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Introduction to gradient descent with pytorch
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sgd-intro.ipynb
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"source": [
"# Introduction to stochastic gradient descent\n",
"\n",
"Max Joseph\n",
"\n",
"fast.ai group, 2020-03-06\n",
"\n",
"![](https://media.giphy.com/media/l0HUp5iTjX4EQ7FjW/giphy.gif)"
]
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"source": [
"import torch\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"import matplotlib\n",
"%config InlineBackend.figure_formats = ['svg']\n",
"matplotlib.rcParams['figure.figsize'] = (6.0, 4.0)"
]
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{
"cell_type": "markdown",
"metadata": {
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"slide_type": "slide"
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"source": [
"## 1D gradient descent\n",
"\n",
"Minimizing $f(x) = x^2/2$ with gradient descent"
]
},
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"source": [
"xvec = torch.linspace(-3, 3)\n",
"\n",
"def f(x):\n",
" return x**2/2\n",
"\n",
"\n",
"plt.plot(xvec, f(xvec))\n",
"plt.xlabel(\"x\")\n",
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{
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"metadata": {
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}
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"source": [
"## Gradient descent:\n",
"\n",
"1. Choose an initial value: $x$\n",
"\n",
"2. Update $x$ to reduce $f(x)$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Choosing an initial value\n",
"\n",
"$x \\leftarrow x_{\\text{init}}$"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
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"x = 1.5\n",
"plt.plot(xvec, f(xvec))\n",
"plt.scatter(x, f(x), c=\"k\")\n",
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}
},
"source": [
"## Update $x$ to reduce $f(x)$: \n",
"\n",
"$x \\leftarrow x - \\alpha \\frac{d}{dx}f(x)$\n",
"\n",
"$\\approx$ take a step of size $\\alpha$ \"downhill\" to reduce $f(x)$"
]
},
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"slide_type": "slide"
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},
"source": [
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},
{
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"execution_count": 4,
"metadata": {
"slideshow": {
"slide_type": "-"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"x=1.50, f(x)=1.12\n",
"x=1.35, f(x)=0.91\n"
]
}
],
"source": [
"def update_x(x, learning_rate=0.1):\n",
" return x - learning_rate * x\n",
"\n",
"print(f\"x={x:.2f}, f(x)={f(x):.2f}\")\n",
"\n",
"x = update_x(x)\n",
"print(f\"x={x:.2f}, f(x)={f(x):.2f}\")"
]
},
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"source": [
"## Do some more updates"
]
},
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"execution_count": 5,
"metadata": {},
"outputs": [
{
"name": "stdout",
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"plt.plot(xvec, f(xvec))\n",
"plt.xlabel(\"x\")\n",
"plt.ylabel(\"f(x)\")\n",
"\n",
"for i in range(100):\n",
" x = update_x(x)\n",
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"\n",
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]
},
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}
},
"source": [
"# Generalizing 1-d gradient descent\n",
"\n",
"Now the input has multiple dimensions.\n",
"\n",
"Want to find a vector that minimizes some function."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"# Linear regression\n",
"\n",
"$y_i = \\pmb{x}'_i \\pmb \\beta + \\epsilon_i$ for $i=1,...,n$\n",
"\n",
"The parameter $\\pmb \\beta$ is **2-dimensional**\n",
"- intercept $\\beta_1$\n",
"- slope $\\beta_2$"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"slideshow": {
"slide_type": "slide"
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"source": [
"beta_true = torch.tensor([-3., 3])\n",
"n = len(xvec)\n",
"X = torch.stack((torch.ones(n), xvec), -1)\n",
"y = X @ beta_true + torch.randn(n)\n",
"\n",
"plt.scatter(xvec, y)\n",
"plt.xlabel(\"x\")\n",
"plt.ylabel(\"y\")\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Minimizing mean squared error\n",
"\n",
"Find values of $\\pmb \\beta$ that minimize mean squared error loss.\n",
"\n",
"$L(\\pmb \\beta) = \\frac{1}{n} \\sum_{i=1}^n (y_i - \\pmb x'_i \\pmb \\beta)^2$"
]
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{
"cell_type": "markdown",
"metadata": {
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},
"source": [
"1. Choose an initial value: \n",
"\n",
"$\\pmb \\beta \\leftarrow \\pmb{\\beta}_{\\text{init}}$\n",
"\n",
"2. Iteratively update to reduce the loss $L(\\pmb \\beta)$: \n",
"\n",
"$\\pmb \\beta \\leftarrow \\pmb \\beta - \\alpha \\nabla_{\\pmb{\\beta}} L(\\pmb \\beta)$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### Gradients $\\approx$ \"slope\" in each direction\n",
"\n",
"$\\nabla_{\\pmb{\\beta}} L(\\pmb \\beta) = \\Big(\\frac{\\partial}{\\partial \\beta_1} L(\\pmb \\beta), \\frac{\\partial}{\\partial \\beta_2} L(\\pmb \\beta) \\Big)$\n",
"\n",
"(Don't worry - pytorch computes this for you!)\n",
"\n",
"![](https://media.giphy.com/media/XyOrJljDNBEpa/giphy.gif)"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"outputs": [],
"source": [
"def mse(yhat, y):\n",
" \"\"\" Mean squared error loss \"\"\"\n",
" return torch.mean((yhat - y)**2)\n",
"\n",
"def update(loss, lr=0.1):\n",
" \"\"\" Update beta via gradient descent \"\"\"\n",
" # compute gradients\n",
" loss.backward()\n",
" \n",
" # update the parameters\n",
" with torch.no_grad():\n",
" beta.sub_(lr * beta.grad)\n",
" beta.grad.zero_()"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Gradient descent"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [],
"source": [
"beta_history = []\n",
"loss_history = []\n",
"\n",
"# 1. initialize parameters\n",
"beta = torch.nn.Parameter(torch.tensor((8, -8.)))\n",
"\n",
"# 2. iteratively update values to reduce loss\n",
"for i in range(100):\n",
" beta_history.append(beta.clone().detach())\n",
" yhat = X @ beta\n",
" loss = mse(yhat, y)\n",
" update(loss, lr=0.1)\n",
" loss_history.append(loss.clone().detach())"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## How did we do?"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
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"source": [
"plt.scatter(xvec, y)\n",
"plt.plot(xvec, X @ beta.detach().numpy(), c=\"r\")\n",
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"source": [
"## What happened to our loss?"
]
},
{
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{
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"beta1_grid = torch.linspace(-10, 10)\n",
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{
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"source": [
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]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### Fun aside - what happens when x is not centered?\n",
"\n",
"Re-run this notebook, but add one to `xvec` at the beginning..."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"# Recap of gradient descent\n",
"\n",
"Goal: minimize a loss function\n",
"\n",
"Approach: \n",
"\n",
"1. Initialize parameters\n",
"2. Iteratively update parameters to reduce loss"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Good news about gradient descent\n",
"\n",
"- No calculus necessary (pytorch does this)\n",
"- Applies in many situations\n",
"\n",
"## Bad news\n",
"\n",
"- Does not scale\n",
"\n",
"e.g., $\\nabla_{\\pmb \\beta} L(\\pmb \\beta) = \\frac{1}{n} \\nabla_{\\pmb \\beta} \\sum_{i=1}^n (y_i - \\pmb x'_i \\pmb \\beta)^2$ for large $n$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"# Stochastic gradient descent (SGD)\n",
"\n",
"Updates are based on one example at a time\n",
"\n",
"(not all examples simultaneously)"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"# SGD for linear regression"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [],
"source": [
"beta_history = []\n",
"loss_history = []\n",
"beta = torch.nn.Parameter(torch.tensor((8, -8.)))\n",
"\n",
"n_epochs = 30\n",
"idx = np.arange(n)\n",
"\n",
"for epoch in range(n_epochs):\n",
" np.random.shuffle(idx)\n",
" \n",
" # use one datapoint at a time to update parameters\n",
" for i in idx:\n",
" beta_history.append(beta.clone().detach())\n",
" yhat = X[i, :] @ beta\n",
" loss = mse(yhat, y[i])\n",
" update(loss, lr=0.01)\n",
" loss_history.append(loss.clone().detach())"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"# What did we learn?"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
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"source": [
"plt.scatter(xvec, y)\n",
"plt.plot(xvec, X @ beta.detach().numpy(), c=\"r\")\n",
"plt.xlabel(\"x\")\n",
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"source": [
"## What happened to our loss?"
]
},
{
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],
"source": [
"plt.plot(loss_history)\n",
"plt.xlabel(\"Number of updates\")\n",
"plt.ylabel(\"MSE\")\n",
"plt.yscale(\"log\")\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"# Visualizing the SGD trajectory"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
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"L
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