, then solve a linear system for a step direction $y$. Next, use (inexact) line search to find a step size $t$, and use $\\hat x + t y$ as the starting point for the next iterate.\n",
    "\n",
    "Projected gradient descent means that the step direction at each iteration is given by solving the quadratic program\n",
    "\n",
    "$$\\min ||y - \\nabla f(x)||^2 \\text{ s.t. } Ay = 0$$\n",
    "\n",
    "That is, we take the gradient and project it onto the nullspace of the feasible set. Then if $x$ is feasible, so is $x + ty$ for any scalar $t$.\n",
    "\n",
    "It is easy to show that the optimal solution to this QP is the vector $y$ that solves the following linear system:\n",
    "\n",
    "$$\\begin{bmatrix}\n",
    "I & A^T \\\\ A & \n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix} y \\\\ \\mu \\end{bmatrix}\n",
    "= \\begin{bmatrix} -\\nabla f(\\hat x) \\\\ 0\n",
    "\\end{bmatrix}$$\n",
    "\n",
    "In Newton's method, we pick the step direction by minimizing a second-degree Taylor approximation to the objective function subject to the equality constraint. It is not hard to show that the step direction solves the following linear system: \n",
    "\n",
    "$$\\begin{bmatrix}\n",
    "\\nabla^2 f(\\hat x) & A^T \\\\ A & \n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix} y \\\\ \\mu \\end{bmatrix}\n",
    "= \\begin{bmatrix} -\\nabla f(\\hat x) \\\\ 0\n",
    "\\end{bmatrix}$$\n",
    "\n",
    "$\\mu$ is a vector of dual variables; in the implementations below it is simply discarded.\n",
    "\n",
    "To find $t$, I use [backtracking line search with the Goldstein&mdash;Armijo condition](https://en.wikipedia.org/wiki/Backtracking_line_search#Algorithm). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 139,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING: using StatsBase.loglikelihood in module Main conflicts with an existing identifier.\n"
     ]
    }
   ],
   "source": [
    "using LinearAlgebra\n",
    "using ForwardDiff: gradient, hessian\n",
    "using JuMP, Ipopt # For comparison\n",
    "using DataFrames, CSV, Plots, StatsBase"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "projectedgradient (generic function with 1 method)"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function projectedgradient(f, x0, A, b; ϵ = 1e-4, α = 0.2, β = 0.5, γ = 4)\n",
    "    if A === nothing\n",
    "        n = length(x0)\n",
    "    else\n",
    "        (m, n) = size(A)\n",
    "        @assert n == length(x0) && m == length(b)\n",
    "    end\n",
    "    \n",
    "    y = ones(n)\n",
    "    \n",
    "    k = 1\n",
    "    while k ≤ 100 && norm(y) > ϵ \n",
    "        fx = f(x0)\n",
    "        g = -gradient(f, x0)\n",
    "\n",
    "        if A === nothing\n",
    "            y = g\n",
    "        else\n",
    "            # min ||y - g|| s.t. Ay = 0 ⟺\n",
    "            left = Matrix( [I(n)     A'             ;\n",
    "                            A        zeros(m, m)    ] )\n",
    "            right = vcat(g, zeros(m))\n",
    "\n",
    "            y = (left\\right)[1:n]\n",
    "        end\n",
    "        \n",
    "        # Step size\n",
    "        t = α\n",
    "        target_slope = - β * g' * y\n",
    "        \n",
    "        i = 1 \n",
    "        while i ≤ 25 && f(x0 + t * y) > fx + t * target_slope\n",
    "            t /= γ\n",
    "            i += 1\n",
    "        end\n",
    "        \n",
    "#         println(\"k = $k\\n  x = $x0\\n  f(x) = $fx\\n  g = $g\\n  i = $i\\n  t = $t\")\n",
    "                \n",
    "        x0 += t * y\n",
    "        k += 1\n",
    "    end\n",
    "    \n",
    "    return x0\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Vector{Float64}:\n",
       "  3.0\n",
       " -1.0"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x) = exp(x[1] + 25*x[2]) + 12 * x[1]^2 + x[2]^2\n",
    "\n",
    "A = [1. 1]\n",
    "b = [2.]\n",
    "x0 = [3., -1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Vector{Float64}:\n",
       "  2.0532670668888526\n",
       " -0.053267066888852216"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "projectedgradient(f, x0, A, b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Compare with JuMP"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "optimize_jump (generic function with 1 method)"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function optimize_jump(f, x0, A, b)\n",
    "    m = Model(Ipopt.Optimizer)\n",
    "    \n",
    "    @variable(m, x[1:length(x0)])\n",
    "    @NLobjective(m, Min, exp(x[1] + 25*x[2]) + 12 * x[1]^2 + x[2]^2)\n",
    "    @constraint(m, A*x .== b)\n",
    "    optimize!(m)\n",
    "    \n",
    "    return value.(x)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "This is Ipopt version 3.13.4, running with linear solver mumps.\n",
      "NOTE: Other linear solvers might be more efficient (see Ipopt documentation).\n",
      "\n",
      "Number of nonzeros in equality constraint Jacobian...:        2\n",
      "Number of nonzeros in inequality constraint Jacobian.:        0\n",
      "Number of nonzeros in Lagrangian Hessian.............:        3\n",
      "\n",
      "Total number of variables............................:        2\n",
      "                     variables with only lower bounds:        0\n",
      "                variables with lower and upper bounds:        0\n",
      "                     variables with only upper bounds:        0\n",
      "Total number of equality constraints.................:        1\n",
      "Total number of inequality constraints...............:        0\n",
      "        inequality constraints with only lower bounds:        0\n",
      "   inequality constraints with lower and upper bounds:        0\n",
      "        inequality constraints with only upper bounds:        0\n",
      "\n",
      "iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls\n",
      "   0  1.0000000e+00 2.00e+00 1.20e+01  -1.0 0.00e+00    -  0.00e+00 0.00e+00   0\n",
      "   1  5.2772538e+01 2.22e-16 1.99e+01  -1.0 2.04e+00    -  1.00e+00 1.00e+00h  1\n",
      "   2  5.2653653e+01 0.00e+00 2.47e+00  -1.0 1.15e-02    -  1.00e+00 1.00e+00f  1\n",
      "   3  5.2651409e+01 0.00e+00 5.34e-02  -1.0 1.87e-03    -  1.00e+00 1.00e+00f  1\n",
      "   4  5.2651408e+01 0.00e+00 2.65e-05  -2.5 4.22e-05    -  1.00e+00 1.00e+00f  1\n",
      "   5  5.2651408e+01 0.00e+00 6.51e-12  -5.7 2.10e-08    -  1.00e+00 1.00e+00f  1\n",
      "\n",
      "Number of Iterations....: 5\n",
      "\n",
      "                                   (scaled)                 (unscaled)\n",
      "Objective...............:   5.2651408325897783e+01    5.2651408325897783e+01\n",
      "Dual infeasibility......:   6.5085714595625177e-12    6.5085714595625177e-12\n",
      "Constraint violation....:   0.0000000000000000e+00    0.0000000000000000e+00\n",
      "Complementarity.........:   0.0000000000000000e+00    0.0000000000000000e+00\n",
      "Overall NLP error.......:   6.5085714595625177e-12    6.5085714595625177e-12\n",
      "\n",
      "\n",
      "Number of objective function evaluations             = 6\n",
      "Number of objective gradient evaluations             = 6\n",
      "Number of equality constraint evaluations            = 6\n",
      "Number of inequality constraint evaluations          = 0\n",
      "Number of equality constraint Jacobian evaluations   = 1\n",
      "Number of inequality constraint Jacobian evaluations = 0\n",
      "Number of Lagrangian Hessian evaluations             = 5\n",
      "Total CPU secs in IPOPT (w/o function evaluations)   =      0.016\n",
      "Total CPU secs in NLP function evaluations           =      0.003\n",
      "\n",
      "EXIT: Optimal Solution Found.\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "2-element Vector{Float64}:\n",
       "  2.053267011026211\n",
       " -0.053267011026210905"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimize_jump(f, x0, A, b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Looks good.\n",
    "\n",
    "### Larger problem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 100\n",
    "m = 25\n",
    "\n",
    "A = rand(m, n)\n",
    "b = rand(m)\n",
    "Q = Symmetric(rand(n, n))^2 # PSD\n",
    "c = randn(n)\n",
    "\n",
    "quadratic(x) = x' * Q * x + c' * x\n",
    "\n",
    "x0 = A\\b;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gradient(quadratic, x0) ≈ 2 * Q * x0 + c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hessian(quadratic, x0) ≈ 2 * Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.387151 seconds (210.08 k allocations: 94.474 MiB, 5.57% gc time, 38.15% compilation time)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "-3.57515374103537"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@time projectedgradient(quadratic, x0, A, b) |> quadratic"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Trivially modifying the step direction system to use the Hessian of `f` instead of `I(n)` yields a Newton's method. Compare"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "newtons (generic function with 1 method)"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function newtons(f, x0, A, b; ϵ = 1e-4, α = 0.2, β = 0.5, γ = 4)\n",
    "    if A === nothing\n",
    "        n = length(x0)\n",
    "    else\n",
    "        (m, n) = size(A)\n",
    "        @assert n == length(x0) && m == length(b)\n",
    "    end\n",
    "    \n",
    "    y = ones(n)\n",
    "    \n",
    "    k = 1\n",
    "    while k ≤ 100 && norm(y) > ϵ \n",
    "        fx = f(x0)\n",
    "        g = -gradient(f, x0)\n",
    "        \n",
    "        if A === nothing\n",
    "            y = hessian(f, x0)\\g\n",
    "        else\n",
    "            # min ∇f'(x) (x + y) + 0.5 (x + y)' H f(x) (x + y) s.t. Ay = 0 ⟺\n",
    "            left = Matrix( [hessian(f, x0)     A'             ;                    # Modification here\n",
    "                            A                  zeros(m, m)    ] )\n",
    "            right = vcat(g, zeros(m))\n",
    "\n",
    "            y = (left\\right)[1:n] \n",
    "        end\n",
    "        \n",
    "        # Step size\n",
    "        t = α\n",
    "        target_slope = - β * g' * y\n",
    "        \n",
    "        i = 1 \n",
    "        while i ≤ 25 && f(x0 + t * y) > fx + t * target_slope\n",
    "            t /= γ\n",
    "            i += 1\n",
    "        end\n",
    "        \n",
    "#         println(\"k = $k\\n  x = $x0\\n  f(x) = $fx\\n  g = $g\\n  i = $i\\n  t = $t\")\n",
    "                \n",
    "        x0 += t * y\n",
    "        k += 1\n",
    "    end\n",
    "    \n",
    "    return x0\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  3.614344 seconds (901.33 k allocations: 628.814 MiB, 2.54% gc time, 11.13% compilation time)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "-3.5823932935130323"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@time newtons(quadratic, x0, A, b) |> quadratic"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Same objective value. However, Newton's method is appreciably slower for this problem because of the large number of variables.\n",
    "\n",
    "In theory, for poorly-conditioned problems like the one below, we might expect an advantage with Newton's, but the advantage turns out to be pretty small. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 3\n",
    "m = 1\n",
    "\n",
    "A = rand(m, n)\n",
    "b = rand(m)\n",
    "Q = Diagonal(diagm(10. .^ [-10, 10, 10])) # PSD\n",
    "c = randn(n)\n",
    "\n",
    "quadratic(x) = x' * Q * x + c' * x\n",
    "\n",
    "x0 = A\\b;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.136576 seconds (355.98 k allocations: 25.758 MiB, 98.47% compilation time)\n",
      "  0.169354 seconds (360.27 k allocations: 25.999 MiB, 12.11% gc time, 98.48% compilation time)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "14.373947082230718"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@time projectedgradient(quadratic, x0, A, b) |> quadratic\n",
    "@time newtons(quadratic, x0, A, b) |> quadratic"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Application\n",
    "\n",
    "Predict admissions outcomes using some data from [Kaggle](https://www.kaggle.com/msjass/admittest). As you can see, the data is not very well distinguished."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data = DataFrame(CSV.File(\"admit.csv\"))\n",
    "scatter(data[!, :gre], data[!, :gpa], color=data[!, :admit])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Logistic regression solution\n",
    "\n",
    "Let $y = 0$ if a student was rejected and $y=1$ if a student was admitted. The logistic model says that \n",
    "\n",
    "$$\\begin{align}\n",
    "P(y = 1) &= \\frac{\\exp(a^T x - b)}{1 + \\exp(a^T x - b)} \\\\\n",
    "P(y = 0) &= \\frac{1}{1 + \\exp(a^T x - b)}= 1 - P(y=1) \n",
    "\\end{align}$$\n",
    "\n",
    "where $x$ is the vector `[GRE, GPA]` and $a$ and $b$ are decision variables.\n",
    "\n",
    "Suppose there were $m$ students total and the first $p$ students were accepted. Then the likelihood of this set of observations is\n",
    "\n",
    "$$P(y | X; a, b) = \\prod_{i=1}^p \\frac{\\exp(a^T x_i - b)}{1 + \\exp(a^T x_i - b)} \\prod_{j=p+1}^m \\frac{1}{1 + \\exp(a^T x_i - b)}$$\n",
    "\n",
    "We want to *maximize* this expression. Taking the log and doing math, it suffices to *minimize* the following:\n",
    "\n",
    "$$-\\log \\left[P(y | X; a, b)\\right] = -\\sum_{i=1}^p a^T x_i - b + \\sum_{j=1}^m \\log\\left({1 + \\exp(a^T x_i - b)}\\right)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 196,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "logP (generic function with 1 method)"
      ]
     },
     "execution_count": 196,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X = data[!, [:gre, :gpa]] |> Array\n",
    "X = standardize(ZScoreTransform, X, dims=1)\n",
    "y = data[!, :admit]\n",
    "\n",
    "logP(a, b; X=X, y=y) = - sum(X[y .== 1, :] * a .- b) + sum(log1p.(exp.(X * a .- b)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 201,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "182.91113955326733"
      ]
     },
     "execution_count": 201,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "logP([0.005, 0.5], 0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 274,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{Float64}:\n",
       " 0.4016544455315323\n",
       " 0.316094030073735\n",
       " 0.8825447971832999"
      ]
     },
     "execution_count": 274,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x) = logP([x[1], x[2]], x[3])\n",
    "\n",
    "a = zeros(2)\n",
    "b = 0.\n",
    "a[1], a[2], b = newtons(f, [0.05, 0.5, 0.5], nothing, nothing)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 275,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "174.75968178833915"
      ]
     },
     "execution_count": 275,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "logP(a, b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 276,
   "metadata": {},
   "outputs": [
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   "source": [
    "# Assume anyone whose rated likelihood is greater than the overall admit rate gets in\n",
    "data[!, :pred] = convert(Vector{Int}, exp.(X * a .- b) ./ (1 .+ exp.(X * a .- b)) .> mean(y))\n",
    "\n",
    "scatter(data[!, :gre], data[!, :gpa], color=data[!, :pred])"
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    "Seems reasonable enough.\n",
    "\n",
    "### Hyperplane classifier solution\n",
    "\n",
    "Let us solve the same problem using a hyperplane classifier, also known as a support vector machine.\n",
    "\n",
    "The optimization problem is\n",
    "\n",
    "$$\\begin{align}\n",
    "\\min_{w, b, z} \\quad&w^T w + \\gamma z^T z \\\\\n",
    "\\text{s.t.}\\quad & w^T x_i + b \\geq 1 - z_i ,& i : y_i = 1 \\\\\n",
    "            & w^T x_j + b \\leq -1 + z_j ,& j : y_j = 0 \\\\\n",
    "&            z \\geq 0\n",
    "\\end{align}$$\n",
    "\n",
    "where $\\gamma$ is a parameter and $z$ is an error vector.\n",
    "\n",
    "The intuition is that we are trying to define an affine transformation $w^T x + b$ that projects the score vector $x$ to the real line so that the projected value is greater than $1$ for admitted students and less than $-1$ for rejected students. There are many ways to do this, but we want the one that does so using the least \"work,\" i.e. with the shortest $w$. It can be shown that this is equivalent to maximizing the margin between the classification boundary $w^T x + b = 0$ and the nearest point on either side. Since a perfect classifier is impossible, we then add a penalty term $\\gamma z^T z$ which adds up the squared distances between the transformed score and $1$ or $-1$ as appropriate. \n",
    "\n",
    "I will use a log barrier to make the problem convex. Notice that this formulation is intrinsically more complex, because it requires generating a decision variable for each data point, whereas the formulation above uses only three decision variables. However, choosing projected gradient instead of Newton's means renders the problem tractable."
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     "data": {
      "text/plain": [
       "f (generic function with 1 method)"
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    "log_barrier(x) = x > 0 ? -log(x) : Inf\n",
    "\n",
    "function hyperplane(w, b, z; t = 1., γ = 5, X=X, y=y)\n",
    "    return t * (w' * w + γ * z' * z) +\n",
    "        sum(log_barrier.( X[y .== 1, :] * w .+ b .- 1 + z[y .== 1])) +\n",
    "        sum(log_barrier.(-X[y .== 0, :] * w .- b .- 1 + z[y .== 0])) +\n",
    "        sum(log_barrier.(z))\n",
    "end\n",
    "\n",
    "f(x; t) = hyperplane([x[1], x[2]], x[3], x[4:end]; t=t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 269,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "t = 1\n",
      "f(x0; t = t) / t = 164881.77024187663\n",
      "t = 4\n",
      "f(x0; t = t) / t = 2123.2834611831017\n",
      "t = 16\n",
      "f(x0; t = t) / t = 1618.9637533553719\n",
      "t = 64\n",
      "f(x0; t = t) / t = 1475.0337179516214\n",
      "t = 256\n",
      "f(x0; t = t) / t = 1438.9432499385462\n",
      "t = 1024\n",
      "f(x0; t = t) / t = 1428.292030555626\n",
      "t = 4096\n",
      "f(x0; t = t) / t = 1425.4358421819563\n",
      "t = 16384\n",
      "f(x0; t = t) / t = 1424.7198166831993\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "3-element Vector{Float64}:\n",
       "  0.04838721754270806\n",
       "  0.037901403552514173\n",
       " -0.0982248457806021"
      ]
     },
     "execution_count": 269,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function minimize()\n",
    "    t = 1\n",
    "    x0 = vcat([0.5, 0.5], 0, 10 .+ rand(length(y)))\n",
    "    \n",
    "    for i in 1:8\n",
    "        @show t\n",
    "        @show f(x0; t=t)/t\n",
    "        x0 = projectedgradient(x->f(x, t=t), x0, nothing, nothing)\n",
    "        t *= 4\n",
    "    end\n",
    "    return x0\n",
    "end\n",
    "\n",
    "w = zeros(2)\n",
    "b = 0.\n",
    "w[1], w[2], b = minimize()[1:3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 281,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 281,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Assume anyone whose transformed score is higher than that of the average student gets in\n",
    "data[!, :pred2] = convert(Vector{Int}, X * w .+ b .> mean(X, dims=1) * w .+ b)\n",
    "\n",
    "scatter(data[!, :gre], data[!, :gpa], color=data[!, :pred2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For this choice of $\\gamma$, the classification boundary is basically identical.\n",
    "\n",
    "In both of these problems, we didn't actually use any equality constraints, so this is just plain old gradient descent."
   ]
  }
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admit	gre	gpa	rank
0	380	3.61	3
1	660	3.67	3
1	800	4	1
1	640	3.19	4
0	520	2.93	4
1	760	3	2
1	560	2.98	1
0	400	3.08	2
1	540	3.39	3
0	700	3.92	2
0	800	4	4
0	440	3.22	1
1	760	4	1
0	700	3.08	2
1	700	4	1
0	480	3.44	3
0	780	3.87	4
0	360	2.56	3
0	800	3.75	2
1	540	3.81	1
0	500	3.17	3
1	660	3.63	2
0	600	2.82	4
0	680	3.19	4
1	760	3.35	2
1	800	3.66	1
1	620	3.61	1
1	520	3.74	4
1	780	3.22	2
0	520	3.29	1
0	540	3.78	4
0	760	3.35	3
0	600	3.4	3
1	800	4	3
0	360	3.14	1
0	400	3.05	2
0	580	3.25	1
0	520	2.9	3
1	500	3.13	2
1	520	2.68	3
0	560	2.42	2
1	580	3.32	2
1	600	3.15	2
0	500	3.31	3
0	700	2.94	2
1	460	3.45	3
1	580	3.46	2
0	500	2.97	4
0	440	2.48	4
0	400	3.35	3
0	640	3.86	3
0	440	3.13	4
0	740	3.37	4
1	680	3.27	2
0	660	3.34	3
1	740	4	3
0	560	3.19	3
0	380	2.94	3
0	400	3.65	2
0	600	2.82	4
1	620	3.18	2
0	560	3.32	4
0	640	3.67	3
1	680	3.85	3
0	580	4	3
0	600	3.59	2
0	740	3.62	4
0	620	3.3	1
0	580	3.69	1
0	800	3.73	1
0	640	4	3
0	300	2.92	4
0	480	3.39	4
0	580	4	2
0	720	3.45	4
0	720	4	3
0	560	3.36	3
1	800	4	3
0	540	3.12	1
1	620	4	1
0	700	2.9	4
0	620	3.07	2
0	500	2.71	2
0	380	2.91	4
1	500	3.6	3
0	520	2.98	2
0	600	3.32	2
0	600	3.48	2
0	700	3.28	1
1	660	4	2
0	700	3.83	2
1	720	3.64	1
0	800	3.9	2
0	580	2.93	2
1	660	3.44	2
0	660	3.33	2
0	640	3.52	4
0	480	3.57	2
0	700	2.88	2
0	400	3.31	3
0	340	3.15	3
0	580	3.57	3
0	380	3.33	4
0	540	3.94	3
1	660	3.95	2
1	740	2.97	2
1	700	3.56	1
0	480	3.13	2
0	400	2.93	3
0	480	3.45	2
0	680	3.08	4
0	420	3.41	4
0	360	3	3
0	600	3.22	1
0	720	3.84	3
0	620	3.99	3
1	440	3.45	2
0	700	3.72	2
1	800	3.7	1
0	340	2.92	3
1	520	3.74	2
1	480	2.67	2
0	520	2.85	3
0	500	2.98	3
0	720	3.88	3
0	540	3.38	4
1	600	3.54	1
0	740	3.74	4
0	540	3.19	2
0	460	3.15	4
1	620	3.17	2
0	640	2.79	2
0	580	3.4	2
0	500	3.08	3
0	560	2.95	2
0	500	3.57	3
0	560	3.33	4
0	700	4	3
0	620	3.4	2
1	600	3.58	1
0	640	3.93	2
1	700	3.52	4
0	620	3.94	4
0	580	3.4	3
0	580	3.4	4
0	380	3.43	3
0	480	3.4	2
0	560	2.71	3
1	480	2.91	1
0	740	3.31	1
1	800	3.74	1
0	400	3.38	2
1	640	3.94	2
0	580	3.46	3
0	620	3.69	3
1	580	2.86	4
0	560	2.52	2
1	480	3.58	1
0	660	3.49	2
0	700	3.82	3
0	600	3.13	2
0	640	3.5	2
1	700	3.56	2
0	520	2.73	2
0	580	3.3	2
0	700	4	1
0	440	3.24	4
0	720	3.77	3
0	500	4	3
0	600	3.62	3
0	400	3.51	3
0	540	2.81	3
0	680	3.48	3
1	800	3.43	2
0	500	3.53	4
1	620	3.37	2
0	520	2.62	2
1	620	3.23	3
0	620	3.33	3
0	300	3.01	3
0	620	3.78	3
0	500	3.88	4
0	700	4	2
1	540	3.84	2
0	500	2.79	4
0	800	3.6	2
0	560	3.61	3
0	580	2.88	2
0	560	3.07	2
0	500	3.35	2
1	640	2.94	2
0	800	3.54	3
0	640	3.76	3
0	380	3.59	4
1	600	3.47	2
0	560	3.59	2
0	660	3.07	3
1	400	3.23	4
0	600	3.63	3
0	580	3.77	4
0	800	3.31	3
1	580	3.2	2
1	700	4	1
0	420	3.92	4
1	600	3.89	1
1	780	3.8	3
0	740	3.54	1
1	640	3.63	1
0	540	3.16	3
0	580	3.5	2
0	740	3.34	4
0	580	3.02	2
0	460	2.87	2
0	640	3.38	3
1	600	3.56	2
1	660	2.91	3
0	340	2.9	1
1	460	3.64	1
0	460	2.98	1
1	560	3.59	2
0	540	3.28	3
0	680	3.99	3
1	480	3.02	1
0	800	3.47	3
0	800	2.9	2
1	720	3.5	3
0	620	3.58	2
0	540	3.02	4
0	480	3.43	2
1	720	3.42	2
0	580	3.29	4
0	600	3.28	3
0	380	3.38	2
0	420	2.67	3
1	800	3.53	1
0	620	3.05	2
1	660	3.49	2
0	480	4	2
0	500	2.86	4
0	700	3.45	3
0	440	2.76	2
1	520	3.81	1
1	680	2.96	3
0	620	3.22	2
0	540	3.04	1
0	800	3.91	3
0	680	3.34	2
0	440	3.17	2
0	680	3.64	3
0	640	3.73	3
0	660	3.31	4
0	620	3.21	4
1	520	4	2
1	540	3.55	4
1	740	3.52	4
0	640	3.35	3
1	520	3.3	2
1	620	3.95	3
0	520	3.51	2
0	640	3.81	2
0	680	3.11	2
0	440	3.15	2
1	520	3.19	3
1	620	3.95	3
1	520	3.9	3
0	380	3.34	3
0	560	3.24	4
1	600	3.64	3
1	680	3.46	2
0	500	2.81	3
1	640	3.95	2
0	540	3.33	3
1	680	3.67	2
0	660	3.32	1
0	520	3.12	2
1	600	2.98	2
0	460	3.77	3
1	580	3.58	1
1	680	3	4
1	660	3.14	2
0	660	3.94	2
0	360	3.27	3
0	660	3.45	4
0	520	3.1	4
1	440	3.39	2
0	600	3.31	4
1	800	3.22	1
1	660	3.7	4
0	800	3.15	4
0	420	2.26	4
1	620	3.45	2
0	800	2.78	2
0	680	3.7	2
0	800	3.97	1
0	480	2.55	1
0	520	3.25	3
0	560	3.16	1
0	460	3.07	2
0	540	3.5	2
0	720	3.4	3