udacityが提供しているこのチュートリアルを利用してみる。
gitをcloneする¶
cd git
!git clone https://github.com/udacity/deep-learning-v2-pytorch.git
cd deep-learning-v2-pytorch
cd intro-neural-networks/gradient-descent
Implementing the Gradient Descent Algorithm¶
In this lab, we’ll implement the basic functions of the Gradient Descent algorithm to find the boundary in a small dataset. First, we’ll start with some functions that will help us plot and visualize the data.
このラボでは、小規模データ・セットの境界を検出するために、勾配降下法アルゴリズムの基本的な関数を実装する。先ずは、データをプロット・視覚化するための関数作成から始める。
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
plt.rcParams['figure.figsize'] = 12, 8
plt.rcParams["font.size"] = "17"
#Some helper functions for plotting and drawing lines
def plot_points(X, y):
admitted = X[np.argwhere(y==1)]
rejected = X[np.argwhere(y==0)]
plt.scatter([s[0][0] for s in rejected], [s[0][1] for s in rejected], s = 25, color = 'blue', edgecolor = 'k')
plt.scatter([s[0][0] for s in admitted], [s[0][1] for s in admitted], s = 25, color = 'red', edgecolor = 'k')
def display(m, b, color='g--'):
plt.xlim(-0.05,1.05)
plt.ylim(-0.05,1.05)
x = np.arange(-10, 10, 0.1)
plt.plot(x, m*x+b, color)
data = pd.read_csv('data.csv', header=None)
X = np.array(data[[0,1]])
y = np.array(data[2])
plot_points(X,y)
plt.show()
Implementing the basic functions¶
Here is your turn to shine. Implement the following formulas, as explained in the text.
今度はあなたが輝く番です。本文中で説明されている通り以下の関数を実装してみましょう。
-
Sigmoid activation function
シグモイド活性化関数
$$\sigma(x) = \frac{1}{1+e^{-x}}$$ -
Output (prediction) formula
出力(予測)式
$$\hat{y} = \sigma(w_1 x_1 + w_2 x_2 + b)$$ -
Error function
エラー関数
$$Error(y, \hat{y}) = – y \log(\hat{y}) – (1-y) \log(1-\hat{y})$$ -
The function that updates the weights
重みを更新する関数
$$ w_i \longrightarrow w_i + \alpha (y – \hat{y}) x_i$$
$$ b \longrightarrow b + \alpha (y – \hat{y})$$
# Activation (sigmoid) function
def sigmoid(x):
return 1 / (1 + np.exp(-x))
def output_formula(features, weights, bias):
return sigmoid(np.dot(features, weights) + bias)
def error_formula(y, output):
return - y*np.log(output) - (1 - y) * np.log(1-output)
def update_weights(x, y, weights, bias, learnrate):
output = output_formula(x, weights, bias)
d_error = y - output
weights += learnrate * d_error * x
bias += learnrate * d_error
return weights, bias
Training function¶
This function will help us iterate the gradient descent algorithm through all the data, for a number of epochs. It will also plot the data, and some of the boundary lines obtained as we run the algorithm.
この関数は、複数エポックにわたって、全データに対して勾配降下法アルゴリズムを繰り返すのに役立つでしょう。また、それは、データと、アルゴリズムを実行して得られる境界線を描画する。
np.random.seed(44)
epochs = 100
learnrate = 0.01
def train(features, targets, epochs, learnrate, graph_lines=False):
errors = []
n_records, n_features = features.shape
last_loss = None
weights = np.random.normal(scale=1 / n_features**.5, size=n_features)
bias = 0
for e in range(epochs):
del_w = np.zeros(weights.shape)
for x, y in zip(features, targets):
output = output_formula(x, weights, bias)
error = error_formula(y, output)
weights, bias = update_weights(x, y, weights, bias, learnrate)
# Printing out the log-loss error on the training set
out = output_formula(features, weights, bias)
loss = np.mean(error_formula(targets, out))
errors.append(loss)
if e % (epochs / 10) == 0:
print("\n========== Epoch", e,"==========")
if last_loss and last_loss < loss:
print("Train loss: ", loss, " WARNING - Loss Increasing")
else:
print("Train loss: ", loss)
last_loss = loss
predictions = out > 0.5
accuracy = np.mean(predictions == targets)
print("Accuracy: ", accuracy)
if graph_lines and e % (epochs / 100) == 0:
display(-weights[0]/weights[1], -bias/weights[1])
# Plotting the solution boundary
plt.title("Solution boundary")
display(-weights[0]/weights[1], -bias/weights[1], 'black')
# Plotting the data
plot_points(features, targets)
plt.show()
# Plotting the error
plt.title("Error Plot")
plt.xlabel('Number of epochs')
plt.ylabel('Error')
plt.plot(errors)
plt.show()
Time to train the algorithm!¶
When we run the function, we’ll obtain the following:
関数を実行すると以下のような事項を得る。
- 10 updates with the current training loss and accuracy
最新学習損失・正確度付きエポック10回毎更新 - A plot of the data and some of the boundary lines obtained. The final one is in black. Notice how the lines get closer and closer to the best fit, as we go through more epochs.
得られたデータと幾つかの境界線の描画。最後の物については黒色。エポックが増す毎に、その線が、どのようにして最良適合直線に近付いていくのかに注目する。 - A plot of the error function. Notice how it decreases as we go through more epochs.
誤差関数のプロット。エポックが増す毎に、それが減少していくことに注目する。
train(X, y, epochs, learnrate, True)