使用python从头开始实现一个神经网络

發表於 2015-09-27   |   分類於 机器学习   |  

导入相关包

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# Package imports
import matplotlib.pyplot as plt
import numpy as np
from numpy import arange
import sklearn
import sklearn.datasets
import sklearn.linear_model
import matplotlib
# Display plots inline and change default figure size
matplotlib.rcParams['figure.figsize'] = (10.0, 8.0)

生成数据集

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# Generate a dataset and plot it
# 生成两个交错的半圆的数据集
np.random.seed(0)
X, y = sklearn.datasets.make_moons(200, noise=0.20)
plt.scatter(X[:,0], X[:,1], s=40, c=y, cmap=plt.cm.Spectral)

绘制决策边界

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# Helper function to plot a decision boundary.
# If you don't fully understand this function don't worry, it just generates the contour plot below.
def plot_decision_boundary(pred_func):
    # Set min and max values and give it some padding
    x_min, x_max = X[:, 0].min() - .5, X[:, 0].max() + .5
    y_min, y_max = X[:, 1].min() - .5, X[:, 1].max() + .5
    h = 0.01
    # Generate a grid of points with distance h between them
    xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))
    # Predict the function value for the whole gid
    Z = pred_func(np.c_[xx.ravel(), yy.ravel()])
    Z = Z.reshape(xx.shape)
    # Plot the contour and training examples
    plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)
    plt.scatter(X[:, 0], X[:, 1], c=y, cmap=plt.cm.Spectral)

logistic rgeression classifier 无法解决问题

模型训练

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# Train the logistic rgeression classifier
clf = sklearn.linear_model.LogisticRegressionCV()
clf.fit(X, y)

查看结果

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# Plot the decision boundary
plot_decision_boundary(lambda x: clf.predict(x))
plt.title("Logistic Regression")
plt.show()

神经网络的初始化参数

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num_examples = len(X) # training set size
nn_input_dim = 2 # input layer dimensionality
nn_output_dim = 2 # output layer dimensionality
# Gradient descent parameters (I picked these by hand)
epsilon = 0.01 # learning rate for gradient descent
reg_lambda = 0.01 # regularization strength

计算损失函数

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# Helper function to evaluate the total loss on the dataset
def calculate_loss(model):
    W1, b1, W2, b2 = model['W1'], model['b1'], model['W2'], model['b2']
    # Forward propagation to calculate our predictions
    z1 = X.dot(W1) + b1
    a1 = np.tanh(z1)
    z2 = a1.dot(W2) + b2
    exp_scores = np.exp(z2)
    probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)
    # Calculating the loss
    corect_logprobs = -np.log(probs[range(num_examples), y])
    data_loss = np.sum(corect_logprobs)
    # Add regulatization term to loss (optional)
    data_loss += reg_lambda/2 * (np.sum(np.square(W1)) + np.sum(np.square(W2)))
    return 1./num_examples * data_loss

预测

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# Helper function to predict an output (0 or 1)
def predict(model, x):
    W1, b1, W2, b2 = model['W1'], model['b1'], model['W2'], model['b2']
    # Forward propagation
    z1 = x.dot(W1) + b1
    a1 = np.tanh(z1)
    z2 = a1.dot(W2) + b2
    exp_scores = np.exp(z2)
    probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)
    return np.argmax(probs, axis=1)

模型训练

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# This function learns parameters for the neural network and returns the model.
# - nn_hdim: Number of nodes in the hidden layer
# - num_passes: Number of passes through the training data for gradient descent
# - print_loss: If True, print the loss every 1000 iterations
def build_model(nn_hdim, num_passes=20000, print_loss=False):
    
    # Initialize the parameters to random values. We need to learn these.
    np.random.seed(0)
    W1 = np.random.randn(nn_input_dim, nn_hdim) / np.sqrt(nn_input_dim)
    b1 = np.zeros((1, nn_hdim))
    W2 = np.random.randn(nn_hdim, nn_output_dim) / np.sqrt(nn_hdim)
    b2 = np.zeros((1, nn_output_dim))
    # This is what we return at the end
    model = {}
    
    # Gradient descent. For each batch...
    for i in xrange(0, num_passes):
        # Forward propagation
        z1 = X.dot(W1) + b1
        a1 = np.tanh(z1)
        z2 = a1.dot(W2) + b2
        exp_scores = np.exp(z2)
        probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)
        # Backpropagation
        delta3 = probs
        delta3[range(num_examples), y] -= 1
        dW2 = (a1.T).dot(delta3)
        db2 = np.sum(delta3, axis=0, keepdims=True)
        delta2 = delta3.dot(W2.T) * (1 - np.power(a1, 2))
        dW1 = np.dot(X.T, delta2)
        db1 = np.sum(delta2, axis=0)
        # Add regularization terms (b1 and b2 don't have regularization terms)
        dW2 += reg_lambda * W2
        dW1 += reg_lambda * W1
        # Gradient descent parameter update
        W1 += -epsilon * dW1
        b1 += -epsilon * db1
        W2 += -epsilon * dW2
        b2 += -epsilon * db2
        
        # Assign new parameters to the model
        model = { 'W1': W1, 'b1': b1, 'W2': W2, 'b2': b2}
        
        # Optionally print the loss.
        # This is expensive because it uses the whole dataset, so we don't want to do it too often.
        if print_loss and i % 1000 == 0:
          print "Loss after iteration %i: %f" %(i, calculate_loss(model))
    
    return model

模型实例

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# Build a model with a 3-dimensional hidden layer
model = build_model(3, print_loss=True)
# Plot the decision boundary
plot_decision_boundary(lambda x: predict(model, x))
plt.title("Decision Boundary for hidden layer size 3")
plt.show()

不同的隐含层规模

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plt.figure(figsize=(16, 32))
hidden_layer_dimensions = [1, 2, 3, 4, 5, 20, 50]
for i, nn_hdim in enumerate(hidden_layer_dimensions):
    plt.subplot(5, 2, i+1)
    plt.title('Hidden Layer size %d' % nn_hdim)
    model = build_model(nn_hdim)
    plot_decision_boundary(lambda x: predict(model, x))
plt.show()

其他值得关注的点

查看tanh函数

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# http://stackoverflow.com/questions/7267226/range-for-floats
# 查看tanh
X=arange(-10,10,0.05)
print X.shape
y=np.tanh(X)
plt.scatter(X, y, c='r')
plt.show()

softmax函数

http://blog.csdn.net/kevinew/article/details/9407367

关于隐含层的规模

But higher dimensionality comes at a cost. First, more computation is required to make predictions and learn the network parameters.A bigger number of parameters also means we become more prone to overfitting our data.

过大的隐含层规模有两个缺点，首先是建完的模型有更多的参数需要运算;另外隐含层过大非常容易导致数据过拟合。

关于合适的隐含层规模(hidden layer size)，《Matlab在数学建模中的应用》有一小节简单做了讨论，提供了两个经验公式,当然这里的结论只是仅供参考而已。

$$ l=\sqrt{m+n}+a $$

$$ l=\sqrt{0.43mn+0.12n^{2}+2.54m+0.77n+0.35+0.51} $$

其中m,n分别为输入层规模与输出层规模，a为1～10之间的常数,l为需要的隐含层规模。

关于反向传播的细节

I won’t go into detail how backpropagation works, but there are many excellent explanations (here or here) floating around the web.

http://colah.github.io/posts/2015-08-Backprop/

http://cs231n.github.io/optimization-2/

变学习速率

So if you are serious you’ll want to use one of these, and ideally you would also decay the learning rate over time.

http://cs231n.github.io/neural-networks-3/#anneal

参考资料

[1] 从头开始实现神经网络：入门
[2] Implementing a Neural Network from Scratch in Python – An Introduction
[3] nn-from-scratch (github)
[4] sklearn的多层感知器

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