目錄

• 翻譯
• • 從頭開始建立神經(jīng)網(wǎng)絡-簡介
  • 導包和配置
  • 生成一個數(shù)據(jù)集
  • 實現(xiàn)用來展示決策邊界的輔助函數(shù)
  • Logistic Regression
  • 訓練一個神經(jīng)網(wǎng)絡
  • • 我們的神經(jīng)網(wǎng)絡如何進行預測
    • 學習神經(jīng)網(wǎng)絡的參數(shù)
    • 實現(xiàn)神經(jīng)網(wǎng)絡
  • 訓練一個隱層有3個神經(jīng)元的神經(jīng)網(wǎng)絡
  • 驗證隱層神經(jīng)元個數(shù)對神經(jīng)網(wǎng)絡的影響
  • 練習
• 練習題解答
• • 1. Minibatch gradient
  • 2.Annealing learning rate
  • 3.其他激活函數(shù)
  • • Sigmoid Activation
    • ReLU Activation
  • 4.Three Classes
  • 5.Extend the network to 4 layers

翻譯

這篇文章是在完成吳恩達的深度學習課程作業(yè)的時候，在參考資料中看到的，感覺寫的不錯；這里翻譯一下內(nèi)容來加深自己的理解，同時在后面也完成了作者留下的一些作業(yè)，來提高自己對神經(jīng)網(wǎng)絡的認識。翻譯部分的原內(nèi)容請轉到原文傳送門。

從頭開始建立神經(jīng)網(wǎng)絡-簡介

在這篇文章中我們將從頭開始實現(xiàn)一個非常簡單的3層神經(jīng)網(wǎng)絡。這里并不會對所需的所有數(shù)學知識進行推導，但我會試著從直覺上來解釋我們在做什么。我也會給你指出去哪里查看你想要的細節(jié)。

這里我假設你已經(jīng)具備了基礎的微積分基礎和機器學的概念，舉例來說就是，你已經(jīng)知道什么是回歸問題，而什么是分類問題。更理想的是你也已經(jīng)知道了一些有關像梯度下降這樣的優(yōu)化方法。不過就算上面提到的東西你一無所知，這篇文章也能給你帶來樂趣。

為什么我想要從頭開始建立一個神經(jīng)網(wǎng)絡呢？就算是你將來打算使用像PyBrain這樣的框架來實現(xiàn)你的神經(jīng)網(wǎng)絡，那么有過至少一次從頭開始實現(xiàn)一個神經(jīng)網(wǎng)絡的經(jīng)驗能夠讓你明白神經(jīng)網(wǎng)絡是如何工作的；而明白神經(jīng)網(wǎng)絡的工作原理對于設計一個有效的模型是至關重要的。

還有一個需要注意的問題是這里的代碼并不是很高效，因為我想讓這些代碼更加易于理解。在后面的文章中我會使用Theano來實現(xiàn)一個高效的神經(jīng)網(wǎng)絡。（在我看的時候已經(jīng)實現(xiàn)好了，–>傳送門!!!所謂的高效就是用GPU）。

導包和配置

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

生成一個數(shù)據(jù)集

要訓練模型，首先就要生成一個數(shù)據(jù)集。很慶幸，scikit-learn 有很多有用的數(shù)據(jù)集生成器，我們這里直接使用make_moons這個函數(shù)來生成我們的數(shù)據(jù)集。

# Package imports # Generate a dataset and plot it 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)

生成的數(shù)據(jù)集結果如下

我們生成的數(shù)據(jù)集有兩個類別，分別用紅色和藍色的點來表示。你可以用這樣的場景來描述這個數(shù)據(jù)集：藍色的代表是男病人、紅色代表女病人，而x和y的值代表了兩項醫(yī)療指標。

我們的目標就是訓練一個能夠通過給定的x和y的值來正確識別紅色和藍色這兩個類別（男人或者女人）。注意我們生成的數(shù)據(jù)集不是線性可分的，也就是說我們無法畫一條直線來區(qū)分這兩個類別。這就意味著像邏輯回歸這樣的線性分類器無法識別這個數(shù)據(jù)集中的模式，除非你手工制造一些適合該模型的非線性的特征（例如：多項式）。

事實上，這是神經(jīng)網(wǎng)絡的一個主要優(yōu)勢。你無需擔心特征工程。隱層的神經(jīng)元將會幫你提取有用的特征。

實現(xiàn)用來展示決策邊界的輔助函數(shù)

# 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 paddingx_min, x_max = X[:, 0].min() - .5, X[:, 0].max() + .5y_min, y_max = X[:, 1].min() - .5, X[:, 1].max() + .5h = 0.01# Generate a grid of points with distance h between themxx, yy = np.meshgrid(np.arange(x_min, x_max, h),np.arange(y_min, y_max, h))# Predict the function value for the whole gidZ = pred_func(np.c_[xx.ravel(), yy.ravel()])Z = Z.reshape(xx.shape)# Plot the contour and training examplesplt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)plt.scatter(X[:, 0], X[:, 1], c=y, cmap=plt.cm.Spectral)

Logistic Regression

為了證明我的觀點，我將訓練一個邏輯回歸分類器。這個分類器接收x，y的輸入，然后輸出對應的分類（0 或者 1）。為了簡便，這里直接使用scikit-learn提供的模型來實現(xiàn)。

# Train the logistic regression classifier clf = sklearn.linear_model.LogisticRegressionCV(cv=5) clf.fit(X, y) # Plot the decision boundary plot_decision_boundary(lambda x: clf.predict(x)) plt.title("Logistic Regression")

運行結果如下

圖像顯示了我們的邏輯回歸算法所學習到的決策邊界，它已經(jīng)盡力地用一條直線來分割我們的數(shù)據(jù)集了，但是它永遠無法捕捉數(shù)據(jù)集中的“月牙”形狀。

訓練一個神經(jīng)網(wǎng)絡

我們現(xiàn)在開始建立一個3層的神經(jīng)網(wǎng)絡：一個輸入層、一個隱層、一個輸出層。輸入層的神經(jīng)元的個數(shù)由數(shù)據(jù)的特征緯度來決定，所以這里是2（x和y）。輸出層的神經(jīng)元個數(shù)由類別數(shù)來決定，也是2（有兩個類別）（由于我們只有兩個類別，因此我們完全可以用一個神經(jīng)元來作為輸出層，分別輸出1和0來代表兩個分類。但是用兩個神經(jīng)元可以為后面做多分類任務的擴展帶來便利）。我們輸入x和y，神經(jīng)網(wǎng)絡會輸出兩個概率值，一個代表類別0的概率（“女人”），另一個代表類別為1的概率（“男人”）。神經(jīng)網(wǎng)絡的結構如下圖所示：

隱層的神經(jīng)元的個數(shù)是由我們來指定的。我們放入隱層的神經(jīng)元的個數(shù)越多，我們就能夠模擬越復雜的函數(shù)。但是大量的隱層神經(jīng)元增加了代價。首先，在預測和訓練的過程中就需要跟高的計算能力。同時大量的參數(shù)就意味著更容易發(fā)生過擬合。

如何來決定隱層的大小？沒有什么一般性的知道方針，它由你所處理的問題不同而決定，同時這也是一種藝術！（調(diào)參的藝術！！！）后面我們會對隱層數(shù)量進行改變，來看看它是如何影響我們的輸出結果的。

同時我們也需要為我們的隱層挑選一個合適的激活函數(shù)。激活函數(shù)的作用就是把該層的輸入的線性組合某種變換。一個非線性的激活函數(shù)能夠讓我們做出非線性的預測。通常的激活函數(shù)選擇有tanh和sigmoid以及ReLUs。這里我們選擇使用tanh，因為它在很多場景下都表現(xiàn)的很好。這些函數(shù)的一個很好的屬性是它們的倒數(shù)可以使用原函數(shù)的值來計算。那tanh(x)來舉例，tanh(x)的導數(shù)是1-tanh²(x)。這個特性很有用，因為這使得我們能夠只計算一次tanh(x)的值，然后利用這個值來計算導數(shù)，減少了很多計算量。

因為我們想要我們的神經(jīng)網(wǎng)絡輸出概率，所以輸出層的激活函數(shù)需要使用softmax，它提供了將得分轉換為概率的途徑。如果你對logistic 函數(shù)很熟悉，那么你可以將softmax函數(shù)看作它在多分類問題上的擴展。

我們的神經(jīng)網(wǎng)絡如何進行預測

我們的神經(jīng)網(wǎng)絡通過前想傳播來進行預測，也就是一堆的矩陣乘法和對我們所定義的激活函數(shù)的應用。假設x是一個2維向量，那么我們通過如下的方式來計算$\hat{y}$：
z₁=xW₁+b₁
a₁=tanh(z₁)
z₂=a₁W₂+b₂
a₂=$\hat{y}$=softmax(z₂)
z_i是第i層的輸入，a_i是第i層經(jīng)過激活函數(shù)的作用后的輸出。W₁，b₁，W₂，b₂是神經(jīng)網(wǎng)絡的參數(shù)，需要我們從數(shù)據(jù)集中來學習。可以把它們看作是神經(jīng)網(wǎng)絡不同層之間的數(shù)據(jù)傳輸矩陣。通過矩陣乘法的定義，我們可以決定這些矩陣的緯度。假如隱層有500個神經(jīng)元，那么$W_1\in {\mathbb{R}}^{2\times 500}$，$b_1\in {\mathbb{R}}^{500}$，$W_2\in {\mathbb{R}}^{500\times 2}$，$b_2\in {\mathbb{R}}^2$。現(xiàn)在你應該能夠發(fā)現(xiàn)為什么隱層的神經(jīng)元的個數(shù)越多，我們的參數(shù)就越多了。

學習神經(jīng)網(wǎng)絡的參數(shù)

對參數(shù)進行學習，也就是尋找能夠最小化訓練集上的誤差的(W₁,b₁,W₂,b₂)。所以問題就變成了如何來定義這個誤差？我們會定義一個損失函數(shù)來描述這個誤差。輸出使用softmax，那么對應的損失函數(shù)通常定義為cross-entropy-loss(也叫做負log likelihood)。假如我們擁有N個訓練數(shù)據(jù)和C個類別，那么我們所預測的$\hat{y}$與真正的標簽y之間的損失函數(shù)定義為：
$$L(y,\hat{y})=?\frac{1}{N}\sum _{n\in N}\sum _{i\in C}{y}_{n,i}log{\hat{y}}_{n,i}$$

這個公式開起來很復雜，但實際上它做的事情就是如果我們在某個樣本上預測錯誤，就累計該錯誤。然后在整個樣本上求和。y（實際值）和$\hat{y}$（預測值）相差越大，損失函數(shù)的值就會越大。最小化損失函數(shù)，其實就是在數(shù)據(jù)集上最大話似然函數(shù)。

我們使用梯度下降法來最小化損失函數(shù)，這里使用最常見的一種梯度下降法，也就是通常的使用固定學習率的批量梯度下降法。它的變種SGD和minibatch-gradient-descent在實際應用中會表現(xiàn)更好。所以如果你是來真的，那么在他們之中選擇一個是正確的選擇。同時更理想的是你要使用隨著時間衰減的學習率。

作為輸入，梯度下降法需要損失函數(shù)對我們的參數(shù)的的梯度（導數(shù)的向量）:$\frac{?L}{?W_1}$,$\frac{?L}{?b_1}$,$\frac{?L}{?W_2}$,$\frac{?L}{?b_2}$。我們使用著名的誤差逆?zhèn)鞑ニ惴▉砬筮@些梯度。這里我不會細說誤差逆?zhèn)鞑ニ惴ㄊ侨绾喂ぷ鞯?#xff0c;你可以參考這些在網(wǎng)上流傳很廣的解釋傳送門、傳送門。

使用誤差逆?zhèn)鞑ニ惴?#xff0c;我們得到如下的計算公式（信我，我算的是對的）：
$${\delta }_3=\hat{y}?y$$
$${\delta }_2=(1?tan{h}^2{z}_1)°{\delta }_3{W}_2^T$$
$$\frac{?L}{?W_2}=a_1^T{\delta }_3$$
$$\frac{?L}{?b_2}={\delta }_3$$
$$\frac{?L}{?W_1}=x^T{\delta }_2$$
$$\frac{?L}{?b_1}={\delta }_2$$

實現(xiàn)神經(jīng)網(wǎng)絡

現(xiàn)在我們就要來實現(xiàn)一個神經(jīng)網(wǎng)絡了。我們先來定義一些有用的參數(shù)。

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

首先來實現(xiàn)一損失函數(shù)的計算，用它來評價我們的模型的表現(xiàn)。

# 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 predictionsz1 = X.dot(W1) + b1a1 = np.tanh(z1)z2 = a1.dot(W2) + b2exp_scores = np.exp(z2)probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)# Calculating the losscorect_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

同樣我們還要定義一個用來做預測的輔助函數(shù)。它運行我們定義的前向傳播算法，輸出概率最高的類別作為預測結果。

# 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 propagationz1 = x.dot(W1) + b1a1 = np.tanh(z1)z2 = a1.dot(W2) + b2exp_scores = np.exp(z2)probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)return np.argmax(probs, axis=1)

最后就是定義一個訓練模型的函數(shù)。實現(xiàn)了固定學習率的批量梯度下降算法。

#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 endmodel = {}# Gradient descent. For each batch...for i in range(0, num_passes):# Forward propagationz1 = X.dot(W1) + b1a1 = np.tanh(z1)z2 = a1.dot(W2) + b2exp_scores = np.exp(z2)probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)# Backpropagationdelta3 = probsdelta3[range(num_examples), y] -= 1dW2 = (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 * W2dW1 += reg_lambda * W1# Gradient descent parameter updateW1 += -epsilon * dW1b1 += -epsilon * db1W2 += -epsilon * dW2b2 += -epsilon * db2# Assign new parameters to the modelmodel = { '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

訓練一個隱層有3個神經(jīng)元的神經(jīng)網(wǎng)絡

# 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")

結果

哈!結果看起來相當不錯。我們的神經(jīng)網(wǎng)絡可以找到區(qū)分這兩個類別的決策邊界。

驗證隱層神經(jīng)元個數(shù)對神經(jīng)網(wǎng)絡的影響

在上面的例子中我們指定了隱層的神經(jīng)元的數(shù)量為3。現(xiàn)在讓我們來看看不同的神經(jīng)元個數(shù)對于輸出結果的影響。

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()

我們可以看到，低維的隱層能夠很好地捕捉數(shù)據(jù)的變化趨勢。高維的隱層有過擬合的趨向；高緯度的隱層“記憶”了訓練集中的所有數(shù)據(jù)，從而降低了在整個形狀上的泛華能力。加入我們使用獨立的測試集來衡量我們的模型（這是你應該做的！！）。小隱層的神經(jīng)網(wǎng)絡將會因為更強的泛華能力而表現(xiàn)的更好。我們可以通過更強的正則化來抵消過擬合，但為隱藏層選擇正確的大小是一種更“經(jīng)濟”的解決方案。

練習

用小批量梯度下降算法來替代批量梯度下降算法(參考)來訓練模型，小批量梯度下降算法通常表現(xiàn)的更好。

這里我們使用了固定的學習率$?$，為學習率實現(xiàn)一種退火的策略（也就是衰減策略）參考

我們使用了tanh來作為隱層的激活函數(shù)。嘗試一下其他的激活函數(shù)。

擴展神經(jīng)網(wǎng)絡為3分類神經(jīng)網(wǎng)絡（自己建立一個數(shù)據(jù)集）

擴展神經(jīng)網(wǎng)絡為一個4層的神經(jīng)網(wǎng)絡。嘗試一些不同的隱層的大小。

練習題解答

1. Minibatch gradient

只需要修改一下模型的訓練函數(shù)

import random def build_model_batch(nn_hdim, num_passes=50000, print_loss=False, batch_size=50):# 這里的batch_size就是小批量的大小# 建立一個訓練集的索引列表indexes = [index for index in range(num_examples)]# 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 endmodel = {}# Gradient descent. For each batch...for i in range(0, num_passes):# 隨機從訓練集中拿出 batch_size個數(shù)據(jù)進行訓練train_indexes = random.sample(indexes, batch_size)X_TRAIN = X[train_indexes, :]y_train = y[train_indexes]# Forward propagationz1 = X_TRAIN.dot(W1) + b1a1 = np.tanh(z1)z2 = a1.dot(W2) + b2exp_scores = np.exp(z2)probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)# Backpropagationdelta3 = probsdelta3[range(batch_size), y_train] -= 1dW2 = (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_TRAIN.T, delta2)db1 = np.sum(delta2, axis=0)# Add regularization terms (b1 and b2 don't # have regularization terms)dW2 += reg_lambda * W2dW1 += reg_lambda * W1# Gradient descent parameter updateW1 += -epsilon * dW1b1 += -epsilon * db1W2 += -epsilon * dW2b2 += -epsilon * db2# Assign new parameters to the modelmodel = { '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

用該方式訓練模型

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

結果

2.Annealing learning rate

退火有利于跳出局部最小！這里實現(xiàn)一個最簡單的退火方式。

# 初始的學習率 max_epsilon = 0.01 # 終止的學習率 min_epsilon = 0.001 def build_model_annealing(nn_hdim, num_passes=80000, print_loss=False, explore=100):# explore就是退火周期，每explore次迭代，退火一次# 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 endmodel = {}# 初始化學習率為最大tem_epsilon = max_epsilon# Gradient descent. For each batch...for i in range(0, num_passes):# 進行退火if tem_epsilon > min_epsilon and i % explore == 0:tem_epsilon -= (max_epsilon - min_epsilon) / exploretem_epsilon = max(tem_epsilon, min_epsilon)# Forward propagationz1 = X.dot(W1) + b1a1 = np.tanh(z1)z2 = a1.dot(W2) + b2exp_scores = np.exp(z2)probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)# Backpropagationdelta3 = probsdelta3[range(num_examples), y] -= 1dW2 = (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)dW2 += reg_lambda * W2dW1 += reg_lambda * W1# Gradient descent parameter updateW1 += -tem_epsilon * dW1b1 += -tem_epsilon * db1W2 += -tem_epsilon * dW2b2 += -tem_epsilon * db2# Assign new parameters to the modelmodel = { 'W1': W1, 'b1': b1, 'W2': W2, 'b2': b2}if print_loss and i % 1000 == 0:print("Loss after iteration %i: %f，the now learning rate is %s" %(i, calculate_loss(model), tem_epsilon))return model # Build a model with a 3-dimensional hidden layer with Annealing model = build_model_annealing(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 with Annealing")

運行結果

3.其他激活函數(shù)

更改了激活函數(shù)以后，計算損失、預測和模型訓練方法都需要進行修改。其實最合理的是增加一個激活函數(shù)的參數(shù)，我這里偷懶就直接復制了·

Sigmoid Activation

# 定義sigmoid函數(shù) def sigmoid(z):s = 1.0 / (1 + np.exp(-z))return s # 繪制圖像 def draw_sigmoid():fig = plt.figure(figsize=(6,4))ax = fig.add_subplot(111)x=np.linspace(-6,6,1000) #這個表示在-5到5之間生成1000個x值y=sigmoid(x) #對上述生成的1000個數(shù)循環(huán)用sigmoid公式求對應的yplt.xlim((-6,6))plt.ylim((0.00,1.00))plt.yticks([0,0.5,1.0],[0,0.5,1.0]) #設置y軸顯示的刻度plt.plot(x,y,color='darkblue') #用上述生成的1000個xy值對生成1000個點ax=plt.gca()ax.spines['right'].set_color('none') #刪除右邊框設為無ax.spines['top'].set_color('none') #刪除上邊框設為無ax.xaxis.set_ticks_position('bottom')ax.spines['bottom'].set_position(('data', 0)) #調(diào)整x軸位置ax.yaxis.set_ticks_position('left')ax.spines['left'].set_position(('data', 0)) #調(diào)整y軸位置plt.xlabel("sigmoid")plt.show() draw_sigmoid()

運行結果

激活函數(shù)為sigmoid的模型實現(xiàn)

# Helper function to evaluate the total loss on the dataset # sigmoid edition def calculate_loss_sigmoid(model):W1, b1, W2, b2 = model['W1'], model['b1'], model['W2'], model['b2']# Forward propagation to calculate our predictionsz1 = X.dot(W1) + b1# 修改了激活函數(shù)a1 = sigmoid(z1)z2 = a1.dot(W2) + b2exp_scores = np.exp(z2)probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)# Calculating the losscorect_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 # Helper function to predict an output (0 or 1) # sigmoid edtion def predict_sigmoid(model, x):W1, b1, W2, b2 = model['W1'], model['b1'], model['W2'], model['b2']# Forward propagationz1 = x.dot(W1) + b1# 修改了激活函數(shù)a1 = sigmoid(z1)z2 = a1.dot(W2) + b2exp_scores = np.exp(z2)probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)return np.argmax(probs, axis=1) # sigmoid edition of building model def build_model_sigmoid(nn_hdim, num_passes=50000, 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 endmodel = {}# Gradient descent. For each batch...for i in range(0, num_passes):# Forward propagationz1 = X.dot(W1) + b1# 修改了激活函數(shù)a1 = sigmoid(z1)z2 = a1.dot(W2) + b2exp_scores = np.exp(z2)probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)# Backpropagationdelta3 = probsdelta3[range(num_examples), y] -= 1dW2 = (a1.T).dot(delta3)db2 = np.sum(delta3, axis=0, keepdims=True)# 這里的導數(shù)變成了sigmoid的導數(shù)，即a1*(1-a1)delta2 = delta3.dot(W2.T)*a1*(1-a1)dW1 = np.dot(X.T, delta2)db1 = np.sum(delta2, axis=0)dW2 += reg_lambda * W2dW1 += reg_lambda * W1# Gradient descent parameter updateW1 += -epsilon * dW1b1 += -epsilon * db1W2 += -epsilon * dW2b2 += -epsilon * db2# Assign new parameters to the modelmodel = { 'W1': W1, 'b1': b1, 'W2': W2, 'b2': b2}# 改為用sigmoid計算損失if print_loss and i % 1000 == 0:print("Loss after iteration %i: %f" %(i, calculate_loss_sigmoid(model)))return model # Build a model with a 3-dimensional hidden layer with sigmoid model = build_model_sigmoid(3, print_loss=True)# Plot the decision boundary plot_decision_boundary(lambda x: predict_sigmoid(model, x)) plt.title("Decision Boundary for hidden layer size 3 with sigmoid")

運行結果

ReLU Activation

ReLU圖像

def draw_ReLU():fig = plt.figure(figsize=(6, 4))ax = fig.add_subplot(111)x = np.arange(-10, 10)y = np.where(x>0, x, 0)plt.xlim(-11, 11)plt.ylim(-11, 11)ax.spines['top'].set_color('none')ax.spines['right'].set_color('none')ax.xaxis.set_ticks_position('bottom')ax.spines['bottom'].set_position(('data', 0))ax.set_xticks([-10, -5, 0, 5, 10])ax.yaxis.set_ticks_position('left')ax.spines['left'].set_position(('data', 0))ax.set_yticks([-10, -5, 5, 10])plt.plot(x, y, label="ReLU", color="blue")plt.legend()plt.show() draw_ReLU()

運行結果

激活函數(shù)為ReLU的模型實現(xiàn)

# Helper function to evaluate the total loss on the dataset # ReLU edition def calculate_loss_ReLU(model):W1, b1, W2, b2 = model['W1'], model['b1'], model['W2'], model['b2']# Forward propagation to calculate our predictionsz1 = X.dot(W1) + b1# 激活函數(shù)改為了ReLUa1 = np.where(z1>0, z1, 0)z2 = a1.dot(W2) + b2exp_scores = np.exp(z2)probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)# Calculating the losscorect_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 # Helper function to predict an output (0 or 1) # ReLU edtion def predict_ReLU(model, x):W1, b1, W2, b2 = model['W1'], model['b1'], model['W2'], model['b2']# Forward propagationz1 = x.dot(W1) + b1# 激活函數(shù)改為了ReLUa1 = np.where(z1>0, z1, 0)z2 = a1.dot(W2) + b2exp_scores = np.exp(z2)probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)return np.argmax(probs, axis=1) # ReLU edition of building model def build_model_ReLU(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 endmodel = {}# Gradient descent. For each batch...for i in range(0, num_passes):# Forward propagationz1 = X.dot(W1) + b1# 激活函數(shù)改為ReLUa1 = np.where(z1>0, z1,0)z2 = a1.dot(W2) + b2exp_scores = np.exp(z2)probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)# Backpropagationdelta3 = probsdelta3[range(num_examples), y] -= 1dW2 = (a1.T).dot(delta3)db2 = np.sum(delta3, axis=0, keepdims=True)delta2 = delta3.dot(W2.T)# ReLu求導delta2 = np.where(z1 >0, delta2, 0)dW1 = np.dot(X.T, delta2)db1 = np.sum(delta2, axis=0)dW2 += reg_lambda * W2dW1 += reg_lambda * W1# Gradient descent parameter updateW1 += -epsilon * dW1b1 += -epsilon * db1W2 += -epsilon * dW2b2 += -epsilon * db2# Assign new parameters to the modelmodel = { 'W1': W1, 'b1': b1, 'W2': W2, 'b2': b2}# 改為ReLU計算損失if print_loss and i % 1000 == 0:print("Loss after iteration %i: %f" %(i, calculate_loss_ReLU(model)))return model # Build a model with a 3-dimensional hidden layer with sigmoid model = build_model_ReLU(3, print_loss=True)# Plot the decision boundary plot_decision_boundary(lambda x: predict_ReLU(model, x)) plt.title("Decision Boundary for hidden layer size 3 with ReLU")

運行結果

4.Three Classes

先搞一個數(shù)據(jù)集
這里也用scikit-learn來生成

# Generate a dataset and plot it X1, y1 = sklearn.datasets.make_classification(n_samples=300, n_features=2,n_redundant=0, n_informative=2,n_clusters_per_class=1,n_classes=3,random_state=29) plt.scatter(X1[:,0], X1[:,1], s=40, c=y1, cmap=plt.cm.Spectral)

運行結果

初始化三分類問題的參數(shù)

num_examples1 = len(X1) # training set size nn_input_dim1 = 2 # input layer dimensionality nn_output_dim1 = 3 # output layer dimensionality

實現(xiàn)三分類神經(jīng)網(wǎng)絡
由于輸出采用的是softmax，所以基本不用修改。只是修改一下數(shù)據(jù)集和輸出層的神經(jīng)元個數(shù)。

# Helper function to evaluate the total loss on the dataset # 這里只是簡單的把數(shù)據(jù)集更換了 def calculate_loss1(model):W1, b1, W2, b2 = model['W1'], model['b1'], model['W2'], model['b2']# Forward propagation to calculate our predictionsz1 = X1.dot(W1) + b1a1 = np.tanh(z1)z2 = a1.dot(W2) + b2exp_scores = np.exp(z2)probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)# Calculating the losscorect_logprobs = -np.log(probs[range(num_examples1), y1])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_examples1 * data_loss # Helper function to predict an output (0 or 1) # 這個就沒改··· def predict1(model, x):W1, b1, W2, b2 = model['W1'], model['b1'], model['W2'], model['b2']# Forward propagationz1 = x.dot(W1) + b1a1 = np.tanh(z1)z2 = a1.dot(W2) + b2exp_scores = np.exp(z2)probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)return np.argmax(probs, axis=1) def build_model1(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_dim1, nn_hdim) / np.sqrt(nn_input_dim1)b1 = np.zeros((1, nn_hdim))W2 = np.random.randn(nn_hdim, nn_output_dim1) / np.sqrt(nn_hdim)b2 = np.zeros((1, nn_output_dim1))# This is what we return at the endmodel = {}# Gradient descent. For each batch...for i in range(0, num_passes):# Forward propagationz1 = X1.dot(W1) + b1a1 = np.tanh(z1)z2 = a1.dot(W2) + b2exp_scores = np.exp(z2)probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)# Backpropagationdelta3 = probsdelta3[range(num_examples1), y1] -= 1dW2 = (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(X1.T, delta2)db1 = np.sum(delta2, axis=0)dW2 += reg_lambda * W2dW1 += reg_lambda * W1# Gradient descent parameter updateW1 += -epsilon * dW1b1 += -epsilon * db1W2 += -epsilon * dW2b2 += -epsilon * db2# Assign new parameters to the modelmodel = { 'W1': W1, 'b1': b1, 'W2': W2, 'b2': b2}if print_loss and i % 1000 == 0:print("Loss after iteration %i: %f" %(i, calculate_loss1(model)))return model

由于數(shù)據(jù)集發(fā)生了變化，作者的繪制邊界的函數(shù)中的數(shù)據(jù)集都是寫死在函數(shù)里的，為了偷懶我就復制一個新的，改了改

def plot_decision_boundary1(pred_func):# Set min and max values and give it some paddingx_min, x_max = X1[:, 0].min() - .5, X1[:, 0].max() + .5y_min, y_max = X1[:, 1].min() - .5, X1[:, 1].max() + .5h = 0.01# Generate a grid of points with distance h between themxx, yy = np.meshgrid(np.arange(x_min, x_max, h),np.arange(y_min, y_max, h))# Predict the function value for the whole gidZ = pred_func(np.c_[xx.ravel(), yy.ravel()])Z = Z.reshape(xx.shape)# Plot the contour and training examplesplt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)plt.scatter(X1[:, 0], X1[:, 1], c=y1, cmap=plt.cm.Spectral) # Build a model with a 5-dimensional hidden layer model = build_model1(5, print_loss=True)# Plot the decision boundary plot_decision_boundary1(lambda x: predict1(model, x)) plt.title("Decision Boundary for hidden layer size 3")

運行結果

5.Extend the network to 4 layers

三個函數(shù)都增加了一個隱層而已~

def calculate_loss2(model):W1, b1, W2, b2, W3, b3 = model['W1'], model['b1'], model['W2'], model['b2'], model['W3'], model['b3']# Forward propagation to calculate our predictionsz1 = X.dot(W1) + b1a1 = np.tanh(z1)z2 = a1.dot(W2) + b2a2 = np.tanh(z2)z3 = a2.dot(W3) + b3exp_scores = np.exp(z3)probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)# Calculating the losscorect_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 def predict2(model, x):W1, b1, W2, b2, W3, b3 = model['W1'], model['b1'], model['W2'], model['b2'], model['W3'], model['b3']# Forward propagationz1 = x.dot(W1) + b1a1 = np.tanh(z1)z2 = a1.dot(W2) + b2a2 = np.tanh(z2)z3 = a2.dot(W3) + b3exp_scores = np.exp(z3)probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)return np.argmax(probs, axis=1) def build_model2(nn_hdim, nn_hdim1, 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_dim1)b1 = np.zeros((1, nn_hdim))W2 = np.random.randn(nn_hdim, nn_hdim1) / np.sqrt(nn_hdim)b2 = np.zeros((1, nn_hdim1))W3 = np.random.randn(nn_hdim1, nn_output_dim) / np.sqrt(nn_hdim1)b3 = np.zeros((1, nn_output_dim))# This is what we return at the endmodel = {}# Gradient descent. For each batch...for i in range(0, num_passes):# Forward propagationz1 = X.dot(W1) + b1a1 = np.tanh(z1)z2 = a1.dot(W2) + b2a2 = np.tanh(z2)z3 = a2.dot(W3) + b3exp_scores = np.exp(z3)probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)# Backpropagationdelta4 = probsdelta4[range(num_examples), y] -= 1dW3 = (a2.T).dot(delta4)db3 = np.sum(delta4, axis=0, keepdims=True)delta3 = delta4.dot(W3.T) * (1 - np.power(a2, 2))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, keepdims=True)dW3 += reg_lambda * W3dW2 += reg_lambda * W2dW1 += reg_lambda * W1# Gradient descent parameter updateW1 += -epsilon * dW1b1 += -epsilon * db1W2 += -epsilon * dW2b2 += -epsilon * db2W3 += -epsilon * dW3b3 += -epsilon * db3# Assign new parameters to the modelmodel = { 'W1': W1, 'b1': b1, 'W2': W2, 'b2': b2, 'W3': W3, 'b3': b3}if print_loss and i % 1000 == 0:print("Loss after iteration %i: %f" %(i, calculate_loss2(model)))return model # Build a model with a 3_4-dimensional hidden layer model = build_model2(3,4, print_loss=True)# Plot the decision boundary plot_decision_boundary(lambda x: predict2(model, x)) plt.title("Decision Boundary for hidden layer size 3-4")

運行結果
我嘗試了一下，感覺3，4就不錯。

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