MNIST 数字的神经网络根本没有学习——反向传播的问题

Question

很长一段时间后，我仍然无法在没有任何错误的情况下运行我的 nn。这个玩具 nn 的准确度是惊人的 1-2%（隐藏层中的 60 个神经元，100 个 epoch，0.3 学习率，tanh 激活，通过 TF 下载的 MNIST 数据集）——所以基本上它根本没有学习。看了这么久的视频/关于反向传播的帖子后，我仍然无法修复它。 所以我的错误必须在标有两条##### 行的部分之间。我认为我对衍生品的总体理解很好，但我无法将这些知识与反向传播联系起来。 如果反向传播基础是正确的，那么错误一定是axis = 0/1，因为我也无法理解，如何确定我将在哪个轴上工作。

另外，我有一种强烈的感觉，dZ2 = A2 - Y 可能是错的，应该是dZ2 = Y - A2，但是在更正之后，nn 开始只猜一个数字。

（是的，反向传播本身我没有写，我在网上找到的）

#importing data and normalizing it
#"x_test" will be my X
#"y_test" will be my Y

import tensorflow as tf
(traindataX, traindataY), (testdataX, testdataY) = tf.keras.datasets.mnist.load_data()
x_test = testdataX.reshape(testdataX.shape[0], testdataX.shape[1]**2).astype('float32')
x_test = x_test / 255

y_test = testdataY
y_test = np.eye(10)[y_test]

#Activation functions:
def tanh(z):
    a = (np.exp(z)-np.exp(-z))/(np.exp(z)+np.exp(-z))
    return a
###############################################################################START
def softmax(z):
    smExp = np.exp(z - np.max(z, axis=0))
    out = smExp / np.sum(smExp, axis=0)
    return out
###############################################################################STOP

def neural_network(num_hid, epochs, 
                  learning_rate, X, Y):
    #num_hid - number of neurons in the hidden layer
    #X - dataX - shape (10000, 784)
    #Y - labels - shape (10000, 10)

    #inicialization
    W1 = np.random.randn(784, num_hid) * 0.01
    W2 = np.random.randn(num_hid, 10) * 0.01
    b1 = np.zeros((1, num_hid))
    b2 = np.zeros((1, 10))
    correct = 0

    for x in range(1, epochs+1):
        #feedforward
        Z1 = np.dot(X, W1) + b1
        A1 = tanh(Z1)
        Z2 = np.dot(A1, W2) + b2
        A2 = softmax(Z2)


        ###############################################################################START
        m = X.shape[1] #-> 784
        loss = - np.sum((Y * np.log(A2)), axis=0, keepdims=True)
        cost = np.sum(loss, axis=1) / m

        #backpropagation
        dZ2 = A2 - Y
        dW2 = (1/m)*np.dot(A1.T, dZ2)
        db2 = (1/m)*np.sum(dZ2, axis = 1, keepdims = True)
        dZ1 = np.multiply(np.dot(dZ2, W2.T), 1 - np.power(A1, 2))
        dW1 = (1/m)*np.dot(X.T, dZ1)
        db1 = (1/m)*np.sum(dZ1, axis = 1, keepdims = True)
        ###############################################################################STOP


        #parameters update - gradient descent
        W1 = W1 - dW1*learning_rate
        b1 = b1 - db1*learning_rate
        W2 = W2 - dW2*learning_rate
        b2 = b2 - db2*learning_rate


        for i in range(np.shape(Y)[1]):
            guess = np.argmax(A2[i, :])
            ans = np.argmax(Y[i, :])
            print(str(x) + " " + str(i) + ". " +"guess: ", guess, "| ans: ", ans)
            if guess == ans:
                correct = correct + 1;

    accuracy = (correct/np.shape(Y)[0]) * 100

Answer 1

卢卡斯， 刷新基础的好问题。我对您的代码进行了一些修复：

• 米的计算
• 转置了所有权重和偏差（无法正确解释，但否则无法正常工作）。
• 更改了准确度（和损失，未使用）的计算。

请参阅下面的更正代码。使用您的原始参数可以达到 90% 的准确度：

def neural_network(num_hid, epochs, learning_rate, X, Y):
#num_hid - number of neurons in the hidden layer
#X - dataX - shape (10000, 784)
#Y - labels - shape (10000, 10)

#inicialization
# W1 = np.random.randn(784, num_hid) * 0.01
# W2 = np.random.randn(num_hid, 10) * 0.01
# b1 = np.zeros((1, num_hid))
# b2 = np.zeros((1, 10))
W1 = np.random.randn(num_hid, 784) * 0.01
W2 = np.random.randn(10, num_hid) * 0.01
b1 = np.zeros((num_hid, 1))
b2 = np.zeros((10, 1))

for x in range(1, epochs+1):
    correct = 0  # moved inside cycle
    #feedforward
    # Z1 = np.dot(X, W1) + b1
    Z1 = np.dot(W1, X.T) + b1
    A1 = tanh(Z1)
    # Z2 = np.dot(A1, W2) + b2
    Z2 = np.dot(W2, A1) + b2
    A2 = softmax(Z2)

    ###############################################################################START
    m = X.shape[0] #-> 784  # SHOULD BE NUMBER OF SAMPLES IN THE BATCH
    # loss = - np.sum((Y * np.log(A2)), axis=0, keepdims=True)
    loss = - np.sum((Y.T * np.log(A2)), axis=0, keepdims=True)
    cost = np.sum(loss, axis=1) / m

    #backpropagation
    # dZ2 = A2 - Y
    # dW2 = (1/m)*np.dot(A1.T, dZ2)
    # db2 = (1/m)*np.sum(dZ2, axis = 1, keepdims = True)
    # dZ1 = np.multiply(np.dot(dZ2, W2.T), 1 - np.power(A1, 2))
    # dW1 = (1/m)*np.dot(X.T, dZ1)
    dZ2 = A2 - Y.T
    dW2 = (1/m)*np.dot(dZ2, A1.T)
    db2 = (1/m)*np.sum(dZ2, axis = 1, keepdims = True)
    dZ1 = np.multiply(np.dot(W2.T, dZ2), 1 - np.power(A1, 2))
    dW1 = (1/m)*np.dot(dZ1, X)

    db1 = (1/m)*np.sum(dZ1, axis = 1, keepdims = True)
    ###############################################################################STOP

    #parameters update - gradient descent
    W1 = W1 - dW1*learning_rate
    b1 = b1 - db1*learning_rate
    W2 = W2 - dW2*learning_rate
    b2 = b2 - db2*learning_rate

    guess = np.argmax(A2, axis=0)  # axis fixed
    ans = np.argmax(Y, axis=1)  # axis fixed
    # print (guess.shape, ans.shape)
    correct += sum (guess==ans)

    #     #print(str(x) + " " + str(i) + ". " +"guess: ", guess, "| ans: ", ans)
    #     if guess == ans:
    #         correct = correct + 1;
    accuracy = correct / x_test.shape[0]
    print (f"Epoch {x}. accuracy = {accuracy*100:.2f}%")


neural_network (64, 100, 0.3, x_test, y_test)

Epoch 1. accuracy = 14.93%
Epoch 2. accuracy = 34.70%
Epoch 3. accuracy = 47.41%
(...)
Epoch 98. accuracy = 89.29%
Epoch 99. accuracy = 89.33%
Epoch 100. accuracy = 89.37%

Answer 2

这可能是因为您应该通过将 X 除以 255（255 是最大像素值）在 0 和 1 的值之间标准化输入。您还应该将 Y one hot 编码为一系列大小为 10 的向量。我认为你的反向传播是正确的，但你应该实施梯度检查来仔细检查。

Answer 3

您以错误的方式计算准确性。 对于一个时期的每次迭代，第一个正确的变量应该初始化为 0，第二个 如果 y.shape 是 (10000, 10)，那么为了计算精度，循环应该是 for i in range(np.shape(Y)[0]) 而不是 for i in range(np.shape(Y)[1]) ，第一个将迭代 10,000 次，第二个将迭代 10 次。

更好的方法是使用 NumPy 计算正确猜测的数量correct = np.sum(np.argmax(A2,axis=1) == np.argmax(Y,axis=1))

你的学习率太高了，我可以通过将学习率设置为 0.003 50 个 epoch 和 60 个隐藏神经元来达到 50% 的准确率

def neural_network(num_hid, epochs, 
                  learning_rate, X, Y):
    #num_hid - number of neurons in the hidden layer
    #X - dataX - shape (10000, 784)
    #Y - labels - shape (10000, 10)

    #inicialization
    W1 = np.random.randn(784, num_hid) * 0.01
    W2 = np.random.randn(num_hid, 10) * 0.01
    b1 = np.zeros((1, num_hid))
    b2 = np.zeros((1, 10))
    correct = 0

    for x in range(1, epochs+1):
        #feedforward
        Z1 = np.dot(X, W1) + b1
        A1 = tanh(Z1)
        Z2 = np.dot(A1, W2) + b2
        A2 = softmax(Z2)


        ###############################################################################START
        m = X.shape[1] #-> 784
        loss = - np.sum((Y * np.log(A2)), axis=0, keepdims=True)
        cost = np.sum(loss, axis=1) / m

        #backpropagation
        dZ2 = A2 - Y
        dW2 = (1/m)*np.dot(A1.T, dZ2)
        db2 = (1/m)*np.sum(dZ2, axis = 1, keepdims = True)
        dZ1 = np.multiply(np.dot(dZ2, W2.T), 1 - np.power(A1, 2))
        dW1 = (1/m)*np.dot(X.T, dZ1)
        db1 = (1/m)*np.sum(dZ1, axis = 1, keepdims = True)
        ###############################################################################STOP


        #parameters update - gradient descent
        W1 = W1 - dW1*learning_rate
        b1 = b1 - db1*learning_rate
        W2 = W2 - dW2*learning_rate
        b2 = b2 - db2*learning_rate

        correct = 0
        for i in range(np.shape(Y)[0]):
            guess = np.argmax(A2[i, :])
            ans = np.argmax(Y[i, :])
#             print(str(x) + " " + str(i) + ". " +"guess: ", guess, "| ans: ", ans)
            if guess == ans:
                correct = correct + 1

#         correct = np.sum(np.argmax(A2,axis=1) == np.argmax(Y,axis=1))
#         print(correct)
        accuracy = (correct/np.shape(Y)[0]) * 100
        print(accuracy)

您需要进行实验，为了获得良好的准确性，请尝试调整隐藏层数、时期和学习率。