schuetoliver
/
handson-ml
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Merge pull request #291 from daniel-s-ingram/master 

Visualization of differences between gradient descent methods
main
Aurélien Geron 2018-09-17 11:40:59 +02:00 committed by GitHub
parent c224d8e74c 3651a2608b
commit c13d4c322b
2 changed files with 374 additions and 0 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

245
gradient_descent_comparison.ipynb Normal file
View File

@ -0,0 +1,245 @@
{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"from __future__ import print_function, division, unicode_literals\n",
"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": 15,
"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": 16,
"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'$y$')\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'))\n",
" cost_ax.legend(('Normal Equation', 'BGD', 'SGD', 'MBGD'))"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [],
"source": [
"animation = FuncAnimation(fig, animate, frames=n_iter)\n",
"plt.show()"
]
}
],
"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": "3.5.2"
}
},
"nbformat": 4,
"nbformat_minor": 2
}

129
gradient_descent_comparison.py Normal file
View File

@ -0,0 +1,129 @@
from __future__ import print_function, division, unicode_literals
import sys
from math import sqrt
import numpy as np
import matplotlib.pyplot as plt
m = 100
X = 2*np.random.rand(m, 1)
X_b = np.c_[np.ones((m, 1)), X]
y = 4 + 3*X + np.random.rand(m, 1)
fig = plt.figure(figsize=(10, 5))
data_ax = fig.add_subplot(121)
cost_ax = fig.add_subplot(122)
def batch_gradient_descent():
n_iterations = 1000
learning_rate = 0.05
thetas = np.random.randn(2, 1)
thetas_path = [thetas]
for i in range(n_iterations):
gradients = 2*X_b.T.dot(X_b.dot(thetas) - y)/m
thetas = thetas - learning_rate*gradients
thetas_path.append(thetas)
return thetas_path
def stochastic_gradient_descent():
n_epochs = 50
t0, t1 = 5, 50
thetas = np.random.randn(2, 1)
thetas_path = [thetas]
for epoch in range(n_epochs):
for i in range(m):
random_index = np.random.randint(m)
xi = X_b[random_index:random_index+1]
yi = y[random_index:random_index+1]
gradients = 2*xi.T.dot(xi.dot(thetas) - yi)
eta = learning_schedule(epoch*m + i, t0, t1)
thetas = thetas - eta*gradients
thetas_path.append(thetas)
return thetas_path
def mini_batch_gradient_descent():
n_iterations = 50
minibatch_size = 20
t0, t1 = 200, 1000
thetas = np.random.randn(2, 1)
thetas_path = [thetas]
t = 0
for epoch in range(n_iterations):
shuffled_indices = np.random.permutation(m)
X_b_shuffled = X_b[shuffled_indices]
y_shuffled = y[shuffled_indices]
for i in range(0, m, minibatch_size):
t += 1
xi = X_b_shuffled[i:i+minibatch_size]
yi = y_shuffled[i:i+minibatch_size]
gradients = 2*xi.T.dot(xi.dot(thetas) - yi)/minibatch_size
eta = learning_schedule(t, t0, t1)
thetas = thetas - eta*gradients
thetas_path.append(thetas)
return thetas_path
def compute_mse(theta):
return np.sum((np.dot(X_b, theta) - y)**2)/m
def learning_schedule(t, t0, t1):
return t0/(t+t1)
if __name__ == '__main__':
plt.ion()
theta0, theta1 = np.meshgrid(np.arange(0, 5, 0.1), np.arange(0, 5, 0.1))
r, c = theta0.shape
cost_map = np.array([[0 for _ in range(c)] for _ in range(r)])
for i in range(r):
for j in range(c):
theta = np.array([theta0[i,j], theta1[i,j]])
cost_map[i,j] = compute_mse(theta)
exact_solution = np.linalg.inv(X_b.T.dot(X_b)).dot(X_b.T).dot(y)
bgd_thetas = np.array(batch_gradient_descent())
sgd_thetas = np.array(stochastic_gradient_descent())
mbgd_thetas = np.array(mini_batch_gradient_descent())
bgd_len = len(bgd_thetas)
sgd_len = len(sgd_thetas)
mbgd_len = len(mbgd_thetas)
n_iter = min(bgd_len, sgd_len, mbgd_len)
cost_ax.plot(exact_solution[0,0], exact_solution[1,0], 'y*')
cost_img = cost_ax.pcolor(theta0, theta1, cost_map)
fig.colorbar(cost_img)
for i in range(n_iter):
data_ax.cla()
cost_ax.cla()
data_ax.plot(X, y, 'k.')
cost_ax.plot(exact_solution[0,0], exact_solution[1,0], 'y*')
cost_ax.pcolor(theta0, theta1, cost_map)
data_ax.plot(X, X_b.dot(bgd_thetas[i,:]), 'r-')
cost_ax.plot(bgd_thetas[:i,0], bgd_thetas[:i,1], 'r--')
data_ax.plot(X, X_b.dot(sgd_thetas[i,:]), 'g-')
cost_ax.plot(sgd_thetas[:i,0], sgd_thetas[:i,1], 'g--')
data_ax.plot(X, X_b.dot(mbgd_thetas[i,:]), 'b-')
cost_ax.plot(mbgd_thetas[:i,0], mbgd_thetas[:i,1], 'b--')
data_ax.set_xlim([0, 2])
data_ax.set_ylim([0, 15])
cost_ax.set_xlim([0, 5])
cost_ax.set_ylim([0, 5])
data_ax.set_xlabel(r'$x_1$')
data_ax.set_ylabel(r'$y$')
cost_ax.set_xlabel(r'$\theta_0$')
cost_ax.set_ylabel(r'$\theta_1$')
data_ax.legend(('Data', 'BGD', 'SGD', 'MBGD'))
cost_ax.legend(('Normal Equation', 'BGD', 'SGD', 'MBGD'))
plt.pause(1e-5)