{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import os\n",
    "import sys\n",
    "import glob\n",
    "import operator as op\n",
    "import itertools as it\n",
    "from functools import reduce, partial\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "from pandas import DataFrame, Series\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "sns.set_context(\"notebook\", font_scale=1.5)\n",
    "%matplotlib inline\n",
    "\n",
    "from sympy import symbols, hessian, Function, N"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Algorithms for Optimization and Root Finding for Multivariate Problems"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Note**: much of the following notes are taken from Nodecal, J., and S. J. Wright. \"Numerical optimization.\" (2006). It is available online via the library."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Convexity\n",
    "\n",
    "A subset $A\\subset \\mathbb{R}^n$ is *convex* if for any two points $x,y\\in A$, the line segment:\n",
    "\n",
    "$$tx + (1-t)y \\;\\;\\;\\;\\;\\; t\\in [0,1]$$\n",
    "is also in $A$\n",
    "\n",
    "A function $f:\\mathbb{R}^n \\rightarrow \\mathbb{R}$ is *convex* if its domain $D$ is a convex set and for any two points $x,y\\in D$, the graph of $f$ (a subset of $\\mathbb{R}^{n+1})$ lies below the line:\n",
    "\n",
    "$$tf(x) + (1-t)f(y)\\;\\;\\;\\;\\;t\\in [0,1]$$\n",
    "\n",
    "i.e. \n",
    "$$f(tx+(1-t)y) \\leq tf(x) + (1-t)f(y)\\;\\;\\;\\;\\;t\\in [0,1]$$\n",
    "\n",
    "#### Convexity guarantees that if an optimizer converges, it converges to the global minimum.\n",
    "\n",
    "Luckily, we often encounter convex problems in statistics."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Line Search Methods\n",
    "\n",
    "There are essentially two classes of multivariate optimization methods. We'll cover line search methods, but refer the reader to Nodecal and Wright for discussion of 'trust region methods'. We should note that all of these methods require that we are 'close' to the minimum (maximum) we are seeking, and that 'noisy' functions or ill-behaved functions are beyond our scope.\n",
    "\n",
    "A line search method is exactly as it sounds - we search on a line (in $n$ dimensional space) and try to find a minimum.  We start with an initial point, and use an iterative method:\n",
    "\n",
    "$$x_{k+1} = x_k + \\alpha_k p_k$$\n",
    "\n",
    "where $\\alpha_k$ is the *step size* and $p_k$ is the search direction. These are the critical choices that change the behavior of the search. \n",
    "\n",
    "#### Step Size\n",
    "\n",
    "Ideally, (given a choice of direction, $p_k$) we would want to minimize:\n",
    "\n",
    "$$\\varphi(\\alpha) = f(x_k + \\alpha p_k)$$\n",
    "\n",
    "with respect to $\\alpha$. This is usually computationally intensive, so in practice, a sequence of $\\alpha$ candidates are generated, and then the 'best' is chosen according to some 'conditions'. We won't be going into detail regarding these. The important thing to know is that they ensure that $f$ decreases sufficiently, according to some conditions. Interested students should see Nodecal."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Steepest Descent\n",
    "\n",
    "In steepest descent, one chooses $p_k=\\nabla f_k = \\nabla f(x_k)$. It is so named, because the gradient points in the direction of steepest ascent, thus, $-\\nabla f_k$ will point in the direction of steepest descent. We'll consider this method in its ideal case, that of a quadratic:\n",
    "\n",
    "$$f(x) = \\frac12 x^TQx - b^Tx$$\n",
    "\n",
    "where $Q$ is positive-definite and symmetric. Note that:\n",
    "\n",
    "$$\\nabla f = Qx -b$$\n",
    "\n",
    "so the minimum occurs at $x$ such that\n",
    "\n",
    "$$Qx= b$$\n",
    "\n",
    "Clearly, we can solve this easily, but let's walk through the algorithm and first find the (ideal) step length:\n",
    "\n",
    "$$f(x_k - \\alpha \\nabla f_k) = \\frac12\\left(x_k - \\alpha \\nabla f_k\\right)^TQ\\left(x_k - \\alpha \\nabla f_k\\right) - b^T \\left(x_k - \\alpha \\nabla f_k\\right) $$\n",
    "\n",
    "If we differentiate this with respect to $\\alpha$ and find the zero, we obtain:\n",
    "\n",
    "$$\\alpha_k = \\frac{\\nabla f_k^T\\nabla f_k}{\\nabla f_k^TQ\\nabla f_k}$$\n",
    "\n",
    "Thus, \n",
    "\n",
    "$$x_{k+1} = x_k - \\frac{\\nabla f_k^T\\nabla f_k}{\\nabla f_k^TQ\\nabla f_k} \\nabla f_k$$\n",
    "\n",
    "But we know that $\\nabla f_k = Qx_k -b$, so we have a closed form solution for $x_{k+1}$. This allows us to compute an error bound. Again, details can be found in the text, but here is the result:\n",
    "\n",
    "$$||x_{k+1} - x^*||_Q^2 \\leq \\left(\\frac{\\lambda_n - \\lambda_1}{\\lambda_n+\\lambda_1}\\right)^2 ||x_{k} - x^*||_Q^2$$\n",
    "\n",
    "where $0<\\lambda_1\\leq ... \\leq \\lambda_n$ and $x^*$ denotes the minimizer.\n",
    "\n",
    "Now, if $\\lambda_1=...=\\lambda_n = \\lambda$, then $Q=\\lambda I$, the algorithm converges in one step. Geometrically, the contours are ellipsoids, the value of $\\frac{\\lambda_n}{\\lambda_1}$ elongates the axes and causes the steps to 'zig-zag'. Because of this, convergence slows as $\\frac{\\lambda_n}{\\lambda_1}$ increases.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Newton's Method\n",
    "\n",
    "Newton's method is another line-search, and here \n",
    "\n",
    "$$p_k = -H^{-1}\\nabla f_k$$\n",
    "\n",
    "Note that if the Hessian is not positive definite, this may not always be a descent direction.\n",
    "\n",
    "In the neighborhood of a local minimum, the Hessian *will* be positive definite. Now, if $x_0$ is 'close enough' to the minimizer $x^*$, the step size $\\alpha_k =1$ gives quadratic convergence. \n",
    "\n",
    "The advantage of multiplying the gradient by the inverse of the Hessian is that the gradient is corrected for curvature, and the new direction points toward the minimum."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "\n",
    "#def Quad(x):\n",
    "#    return (x[1:])*np.sin(x[:-1])**2.0)\n",
    "\n",
    "#def DQuad(x,y):\n",
    "#    return (np.array([np.cos(x)*np.sin(y)**2.0,2.0*np.sin(x)*np.cos(y)**2.0]))\n",
    "\n",
    "def Quad(x):\n",
    "    return ((x[1:])**2.0 + 5*(x[:-1])**2.0)\n",
    "\n",
    "def DQuad(x,y):\n",
    "    return (np.array([2.0*x,10.0*y]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x = np.linspace(-20,20, 100)\n",
    "y = np.linspace(-20,20, 100)\n",
    "X, Y = np.meshgrid(x, y)\n",
    "Z = Quad(np.vstack([X.ravel(), Y.ravel()])).reshape((100,100))\n",
    "Hinv=-np.array([[0.5,0],[0,0.1]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
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K7NRDatl8FEXxCSDSpdwSFAGVAsxU+65Q/29/jPd1LYpSXY4S7ikTJbTgFS5lbkbZNjys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\nQyyRwelRsb4TblnGpsYmnIxNIpnne3+tFnC+o55lkIm7pAk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qI2KArUmzaAvWQgLh4uwq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\n/wP0eh9TBEZzXAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x116d96eb8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(12,4))\n",
    "plt.subplot(121)\n",
    "\n",
    "plt.contour(X,Y,Z);\n",
    "plt.title(\"Steepest Descent\");\n",
    "step=-0.25\n",
    "X0 = 10.0\n",
    "Y0 = 1.0\n",
    "\n",
    "Ngrad=Hinv.dot(DQuad(X0,Y0))\n",
    "\n",
    "sgrad = step*DQuad(X0,Y0)\n",
    "plt.quiver(X0,Y0,sgrad[0],sgrad[1],color='red',angles='xy',scale_units='xy',scale=1);\n",
    "X1 = X0 + sgrad[0]\n",
    "Y1 = Y0 + sgrad[1]\n",
    "sgrad = step*DQuad(X1,Y1)\n",
    "plt.quiver(X1,Y1,sgrad[0],sgrad[1],color='green',angles='xy',scale_units='xy',scale=1);\n",
    "X2 = X1 + sgrad[0]\n",
    "Y2 = Y1 + sgrad[1]\n",
    "sgrad = step*DQuad(X2,Y2)\n",
    "plt.quiver(X2,Y2,sgrad[0],sgrad[1],color='purple',angles='xy',scale_units='xy',scale=1);\n",
    "\n",
    "\n",
    "plt.subplot(122)\n",
    "plt.contour(X,Y,Z);\n",
    "plt.title(\"Newton's Method\")\n",
    "plt.quiver(X0,Y0,Ngrad[0],Ngrad[1],color='purple',angles='xy',scale_units='xy',scale=1);\n",
    "\n",
    "#Compute Hessian and plot again.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Coordinate Descent\n",
    "\n",
    "Another method is called 'coordinate' descent, and it involves searching along coordinate directions (cyclically), i.e.:\n",
    "\n",
    "$$p_{mk} = e_{k} \\;\\;\\;\\;\\;\\; k=1,...,n$$\n",
    "\n",
    "where $m$ is the number of steps.\n",
    "\n",
    "The main advantage is that $\\nabla f$ is not required. It can behave reasonably well, if coordinates are not tightly coupled. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Newton CG Algorithm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Features:\n",
    "\n",
    "* Minimizes a 'true' quadratic on $\\mathbb{R}^n$ in $n$ steps\n",
    "* Does NOT require storage or inversion of an $n \\times n$ matrix."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We begin with $:\\mathbb{R}^n\\rightarrow \\mathbb{R}$.  Take a quadratic approximation to $f$:\n",
    "\n",
    "$$f(x) \\approx \\frac12 x^T H x + b^Tx + c$$\n",
    "\n",
    "Note that in the neighborhood of a minimum, $H$ will be positive-definite (and symmetric). (If we are maximizing, just consider $-H$).\n",
    "\n",
    "This reduces the optimization problem to finding the zeros of \n",
    "\n",
    "$$Hx = -b$$\n",
    "\n",
    "This is a linear problem, which is nice.  The dimension $n$ may be very large - which is not so nice."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### General Inner Product"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Recall the axiomatic definition of an inner product $<,>_A$:\n",
    "\n",
    "* For any two vectors $v,w$ we have\n",
    "$$<v,w>_A = <w,v>_A$$\n",
    "* For any vector $v$\n",
    "$$<v,v>_A \\;\\geq 0$$\n",
    "   with equality $\\iff$ $v=0$.\n",
    "* For $c\\in\\mathbb{R}$ and $u,v,w\\in\\mathbb{R}^n$, we have\n",
    "$$<cv+w,u> = c<v,u> + <w,u>$$\n",
    "\n",
    "These properties are known as symmetric, positive definite and bilinear, respectively.\n",
    "\n",
    "Fact: If we denote the standard inner product on $\\mathbb{R}^n$ as $<,>$ (this is the 'dot product'), any symmetric, positive definite $n\\times n$ matrix $A$ defines an inner product on $\\mathbb{R}^n$ via:\n",
    "\n",
    "$$<v,w>_A \\; = <v,Aw> = v^TAw$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Just as with the standard inner product, general inner products define for us a notion of 'orthogonality'. Recall that with respect to the standard product, 2 vectors are orthogonal if their product vanishes. The same applies to $<,>_A$:\n",
    "\n",
    "$$<v,w>_A = 0 $$\n",
    "\n",
    "means that $v$ and $w$ are orthogonal under the inner product induced by $A$.  Equivalently, if $v,w$ are orthogonal under $A$, we have:\n",
    "\n",
    "$$v^TAw = 0$$\n",
    "\n",
    "This is also called *conjugate* (thus the name of the method)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Conjugate Vectors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose we have a set of $n$ vectors $p_1,...,p_n$ that are mutually conjugate.  These vectors form a basis of $\\mathbb{R}^n$.  Getting back to the problem at hand, this means that our solution vector $x$ to the linear problem may be written as follows:\n",
    "\n",
    "$$x = \\sum\\limits_{i=1}^n \\alpha_i p_i$$\n",
    "\n",
    "So, finding $x$ reduces to finding a conjugate basis and the coefficients for $x$ in that basis."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we let $A=H$,note that:\n",
    "\n",
    "$${p}_k^{T} {-b}={p}_k^{T} {A}{x}$$\n",
    "\n",
    "and because $x = \\sum\\limits_{i=1}^n \\alpha_i p_i$, we have:\n",
    "$$p^TAx = \\sum\\limits_{i=1}^n \\alpha_i p^TA p_i$$\n",
    "\n",
    "we can solve for $\\alpha_k$:\n",
    "\n",
    "$$\\alpha_k = \\frac{{p}_k^{T}{(-b)}}{{p}_k^{T} {A}{p}_k} = -\\frac{\\langle {p}_k, {b}\\rangle}{\\,\\,\\,\\langle {p}_k,  {p}_k\\rangle_{A}} = -\\frac{\\langle{p}_k, {b}\\rangle}{\\,\\,\\,\\|{p}_k\\|_{A}^2}.$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, all we need are the $p_k$'s."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A nice initial guess would be the gradient at some initial point $x_1$.  So, we set $p_1 = \\nabla f(x_1)$. Then set:\n",
    "\n",
    "$$x_2 = x_1 + \\alpha_1p_1$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This should look familiar.  In fact, it is gradient descent.  For $p_2$, we want $p_1$ and $p_2$ to be conjugate (under $A$).  That just means orthogonal under the inner product induced by $A$.  We set \n",
    "\n",
    "$$p_2 = \\nabla f(x_2) - \\frac{p_1^TA\\nabla f(x_2)}{{p}_1^{T}{A}{p}_1} {p}_1$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I.e. We take the gradient at $x_1$ and subtract its projection onto $p_1$.  This is the same as Gram-Schmidt orthogonalization.\n",
    "\n",
    "The $k^{th}$ conjugate vector is:\n",
    "\n",
    "$$p_{k} = \\nabla f(x_k) - \\sum\\limits_{i=1}^{k-1}\\frac{p_i^T A \\nabla f(x_k)}{p_i^TAp_i} p_i$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The 'trick' is that in general, we do not need all $n$ conjugate vectors. In fact, it turns out that $\\nabla f(x_k) = b-Ax_k$ is conjugate to all the $p_i$ for $i=1,...,k-2$. Therefore, we need only the last term in the sum."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Convergence rate is dependent on sparsity and condition number of $A$.  Worst case is $n^2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### BFGS - Broyden–Fletcher–Goldfarb–Shanno"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "BFGS is a 'quasi' Newton method of optimization.  Such methods are variants of the Newton method, where the Hessian $H$ is replaced by some approximation.  We we wish to solve the equation:\n",
    "\n",
    "$$B_k{p}_k = -\\nabla f({x}_k)$$\n",
    "\n",
    "for $p_k$. This gives our search direction, and the next candidate point is given by:\n",
    "\n",
    "$$x_{k+1} = x_k + \\alpha_k p_k$$.\n",
    "\n",
    "where $\\alpha_k$ is a step size."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "At each step, we require that the new approximate $H$ meets the secant condition:\n",
    "\n",
    "$$B_{k+1}(x_{k+1}-x_k) = \\nabla f(x_{k+1}) -\\nabla f(x_k)$$\n",
    "\n",
    "There is a unique, rank one update that satisfies the above:\n",
    "\n",
    "$$B_{k+1} = B_k + c_k v_kv_k^T$$\n",
    "\n",
    "where\n",
    "\n",
    "$$ c_k = -\\frac{1}{\\left(B_k(x_{k+1}-x_k) - (\\nabla f(x_{k+1})-\\nabla f(x_k)\\right)^T (x_{k+1}-x_k) }$$ \n",
    "\n",
    "and\n",
    "\n",
    "$$v_k = B_k(x_{k+1}-x_k) - (\\nabla f(x_{k+1})-\\nabla f(x_k))$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that the update does NOT preserve positive definiteness if $c_k<0$.  In this case, there are several options for the rank one correction, but we will not address them here.  Instead, we will describe the BFGS method, which almost always guarantees a positive-definite correction.  Specifically:\n",
    "\n",
    "$$B_{k+1} = B_k + b_k g_k g_k^T + c_k B_k d_k d_k^TB_k$$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where we have introduced the shorthand:\n",
    "\n",
    "$$g_k = \\nabla f(x_{k+1}) - \\nabla f(x_k) \\;\\;\\;\\;\\;\\;\\;\\ \\mathrm{ and }\\;\\;\\;\\;\\;\\;\\; d_k = x_{k+1} - x_k$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we set:\n",
    "\n",
    "$$b_k = \\frac{1}{g_k^Td_k} \\;\\;\\;\\;\\; \\mathrm{ and } \\;\\;\\;\\;\\; c_k = \\frac{1}{d_k^TB_kd_k}$$\n",
    "\n",
    "we satisfy the secant condition."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Nelder-Mead Simplex"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "While Newton's method is considered a 'second order method' (requires the second derivative), and quasi-Newton methods are first order (require only first derivatives), Nelder-Mead is a zero-order method.  I.e. NM requires only the function itself - no derivatives.\n",
    "\n",
    "For $f:\\mathbb{R}^n\\rightarrow \\mathbb{R}$, the algorithm computes the values of the function on a simplex of dimension $n$, constructed from $n+1$ vertices.  For a univariate function, the simplex is a line segment.  In two dimensions, the simplex is a triangle, in 3D, a tetrahedral solid, and so on.\n",
    "\n",
    "The algorithm begins with $n+1$ starting points and then the follwing steps are repeated until convergence:\n",
    "\n",
    "* Compute the function at each of the points\n",
    "* Sort the function values so that \n",
    "$$f(x_1)\\leq ...\\leq f(x_{n+1})$$\n",
    "* Compute the centroid $x_c$ of the n-dimensional region defined by $x_1,...,x_n$\n",
    "* Reflect $x_{n+1}$ about the centroid to get $x_r$\n",
    "$$x_r = x_c + \\alpha (x_c - x_{n+1})$$\n",
    "* Create a new simplex according to the following rules:\n",
    "    - If $f(x_1)\\leq f(x_r) < f(x_n)$, replace $x_{n+1}$ with $x_r$\n",
    "    - If $f(x_r)<f(x_1)$, expand the simplex through $x_r$:\n",
    "    $$x_e = x_c + \\gamma (x_c - x_{n+1})$$\n",
    "      If $f(x_e)<f(x_r)$, replace $x_{n+1}$ with $x_e$, otherwise, replace $x_{n+1}$ with $x_r$\n",
    "    - If $f({x}_{r}) \\geq f({x}_{n})$, compute $x_p = x_c + \\rho(x_c - x_{n+1})$.  If $f({x}_{p}) < f({x}_{n+1})$, replace $x_{n+1}$ with $x_p$\n",
    "    - If all else fails, replace *all* points except $x_1$ according to\n",
    "    $$x_i = {x}_{1} + \\sigma({x}_{i} - {x}_{1})$$\n",
    "    \n",
    "The default values of $\\alpha, \\gamma,\\rho$ and $\\sigma$ in scipy are not listed in the documentation, nor are they inputs to the function."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Powell's Method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Powell's method is another derivative-free optimization method that is similar to conjugate-gradient.  The algorithm steps are as follows:\n",
    "\n",
    "Begin with a point $p_0$ (an initial guess) and a set of vectors $\\xi_1,...,\\xi_n$, initially the standard basis of $\\mathbb{R}^n$.  \n",
    "\n",
    "- Compute for $i=1,...,n$, find $\\lambda_i$ that minimizes $f(p_{i-1} +\\lambda_i \\xi_i)$ and set $p_i = p_{i-1} + \\lambda_i\\xi_i$\n",
    "- For $i=1,...,n-1$, replace $\\xi_{i}$ with $\\xi_{i+1}$ and then replace $\\xi_n$ with $p_n - p_0$\n",
    "- Choose $\\lambda$ so that $f(p_0 + \\lambda(p_n-p_0)$ is minimum and replace $p_0$ with $p_0 + \\lambda(p_n-p_0)$\n",
    "\n",
    "Essentially, the algorithm performs line searches and tries to find fruitful directions to search."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solvers"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Levenberg-Marquardt (Damped Least Squares)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Recall the least squares problem:\n",
    "    \n",
    "Given a set of data points $(x_i, y_i)$ where $x_i$'s are independent variables (in $\\mathbb{R}^n$ and the $y_i$'s are response variables (in $\\mathbb{R}$), find the parameter values of $\\beta$ for the model $f(x;\\beta)$ so that\n",
    "\n",
    "$$S(\\beta) = \\sum\\limits_{i=1}^m \\left(y_i - f(x_i;\\beta)\\right)^2$$\n",
    "is minimized.\n",
    "\n",
    "If we were to use Newton's method, our update step would look like:\n",
    "\n",
    "$$\\beta_{k+1} = \\beta_k - H^{-1}\\nabla S(\\beta_k)$$\n",
    "\n",
    "Gradient descent, on the other hand, would yield:\n",
    "\n",
    "$$\\beta_{k+1} = \\beta_k - \\gamma\\nabla S(\\beta_k)$$\n",
    "\n",
    "Levenberg-Marquardt adaptively switches between Newton's method and gradient descent.  \n",
    "\n",
    "$$\\beta_{k+1} = \\beta_k - (H + \\lambda I)^{-1}\\nabla S(\\beta_k)$$\n",
    "\n",
    "When $\\lambda$ is small, the update is essentially Newton-Gauss, while for $\\lambda$ large, the update is gradient descent."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Newton-Krylov"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The notion of a Krylov space comes from the Cayley-Hamilton theorem (CH).  CH states that a matrix $A$ satisfies its characteristic polynomial.  A direct corollary is that $A^{-1}$ may be written as a linear combination of powers of the matrix (where the highest power is $n-1$).  \n",
    "\n",
    "The Krylov space of order $r$ generated by an $n\\times n$ matrix $A$ and an $n$-dimensional vector $b$ is given by:\n",
    "\n",
    "$$\\mathcal{K}_r(A,b) = \\operatorname{span} \\, \\{ b, Ab, A^2b, \\ldots, A^{r-1}b \\}$$\n",
    "\n",
    "These are actually the subspaces spanned by the conjugate vectors we mentioned in Newton-CG, so, technically speaking, Newton-CG is a Krylov method.\n",
    "\n",
    "Now, the scipy.optimize newton-krylov solver is what is known as a 'Jacobian Free Newton Krylov'.  It is a very efficient algorithm for solving *large* $n\\times n$ non-linear systems.  We won't go into detail of the algorithm's steps, as this is really more applicable to problems in physics and non-linear dynamics."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## GLM Estimation and IRLS"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Recall generalized linear models are models with the following components:\n",
    "\n",
    "* A linear predictor $\\eta = X\\beta$\n",
    "* A response variable with distribution in the exponential family\n",
    "* An invertible 'link' function $g$ such that\n",
    "$$E(Y) = \\mu = g^{-1}(\\eta)$$\n",
    "\n",
    "We may write the log-likelihood:\n",
    "\n",
    "$$\\ell(\\eta) = \\sum\\limits_{i=1}^m (y_i \\log(\\eta_i) + (\\eta_i - y_i)\\log(1-\\eta_i) $$\n",
    "\n",
    "where $\\eta_i = \\eta(x_i,\\beta)$.\n",
    "\n",
    "Differentiating, we obtain:\n",
    "\n",
    "$$\\frac{\\partial L}{\\partial \\beta} = \\frac{\\partial \\eta}{\\partial \\beta}^T\\frac{\\partial L}{\\partial \\eta} = 0$$\n",
    "\n",
    "Written slightly differently than we have in the previous sections, the Newton update to find $\\beta$ would be:\n",
    "\n",
    "$$-\\frac{\\partial^2 L}{\\partial \\beta \\beta^T} \\left(\\beta_{k+1} -\\beta_k\\right) = \\frac{\\partial \\eta}{\\partial \\beta}^T\\frac{\\partial L}{\\partial \\eta}$$\n",
    "\n",
    "Now, if we compute:\n",
    "\n",
    "$$-\\frac{\\partial^2 L}{\\partial \\beta \\beta^T} = \\sum \\frac{\\partial L}{\\partial \\eta_i}\\frac{\\partial^2 \\eta_i}{\\partial \\beta \\beta^T} - \\frac{\\partial \\eta}{\\partial \\beta}^T \\frac{\\partial^2 L}{\\partial \\eta \\eta^T}  \\frac{\\partial \\eta}{\\partial \\beta}$$\n",
    "\n",
    "Taking expected values on the right hand side and noting:\n",
    "\n",
    "$$E\\left(\\frac{\\partial L}{\\partial \\eta_i} \\right) = 0$$\n",
    "\n",
    "and \n",
    "\n",
    "$$E\\left(-\\frac{\\partial^2 L}{\\partial \\eta \\eta^T} \\right) = E\\left(\\frac{\\partial L}{\\partial \\eta}\\frac{\\partial L}{\\partial \\eta}^T\\right) \\equiv A$$\n",
    "\n",
    "So if we replace the Hessian in Newton's method with its expected value, we obtain:\n",
    "\n",
    "$$\\frac{\\partial \\eta}{\\partial \\beta}^TA\\frac{\\partial \\eta}{\\partial \\beta}\\left(\\beta_{k+1} -\\beta_k\\right) = \\frac{\\partial \\eta}{\\partial \\beta}^T\\frac{\\partial L}{\\partial \\eta} $$\n",
    "\n",
    "Now, these actually have the form of the normal equations for a weighted least squares problem.  \n",
    "\n",
    "$$\\min_{\\beta_{k+1}}\\left(A^{-1}\\frac{\\partial L}{\\partial \\eta} + \\frac{\\partial \\eta}{\\partial \\beta}\\left(\\beta_{k+1} -\\beta_k\\right)\\right)^T A \\left(A^{-1}\\frac{\\partial L}{\\partial \\eta} + \\frac{\\partial \\eta}{\\partial \\beta}\\left(\\beta_{k+1} -\\beta_k\\right)\\right)$$\n",
    "\n",
    "$A$ is a weight matrix, and changes with iteration - thus this technique is *iteratively reweighted least squares*."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Constrained Optimization and Lagrange Multipliers"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Often, we want to optimize a function subject to a constraint or multiple constraints.  The most common analytical technique for this is called 'Lagrange multipliers'.  The theory is based on the following:\n",
    "\n",
    "If we wish to optimize a function $f(x,y)$ subject to the constraint $g(x,y)=c$, we are really looking for points at which the gradient of $f$ and the gradient of $g$ are in the same direction.  This amounts to:\n",
    "\n",
    "$$\\nabla_{(x,y)}f = \\lambda \\nabla_{(x,y)}g$$\n",
    "\n",
    "(often, this is written with a (-) sign in front of $\\lambda$). The 2-d problem above defines two equations in three unknowns.  The original constraint, $g(x,y)=c$ yields a third equation.  Additional constraints are handled by finding:\n",
    "\n",
    "$$\\nabla_{(x,y)}f = \\lambda_1 \\nabla_{(x,y)}g_1 + ... + \\lambda_k \\nabla_{(x,y)}g_k$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Lagrange Multipliers](./Lagrange_multiplier.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "\n",
    "The generalization to functions on $\\mathbb{R}^n$ is also trivial:\n",
    "\n",
    "$$\\nabla_{x}f = \\lambda \\nabla_{x}g$$"
   ]
  },
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