277 lines
7.6 KiB
Plaintext
277 lines
7.6 KiB
Plaintext
{
|
|
"cells": [
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# Comparison of Batch, Mini-Batch and Stochastic Gradient Descent"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"This notebook displays an animation comparing Batch, Mini-Batch and Stochastic Gradient Descent (introduced in Chapter 4). Thanks to [Daniel Ingram](https://github.com/daniel-s-ingram) who contributed this notebook."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"<table align=\"left\">\n",
|
|
" <td>\n",
|
|
" <a target=\"_blank\" href=\"https://colab.research.google.com/github/ageron/handson-ml2/blob/master/extra_gradient_descent_comparison.ipynb\"><img src=\"https://www.tensorflow.org/images/colab_logo_32px.png\" />Run in Google Colab</a>\n",
|
|
" </td>\n",
|
|
"</table>"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 1,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"import numpy as np\n",
|
|
"\n",
|
|
"%matplotlib nbagg\n",
|
|
"import matplotlib.pyplot as plt\n",
|
|
"from matplotlib.animation import FuncAnimation"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 2,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"m = 100\n",
|
|
"X = 2*np.random.rand(m, 1)\n",
|
|
"X_b = np.c_[np.ones((m, 1)), X]\n",
|
|
"y = 4 + 3*X + np.random.rand(m, 1)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 3,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"def batch_gradient_descent():\n",
|
|
" n_iterations = 1000\n",
|
|
" learning_rate = 0.05\n",
|
|
" thetas = np.random.randn(2, 1)\n",
|
|
" thetas_path = [thetas]\n",
|
|
" for i in range(n_iterations):\n",
|
|
" gradients = 2*X_b.T.dot(X_b.dot(thetas) - y)/m\n",
|
|
" thetas = thetas - learning_rate*gradients\n",
|
|
" thetas_path.append(thetas)\n",
|
|
"\n",
|
|
" return thetas_path"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 4,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"def stochastic_gradient_descent():\n",
|
|
" n_epochs = 50\n",
|
|
" t0, t1 = 5, 50\n",
|
|
" thetas = np.random.randn(2, 1)\n",
|
|
" thetas_path = [thetas]\n",
|
|
" for epoch in range(n_epochs):\n",
|
|
" for i in range(m):\n",
|
|
" random_index = np.random.randint(m)\n",
|
|
" xi = X_b[random_index:random_index+1]\n",
|
|
" yi = y[random_index:random_index+1]\n",
|
|
" gradients = 2*xi.T.dot(xi.dot(thetas) - yi)\n",
|
|
" eta = learning_schedule(epoch*m + i, t0, t1)\n",
|
|
" thetas = thetas - eta*gradients\n",
|
|
" thetas_path.append(thetas)\n",
|
|
"\n",
|
|
" return thetas_path"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 5,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"def mini_batch_gradient_descent():\n",
|
|
" n_iterations = 50\n",
|
|
" minibatch_size = 20\n",
|
|
" t0, t1 = 200, 1000\n",
|
|
" thetas = np.random.randn(2, 1)\n",
|
|
" thetas_path = [thetas]\n",
|
|
" t = 0\n",
|
|
" for epoch in range(n_iterations):\n",
|
|
" shuffled_indices = np.random.permutation(m)\n",
|
|
" X_b_shuffled = X_b[shuffled_indices]\n",
|
|
" y_shuffled = y[shuffled_indices]\n",
|
|
" for i in range(0, m, minibatch_size):\n",
|
|
" t += 1\n",
|
|
" xi = X_b_shuffled[i:i+minibatch_size]\n",
|
|
" yi = y_shuffled[i:i+minibatch_size]\n",
|
|
" gradients = 2*xi.T.dot(xi.dot(thetas) - yi)/minibatch_size\n",
|
|
" eta = learning_schedule(t, t0, t1)\n",
|
|
" thetas = thetas - eta*gradients\n",
|
|
" thetas_path.append(thetas)\n",
|
|
"\n",
|
|
" return thetas_path"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 6,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"def compute_mse(theta):\n",
|
|
" return np.sum((np.dot(X_b, theta) - y)**2)/m"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 7,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"def learning_schedule(t, t0, t1):\n",
|
|
" return t0/(t+t1)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 8,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"theta0, theta1 = np.meshgrid(np.arange(0, 5, 0.1), np.arange(0, 5, 0.1))\n",
|
|
"r, c = theta0.shape\n",
|
|
"cost_map = np.array([[0 for _ in range(c)] for _ in range(r)])\n",
|
|
"for i in range(r):\n",
|
|
" for j in range(c):\n",
|
|
" theta = np.array([theta0[i,j], theta1[i,j]])\n",
|
|
" cost_map[i,j] = compute_mse(theta)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 9,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"exact_solution = np.linalg.inv(X_b.T.dot(X_b)).dot(X_b.T).dot(y)\n",
|
|
"bgd_thetas = np.array(batch_gradient_descent())\n",
|
|
"sgd_thetas = np.array(stochastic_gradient_descent())\n",
|
|
"mbgd_thetas = np.array(mini_batch_gradient_descent())"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 10,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"bgd_len = len(bgd_thetas)\n",
|
|
"sgd_len = len(sgd_thetas)\n",
|
|
"mbgd_len = len(mbgd_thetas)\n",
|
|
"n_iter = min(bgd_len, sgd_len, mbgd_len)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 11,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"fig = plt.figure(figsize=(10, 5))\n",
|
|
"data_ax = fig.add_subplot(121)\n",
|
|
"cost_ax = fig.add_subplot(122)\n",
|
|
"\n",
|
|
"cost_ax.plot(exact_solution[0,0], exact_solution[1,0], 'y*')\n",
|
|
"cost_img = cost_ax.pcolor(theta0, theta1, cost_map)\n",
|
|
"fig.colorbar(cost_img)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 12,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"def animate(i):\n",
|
|
" data_ax.cla()\n",
|
|
" cost_ax.cla()\n",
|
|
"\n",
|
|
" data_ax.plot(X, y, 'k.')\n",
|
|
"\n",
|
|
" cost_ax.plot(exact_solution[0,0], exact_solution[1,0], 'y*')\n",
|
|
" cost_ax.pcolor(theta0, theta1, cost_map)\n",
|
|
"\n",
|
|
" data_ax.plot(X, X_b.dot(bgd_thetas[i,:]), 'r-')\n",
|
|
" cost_ax.plot(bgd_thetas[:i,0], bgd_thetas[:i,1], 'r--')\n",
|
|
"\n",
|
|
" data_ax.plot(X, X_b.dot(sgd_thetas[i,:]), 'g-')\n",
|
|
" cost_ax.plot(sgd_thetas[:i,0], sgd_thetas[:i,1], 'g--')\n",
|
|
"\n",
|
|
" data_ax.plot(X, X_b.dot(mbgd_thetas[i,:]), 'b-')\n",
|
|
" cost_ax.plot(mbgd_thetas[:i,0], mbgd_thetas[:i,1], 'b--')\n",
|
|
"\n",
|
|
" data_ax.set_xlim([0, 2])\n",
|
|
" data_ax.set_ylim([0, 15])\n",
|
|
" cost_ax.set_xlim([0, 5])\n",
|
|
" cost_ax.set_ylim([0, 5])\n",
|
|
"\n",
|
|
" data_ax.set_xlabel(r'$x_1$')\n",
|
|
" data_ax.set_ylabel(r'$y$', rotation=0)\n",
|
|
" cost_ax.set_xlabel(r'$\\theta_0$')\n",
|
|
" cost_ax.set_ylabel(r'$\\theta_1$')\n",
|
|
"\n",
|
|
" data_ax.legend(('Data', 'BGD', 'SGD', 'MBGD'), loc=\"upper left\")\n",
|
|
" cost_ax.legend(('Normal Equation', 'BGD', 'SGD', 'MBGD'), loc=\"upper left\")"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 13,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"animation = FuncAnimation(fig, animate, frames=n_iter)\n",
|
|
"plt.show()"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": []
|
|
}
|
|
],
|
|
"metadata": {
|
|
"kernelspec": {
|
|
"display_name": "Python 3",
|
|
"language": "python",
|
|
"name": "python3"
|
|
},
|
|
"language_info": {
|
|
"codemirror_mode": {
|
|
"name": "ipython",
|
|
"version": 3
|
|
},
|
|
"file_extension": ".py",
|
|
"mimetype": "text/x-python",
|
|
"name": "python",
|
|
"nbconvert_exporter": "python",
|
|
"pygments_lexer": "ipython3",
|
|
"version": "3.7.9"
|
|
}
|
|
},
|
|
"nbformat": 4,
|
|
"nbformat_minor": 4
|
|
}
|