2_automatic_differentiation.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "e82669e5",
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots, LaTeXStrings, BenchmarkTools"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "49bca20c",
   "metadata": {},
   "source": [
    "## The need for differentiation\n",
    "\n",
    "Computing derivatives is a ubiquitous operation in scientific computing. Examples:\n",
    "- Optimization\n",
    "    - Variational inference\n",
    "    - Parameter fitting (Deep Learning)\n",
    "- Sensitivity analysis of Differential Equations\n",
    "- more..."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7c854f15",
   "metadata": {},
   "source": [
    "## The alternatives\n",
    "\n",
    "You have a program computing some quantity (running toy example?). Now you want to also compute the derivative of the output with respect to some parameter. What do you do?\n",
    "- **Symbolic differentiation** - Requires computation by hand. It's error-prone, and formulas explode in size because of the chain rule. Exact but expensive\n",
    "- **Finite differences** -> Tradeoff between numerical cancellation and stepsize. Cheap but inaccurate"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "00f1b385",
   "metadata": {},
   "source": [
    "## Introduction to Automatic Differentiation\n",
    "\n",
    "Think of a computer program as a (arbitrary complicated) composition of functions\n",
    "$$f(p) = (f_n \\circ f_{n-1} \\circ \\dots \\circ f_2 \\circ f_1)(p)$$\n",
    "which we suppose here to be nice enough (differentiable). Here $p$ is some parameter.\n",
    "\n",
    "We are interested in computing $f'$. By the chain rule:\n",
    "$$f'(p) = f_n'(f_{n-1}(\\ldots f_1(p)))\\cdot f_{n-1}'(f_{n-2}(\\ldots f_1(p))) \\cdots f_2'(f_1(p))\\cdot f_1'(p) $$\n",
    "\n",
    "Look at this formula and notice that a lot of stuff can be re-used (same idea as dynamic programming!).\n",
    "\n",
    "Idea: differentiate each step through the program following the chain rule\n",
    "\n",
    "#### Toy example:\n",
    "$$f(p) = \\log[1-sin^2(4p)]$$\n",
    "\n",
    "Introduce **dual numbers**, leverage Julia's **type dispatch** system (-> you *almost* don't have to re-write your original code)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "a343e45a",
   "metadata": {},
   "outputs": [],
   "source": [
    "struct Dual <: Number\n",
    "    val::Float64  # value\n",
    "    der::Float64  # derivative\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "11676594",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Sum Rule\n",
    "Base.:+(f::Dual, g::Dual) = Dual(f.val + g.val, f.der + g.der)\n",
    "\n",
    "# Product Rule\n",
    "Base.:*(f::Dual, g::Dual) = Dual(f.val*g.val, f.der*g.val + f.val*g.der)\n",
    "\n",
    "# Difference Rule\n",
    "Base.:-(f::Dual, g::Dual) = Dual(f.val - g.val, f.der - g.der)\n",
    "\n",
    "# Quotient Rule\n",
    "Base.:/(f::Dual, g::Dual) = Dual(f.val/g.val, (f.der*g.val - f.val*g.der)/(g.val^2))\n",
    "\n",
    "# Derivative of sin, log\n",
    "Base.sin(f::Dual) = Dual(sin(f.val), cos(f.val)*f.der)\n",
    "Base.log(f::Dual) = Dual(log(f.val), 1/f.val*f.der)\n",
    "\n",
    "# Raise to power by repeatedly squaring\n",
    "Base.:^(f::Dual, n::Integer) = Base.power_by_squaring(f, n)\n",
    "\n",
    "Base.convert(::Type{Dual}, x::Real) = Dual(x, zero(x))\n",
    "Base.promote_rule(::Type{Dual}, ::Type{<:Number}) = Dual"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "b9549bbc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "fprime (generic function with 1 method)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(p) = log(1-sin(4p)^2)\n",
    "fprime(p) = -8sin(4p)*cos(4p) / (1 - sin(4p)^2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "dfc7308d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-0.9916355493170909, -10.417338533259999)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = rand()\n",
    "f(p), fprime(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "eebd4a94",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dual(-0.9916355493170909, -10.417338533259999)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(Dual(p,1.0))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "784e15c8",
   "metadata": {},
   "source": [
    "We are ready to write the code for a generic derivative:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "b0b4c6e1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "derivative (generic function with 1 method)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "derivative(f::Function, p) = f(Dual(p, 1.0)).der "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "3fe08362",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "derivative(f, p) = -10.417338533259999\n",
      "fprime(p) = -10.417338533259999\n"
     ]
    }
   ],
   "source": [
    "@show derivative(f, p)\n",
    "@show fprime(p);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "463bf2fa",
   "metadata": {},
   "source": [
    "In practice, differentiation rules are already stored in libraries, no need to write them all ourselves! (see e.g. https://github.com/JuliaDiff/DiffRules.jl/blob/master/src/rules.jl)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "281b526c",
   "metadata": {},
   "source": [
    "## Examples of Forward Mode AD in Julia"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1dd18331",
   "metadata": {},
   "source": [
    "### Newton's method without having to manually provide the derivative\n",
    "\n",
    "Newton's method is a simple solver for non-linear equation, here we consider the single-variable case. \n",
    "\n",
    "Given a function in the form $f$, the goal is to compute the value $x^*$ such that $f(x^*)=0$. \n",
    "Starting from a random guess $x^{(t=0)}$, the method is iterative and goes as\n",
    "$$x^{(t+1)} = \\frac{f(x^{(t)})}{f'(x^{(t)})} - x^{(t)}.$$\n",
    "\n",
    "Here we use AD to compute $f'(x^{(t)})$ automatically, without having to provide the function $f'$ explicitly.\n",
    "The example is for $f(x)=cos(x) - x$, i.e. we look for $x^* : cos(x^*)=x^*$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "38248e0d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "newton (generic function with 1 method)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function newton(f; x0=rand())\n",
    "    x = x0\n",
    "    for it in 1:100\n",
    "        fprime = derivative(f, x)\n",
    "        xnew = x - f(x) / fprime\n",
    "        abs(x - xnew) < 1e-12 && return x\n",
    "        x, xnew = xnew, x\n",
    "    end\n",
    "    x\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "10234603",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.7390851332151607"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Base.cos(f::Dual) = Dual(cos(f.val), -sin(f.val)*f.der)\n",
    "\n",
    "f(x) = cos(x) - x\n",
    "xstar = newton(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "580f4ac4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.7390851332151607, 0.7390851332151607)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cos(xstar), xstar"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9a081ba3",
   "metadata": {},
   "source": [
    "### Observables of Ising model on random regular lattice\n",
    "\n",
    "Here we consider the Ising model from statistical mechanics which involves classical \"spins\" $\\{\\sigma_i\\}_{i=1}^N$ interacting according to the Hamiltonian\n",
    "$$H(\\underline \\sigma)=-J\\sum_{(ij)}\\sigma_i\\sigma_j - h\\sum_i\\sigma_i$$\n",
    "where $(ij)$ are the edges of a [Random regular lattice](wikipedia.org/) of degree $k$ and size $N\\to\\infty$. The goal is to compute the average magnetization $m=\\frac{1}{N}\\sum_{i=1}^N \\langle\\sigma_i\\rangle$, which is done by iterating the following equation (Belief Propagation)\n",
    "$$u=h + (k-1)\\tanh^{-1}[\\tanh(u)tanh(J)].$$\n",
    "On a fixed point $u^*$, the magnetization is obtained as\n",
    "$$m = \\tanh\\left[h + (u^*-h)\\frac{k}{k-1}\\right].$$\n",
    "\n",
    "We first write a program to compute $m$ at $h=0$, then use automatic differentiation to compute the magnetic susceptibility\n",
    "$$\\chi = \\frac{d m}{d h}\\left.\\right|_{h=0}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "042de45a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "magnetiz (generic function with 1 method)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function magnetiz(J, h, k)\n",
    "    u = one(h)\n",
    "    for _ in 1:10^4\n",
    "        unew = h + (k-1) * atanh(tanh(u)*tanh(J))\n",
    "        u = unew\n",
    "    end\n",
    "    return tanh((h + (u-h)*k/(k-1)))\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "a3d241b4",
   "metadata": {},
   "outputs": [
    {
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       "</svg>\n"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k = 3\n",
    "h = 0.0\n",
    "J = LinRange(0.2, 1.0, 100)\n",
    "m = [magnetiz(JJ, h, k) for JJ in J]\n",
    "\n",
    "plot(J, m, xlabel=L\"J\", ylabel=L\"m\", grid=false, size=(400,300))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "3bd0312c",
   "metadata": {},
   "outputs": [],
   "source": [
    "Base.tanh(f::Dual) = Dual(tanh(f.val), (1-tanh(f.val)^2) * f.der)\n",
    "Base.atanh(f::Dual) = Dual(atanh(f.val), inv(1-f.val^2) * f.der)\n",
    "\n",
    "χ = [derivative(hh -> magnetiz(JJ, hh, k), h) for JJ in J];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "162899ef",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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       "</svg>\n"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(J, χ, xlabel=\"J\", ylabel=\"χ\", grid=false, size=(400,300))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dca56ac0",
   "metadata": {},
   "source": [
    "### Benchmark performance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "410061dc",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  95.968 ms (203 allocations: 4.06 KiB)\n"
     ]
    }
   ],
   "source": [
    "@btime [magnetiz(JJ, h, k) for JJ in $J];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "e5ef3210",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  115.418 ms (503 allocations: 11.88 KiB)\n"
     ]
    }
   ],
   "source": [
    "@btime [derivative(hh -> magnetiz(JJ, hh, k), h) for JJ in $J];"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0ddc9129",
   "metadata": {},
   "source": [
    "# Acknowledgement\n",
    "The minimal dual number implementation is from https://book.sciml.ai/notes/08-Forward-Mode_Automatic_Differentiation_(AD)_via_High_Dimensional_Algebras/ and https://www.youtube.com/watch?v=vAp6nUMrKYg"
   ]
  }
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