Merge pull request #291 from daniel-s-ingram/master
Visualization of differences between gradient descent methodsmain
Aurélien Geron
GitHub
commit
c13d4c322b
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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [],
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"source": [
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"from __future__ import print_function, division, unicode_literals\n",
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"import numpy as np\n",
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"\n",
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"%matplotlib nbagg\n",
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"import matplotlib.pyplot as plt\n",
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"from matplotlib.animation import FuncAnimation"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [],
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"source": [
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"m = 100\n",
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"X = 2*np.random.rand(m, 1)\n",
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"X_b = np.c_[np.ones((m, 1)), X]\n",
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"y = 4 + 3*X + np.random.rand(m, 1)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
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"outputs": [],
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"source": [
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"def batch_gradient_descent():\n",
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" n_iterations = 1000\n",
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" learning_rate = 0.05\n",
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" thetas = np.random.randn(2, 1)\n",
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" thetas_path = [thetas]\n",
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" for i in range(n_iterations):\n",
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" gradients = 2*X_b.T.dot(X_b.dot(thetas) - y)/m\n",
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" thetas = thetas - learning_rate*gradients\n",
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" thetas_path.append(thetas)\n",
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"\n",
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" return thetas_path"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
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"outputs": [],
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"source": [
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"def stochastic_gradient_descent():\n",
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" n_epochs = 50\n",
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" t0, t1 = 5, 50\n",
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" thetas = np.random.randn(2, 1)\n",
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" thetas_path = [thetas]\n",
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" for epoch in range(n_epochs):\n",
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" for i in range(m):\n",
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" random_index = np.random.randint(m)\n",
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" xi = X_b[random_index:random_index+1]\n",
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" yi = y[random_index:random_index+1]\n",
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" gradients = 2*xi.T.dot(xi.dot(thetas) - yi)\n",
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" eta = learning_schedule(epoch*m + i, t0, t1)\n",
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" thetas = thetas - eta*gradients\n",
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" thetas_path.append(thetas)\n",
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"\n",
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" return thetas_path"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {},
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"outputs": [],
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"source": [
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"def mini_batch_gradient_descent():\n",
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" n_iterations = 50\n",
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" minibatch_size = 20\n",
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" t0, t1 = 200, 1000\n",
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" thetas = np.random.randn(2, 1)\n",
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" thetas_path = [thetas]\n",
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" t = 0\n",
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" for epoch in range(n_iterations):\n",
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" shuffled_indices = np.random.permutation(m)\n",
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" X_b_shuffled = X_b[shuffled_indices]\n",
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" y_shuffled = y[shuffled_indices]\n",
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" for i in range(0, m, minibatch_size):\n",
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" t += 1\n",
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" xi = X_b_shuffled[i:i+minibatch_size]\n",
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" yi = y_shuffled[i:i+minibatch_size]\n",
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" gradients = 2*xi.T.dot(xi.dot(thetas) - yi)/minibatch_size\n",
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" eta = learning_schedule(t, t0, t1)\n",
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" thetas = thetas - eta*gradients\n",
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" thetas_path.append(thetas)\n",
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"\n",
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" return thetas_path"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
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"outputs": [],
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"source": [
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"def compute_mse(theta):\n",
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" return np.sum((np.dot(X_b, theta) - y)**2)/m"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [],
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"source": [
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"def learning_schedule(t, t0, t1):\n",
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" return t0/(t+t1)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {},
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"outputs": [],
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"source": [
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"theta0, theta1 = np.meshgrid(np.arange(0, 5, 0.1), np.arange(0, 5, 0.1))\n",
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"r, c = theta0.shape\n",
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"cost_map = np.array([[0 for _ in range(c)] for _ in range(r)])\n",
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"for i in range(r):\n",
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" for j in range(c):\n",
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" theta = np.array([theta0[i,j], theta1[i,j]])\n",
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" cost_map[i,j] = compute_mse(theta)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"metadata": {},
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"outputs": [],
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"source": [
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"exact_solution = np.linalg.inv(X_b.T.dot(X_b)).dot(X_b.T).dot(y)\n",
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"bgd_thetas = np.array(batch_gradient_descent())\n",
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"sgd_thetas = np.array(stochastic_gradient_descent())\n",
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"mbgd_thetas = np.array(mini_batch_gradient_descent())"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"metadata": {},
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"outputs": [],
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"source": [
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"bgd_len = len(bgd_thetas)\n",
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"sgd_len = len(sgd_thetas)\n",
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"mbgd_len = len(mbgd_thetas)\n",
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"n_iter = min(bgd_len, sgd_len, mbgd_len)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 15,
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"metadata": {},
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"outputs": [],
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"source": [
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"fig = plt.figure(figsize=(10, 5))\n",
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"data_ax = fig.add_subplot(121)\n",
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"cost_ax = fig.add_subplot(122)\n",
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"\n",
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"cost_ax.plot(exact_solution[0,0], exact_solution[1,0], 'y*')\n",
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"cost_img = cost_ax.pcolor(theta0, theta1, cost_map)\n",
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"fig.colorbar(cost_img)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 16,
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"metadata": {},
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"outputs": [],
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"source": [
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"def animate(i):\n",
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" data_ax.cla()\n",
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" cost_ax.cla()\n",
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"\n",
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" data_ax.plot(X, y, 'k.')\n",
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"\n",
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" cost_ax.plot(exact_solution[0,0], exact_solution[1,0], 'y*')\n",
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" cost_ax.pcolor(theta0, theta1, cost_map)\n",
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"\n",
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" data_ax.plot(X, X_b.dot(bgd_thetas[i,:]), 'r-')\n",
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" cost_ax.plot(bgd_thetas[:i,0], bgd_thetas[:i,1], 'r--')\n",
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"\n",
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" data_ax.plot(X, X_b.dot(sgd_thetas[i,:]), 'g-')\n",
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" cost_ax.plot(sgd_thetas[:i,0], sgd_thetas[:i,1], 'g--')\n",
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"\n",
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" data_ax.plot(X, X_b.dot(mbgd_thetas[i,:]), 'b-')\n",
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" cost_ax.plot(mbgd_thetas[:i,0], mbgd_thetas[:i,1], 'b--')\n",
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"\n",
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" data_ax.set_xlim([0, 2])\n",
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" data_ax.set_ylim([0, 15])\n",
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" cost_ax.set_xlim([0, 5])\n",
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" cost_ax.set_ylim([0, 5])\n",
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"\n",
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" data_ax.set_xlabel(r'$x_1$')\n",
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" data_ax.set_ylabel(r'$y$')\n",
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" cost_ax.set_xlabel(r'$\\theta_0$')\n",
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" cost_ax.set_ylabel(r'$\\theta_1$')\n",
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"\n",
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" data_ax.legend(('Data', 'BGD', 'SGD', 'MBGD'))\n",
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" cost_ax.legend(('Normal Equation', 'BGD', 'SGD', 'MBGD'))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 17,
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"metadata": {},
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"outputs": [],
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"source": [
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"animation = FuncAnimation(fig, animate, frames=n_iter)\n",
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"plt.show()"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.5.2"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}
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@ -0,0 +1,129 @@
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from __future__ import print_function, division, unicode_literals
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import sys
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from math import sqrt
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import numpy as np
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import matplotlib.pyplot as plt
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m = 100
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X = 2*np.random.rand(m, 1)
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X_b = np.c_[np.ones((m, 1)), X]
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y = 4 + 3*X + np.random.rand(m, 1)
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fig = plt.figure(figsize=(10, 5))
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data_ax = fig.add_subplot(121)
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cost_ax = fig.add_subplot(122)
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def batch_gradient_descent():
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n_iterations = 1000
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learning_rate = 0.05
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thetas = np.random.randn(2, 1)
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thetas_path = [thetas]
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for i in range(n_iterations):
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gradients = 2*X_b.T.dot(X_b.dot(thetas) - y)/m
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thetas = thetas - learning_rate*gradients
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thetas_path.append(thetas)
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return thetas_path
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def stochastic_gradient_descent():
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n_epochs = 50
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t0, t1 = 5, 50
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thetas = np.random.randn(2, 1)
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thetas_path = [thetas]
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for epoch in range(n_epochs):
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for i in range(m):
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random_index = np.random.randint(m)
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xi = X_b[random_index:random_index+1]
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yi = y[random_index:random_index+1]
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gradients = 2*xi.T.dot(xi.dot(thetas) - yi)
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eta = learning_schedule(epoch*m + i, t0, t1)
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thetas = thetas - eta*gradients
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thetas_path.append(thetas)
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return thetas_path
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def mini_batch_gradient_descent():
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n_iterations = 50
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minibatch_size = 20
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t0, t1 = 200, 1000
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thetas = np.random.randn(2, 1)
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thetas_path = [thetas]
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t = 0
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for epoch in range(n_iterations):
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shuffled_indices = np.random.permutation(m)
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X_b_shuffled = X_b[shuffled_indices]
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y_shuffled = y[shuffled_indices]
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for i in range(0, m, minibatch_size):
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t += 1
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xi = X_b_shuffled[i:i+minibatch_size]
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yi = y_shuffled[i:i+minibatch_size]
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gradients = 2*xi.T.dot(xi.dot(thetas) - yi)/minibatch_size
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eta = learning_schedule(t, t0, t1)
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thetas = thetas - eta*gradients
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thetas_path.append(thetas)
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return thetas_path
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def compute_mse(theta):
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return np.sum((np.dot(X_b, theta) - y)**2)/m
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def learning_schedule(t, t0, t1):
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return t0/(t+t1)
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if __name__ == '__main__':
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plt.ion()
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theta0, theta1 = np.meshgrid(np.arange(0, 5, 0.1), np.arange(0, 5, 0.1))
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r, c = theta0.shape
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cost_map = np.array([[0 for _ in range(c)] for _ in range(r)])
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for i in range(r):
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for j in range(c):
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theta = np.array([theta0[i,j], theta1[i,j]])
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cost_map[i,j] = compute_mse(theta)
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exact_solution = np.linalg.inv(X_b.T.dot(X_b)).dot(X_b.T).dot(y)
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bgd_thetas = np.array(batch_gradient_descent())
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sgd_thetas = np.array(stochastic_gradient_descent())
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mbgd_thetas = np.array(mini_batch_gradient_descent())
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bgd_len = len(bgd_thetas)
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sgd_len = len(sgd_thetas)
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mbgd_len = len(mbgd_thetas)
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n_iter = min(bgd_len, sgd_len, mbgd_len)
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cost_ax.plot(exact_solution[0,0], exact_solution[1,0], 'y*')
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cost_img = cost_ax.pcolor(theta0, theta1, cost_map)
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fig.colorbar(cost_img)
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for i in range(n_iter):
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data_ax.cla()
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cost_ax.cla()
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data_ax.plot(X, y, 'k.')
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cost_ax.plot(exact_solution[0,0], exact_solution[1,0], 'y*')
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cost_ax.pcolor(theta0, theta1, cost_map)
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data_ax.plot(X, X_b.dot(bgd_thetas[i,:]), 'r-')
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cost_ax.plot(bgd_thetas[:i,0], bgd_thetas[:i,1], 'r--')
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data_ax.plot(X, X_b.dot(sgd_thetas[i,:]), 'g-')
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cost_ax.plot(sgd_thetas[:i,0], sgd_thetas[:i,1], 'g--')
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data_ax.plot(X, X_b.dot(mbgd_thetas[i,:]), 'b-')
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cost_ax.plot(mbgd_thetas[:i,0], mbgd_thetas[:i,1], 'b--')
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data_ax.set_xlim([0, 2])
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data_ax.set_ylim([0, 15])
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cost_ax.set_xlim([0, 5])
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cost_ax.set_ylim([0, 5])
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data_ax.set_xlabel(r'$x_1$')
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data_ax.set_ylabel(r'$y$')
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cost_ax.set_xlabel(r'$\theta_0$')
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cost_ax.set_ylabel(r'$\theta_1$')
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data_ax.legend(('Data', 'BGD', 'SGD', 'MBGD'))
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cost_ax.legend(('Normal Equation', 'BGD', 'SGD', 'MBGD'))
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plt.pause(1e-5)
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