11_Projected_Gradient_Descent.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Projected Gradient-Descent"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "heading_collapsed": true
   },
   "source": [
    "## Überblick"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   },
   "source": [
    "Bisher haben wir Gradient-Descent für nicht restringierte \n",
    "Optimierungsprobleme untersucht.\n",
    "In diesem Abschnitt modifizieren wir Gradient-Descent so, dass wir es auch auf eine\n",
    "bestimmte Klasse von restringierten Optimierungsproblemen anwenden können. Dazu \n",
    "kombinieren wir Gradient-Descent mit einer geeigneten Projektion auf\n",
    "die zulässige Parametermenge $X\\subset\\mathbb{R}^d$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "heading_collapsed": true
   },
   "source": [
    "## Grundlagen"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hidden": true
   },
   "source": [
    "In den Beispielen oben haben wir bereits Projected-Gradient-Descent benutzt, um  für $f\\in C^1(\\mathbb{R}^d)$  restringierte Optimierungsproblem näherungsweise zu lösen:\n",
    "  \\begin{align*}\n",
    "  x_\\ast  &= \\mathrm{argmin}_{x\\in X}f(x), \\quad X\\subset\\mathbb{R}^d\\\\\n",
    "  y_{t+1} &= x_{t} - \\gamma_{t} f'(x_{t})\\\\\n",
    "  x_{t+1} &= \\mathrm{argmin}_{x\\in X}\\|x - y_{t+1}\\|\n",
    "  \\end{align*}\n",
    "  \n",
    "Ist $x_{t+1}$ überhaupt wohldefiniert (Existenz, Eindeutigkeit)?"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "**Satz:** Ist $V$ eine Hilbert-Raum, $\\emptyset \\neq X\\subset V$ konvex und abgeschlossen, dann existiert genau ein $y\\in X$ (die Projektion von $x$ auf $X$) mit\n",
    "  \\begin{equation*} \n",
    "  \\|x - y\\| = \\inf_{z\\in X}\\|z - y\\|. \n",
    "  \\end{equation*}\n",
    "  Für $x\\in V$ ist $y\\in X$ die zugehörige Projektion, genau dann wenn\n",
    "  \\begin{equation*} \n",
    "  (x-y, z-y) \\leq 0 \\quad \\forall z\\in X.\n",
    "  \\end{equation*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hidden": true
   },
   "source": [
    "**Beweis:**\n",
    "\n",
    "- Existenz von $y$:\n",
    "  \n",
    "  - sei $y_i\\in X$ eine Folge mit\n",
    "      \\begin{equation*} \n",
    "      \\lim_{i\\to\\infty}\\|x - y_i\\| = \\inf_{z\\in X}\\|x - z\\| =:d\n",
    "      \\end{equation*}\n",
    "    \n",
    "  - wegen $\\|v\\|^2 = (v,v)$ gilt die Parallelogrammgleichung\n",
    "      \\begin{equation*} \n",
    "      \\|u+v\\|^{2} + \\|u-v\\|^{2} = 2\\big(\\|u\\|^{2}+\\|v\\|^{2}\\big)\n",
    "      \\end{equation*}\n",
    "    also\n",
    "      \\begin{equation*} \n",
    "      \\|u-v\\|^{2}=2\\big(\\|u\\|^{2}+\\|v\\|^{2}\\big)-\\|u+v\\|^{2}\n",
    "      \\end{equation*}\n",
    "      \n",
    "  - setzen wir nun\n",
    "      \\begin{equation*} \n",
    "      u = y_i -x, \\quad v = y_j - x\n",
    "      \\end{equation*}\n",
    "    dann folgt\n",
    "      \\begin{equation*} \n",
    "      \\|y_{i}-y_{j}\\|^{2}=\n",
    "      2\\big(\\|y_{i}-x\\|^{2}+\\|y_{j}-x\\|^{2}\\big)-\\|y_{i}+y_{j}-2 x\\|^{2}\n",
    "      \\end{equation*}\n",
    "      \n",
    "  - wegen\n",
    "      \\begin{equation*} \n",
    "        \\|y_{i}+y_{j}-2 x\\|^{2} \n",
    "        =4\\big\\|\\underbrace{\\frac{y_{i}+y_{j}}{2}}_{\\in X}-x\\big\\|^2 \\geq 4 d^{2}\n",
    "      \\end{equation*}\n",
    "    gilt\n",
    "      \\begin{equation*} \n",
    "        \\|y_{i}-y_{j}\\|^{2}  \\leq 2\\big(\\|y_{i}-x\\|^{2}+\\|y_{j}-x\\|^{2}\\big)-4 d^{2}\n",
    "      \\end{equation*}\n",
    "      \n",
    "  - andererseits gibt es für alle $\\varepsilon > 0$ ein $N_\\varepsilon \\in  \\mathbb{N}$ mit\n",
    "      \\begin{equation*} \n",
    "      \\|y_{k}-x\\|^{2}  \\leq d^2 + \\varepsilon  \\quad \\forall k\\geq N_\\varepsilon \n",
    "      \\end{equation*}\n",
    "    und somit\n",
    "      \\begin{equation*} \n",
    "        \\|y_{i}-y_{j}\\|^{2}  \n",
    "        \\leq 2\\big(d^2 + \\varepsilon + d^2 + \\varepsilon \\big)-4 d^{2}\n",
    "        = 4 \\varepsilon\n",
    "        \\quad \\forall i,j\\geq N_\\varepsilon \n",
    "      \\end{equation*}\n",
    "      \n",
    "  - damit ist $y_i$ eine Cauchy-Folge\n",
    "    \n",
    "  - da $V$ vollständig ist gibt es ein $y\\in V$ mit $y_i \\xrightarrow{i\\to\\infty}y$\n",
    "    \n",
    "  - wegen $y_i \\in X$, $X$ abgeschlossen, ist dann auch $y\\in X$\n",
    "    \n",
    "- Eindeutigkeit von $y$:\n",
    "  \n",
    "  - für $y, \\tilde{y} \\in X$ gelte\n",
    "      \\begin{equation*} \n",
    "      d = \\inf_{z\\in X}\\|z - x\\| = \\|y - x\\| = \\|\\tilde{y} - x\\|\n",
    "      \\end{equation*}    \n",
    "    \n",
    "  - da $X$ konvex ist gilt\n",
    "      \\begin{equation*} \n",
    "      \\bar{y} = \\frac{y + \\tilde{y}}{2} \\in X \n",
    "      \\end{equation*}\n",
    "    und mit der Parallelogrammgleichung folgt\n",
    "      \\begin{equation*} \n",
    "        \\begin{aligned}\n",
    "        \\|\\bar{y} - x\\|^2 \n",
    "        &=\\big\\|\\frac{y+\\tilde{y}}{2}-x\\big\\|^{2} \\\\\n",
    "        &=\\frac{1}{4} \\| \\underbrace{y-x}_{u}+\\underbrace{\\tilde{y}-x}_{v} \\|^{2} \\\\\n",
    "        &=\\frac{1}{4}\\|u+v\\|^{2} \\\\\n",
    "        &=\\frac{1}{2}(\\|u\\|^{2}+\\|v\\|^{2})-\\frac{1}{4}\\|u-v\\|^{2} \\\\\n",
    "        &=\\frac{1}{2}(\\|y-x\\|^{2}+\\|\\tilde{y}-x\\|^{2})-\\frac{1}{4}\\|y-\\tilde{y}\\|^{2} \\\\\n",
    "        &=d^{2}-\\frac{1}{4}\\|y-\\tilde{y}\\|^{2}\n",
    "        \\end{aligned}\n",
    "      \\end{equation*}\n",
    "   \n",
    "  - für $y \\neq \\tilde{y}$ ist\n",
    "      \\begin{equation*} \n",
    "      \\|\\bar{y} - x\\|^2 < d^2\n",
    "      \\end{equation*}\n",
    "    was ein Widerspruch zu $d = \\inf_{z\\in X}\\|z - x\\|$ ist    \n",
    "\n",
    "- Ungleichung:\n",
    "  \n",
    "  - es sei $y\\in X$ der eindeutige Minimierer und $d = \\|y - x\\| = \\inf_{z\\in X}\\|z - x\\|$:\n",
    "    \n",
    "    - da $X$ konvex ist gilt für alle $z\\in X$ und alle $\\lambda \\in (0,1)$\n",
    "        \\begin{equation*} \n",
    "        (1-\\lambda)y + \\lambda z = y + \\lambda(z-y) \\in X\n",
    "        \\end{equation*}\n",
    "      \n",
    "    - damit folgt\n",
    "        \\begin{align*}\n",
    "        \\|y - x\\|^2 = d^2\n",
    "        & \\leq \\|y + \\lambda(z-y) - x\\|^2 \\\\\n",
    "        & = \\|y - x\\|^2 +  \\lambda^2\\|z - y\\|^2 + 2  \\lambda(y-x, z-y)\n",
    "        \\end{align*}\n",
    "      also\n",
    "        \\begin{equation*} \n",
    "        2  \\lambda(x-y, z-y) \\leq \\lambda^2\\|z - y\\|^2\n",
    "        \\end{equation*}\n",
    "      und wegen $\\lambda > 0$\n",
    "        \\begin{equation*} \n",
    "        (x-y, z-y) \\leq \\frac{\\lambda}{2}\\|z - y\\|^2\n",
    "        \\end{equation*}\n",
    "        \n",
    "    - durch Grenzübergang $\\lambda \\downarrow 0$ erhalten wir schließlich\n",
    "        \\begin{equation*} \n",
    "        (x-y, z-y) \\leq 0 \\quad \\forall z\\in X\n",
    "        \\end{equation*}\n",
    "        \n",
    "  - es sei nun $y\\in X$ mit $(x-y, z-y) \\leq 0 \\ \\forall z\\in X$:\n",
    "    \n",
    "    - für $z\\in X$, $z\\neq y$, folgt dann\n",
    "        \\begin{align*} \n",
    "        \\|x - z\\|^2\n",
    "        &= \\|x - y  +  y - z\\|^2\\\\\n",
    "        &= \\|x - y\\|^2  +  \\underbrace{\\|y - z\\|^2}_{> 0} \n",
    "        \\:\\underbrace{- 2(x-y, z-y)}_{\\geq 0} \\\\\n",
    "        &> \\|x - y\\|^2\n",
    "        \\end{align*}\n",
    "        \n",
    "    - somit ist $y$ Minimierer von $\\|x - z\\|$, $z\\in X$\n",
    "        \n",
    "$\\square$"
   ]
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   "source": [
    "Mit diesem Ergebnis machen die folgenden Definitionen Sinn.\n",
    "\n",
    "\n",
    "**Definition:** Ist $V$ eine Hilbert-Raum, $\\emptyset \\neq X\\subset V$ konvex und abgeschlossen, dann ist\n",
    "  \\begin{equation*} \n",
    "  d_X:V\\to \\mathbb{R}, \\quad d_X(x) = \\inf_{z\\in X}\\|z - x\\|\n",
    "  \\end{equation*}\n",
    "  der *Abstand* von $x$ zu $X$ und\n",
    "  \\begin{equation*} \n",
    "  \\Pi_X:V\\to X, \\quad \\Pi_X(x) = \\mathrm{argmin}_{z\\in X}\\|z - x\\|\n",
    "  \\end{equation*}\n",
    "  die *Projektion* von $x$ auf $X$."
   ]
  },
  {
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   "metadata": {
    "hidden": true
   },
   "source": [
    "**Bemerkung:**\n",
    "\n",
    "- wir lassen bei $d,\\Pi$ den Index $X$ weg\n",
    "  \n",
    "- $\\Pi$ ist wohldefiniert, wenn $V$ vollständig und die Norm auf $V$ strikt konvex ist\n",
    "  \n",
    "- wird die Norm von einem Skalarprodukt induziert, dann ist sie immer strikt konvex \n",
    "  \n",
    "- ist dies nicht der Fall, kann man einfache Gegenbeispiele konstruieren"
   ]
  },
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   "metadata": {
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   "source": [
    "**Beispiel:**\n",
    "\n",
    "  - betrachte\n",
    "      \\begin{equation*} \n",
    "      V = \\mathbb{R}^2, \n",
    "      \\quad \\|x\\|_\\infty = \\max(|x_1|,|x_2|),\n",
    "      \\end{equation*}\n",
    "\n",
    "  - $(V, \\|\\cdot\\|_\\infty)$ ist ein Banachraum, also ein vollständiger\n",
    "      normierter Vektorraum\n",
    "      \n",
    "  - für die konvexe Menge\n",
    "      \\begin{equation*} \n",
    "      X = [-1,1]\\times[-1,1]\n",
    "      \\end{equation*}\n",
    "    und den Punkt $x = (2,0)^T$ erhalten wir\n",
    "      \\begin{equation*} \n",
    "      d(x) = \\inf_{z\\in X}\\|z - x\\|_\\infty = \\inf_{z\\in X} \\max(|z_1 - 2|,|z_2|)\n",
    "      \\end{equation*}\n",
    "    \n",
    "  - wegen $z_1 \\in [-1,1]$ ist für alle $z\\in X$ \n",
    "      \\begin{equation*} \n",
    "      |z_1 - 2|\\geq 1,\n",
    "      \\end{equation*}\n",
    "    s.d. $d(x) \\geq 1$\n",
    "    \n",
    "  - andererseits gilt für alle $z$ mit $z_1=1$ und $z_2 \\in [-1,1]$\n",
    "      \\begin{equation*} \n",
    "      \\|z - x\\|_\\infty = \\max(|z_1 - 2|,|z_2|) = 1,\n",
    "      \\end{equation*}\n",
    "    der minimale Abstand $d(x)=1$ wird also für alle diese $z$ angenommen\n",
    "   \n",
    "  - $d(x)$ ist also nach wie vor wohldefiniert, $\\Pi(x)$ wegen der\n",
    "    fehlenden Eindeutigkeit dagegen nicht"
   ]
  },
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   "metadata": {
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   "source": [
    "Jetzt betrachten wir einige wichtige Eigenschaften von $d$ und $\\Pi$.\n",
    "\n",
    "\n",
    "**Lemma:** Sind $V,X,d,\\Pi$ wie oben definiert, dann gilt:\n",
    "\n",
    "- $d:V\\to \\mathbb{R}$ ist konvex\n",
    "  \n",
    "- $\\big(y - \\Pi(y), x - \\Pi(y) \\big) \\leq 0 \\quad \\forall x\\in X, \\ \\forall y\\in V$\n",
    "  \n",
    "- $\\|\\Pi(y) - \\Pi(x)\\| \\leq \\|y-x\\| \\quad \\forall x,y \\in V$\n",
    "  \n",
    "- $\\|x - \\Pi(y)\\|^2 + \\|y - \\Pi(y) \\|^2 \\leq \\|y-x\\|^2 \\quad \\forall x\\in X, \\ \\forall y\\in V$"
   ]
  },
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   "metadata": {
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   },
   "source": [
    "**Beweis:**\n",
    "\n",
    "- Teil 1:\n",
    "  \n",
    "  - $x,y\\in V$, $\\bar{x} = \\Pi(x)$, $\\bar{y} = \\Pi(y)$\n",
    "    \n",
    "  - $\\bar{x}, \\bar{y}\\in X$ und da $X$ konvex ist gilt \n",
    "      \\begin{equation*} \n",
    "      \\bar{z} = (1-\\lambda)\\bar{x} + \\lambda \\bar{y} \\in X \\quad \\forall \\lambda \\in [0,1]\n",
    "      \\end{equation*}\n",
    "      \n",
    "  - mit $u=(1-\\lambda)x + \\lambda y$ folgt\n",
    "      \\begin{align*}\n",
    "      d\\big((1-\\lambda)x + \\lambda y\\big)\n",
    "      & = \\inf_{z\\in X} \\|z - u\\| \\\\\n",
    "      & \\leq \\|u - \\bar{z}\\| \\\\\n",
    "      & = \\|(1-\\lambda)x + \\lambda y - (1-\\lambda)\\bar{x} - \\lambda \\bar{y}\\| \\\\\n",
    "      & = \\|(1-\\lambda)(x-\\bar{x}) + \\lambda (y-\\bar{y})\\| \\\\\n",
    "      & \\leq (1-\\lambda)\\|x-\\bar{x}\\| + \\lambda \\|y-\\bar{y}\\| \\\\\n",
    "      & = (1-\\lambda) d(x) + \\lambda d(y)\n",
    "      \\end{align*}\n",
    "\n",
    "- Teil 2:\n",
    "  \n",
    "  - nach dem Satz von oben gilt für $y\\in V$\n",
    "      \\begin{equation*} \n",
    "      \\bar{y} \\in X, \\quad \\|y-\\bar{y}\\|=\\inf _{x \\in X}\\|y-x\\| \n",
    "      \\quad \\Leftrightarrow \\quad\n",
    "      (y-\\bar{y}, x-\\bar{y}) \\leq 0 \\quad \\forall x \\in X\n",
    "      \\end{equation*}\n",
    "\n",
    "  - mit $\\bar{y}=\\Pi(y)$ folgt die Behauptung\n",
    "    \n",
    "- Teil 3:\n",
    "  \n",
    "  - es sei wieder $x,y\\in V$, $\\bar{x} = \\Pi(x)$, $\\bar{y} = \\Pi(y)$    \n",
    "    \n",
    "  - nach Teil 2 gilt\n",
    "      \\begin{equation*} \n",
    "      (x-\\bar{x}, z-\\bar{x}) \\leq 0,\n",
    "      \\quad\n",
    "      (y-\\bar{y}, z-\\bar{y}) \\leq 0 \n",
    "      \\quad \n",
    "      \\forall z \\in X\n",
    "      \\end{equation*}\n",
    "    \n",
    "  - mit $z = \\bar{y}$ folgt aus der ersten Ungleichung\n",
    "      \\begin{equation*} \n",
    "      (x-\\bar{x}, \\bar{y}-\\bar{x}) \\leq 0\n",
    "      \\quad\\Rightarrow\\quad\n",
    "      (\\bar{x}-x, \\bar{x}-\\bar{y}) \\leq 0\n",
    "      \\end{equation*}\n",
    "    und analog mit $z = \\bar{x}$ in der zweiten Ungleichung\n",
    "      \\begin{equation*} \n",
    "      (y-\\bar{y}, \\bar{x}-\\bar{y}) \\leq 0\n",
    "      \\end{equation*}\n",
    "    \n",
    "  - Addition der beiden Ungleichungen liefert\n",
    "      \\begin{equation*} \n",
    "      (\\bar{x}-x + y-\\bar{y}, \\bar{x}-\\bar{y}) \\leq 0\n",
    "      \\end{equation*}\n",
    "    also\n",
    "      \\begin{equation*} \n",
    "      -(x-y, \\bar{x}-\\bar{y}) + \\|\\bar{x}-\\bar{y}\\|^2 \\leq 0\n",
    "      \\end{equation*}\n",
    "      \n",
    "  - mit Cauchy-Schwarz folgt\n",
    "      \\begin{equation*} \n",
    "      \\|\\bar{x}-\\bar{y}\\|^2 \n",
    "      \\leq (x-y, \\bar{x}-\\bar{y})\n",
    "      \\leq \\|x-y\\| \\ \\|\\bar{x}-\\bar{y}\\|\n",
    "      \\end{equation*}\n",
    "    und somit\n",
    "      \\begin{equation*} \n",
    "      \\|\\bar{x}-\\bar{y}\\| \\leq \\|x-y\\|\n",
    "      \\end{equation*}\n",
    "      \n",
    "- Teil 4:\n",
    "  \n",
    "  - $\\forall v,w \\in V$ gilt\n",
    "      \\begin{equation*} \n",
    "      \\|v-w\\|^2 = (v-w, v-w) = \\|v\\|^2 + \\|w\\|^2 - 2(v,w)\n",
    "      \\end{equation*}\n",
    "    also\n",
    "      \\begin{equation*} \n",
    "      \\|v\\|^2 + \\|w\\|^2 - \\|v-w\\|^2 =  2(v,w)\n",
    "      \\end{equation*}\n",
    "      \n",
    "  - mit $v = x-\\bar{y}$, $w = y-\\bar{y}$ folgt\n",
    "      \\begin{equation*} \n",
    "      \\|x-\\bar{y}\\|^2 + \\|y-\\bar{y}\\|^2 - \\|x-y\\|^2\n",
    "      = 2(x-\\bar{y}, y-\\bar{y})\n",
    "      \\leq 0\n",
    "      \\end{equation*}\n",
    "      \n",
    "$\\square$"
   ]
  },
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   "metadata": {
    "heading_collapsed": true
   },
   "source": [
    "## Konvergenz"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hidden": true
   },
   "source": [
    "Wir werden jetzt analog zum einfachen Gradientenverfahren Konvergenzaussagen unter verschiedenen Voraussetzungen an $f$ beweisen. Dabei werden wir sehen, dass die asymptotischen Aussagen sich nicht verändern, d.h. die Projektion auf abgeschlossene konvexe Mengen hat keinen negativen Einfluss."
   ]
  },
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   },
   "source": [
    "### Vorüberlegungen"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   },
   "source": [
    "Für $f\\in C^1(\\mathbb{R}^d)$, $X\\subset\\mathbb{R}^d$ \n",
    "  konvex und abgeschlossen, \n",
    "  wenden wir auf das  restringierte Optimierungsproblem\n",
    "  \\begin{equation*} \n",
    "  x_\\ast  = \\mathrm{argmin}_{x\\in X}f(x)\n",
    "  \\end{equation*}\n",
    "  Projected-Gradient-Descent\n",
    "  \\begin{equation*} \n",
    "  y_{t+1} = x_{t} - \\gamma f'(x_{t}),\n",
    "  \\quad\n",
    "  x_{t+1} = \\Pi(y_{t+1})\n",
    "  \\end{equation*}\n",
    "  mit konstanter Schrittweite $\\gamma$ an, wobei $\\Pi$ die\n",
    "  Projektion in $\\mathbb{R}^d$ auf $X$ bezüglich $(\\cdot,\\cdot)_2$ ist.\n",
    "Wegen $y_{t+1} = x_t - \\gamma f'_t$  gilt\n",
    "  \\begin{equation*} \n",
    "  \\|y_{t+1}-x_{*}\\|_{2}^{2} \n",
    "  =\\|x_{t}-x_{*}\\|_{2}^{2}+\\gamma^{2}\\|f_{t}^{\\prime}\\|_{2}^{2}-2\n",
    "  \\gamma f_{t}^{\\prime}(x_{t}-x_{*})\n",
    "  \\end{equation*}\n",
    "  und somit\n",
    "  \\begin{equation*} \n",
    "  f'_{t}\\left(x_{t}-x_{*}\\right) \n",
    "  =\\frac{1}{2 \\gamma}\\left(\\gamma^{2}\\left\\|f_{t}^{\\prime}\\right\\|_{2}^{2}\n",
    "  +\\left\\|x_{t}-x_{*}\\right\\|_{2}^{2}-\\left\\|y_{t+1}-x_{*}\\right\\|_{2}^{2}\\right).\n",
    "  \\end{equation*}\n",
    "Der einzige Unterschied zu Gradient-Descent ist $y_{t+1}$ statt $x_{t+1}$\n",
    "  im letzten Term auf der rechten Seite.\n",
    "  \n",
    "Um diesen Term zu bearbeiten, greifen wir auf die Eigenschaften \n",
    "der Projektion $\\Pi$ zurück.\n",
    "Für $\\Pi$ gilt\n",
    "  \\begin{equation*} \n",
    "  \\|\\Pi(y) - \\Pi(x)\\| \\leq \\|y-x\\| \\quad \\forall x,y \\in V\n",
    "  \\end{equation*}\n",
    "bzw.\n",
    "  \\begin{equation*} \n",
    "  \\Pi(x_\\ast) = x_\\ast\n",
    "  \\end{equation*}\n",
    "und somit\n",
    "  \\begin{equation*} \n",
    "  \\|x_{t+1}-x_{*}\\|_{2}\n",
    "   = \\|\\Pi(y_{t+1}) - \\Pi(x_\\ast)\\|_2\n",
    "  \\leq  \\|y_{t+1}-x_{*}\\|_{2},\n",
    "  \\end{equation*}\n",
    "bzw.\n",
    "  \\begin{equation*} \n",
    "  f'_{t}\\left(x_{t}-x_{*}\\right) \n",
    "  \\leq\\frac{1}{2 \\gamma}\\left(\\gamma^{2}\\left\\|f_{t}^{\\prime}\\right\\|_{2}^{2}\n",
    "  +\\left\\|x_{t}-x_{*}\\right\\|_{2}^{2}-\\left\\|x_{t+1}-x_{*}\\right\\|_{2}^{2}\\right).\n",
    "  \\end{equation*}\n",
    "Wir erhalten also fast das selbe Ergebnis wie bei Gradient-Descent, nur\n",
    "  dass jetzt das \"$=$\" durch \"$\\leq$\" ersetzt wurde."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hidden": true
   },
   "source": [
    "Ist $f$ konvex, dann gilt\n",
    "  \\begin{equation*} \n",
    "  f(y) \\geq f(x) + f'(x)(y-x)\n",
    "  \\end{equation*}\n",
    "und mit $y=x_\\ast$, $x=x_t$\n",
    "  \\begin{equation*} \n",
    "  f_\\ast \\geq f_t + f'_t (x_\\ast-x_t)\n",
    "  \\end{equation*}\n",
    "bzw.\n",
    "  \\begin{align*}\n",
    "  0  \\leq f_t - f_\\ast\n",
    "  & \\leq f'_t(x_t - x_\\ast)\\\\\n",
    "  & = \\frac{1}{2 \\gamma}\\left(\\gamma^{2}\\left\\|f_{t}^{\\prime}\\right\\|_{2}^{2}\n",
    "  +\\left\\|x_{t}-x_{*}\\right\\|_{2}^{2}-\\left\\|y_{t+1}-x_{*}\\right\\|_{2}^{2}\\right)\\\\\n",
    "  & \\leq\\frac{1}{2 \\gamma}\\left(\\gamma^{2}\\left\\|f_{t}^{\\prime}\\right\\|_{2}^{2}\n",
    "  +\\left\\|x_{t}-x_{*}\\right\\|_{2}^{2}-\\left\\|x_{t+1}-x_{*}\\right\\|_{2}^{2}\\right).\n",
    "  \\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hidden": true
   },
   "source": [
    "### Lipschitz-Stetigkeit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hidden": true
   },
   "source": [
    "$f\\in C^1(\\mathbb{R}^d)$ sei konvex und zusätzlich Lipschitz-Stetigkeit \n",
    "  mit Konstante $L_f$, also\n",
    "  \\begin{equation*} \n",
    "  \\|f(y) - f(x)\\| \\leq L_f \\|y - x\\| \n",
    "  \\end{equation*}\n",
    "was äquivalent ist zu\n",
    "  \\begin{equation*} \n",
    "  \\|f'(x)\\| \\leq L_f .\n",
    "  \\end{equation*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hidden": true
   },
   "source": [
    "Wie bei Gradient-Descent folgt\n",
    "  \\begin{align*}\n",
    "  \\sum_{t=0}^{T-1} (f_t - f_\\ast)\n",
    "  &\\leq \n",
    "  \\sum_{t=0}^{T-1} f'_t(x_t - x_\\ast)\\\\\n",
    "  &\\leq \\frac{\\gamma}{2}T L_f^2 \n",
    "  + \\frac{1}{2\\gamma} \n",
    "  \\big( \n",
    "  \\underbrace{\\|x_{0}-x_{*}\\|_{2}^{2}}_{e_0^2}\n",
    "  -\n",
    "  \\underbrace{\\|x_{T}-x_{*}\\|_{2}^{2}}_{\\geq 0}\n",
    "  \\big)\\\\\n",
    "  &\\leq \\frac{\\gamma T L_f^2}{2} + \\frac{e_0^2}{2\\gamma}.\n",
    "  \\end{align*}\n",
    "  \n",
    "\n",
    "**Satz:** $f:\\mathbb{R}^d\\to \\mathbb{R}$, konvex, $C^1$, L-stetig mit Konstante $L_f$\n",
    "  und es existiere $x_\\ast = \\mathrm{argmin}_{x\\in\\mathbb{R}^d}f(x)$.\n",
    "  Mit $\\gamma = \\frac{c}{T^\\omega}$, $\\omega\\in(0,1)$, gilt\n",
    "  \\begin{align*} \n",
    "  \\min_{t=0,\\ldots,T-1}(f_t - f_\\ast)\n",
    "  &\\leq \\frac{1}{T} \\sum_{t=0}^{T-1} (f_t - f_\\ast)\\\\\n",
    "  &= \\mathcal{O}\\Big(\\big(\\frac{1}{T}\\big)^{\\min(\\omega,1-\\omega)}\\Big)\n",
    "  \\quad \n",
    "  \\text{für}\n",
    "  \\quad\n",
    "  T\\to\\infty.\n",
    "  \\end{align*}\n",
    "  Die optimale Ordnung ist $\\frac{1}{2}$ für $\\omega=\\frac{1}{2}$.\n",
    "  Mit $e_0 = \\|x_0 - x_\\ast\\|$, $\\gamma = \\frac{e_0}{L_f\\sqrt{T}}$ gilt außerdem\n",
    "  \\begin{equation*} \n",
    "  \\min_{t=0,\\ldots,T-1}(f_t - f_\\ast)\n",
    "  \\leq \\frac{1}{T} \\sum_{t=0}^{T-1} (f_t - f_\\ast)\n",
    "  \\leq \\frac{L_f e_0}{\\sqrt{T}}.\n",
    "  \\end{equation*}  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hidden": true
   },
   "source": [
    "### $L$-Glattheit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hidden": true
   },
   "source": [
    "Wir leiten zuerst eine neue Version des Descent-Lemmas her.\n",
    "Dazu starten wir mit der $L$-Glattheit von $f$\n",
    "  \\begin{equation*} \n",
    "  f_{t+1} \\leq f_{t}+f_{t}^{\\prime}(x_{t+1}-x_{t})+\\frac{L}{2}\\|x_{t+1}-   x_{t}\\|_{2}^{2}\n",
    "  \\end{equation*}\n",
    "benutzen\n",
    "  \\begin{equation*} \n",
    "  y_{t+1} = x_t - \\gamma f'_t\n",
    "  \\end{equation*}\n",
    "bzw.\n",
    "  \\begin{equation*} \n",
    "  f'_t = \\frac{1}{\\gamma}(x_t - y_{t+1})\n",
    "  \\end{equation*}\n",
    "und erhalten\n",
    "  \\begin{equation*} \n",
    "  f_{t+1} \n",
    "  \\leq f_{t}\n",
    "    - \\frac{1}{\\gamma}\n",
    "    (\\underbrace{y_{t+1} - x_t}_u)^T\n",
    "    (\\underbrace{x_{t+1} - x_t}_v) \n",
    "    + \\frac{L}{2}\\|x_{t+1}-   x_{t}\\|_{2}^{2}.\n",
    "  \\end{equation*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hidden": true
   },
   "source": [
    "Wegen $2u^Tv = 2(u,v)_2 = \\|u\\|_2^2 + \\|v\\|_2^2 - \\|u-v\\|_2^2$ folgt\n",
    "  \\begin{align*}\n",
    "  f_{t+1} \n",
    "  & \\leq f_{t}\n",
    "    - \\frac{1}{2 \\gamma}\n",
    "    \\big(\n",
    "    \\|y_{t+1}-x_{t}\\|_{2}^{2}\n",
    "    + \\|x_{t+1}-x_{t}\\|_{2}^{2}\n",
    "    - \\|y_{t+1}-x_{t+1}\\|_{2}^{2}\n",
    "    \\big)      \n",
    "    +\\frac{L}{2}\\|x_{t+1}-x_{t}\\|_{2}^{2} \\\\\n",
    "  &= f_{t}\n",
    "      -\\frac{1}{2 \\gamma}\n",
    "      \\big(\n",
    "      \\gamma^{2}\\|f_{t}^{\\prime}\\|_{2}^{2}\n",
    "      +\\|x_{t+1}-x_{t}\\|_{2}^{2}\n",
    "      -\\|y_{t+1}-x_{t+1}\\|_{2}^{2}\n",
    "      \\big) \n",
    "      +\\frac{L}{2}\\|x_{t+1}-x_{t}\\|_{2}^{2} \\\\\n",
    "  &= f_{t}\n",
    "      -\\frac{\\gamma}{2}\\|f'_{t}\\|_{2}^{2} \n",
    "      +\\frac{1}{2 \\gamma}\\|y_{t+1}-x_{t+1}\\|_{2}^{2} \n",
    "      +\\frac{1}{2}\\big(L-\\frac{1}{\\gamma}\\big)\\|x_{t+1}-x_{t}\\|_{2}^{2}.\n",
    "  \\end{align*}\n",
    "  \n",
    "Für $0<\\gamma \\leq \\frac{1}{L}$ ist $L-\\frac{1}{\\gamma}\\leq 0$ und wir erhalten das folgende Ergebnis.\n",
    "\n",
    "\n",
    "**Descent-Lemma:** Ist $f:\\mathbb{R}^d\\to \\mathbb{R}$ $L$-glatt und\n",
    "  $0 < \\gamma \\leq \\frac{1}{L}$, dann gilt für Projected-Gradient-Descent\n",
    "  \\begin{equation*} \n",
    "  f_{t+1} \n",
    "  \\leq\n",
    "  f_{t}\n",
    "      -\\frac{\\gamma}{2}\\|f'_{t}\\|_{2}^{2} \n",
    "      +\\frac{1}{2 \\gamma}\\|y_{t+1}-x_{t+1}\\|_{2}^{2} .\n",
    "  \\end{equation*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hidden": true
   },
   "source": [
    "**Bemerkung:**\n",
    "\n",
    "- bei Gradient-Descent war\n",
    "    \\begin{equation*} \n",
    "    f_{t+1} \n",
    "    \\leq f_{t}-\\beta\\|f_{t}^{\\prime}\\|_2^{2},\n",
    "    \\quad\n",
    "    \\beta = \\gamma\\big(1-\\frac{\\gamma L}{2}\\big),\n",
    "    \\end{equation*}\n",
    "  so dass die $f_t$ bei $0<\\gamma < 2/L$ monoton fallend waren\n",
    "    \n",
    "- bei Projected-Gradient-Descent haben wir auf der rechten Seite den\n",
    "  nicht-negativen Zusatzterm\n",
    "    \\begin{equation*} \n",
    "    \\frac{1}{2 \\gamma}\\|y_{t+1}-x_{t+1}\\|_{2}^{2}, \n",
    "    \\end{equation*}\n",
    "  so dass wir zunächst nichts über die Monotonie der $f_t$ aussagen können\n",
    "    \n",
    "- man kann aber trotzdem zeigen, dass die $f_t$ auch hier \n",
    "  (wenn $\\gamma$ kurz genug ist) monoton fallen (Übung)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hidden": true
   },
   "source": [
    "Aus den Vorüberlegungen wissen wir ($f$ konvex)\n",
    "  \\begin{align*}\n",
    "  0  \\leq f_t - f_\\ast\n",
    "  & \\leq f'_t(x_t - x_\\ast)\\\\\n",
    "  & = \\frac{1}{2 \\gamma}(\\gamma^{2}\\|f_{t}^{\\prime}\\|_{2}^{2}\n",
    "  +\\|x_{t}-x_{*}\\|_{2}^{2}-\\|y_{t+1}-x_{*}\\|_{2}^{2}).\n",
    "  \\end{align*}\n",
    "\n",
    "Laut Descent-Lemma ist\n",
    "  \\begin{equation*} \n",
    "  \\|f'_{t}\\|_{2}^{2} \n",
    "  \\leq \\frac{2}{\\gamma}\n",
    "  \\big(\n",
    "  f_{t} - f_{t+1} \n",
    "      +\\frac{1}{2 \\gamma}\\|y_{t+1}-x_{t+1}\\|_{2}^{2}\n",
    "  \\big)\n",
    "  \\end{equation*}\n",
    "und oben eingesetzt folgt\n",
    "  \\begin{equation*} \n",
    "  f_t - f_\\ast\n",
    "  \\leq \n",
    "  f_{t} - f_{t+1} \n",
    "  +\\frac{1}{2 \\gamma}\n",
    "  \\big(\n",
    "  \\|y_{t+1}-x_{t+1}\\|_{2}^{2} + \\|x_{t}-x_{*}\\|_{2}^{2}-\\|y_{t+1}-x_{*}\\|_{2}^{2}\n",
    "  \\big).\n",
    "  \\end{equation*}\n",
    "  \n",
    "Wegen der Eigenschaft\n",
    "  \\begin{equation*} \n",
    "  \\|x - \\Pi(y)\\|^2 + \\|y - \\Pi(y) \\|^2 \\leq \\|y-x\\|^2 \n",
    "  \\quad \\forall x\\in X, \\ \\forall y\\in V\\end{equation*}\n",
    "der Projektion $\\Pi$ und $x_\\ast \\in X$ folgt\n",
    "mit $y = y_{t+1}$, $x=x_\\ast$\n",
    "  \\begin{align*}\n",
    "  \\|y_{t+1}-x_{t+1}\\|_{2}^2\n",
    "  &= \\|y_{t+1} - \\Pi(y_{t+1})\\|_2^2 \\\\\n",
    "  &\\leq \\|y_{t+1} - x_\\ast\\|_2^2  - \\|x_\\ast - \\Pi(y_{t+1})\\|_2^2 \\\\\n",
    "  &  =  \\|y_{t+1} - x_\\ast\\|_2^2  - \\|x_{t+1} - x_\\ast\\|_2^2 \n",
    "  \\end{align*}\n",
    "so dass\n",
    "  \\begin{equation*} \n",
    "  f_t - f_\\ast\n",
    "  \\leq \n",
    "  f_{t} - f_{t+1} \n",
    "  +\\frac{1}{2 \\gamma}\n",
    "  \\big(\n",
    "  \\|x_{t}-x_{*}\\|_{2}^{2}-\\|x_{t+1}-x_{*}\\|_{2}^{2}\n",
    "  \\big).\n",
    "  \\end{equation*}\n",
    "\n",
    "Durch Summation erhalten wir\n",
    "  \\begin{equation*} \n",
    "  \\sum_{t=0}^{T-1}(f_t - f_\\ast)\n",
    "  \\leq\n",
    "  f_0 - f_T   \n",
    "  +\\frac{1}{2 \\gamma}\n",
    "  \\big(\n",
    "  \\|x_{0}-x_{*}\\|_{2}^{2}-\\|x_{T}-x_{*}\\|_{2}^{2}\n",
    "  \\big).\n",
    "  \\end{equation*}\n",
    "  \n",
    "Benutzen wir jetzt noch\n",
    "  \\begin{equation*} \n",
    "  f_0 - f_T = (f_0 - f_\\ast) - (f_T - f_\\ast),\n",
    "  \\quad\n",
    "  \\|x_{T}-x_{*}\\|_{2}^{2} \\geq 0,\n",
    "  \\end{equation*}\n",
    "dann ist\n",
    "  \\begin{equation*} \n",
    "  \\sum_{t=1}^{T}(f_t - f_\\ast)\n",
    "  \\leq \\frac{1}{2 \\gamma} \\|x_{0}-x_{*}\\|_{2}^{2}.\n",
    "  \\end{equation*}  \n",
    "  \n",
    "Nach dem Descent-Lemma und der daran anschließenden Bemerkung gilt\n",
    "  $f_{t+1} \\leq f_t$ und somit\n",
    "  \\begin{equation*} \n",
    "  f_T - f_\\ast\n",
    "  \\leq\\frac{1}{T}\\sum_{t=1}^{T}(f_t - f_\\ast)\n",
    "  \\leq \\frac{1}{2 \\gamma T} \\|x_{0}-x_{*}\\|_{2}^{2}.\n",
    "  \\end{equation*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hidden": true
   },
   "source": [
    "**Satz:** $f:\\mathbb{R}^d\\to \\mathbb{R}$, konvex, $L$-glatt mit Konstante $L$\n",
    "  und es existiere $x_\\ast = \\mathrm{argmin}_{x\\in X}f(x)$. \n",
    "Ist  $0 < \\gamma \\leq \\frac{1}{L}$ dann gilt für Projected-Gradient-Descent\n",
    "  \\begin{equation*} \n",
    "  f_{T}-f_\\ast \n",
    "  \\leq \\frac{1}{2 \\gamma T} \\|x_{0}-x_{*}\\|_{2}^{2}\n",
    "  = \\mathcal{O}\\big( \\frac{1}{T} \\big)\n",
    "  \\quad \\text{für}\\quad  T\\to\\infty .\n",
    "  \\end{equation*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hidden": true
   },
   "source": [
    "### $\\mu$-Konvexität"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hidden": true
   },
   "source": [
    "Ist $f$ $\\mu$-konvex mit $\\mu\\geq 0$, dann gilt\n",
    "  \\begin{equation*} \n",
    "  f_\\ast \\geq f_t + f'_t(x_\\ast-x_t) + \\frac{\\mu}{2} \\|x_\\ast-x_t\\|_2^2\n",
    "  \\end{equation*}\n",
    "bzw.\n",
    "  \\begin{equation*} \n",
    "  f_t - f_\\ast \\leq f'_t(x_t - x_\\ast)   - \\frac{\\mu}{2} \\|x_t - x_\\ast\\|_2^2.\n",
    "  \\end{equation*}\n",
    "Aus den Vorüberlegungen wissen wir\n",
    "  \\begin{equation*} \n",
    "  f'_{t}\\left(x_{t}-x_{*}\\right) \n",
    "  =\\frac{1}{2 \\gamma}\\left(\\gamma^{2}\\left\\|f_{t}^{\\prime}\\right\\|_{2}^{2}\n",
    "  +\\left\\|x_{t}-x_{*}\\right\\|_{2}^{2}-\\left\\|y_{t+1}-x_{*}\\right\\|_{2}^{2}\\right).\n",
    "  \\end{equation*}\n",
    "Wegen\n",
    "  \\begin{equation*} \n",
    "  \\|x - \\Pi(y)\\|^2 + \\|y - \\Pi(y)\\|^2 \\leq \\|x - y\\|^2\n",
    "  \\quad\n",
    "  \\forall x\\in X\n",
    "  \\quad \n",
    "  \\forall y\\in V\n",
    "  \\end{equation*}\n",
    "folgt mit $y = y_{t+1}$, $x=x_\\ast$, $\\Pi(y_{t+1})=x_{t+1}$\n",
    "  \\begin{equation*} \n",
    "  \\|x_\\ast - x_{t+1}\\|_{2}^2 + \\|y_{t+1} - x_{t+1}\\|_{2}^2\n",
    "  \\leq  \\|x_\\ast -  y_{t+1}\\|_{2}^2,\n",
    "  \\end{equation*}\n",
    "also\n",
    "  \\begin{equation*} \n",
    "  - \\|y_{t+1} - x_\\ast\\|_{2}^2 \n",
    "  \\leq - \\|x_{t+1} - x_\\ast\\|_{2}^2 - \\|y_{t+1} - x_{t+1}\\|_{2}^2.\n",
    "  \\end{equation*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hidden": true
   },
   "source": [
    "Oben eingesetzt erhalten wir nun\n",
    "  \\begin{align*}\n",
    "  f_t - f_\\ast \n",
    "  &\\leq f'_t(x_t - x_\\ast) - \\frac{\\mu}{2} \\|x_t - x_\\ast\\|_2^2\\\\\n",
    "  &\\leq \\frac{1}{2 \\gamma}\n",
    "    \\big(\n",
    "    \\gamma^{2} \\|f'_{t}\\|_{2}^{2}  \n",
    "    + \\|x_{t}-x_{*}\\|_{2}^{2}\n",
    "    - \\|x_{t+1}-x_{*}\\|_{2}^{2}\n",
    "    - \\|y_{t+1} - x_{t+1}\\|_{2}^2\n",
    "    \\big)\n",
    "    \\\\\n",
    "    &\\quad - \\frac{\\mu}{2} \\|x_t - x_\\ast\\|_2^2,\n",
    "  \\end{align*}\n",
    "also\n",
    "  \\begin{align*} \n",
    "  2\\gamma (f_t - f_\\ast)\n",
    "  & \\leq\n",
    "  \\gamma^{2} \\|f'_{t}\\|_{2}^{2}  \n",
    "    + (1-\\gamma\\mu)\\|x_{t}-x_{*}\\|_{2}^{2}\n",
    "    \\\\\n",
    "    &\\quad\n",
    "    - \\|x_{t+1}-x_{*}\\|_{2}^{2}\n",
    "    - \\|y_{t+1} - x_{t+1}\\|_{2}^2\n",
    "  \\end{align*}\n",
    "bzw.\n",
    "  \\begin{align*} \n",
    "  \\|x_{t+1}-x_{*}\\|_{2}^{2}\n",
    "  &\\leq\n",
    "  2\\gamma (f_\\ast - f_t)\n",
    "  + \\gamma^{2} \\|f'_{t}\\|_{2}^{2}\n",
    "  - \\|y_{t+1} - x_{t+1}\\|_{2}^2\n",
    "  \\\\\n",
    "  &\\quad\n",
    "  + (1-\\gamma\\mu)\\|x_{t}-x_{*}\\|_{2}^{2}.\n",
    "  \\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hidden": true
   },
   "source": [
    "Die ersten drei Terme auf der rechten Seite schätzen wir wieder mit dem Descent-Lemma ab.\n",
    "Wegen\n",
    "  \\begin{equation*} \n",
    "  f_{t+1} \n",
    "  \\leq\n",
    "  f_{t}\n",
    "      -\\frac{\\gamma}{2}\\|f'_{t}\\|_{2}^{2} \n",
    "      +\\frac{1}{2 \\gamma}\\|y_{t+1}-x_{t+1}\\|_{2}^{2} \n",
    "  \\end{equation*}\n",
    "ist\n",
    "  \\begin{equation*} \n",
    "  f_\\ast - f_t\n",
    "  \\leq\n",
    "  f_{t+1} - f_t\n",
    "  \\leq\n",
    "  -\\frac{\\gamma}{2}\\|f'_{t}\\|_{2}^{2} \n",
    "      +\\frac{1}{2 \\gamma}\\|y_{t+1}-x_{t+1}\\|_{2}^{2} \n",
    "  \\end{equation*}\n",
    "und somit\n",
    "  \\begin{equation*} \n",
    "  \\|x_{t+1}-x_{*}\\|_{2}^{2}\n",
    "  \\leq (1-\\gamma\\mu)\\|x_{t}-x_{*}\\|_{2}^{2}.\n",
    "  \\end{equation*}\n",
    "  \n",
    "Analog zu Gradient-Descent haben wir damit folgendes Ergebnis gezeigt.\n",
    "\n",
    "\n",
    "**Satz:** $f:\\mathbb{R}^d\\to \\mathbb{R}$, $\\mu$-konvex mit $\\mu>0$, $L$-glatt \n",
    "mit Konstante $L$ und es existiere $x_\\ast = \\mathrm{argmin}_{x\\in X}f(x)$.\n",
    "Für  $0 < \\gamma \\leq \\frac{1}{L}$ folgt\n",
    "  \\begin{equation*} \n",
    "  \\|x_{t+1}-x_{*}\\|_{2}^2 \\leq \\rho \\|x_{t}-x_{*}\\|_{2}^2\n",
    "  \\end{equation*}\n",
    "und\n",
    "  \\begin{equation*} \n",
    "  f_T - f_\\ast \\leq \\frac{L}{2} \\rho^T \\|x_{0}-x_{*}\\|_{2}^2\n",
    "  \\end{equation*}\n",
    "mit\n",
    "  \\begin{equation*} \n",
    "  \\rho = 1-\\gamma\\mu \\in [0,1) .\n",
    "  \\end{equation*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Projektionsoperatoren"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In den vorherigen Abschnitten haben wir gesehen, dass wir für Projected-Gradient-Descent\n",
    "exakt die selben asymptotischen Resultate erhalten wie bei gewöhnlichem Gradient-Descent.\n",
    "Das sieht zunächst sehr angenehm aus,\n",
    "für den praktischen Einsatz muss man allerdings bedenken, dass in jedem\n",
    "Schritt eine Projektion\n",
    "  \\begin{equation*} \n",
    "  x_{t+1} = \\Pi_X(y_{t+1}),\n",
    "  \\quad \\Pi_X(x) = \\mathrm{argmin}_{z\\in X}\\|z - x\\|\n",
    "  \\end{equation*}\n",
    "auf die abgeschlossene, konvexe Menge $X$ zu berechnen ist.\n",
    "  \n",
    "Das Berechnen von $\\Pi(x)$ ist also selbst ein restringiertes konvexes\n",
    "  Optimierungsproblem mit einer  \"einfachen\" konvexen Zielfunktion\n",
    "  $x\\to\\|z - x\\|$.\n",
    "Für allgemeine abgeschlossene und konvexe Menge $X$ kann die Lösung\n",
    "  dieses Problems extrem schwierig werden, für spezielle $X$, die in der Praxis häufig auftauchen, kann\n",
    "  man $\\Pi_X$ dagegen explizit angeben.\n",
    "  \n",
    "Wir untersuchen einige $X\\subset \\mathbb{R}^d$ nun genauer. \n",
    "Dabei betrachten wir jeweils Projektionen bezüglich des euklidischen\n",
    "Skalarprodukts $(\\cdot,\\cdot)_2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Kugel in der 2-Norm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- wir betrachten\n",
    "\\begin{equation*} \n",
    "X \n",
    "= \\bar{B}_r(0)_{\\|\\cdot\\|_2}\n",
    "= \\big\\{ x \\ \\big| \\ x\\in\\mathbb{R}^d,\\ \\|x\\|_2\\leq r \\big\\}\n",
    "\\end{equation*}\n",
    "\n",
    "- $X$ ist offensichtlich konvex und abgeschlossen\n",
    "\n",
    "- die Projektion auf $X$ ist\n",
    "\\begin{equation*} \n",
    "\\Pi_X(x) =\n",
    "\\begin{cases}\n",
    "x & \\|x\\|_2 \\leq  r \\\\\n",
    "\\frac{x}{\\|x\\|_2} r & \\|x\\|_2 > r \n",
    "\\end{cases}\n",
    "\\end{equation*}\n",
    "\n",
    "- mit $\\bar{x} = \\Pi_X(x)$ ist zu zeigen, dass\n",
    "\\begin{equation*} \n",
    "(x - \\bar{x}, z - \\bar{x})_2 \\leq 0 \\quad \\forall z \\in X\n",
    "\\end{equation*}\n",
    "\n",
    "- für $x\\in X$ ist $\\bar{x}=x$, also\n",
    "\\begin{equation*} \n",
    "(x - \\bar{x}, z - \\bar{x})_2 \n",
    "= (x - x, z - x)_2 \n",
    "= 0 \\quad \\forall z \\in X\n",
    "\\end{equation*}\n",
    "\n",
    "- für $x\\notin X$, also $\\|x\\|_2>r$, ist $\\bar{x}=\\frac{x}{\\|x\\|_2} r$ und somit\n",
    "\\begin{align*}\n",
    "(x - \\bar{x}, z - \\bar{x})_2 \n",
    "&   =  \\big( x - \\frac{x}{\\|x\\|_2} r, z - \\frac{x}{\\|x\\|_2} r \\big)_2 \\\\\n",
    "&   =  (x,z)_2 + r^2 - \\big( x , \\frac{x}{\\|x\\|_2} r \\big)_2 - \\big(\\frac{x}{\\|x\\|_2} r , z\\big)_2 \\\\\n",
    "&   =  (x,z)_2 + r^2 - \\|x\\|_2 r - \\frac{r}{\\|x\\|_2} (x, z)_2 \\\\\n",
    "&   =  \\big(\\underbrace{1 - \\frac{r}{\\|x\\|_2}}_{>0}\\big)(x,z)_2 + r^2 - r \\|x\\|_2 \\\\\n",
    "& \\leq \\big(1 - \\frac{r}{\\|x\\|_2}\\big)\\|x\\|_2 \\underbrace{\\|z\\|_2}_{\\leq r} + r^2 - r \\|x\\|_2 \\\\\n",
    "& \\leq r \\|x\\|_2 -  r^2 + r^2 - r \\|x\\|_2\\\\\n",
    "&   =  0\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Nicht negativer Kegel"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- der Kegel\n",
    "    \\begin{equation*} \n",
    "    X\n",
    "    = \\big\\{ x \\ \\big| \\ x\\in\\mathbb{R}^d,\\ x_i \\geq 0, \\  i=1,\\ldots,d \\big\\}\n",
    "    \\end{equation*}\n",
    "  ist konvex und abgeschlossen\n",
    "\n",
    "- die Projektion auf $X$ ist definiert durch\n",
    "    \\begin{equation*} \n",
    "    \\bar{x} = \\Pi_X(x),\n",
    "    \\quad\n",
    "    \\bar{x}_i\n",
    "    =\n",
    "    \\begin{cases}\n",
    "    x_i & x_i \\geq 0 \\\\\n",
    "     0  & x_i < 0\n",
    "    \\end{cases}\n",
    "    \\end{equation*}\n",
    "\n",
    "- für $z\\in X$ folgt dann\n",
    "    \\begin{align*}\n",
    "    (x - \\bar{x}, z - \\bar{x})_2 \n",
    "    &   =  \\sum_{i=1}^d (x_i - \\bar{x}_i) (z_i - \\bar{x}_i) \\\\\n",
    "    &   =  \\sum_{x_i<0} \\underbrace{x_i}_{<0}\\ \\underbrace{z_i}_{\\geq 0} \\\\\n",
    "    & \\leq  0\n",
    "    \\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Kugel in der 1-Norm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- wir betrachten\n",
    "  \\begin{equation*} \n",
    "  X_r \n",
    "  = B_r(0)_{\\|\\cdot\\|_1}\n",
    "  = \\big\\{ x \\ \\big| \\ x\\in\\mathbb{R}^d,\\ \\|x\\|_1 = \\sum_{i=1}^d |x_i|\\leq r \\big\\}\n",
    "  \\end{equation*}\n",
    "\n",
    "- $X_r$ ist damit konvex und abgeschlossen und die Projektion $\\Pi$ auf $X_r$ \n",
    "  bezüglich der Norm $\\|\\cdot\\|_2$ ist wohldefiniert:\n",
    "  \\begin{equation*} \n",
    "  \\Pi(v) = \\mathrm{argmin}_{\\|y\\|_1\\leq r} \\|v - y\\|_2\n",
    "  \\end{equation*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- wegen $\\Pi(v) = v$ für $\\|v\\|_1\\leq r$ reicht es, wenn wir ab jetzt Vektoren\n",
    "  \\begin{equation*} \n",
    "  v\\in\\mathbb{R}^d,\n",
    "  \\quad\n",
    "  \\|v\\|_1 > r\n",
    "  \\end{equation*}\n",
    "  betrachten"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- für diese gilt $\\|\\Pi(v)\\|_1 = r$, d.h. die Projektion liegt auf dem Rand von $X_r$:\n",
    "\n",
    "  - Annahme: $\\|\\Pi(v)\\|_1 < r$\n",
    "  \n",
    "  - wegen $\\|v\\|_1 > r > \\|\\Pi(v)\\|_1$ gilt \n",
    "      \\begin{equation*} \n",
    "      \\lambda = \\frac{r - \\|\\Pi(v)\\|_1}{\\|v\\|_1 -\\|\\Pi(v)\\|_1} \\in (0,1)\n",
    "      \\end{equation*}  \n",
    "\n",
    "  - für\n",
    "      \\begin{equation*} \n",
    "      x = (1-\\lambda) \\Pi(v) + \\lambda v \n",
    "      \\end{equation*}\n",
    "    erhalten wir\n",
    "      \\begin{align*}\n",
    "      \\|x\\|_1\n",
    "      &\\leq (1-\\lambda) \\|\\Pi(v)\\|_1 + \\lambda \\|v\\|_1 \\\\\n",
    "      & = \\|\\Pi(v)\\|_1 + \\lambda \\big(\\|v\\|_1 -\\|\\Pi(v)\\|_1\\big)\\\\\n",
    "      & = \\|\\Pi(v)\\|_1 + \\frac{r - \\|\\Pi(v)\\|_1}{\\|v\\|_1 -\\|\\Pi(v)\\|_1} \\big(\\|v\\|_1 -\\|\\Pi(v)\\|_1\\big)\\\\\n",
    "      & = r\n",
    "      \\end{align*}\n",
    "  \n",
    "  - andererseits gilt\n",
    "      \\begin{equation*} \n",
    "      x - v = (1-\\lambda)\\big(\\Pi(v)-v\\big), \\quad \\lambda\\in(0,1) \n",
    "      \\end{equation*}\n",
    "    und somit\n",
    "      \\begin{equation*} \n",
    "      \\|x - v\\|_2 = (1-\\lambda) \\|\\Pi(v)-v\\|_2 < \\|\\Pi(v)-v\\|_2,\n",
    "      \\end{equation*}\n",
    "    was ein Widerspruch zu\n",
    "      \\begin{equation*} \n",
    "      \\Pi(v) = \\mathrm{argmin}_{\\|y\\|_1\\leq r} \\|v - y\\|_2\n",
    "      \\end{equation*}\n",
    "    ist"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- wir werden jetzt die Menge der zu betrachtenden Vektoren $v$ weiter einschränken\n",
    "\n",
    "- ist $L:\\mathbb{R}^d\\to\\mathbb{R}^d$ eine lineare Abbildung mit\n",
    "  \\begin{equation*} \n",
    "  \\|Lw\\|_1 = \\|w\\|_1,\n",
    "  \\quad\n",
    "  \\|Lw\\|_2 = \\|w\\|_2\n",
    "  \\quad \\forall w\\in \\mathbb{R}^d\n",
    "  \\end{equation*}\n",
    "  dann ist $L$ umkehrbar und es gilt für $x = \\Pi(w)$\n",
    "  \\begin{equation*} \n",
    "  Lx = \\Pi(Lw)\n",
    "  \\end{equation*}\n",
    "  bzw.\n",
    "  \\begin{equation*} \n",
    "  x = L^{-1}\\Pi(Lw)\n",
    "  \\end{equation*}  \n",
    "  denn\n",
    "  \\begin{align*}\n",
    "  \\| Lw - Lx\\|_2\n",
    "  &= \\| L(w - x)\\|_2 \\\\\n",
    "  &= \\| w - x\\|_2 \\\\\n",
    "  &= \\min_{\\|y\\|_1\\leq r} \\|w - y\\|_2 \\\\\n",
    "  &= \\min_{\\|y\\|_1\\leq r} \\|Lw - Ly\\|_2 \\\\\n",
    "  &= \\min_{\\|L^{-1}z\\|_1\\leq r} \\|Lw - z\\|_2 \\\\\n",
    "  &= \\min_{\\|z\\|_1\\leq r} \\|Lw - z\\|_2 \\\\\n",
    "  \\end{align*}  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- die Abbildung\n",
    "  \\begin{equation*} \n",
    "  Sw = (s_1 w_1,\\ldots,s_d w_d)^T,\n",
    "  \\quad\n",
    "  s_i = \\pm 1\n",
    "  \\end{equation*}\n",
    "  besitzt diese Eigenschaften:\n",
    "  \n",
    "  - für $v\\in\\mathbb{R}^d$,\n",
    "    \\begin{equation*} \n",
    "    s_i = \n",
    "    \\begin{cases}\n",
    "    1  & v_i \\geq 0\\\\\n",
    "    -1 & v_i<0\n",
    "    \\end{cases},\n",
    "    \\end{equation*}\n",
    "    folgt dann\n",
    "    $Sv = (|v_1|,\\ldots,|v_d|)^T=|v|$ (komponentenweise) und\n",
    "    \\begin{equation*} \n",
    "    x = \\Pi(v) = S^{-1}\\Pi(Sv) = S^{-1}\\Pi(|v|),\n",
    "    \\end{equation*}\n",
    "    so dass es genügt, die Projektion für Vektoren $v$ mit $v_i\\geq 0$ zu betrachten"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- die Abbildung\n",
    "  \\begin{equation*} \n",
    "  Pw = (w_{\\pi(1)},\\ldots,w_{\\pi(d)})^T\n",
    "  \\end{equation*}\n",
    "  wobei $\\pi$ eine beliebige Permutation der Indizes\n",
    "  $1,\\ldots,d$ ist, besitzt ebenfalls die oben beschriebenen Eigenschaften:\n",
    "  \n",
    "  - für $v\\in\\mathbb{R}^d$ betrachten wir jetzt eine Permutation $\\pi$,\n",
    "    die die Komponenten von $v$ absteigend sortiert, also\n",
    "    \\begin{equation*} \n",
    "    v_{\\pi(1)} \\geq v_{\\pi(2)} \\geq \\ldots \\geq v_{\\pi(d)}\n",
    "    \\end{equation*} \n",
    "    und erhalten\n",
    "    \\begin{equation*} \n",
    "    x = \\Pi(v) = P^{-1}\\Pi(Pv),\n",
    "    \\end{equation*}\n",
    "    so dass es genügt, die Projektion für Vektoren $v$ zu betrachten,\n",
    "    deren Komponenten absteigend sortiert sind"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- wir müssen jetzt also nur noch Vektoren $v\\in \\mathbb{R}^d$ betrachten mit\n",
    "    \\begin{equation*} \n",
    "    \\|v\\|_1 > r,\n",
    "    \\quad\n",
    "    v_1 \\geq v_2 \\geq \\ldots \\geq v_d \\geq 0\n",
    "    \\end{equation*}\n",
    "  und wissen bereits, dass\n",
    "    \\begin{equation*} \n",
    "    \\|\\Pi(v)\\|_1 = r\n",
    "    \\end{equation*}\n",
    "  ist"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- unter diesen Voraussetzungen gilt für $x = \\Pi(v)$ auch $x_i\\geq 0$:\n",
    "\n",
    "  - Annahme: es existiere ein $x_k<0$\n",
    "  \n",
    "  - mit\n",
    "      \\begin{equation*} \n",
    "      y = (x_1, \\ldots, x_{k-1}, -x_k, x_{k+1},\\ldots, x_d)^T\n",
    "      \\end{equation*}\n",
    "    ist $\\|y\\|_1 = \\|x\\|_1 = r$ und wegen $x_k < 0 \\leq v_k$\n",
    "      \\begin{align*}\n",
    "      \\|v - y\\|_2^2\n",
    "      &= \\sum_{i=1}^d (v_i - y_i)^2 \\\\\n",
    "      &= (v_k + x_k)^2 + \\sum_{i\\neq k} (v_i - x_i)^2 \\\\\n",
    "      &< (v_k - x_k)^2 + \\sum_{i\\neq k} (v_i - x_i)^2 \\\\\n",
    "      &= \\|v - x\\|_2^2\n",
    "      \\end{align*}\n",
    "    was ein Widerspruch zu $x=\\Pi(v)$ ist\n",
    "  \n",
    "  \n",
    "- somit folgt unter diesen Voraussetzungen an $v$\n",
    "  \\begin{align*} \n",
    "  \\Pi(v) \n",
    "  & = \\mathrm{argmin}_{x\\in Y_r}\\|v - x\\|_2,\n",
    "  \\\\\n",
    "  Y_r &= \\Big\\{x \\ \\big| \\ x\\in\\mathbb{R}^d, \\ x_i\\geq 0, \\ i=1,\\ldots,d,\\ \\sum_{i=1}^d x_i = r \\Big\\},\n",
    "  \\end{align*}\n",
    "  wobei $Y_r$ wieder konvex ist"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- wir zeigen jetzt, dass ein eindeutiger Index $p \\in \\{1,\\ldots,d\\}$ existieren muss\n",
    "  mit $x_i>0$ für $i\\leq p$ und $x_i=0$ für $i>p$:\n",
    "\n",
    "  - $x = \\Pi(v)$ minimiert die konvexe differenzierbare Funktion $f(y) = \\frac{1}{2}\\|y - v\\|_2^2$\n",
    "    über der konvexen Menge $Y_r$, weshalb\n",
    "    \\begin{equation*} \n",
    "    0 \\leq f'(x)(y-x) = (x - v)^T(y-x) \\quad \\forall y\\in Y_r\n",
    "    \\end{equation*}\n",
    "    gelten muss, also\n",
    "    \\begin{equation*} \n",
    "    0 \\leq \\sum_{i=1}^d (x_i - v_i)(y_i-x_i)\\quad \\forall y\\in Y_r\n",
    "    \\end{equation*}\n",
    "    \n",
    "  - nach oben wissen wir, dass $\\|x\\|_1 = r > 0$, $x_i \\geq 0$, $i=1,\\ldots,d$, ist, \n",
    "    d.h. es existiert ein Index $k$ mit $x_k>0$\n",
    "    \n",
    "  - nehmen wir nun an, dass ein Index $j$ existiert mit $x_j=0$ und $x_{j+1} > 0$\n",
    "  \n",
    "  - definieren wir für $\\varepsilon>0$ \n",
    "    \\begin{equation*} \n",
    "    y = (x_1, \\ldots, x_{j-1}, \\varepsilon, x_{j+1} - \\varepsilon, x_{j+2}, \\ldots, x_d)^T\n",
    "    \\end{equation*}\n",
    "    dann ist $y_i \\geq 0$, $i = 1,\\ldots,d$, $\\|y\\|_1 = \\|x\\|_1 = r$, also $y\\in Y_r$, und\n",
    "    \\begin{align*}\n",
    "    \\sum_{i=1}^d (x_i - v_i)(y_i-x_i)\n",
    "    &= (x_j - v_j)(y_j - x_j) + (x_{j+1} - v_{j+1})(y_{j+1} - x_{j+1}) \\\\\n",
    "    &= - v_j\\varepsilon - (x_{j+1} - v_{j+1})\\varepsilon \\\\\n",
    "    &= \\varepsilon (\\underbrace{v_{j+1} - v_j}_{\\leq 0} - \\underbrace{x_{j+1}}_{>0}) \\\\\n",
    "    & < 0\n",
    "    \\end{align*}\n",
    "    was ein Widerspruch zu\n",
    "    \\begin{equation*} \n",
    "    0 \\leq \\sum_{i=1}^d (x_i - v_i)(y_i-x_i)\\quad \\forall y\\in Y_r\n",
    "    \\end{equation*}\n",
    "    ist\n",
    "    \n",
    "  - damit kann der Fall $x_j=0$, $x_{j+1}>0$ nicht auftreten\n",
    "    \n",
    "  - wegen $\\|x\\|_1 = r > 0$ muss $x\\neq 0$ sein, weshalb es einen Index $p$ gibt mit $x_i>0$ für $i\\leq p$ und $x_i = 0$ für $i > p$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- für $x = x(p) =\\Pi(v)$ gilt \n",
    "  \\begin{align*} \n",
    "  x_i \n",
    "  & = \n",
    "  \\begin{cases}\n",
    "  v_i - c_p > 0 & i=1,\\ldots,p \\\\\n",
    "  0 & i=p+1,\\ldots,d\n",
    "  \\end{cases},\n",
    "  \\\\\n",
    "  c_p &= \\frac{1}{p}\\big(\\sum_{i=1}^p v_i  - r \\big)\n",
    "  \\end{align*}\n",
    "\n",
    "  - Annahme: es existiere ein Index $j<p$ mit $x_j - v_j \\neq x_{j+1}-v_{j+1}$\n",
    "  \n",
    "    - ohne Einschränkung können wir $x_j - v_j < x_{j+1}-v_{j+1}$ annehmen\n",
    "\n",
    "    - wegen $j<p$ ist $x_j>0$, so dass für $0<\\varepsilon\\leq x_j$ \n",
    "    \\begin{equation*} \n",
    "    y = (x_1, \\ldots, x_{j-1}, x_j - \\varepsilon, \n",
    "         x_{j+1} + \\varepsilon, x_{j+2}, \\ldots, x_d)^T \n",
    "    \\in Y_r\n",
    "    \\end{equation*}\n",
    "\n",
    "    - analog zum letzten Abschnitt erhalten wir\n",
    "    \\begin{align*}\n",
    "    (x - v)^T(y - x)\n",
    "    &= (x_j - v_j)(y_j - x_j) + (x_{j+1} - v_{j+1})(y_{j+1} - x_{j+1}) \\\\\n",
    "    &= (x_j - v_j)\\varepsilon - (x_{j+1} - v_{j+1})\\varepsilon \\\\\n",
    "    & < 0\n",
    "    \\end{align*}\n",
    "    was erneut ein Widerspruch zu\n",
    "    \\begin{equation*} \n",
    "    0 \\leq (x - v)^T(y - x)\\quad \\forall y\\in Y_r\n",
    "    \\end{equation*}\n",
    "    ist\n",
    "    \n",
    "  - damit ist also \n",
    "    \\begin{equation*} \n",
    "    x_i = \n",
    "    \\begin{cases}\n",
    "    v_i - c_p > 0 & i=1,\\ldots,p \\\\\n",
    "    0 & i=p+1,\\ldots,d\n",
    "    \\end{cases}\n",
    "    \\end{equation*}\n",
    "    \n",
    "  - wegen $x\\in Y_r$ gilt \n",
    "   \\begin{align*} \n",
    "   r \n",
    "   &= \\|x\\|_1 \n",
    "   = \\sum_{i=1}^d |x_i| \n",
    "   = \\sum_{i=1}^p x_i\n",
    "   \\\\\n",
    "   &= \\sum_{i=1}^p (v_i - c_p)\n",
    "   \\\\\n",
    "   &= \\big(\\sum_{i=1}^p v_i\\big) - p c_p\n",
    "   \\end{align*}\n",
    "   und somit\n",
    "   \\begin{equation*} \n",
    "   c_p = \\frac{1}{p}\\big(\\sum_{i=1}^p v_i  - r\\big)\n",
    "   \\end{equation*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- zum Schluss muss jetzt noch der Index $p$ bestimmt werden\n",
    "\n",
    "- es gilt $\\Pi(v)=x(p_\\ast)$ für \n",
    "  \\begin{equation*} \n",
    "  p_\\ast = \\mathrm{argmax}_{p\\in\\{1,\\ldots,d\\}}\\{v_p - c_p > 0\\}\n",
    "  \\end{equation*}\n",
    "  \n",
    "  - für jedes $p\\in\\{1,\\ldots,d\\}$ ist\n",
    "    \\begin{align*} \n",
    "    x_i \n",
    "    & = \n",
    "    \\begin{cases}\n",
    "    v_i - c_p > 0 & i=1,\\ldots,p \\\\\n",
    "    0 & i=p+1,\\ldots,d\n",
    "    \\end{cases},\n",
    "    \\\\\n",
    "    c_p \n",
    "    & = \n",
    "    \\frac{1}{p}\\big(\\sum_{i=1}^p v_i  - r \\big)\n",
    "    \\end{align*}\n",
    "    eindeutige Lösung von\n",
    "    \\begin{equation*} \n",
    "    \\mathrm{argmin}_{z\\in Z_p}\\|z - v\\|_2^2\n",
    "    \\end{equation*}\n",
    "    mit\n",
    "    \\begin{equation*} \n",
    "    Z_p = \\big\\{ z \\ \\big | \\ z\\in\\mathbb{R}^d, z_i\\geq 0, \\ z_i = 0 \\ \\forall i>p, \n",
    "                \\ \\|z\\|_1 = \\sum_{i=1}^p z_i = r \\big\\} \\subset Y_r,\n",
    "    \\end{equation*}\n",
    "    denn $Z_p$ ist abgeschlossen und konvex, $f(z) =  \\|z - v\\|_2^2$ ist konvex \n",
    "    und für alle\n",
    "    $z \\in Z_p$ gilt\n",
    "    \\begin{align*} \n",
    "    f'(x)(z - x)\n",
    "    & = (x - v)^T(z - x)\n",
    "    \\\\\n",
    "    & = \\sum_{i=1}^p (x_i - v_i)(z_i - x_i)\n",
    "    \\\\\n",
    "    &= -c_p \\sum_{i=1}^p (z_i - x_i)\n",
    "    \\\\\n",
    "    &= 0\n",
    "    \\end{align*}\n",
    "    \n",
    "  - Annahme: $p < p_\\ast$\n",
    "  \n",
    "    - dann ist\n",
    "     \\begin{align*} \n",
    "     Z_p \\ni x(p) \n",
    "       = \\Pi(v)\n",
    "       & = \\mathrm{argmin}_{y\\in Y_r}\\|y - v\\|_2^2 \n",
    "       \\\\\n",
    "       &= \\mathrm{argmin}_{z\\in Z_p}\\|z - v\\|_2^2\n",
    "     \\end{align*}\n",
    "     bzw.\n",
    "     \\begin{equation*} \n",
    "     Z_{p_\\ast} \\ni x(p_\\ast) \n",
    "       = \\mathrm{argmin}_{z\\in Z_{p_\\ast}}\\|z - v\\|_2^2\n",
    "     \\end{equation*}\n",
    "    - nach Definition ist $Z_p \\subset Z_{p_\\ast} \\subset Y_r$\n",
    "      und alle Mengen sind konvex und abgeschlossen, so dass\n",
    "      alle Minima eindeutig sind\n",
    "      \n",
    "    - somit ist $x(p) = x(p_\\ast)$\n",
    "    \n",
    "    - nach Konstruktion von $x(p), x(p_\\ast)$ gilt wegen $p < p_\\ast$\n",
    "      \\begin{equation*} \n",
    "      x(p)_{p+1} = 0,\n",
    "      \\quad\n",
    "      x(p_\\ast)_{p+1} \\ne 0,\n",
    "      \\end{equation*}\n",
    "      was zu einem Widerspruch führt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- zusammengefasst erhalten wir damit folgenden Algorithmus zur Berechnung von $x = \\Pi(v)$:\n",
    "\n",
    "  - $v\\in\\mathbb{R}^d$ gegeben\n",
    "  \n",
    "  - ist $\\|v\\|_1 \\leq r$ dann ist $x = v$\n",
    "  \n",
    "  - $\\|v\\|_1 > r$:\n",
    "  \n",
    "    - setze\n",
    "      \\begin{equation*} \n",
    "      w = PSv,\n",
    "      \\end{equation*}\n",
    "      d.h. $w$ enthält die Beträge der Komponenten von $v$ absteigend sortiert\n",
    "    \n",
    "    - bestimme\n",
    "      \\begin{align*} \n",
    "      p_\\ast = \\mathrm{argmax}_{p\\in\\{1,\\ldots,d\\}}\n",
    "      \\big\\{\n",
    "      w_p - \\frac{1}{p}\\big(\\sum_{i=1}^p w_i  - r\\big) > 0\n",
    "      \\big\\}\n",
    "      \\end{align*}\n",
    "      \n",
    "    - berechne $y$ als\n",
    "      \\begin{align*} \n",
    "      y_i \n",
    "      &= \n",
    "      \\begin{cases}\n",
    "      w_i - c_{p_\\ast} > 0 & i=1,\\ldots,p_\\ast \\\\\n",
    "      0 & i=p_\\ast+1,\\ldots,d\n",
    "      \\end{cases},\n",
    "      \\\\\n",
    "      c_p \n",
    "      &= \\frac{1}{p_\\ast}\\big(\\sum_{i=1}^{p_\\ast} w_i  - r \\big)\n",
    "      \\end{align*}\n",
    "      \n",
    "    - damit folgt\n",
    "      \\begin{equation*} \n",
    "      x = S^{-1}P^{-1}y\n",
    "      \\end{equation*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- eine Anwendung in $\\mathbb{R}^2$ bzw. $\\mathbb{R}^3$ liefert folgende Resultate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "code_folding": [
     6,
     36
    ],
    "execution": {
     "iopub.execute_input": "2021-09-24T09:33:38.987272Z",
     "iopub.status.busy": "2021-09-24T09:33:38.981259Z",
     "iopub.status.idle": "2021-09-24T09:33:39.869435Z",
     "shell.execute_reply": "2021-09-24T09:33:39.869704Z"
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def l1pro(v, r = 1.0):\n",
    "    # Projektion auf L1-Kugel mit Radius r bzgl. eukl. Skalarpr.\n",
    "    \n",
    "    if np.linalg.norm(v, ord=1) > r:\n",
    "        d = v.shape[0]\n",
    "\n",
    "        # S, Vorzeichen merken fuer S_inv\n",
    "        s = np.ones(v.shape)\n",
    "        s[v < 0] = -1\n",
    "\n",
    "        # PS anwenden\n",
    "        w  = s * v\n",
    "        ii = w.argsort()[::-1]\n",
    "        w  = w[ii]\n",
    "\n",
    "        # Projektion\n",
    "        p  = np.arange(d) + 1\n",
    "        cp = (w.cumsum() - r)/p\n",
    "        u  = w - cp\n",
    "        jj = np.where(u > 0)[0].max()\n",
    "\n",
    "        ps  = p[jj]\n",
    "        cps = cp[jj]\n",
    "\n",
    "        y = np.zeros(v.shape)\n",
    "        y[:ps] = w[:ps] - cps\n",
    "\n",
    "        # P_inv und S_inv anwenden\n",
    "        x = np.zeros(y.shape)\n",
    "        # P_inv\n",
    "        x[ii] = y\n",
    "        # S_inv = S\n",
    "        x = s*x\n",
    "    else:\n",
    "        x = v\n",
    "        \n",
    "    return x\n",
    "\n",
    "\n",
    "np.random.seed(17)\n",
    "plt.figure(figsize=(5,5))\n",
    "\n",
    "d = 2\n",
    "n = 500\n",
    "l = 1.0\n",
    "r = 2.0\n",
    "\n",
    "for k in range(n):\n",
    "    w = np.random.randn(d)\n",
    "    w = w/np.linalg.norm(w)*(l+r)\n",
    "    \n",
    "    x = l1pro(w, r)\n",
    "\n",
    "    plt.plot(*w, 'b.', markersize = 1)\n",
    "    plt.plot(*np.c_[x, w], 'b', alpha = 0.05)\n",
    "    plt.plot(*x, 'r.', markersize = 1)\n",
    "\n",
    "plt.axis('equal')\n",
    "plt.gca().set_xticks([])\n",
    "plt.gca().set_yticks([]);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-09-24T09:33:39.882533Z",
     "iopub.status.busy": "2021-09-24T09:33:39.880738Z",
     "iopub.status.idle": "2021-09-24T09:33:44.604865Z",
     "shell.execute_reply": "2021-09-24T09:33:44.605113Z"
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x576 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#%matplotlib notebook\n",
    "%matplotlib inline\n",
    "\n",
    "plt.figure(figsize=(8,8))\n",
    "plt.axes(projection='3d')\n",
    "ax = plt.gca()\n",
    "\n",
    "np.random.seed(17)\n",
    "\n",
    "d = 3\n",
    "n = 1000\n",
    "l = 1.0\n",
    "r = 2.0\n",
    "\n",
    "for k in range(n):\n",
    "    w = np.random.randn(d)\n",
    "    w = w/np.linalg.norm(w)*(l + r)\n",
    "    \n",
    "    x = l1pro(w, r)\n",
    "    \n",
    "    ax.plot(*w, 'b.', markersize = 1)\n",
    "    ax.plot(*np.c_[x, w], 'b', alpha = 0.05)\n",
    "    ax.plot(*x, 'r.', markersize = 1)\n",
    "\n",
    "#plt.axis('equal')\n",
    "lr = l + r\n",
    "\n",
    "ax.set_xlim(-lr, lr)\n",
    "ax.set_ylim(-lr, lr)\n",
    "ax.set_zlim(-lr, lr)\n",
    "ax.set_xticks([])\n",
    "ax.set_yticks([])\n",
    "ax.set_zticks([]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Beispiel Tomographie"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Bei den Regularisierungsverfahren hatten wir für unser Tomographieproblem unter anderem die Lasso-Methode kennen gelernt.\n",
    "Dort bestimmt man\n",
    "\\begin{equation*} \n",
    "\\mathrm{argmin}_{w\\in\\mathbb{R}^n}\\frac{1}{2m}\\| X w - y\\|_2^2 + \\alpha \\|w\\|_1,\n",
    "\\end{equation*}\n",
    "wobei $\\alpha > 0$ ein vom Benutzer zu wählender Regularisierungsparameter ist.\n",
    "\n",
    "Dieses nicht restringierte Problem kann in das\n",
    "restringierte Problem\n",
    "\\begin{equation*} \n",
    "\\mathrm{argmin}_{\\|w\\|_1 \\leq r}\\frac{1}{2m}\\| X w - y\\|_2^2 \n",
    "\\end{equation*}\n",
    "überführt werden, wobei die Rolle des Regularisierungsparameters\n",
    "$\\alpha$ jetzt von $r$ übernommen wird:\n",
    "\n",
    "- für $r\\to\\infty$ ist $\\|w\\|_1 \\leq r$ immer weniger\n",
    "  einschränkend, man \"regularisiert\" immer weniger,\n",
    "  was $\\alpha\\to 0$ entspricht\n",
    "  \n",
    "- für $r\\to 0$ bleibt nur $w = 0$ als Lösung, d.h.\n",
    "  man \"regularisiert\" maximal,\n",
    "  was $\\alpha\\to \\infty$ entspricht\n",
    "  \n",
    "Auf das restringierte Problem wenden wir Projected-Gradient-Descent\n",
    "an und benutzen dabei die Projektion auf $B_r(0)_{\\|\\cdot\\|_1}$\n",
    "aus dem letzten Abschnitt."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Zum Vergleich berechnen wir\n",
    "zunächst eine Rekonstruktion ohne Regularisierung."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-09-24T09:33:44.609834Z",
     "iopub.status.busy": "2021-09-24T09:33:44.609486Z",
     "iopub.status.idle": "2021-09-24T09:33:46.200470Z",
     "shell.execute_reply": "2021-09-24T09:33:46.199943Z"
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "np.random.seed(17)\n",
    "\n",
    "from DatenNotebooks.xrtomo12 import *\n",
    "\n",
    "def plotReko(b):\n",
    "    n = int(np.sqrt(b.shape[0]))\n",
    "    \n",
    "    plt.pcolor(b.reshape(n,n))\n",
    "    plt.axis('equal')\n",
    "    plt.axis('off')\n",
    "    \n",
    "    \n",
    "from sklearn import linear_model    \n",
    "X, y, _ = tomo(delta = 0.01)\n",
    "\n",
    "modell = linear_model.LinearRegression(fit_intercept = False)\n",
    "modell.fit(X, y.ravel())\n",
    "\n",
    "w = modell.coef_\n",
    "plotReko(w)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Die Lasso-Implementierung aus Scikit-Learn liefert das folgende Ergebnis."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-09-24T09:33:46.204713Z",
     "iopub.status.busy": "2021-09-24T09:33:46.204327Z",
     "iopub.status.idle": "2021-09-24T09:33:46.292771Z",
     "shell.execute_reply": "2021-09-24T09:33:46.293066Z"
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "alpha = 9.867860e-06,   ||w||_1 = 20.221146\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "lassocv = linear_model.LassoCV(fit_intercept = False, cv = 5)\n",
    "\n",
    "lassocv.fit(X, y)\n",
    "\n",
    "w = lassocv.coef_\n",
    "plotReko(w)\n",
    "\n",
    "print('alpha = {:e},   ||w||_1 = {:f}'.format(lassocv.alpha_, np.linalg.norm(w, 1)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Mit Projected Gradient erhalten wir ein sehr ähnliches Resultat."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-09-24T09:33:46.300045Z",
     "iopub.status.busy": "2021-09-24T09:33:46.299688Z",
     "iopub.status.idle": "2021-09-24T09:33:46.841335Z",
     "shell.execute_reply": "2021-09-24T09:33:46.841830Z"
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "r = 20.000000\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def l(w):\n",
    "    return 0.5*((X.dot(w) - y)**2).sum()\n",
    "    \n",
    "def l1(w):\n",
    "    return X.T.dot(X.dot(w) - y)\n",
    "\n",
    "def GDp(w0, l1, pro, alpha = 1.0, nit = 10):\n",
    "    w  = w0.copy()\n",
    "    w = pro(w)\n",
    "    \n",
    "    ww = [w]\n",
    "    \n",
    "    for k in range(nit):\n",
    "        w = w - alpha * l1(w)\n",
    "        w = pro(w)\n",
    "        ww.append(w)\n",
    "    \n",
    "    return ww\n",
    "\n",
    "\n",
    "w0 = np.zeros(X.shape[1])\n",
    "\n",
    "r = 20\n",
    "pro1 = lambda w : l1pro(w, r)\n",
    "\n",
    "ww = GDp(w0, l1, pro = pro1, alpha = 0.5, nit = 500)\n",
    "\n",
    "plt.figure()\n",
    "plt.semilogy(list(map(l, ww)))\n",
    "plt.xlabel('$k$')\n",
    "plt.ylabel('$||Xw-y||_2^2$', rotation=0)\n",
    "\n",
    "w = ww[-1]\n",
    "plt.figure()\n",
    "plotReko(w)\n",
    "\n",
    "print('r = {:f}'.format(r))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Zusammenfassung"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Für Projected-Gradient-Descent bei restringierten Optimierungsproblemen mit \n",
    "  $X\\in\\mathbb{R}^d$ abgeschlossen und konvex haben wir das selbe\n",
    "  Konvergenzverhalten wie im nicht restringierten Fall nachgewiesen:\n",
    "\n",
    "- $f$ konvex und Lipschitz-stetig, $\\gamma = \\frac{c}{\\sqrt{T}}$, $c>0$:\n",
    "    \\begin{equation*} \n",
    "    \\min_{t=0,\\ldots,T-1}(f_t - f_\\ast) \\leq \\varepsilon\n",
    "    \\quad \\Rightarrow\\quad\n",
    "    T = \\mathcal{O}\\big(\\frac{1}{\\varepsilon^2}\\big)\n",
    "    \\end{equation*}\n",
    "    \n",
    "- $f$ konvex und $L$-glatt, $0 < \\gamma < \\frac{2}{L}$:\n",
    "    \\begin{equation*} \n",
    "    f_{T}-f_\\ast \\leq \\varepsilon\n",
    "    \\quad \\Rightarrow\\quad\n",
    "    T =\\mathcal{O}\\big( \\frac{1}{\\varepsilon} \\big)\n",
    "    \\end{equation*}  \n",
    "\n",
    "- $f$ $\\mu$-konvex mit $\\mu>0$ und $L$-glatt, $0 < \\gamma \\leq \\frac{1}{L}$:\n",
    "    \\begin{equation*} \n",
    "    f_{T}-f_\\ast \\leq \\varepsilon\n",
    "    \\quad \\Rightarrow\\quad\n",
    "    T = \\mathcal{O}\\Big( \\log\\big(\\frac{1}{\\varepsilon}\\big) \\Big)\n",
    "    \\end{equation*}\n",
    "    \n",
    "Das Berechnen der Projektion $\\Pi_X$ auf $X\\in\\mathbb{R}^d$ abgeschlossen und konvex\n",
    "  kann extrem aufwendig sein.\n",
    "Für einige spezielle Mengen $X$ die in der Praxis häufig auftreten \n",
    "  lässt sich $\\Pi_X$ einfach berechnen."
   ]
  }
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