Python 3.x 如何正确实现机器学习MNIST数据集的反向传播？

因此，我使用Michael Nielson的机器学习书作为代码参考（基本相同）：

有关守则：

    def backpropagate(self, image, image_value) :


        # declare two new numpy arrays for the updated weights & biases
        new_biases = [np.zeros(bias.shape) for bias in self.biases]
        new_weights = [np.zeros(weight_matrix.shape) for weight_matrix in self.weights]

        # -------- feed forward --------
        # store all the activations in a list
        activations = [image]

        # declare empty list that will contain all the z vectors
        zs = []
        for bias, weight in zip(self.biases, self.weights) :
            print(bias.shape)
            print(weight.shape)
            print(image.shape)
            z = np.dot(weight, image) + bias
            zs.append(z)
            activation = sigmoid(z)
            activations.append(activation)

        # -------- backward pass --------
        # transpose() returns the numpy array with the rows as columns and columns as rows
        delta = self.cost_derivative(activations[-1], image_value) * sigmoid_prime(zs[-1])
        new_biases[-1] = delta
        new_weights[-1] = np.dot(delta, activations[-2].transpose())

        # l = 1 means the last layer of neurons, l = 2 is the second-last, etc.
        # this takes advantage of Python's ability to use negative indices in lists
        for l in range(2, self.num_layers) :
            z = zs[-1]
            sp = sigmoid_prime(z)
            delta = np.dot(self.weights[-l+1].transpose(), delta) * sp
            new_biases[-l] = delta
            new_weights[-l] = np.dot(delta, activations[-l-1].transpose())
        return (new_biases, new_weights)

在发生此错误之前，我的算法只能到达第一轮反向传播：

  File "D:/Programming/Python/DPUDS/DPUDS_Projects/Fall_2017/MNIST/network.py", line 97, in stochastic_gradient_descent
    self.update_mini_batch(mini_batch, learning_rate)
  File "D:/Programming/Python/DPUDS/DPUDS_Projects/Fall_2017/MNIST/network.py", line 117, in update_mini_batch
    delta_biases, delta_weights = self.backpropagate(image, image_value)
  File "D:/Programming/Python/DPUDS/DPUDS_Projects/Fall_2017/MNIST/network.py", line 160, in backpropagate
    z = np.dot(weight, activation) + bias
ValueError: shapes (30,50000) and (784,1) not aligned: 50000 (dim 1) != 784 (dim 0)

我明白为什么这是个错误。权重中的列数与像素图像中的行数不匹配，因此我无法进行矩阵乘法。这就是我感到困惑的地方——反向传播中使用了30个神经元，每个神经元都有50000张图像正在被评估。我的理解是，50000个像素中的每一个都应该附加784个权重，每个像素对应一个权重。但当我相应地修改代码时：

        count = 0
        for bias, weight in zip(self.biases, self.weights) :
            print(bias.shape)
            print(weight[count].shape)
            print(image.shape)
            z = np.dot(weight[count], image) + bias
            zs.append(z)
            activation = sigmoid(z)
            activations.append(activation)
            count += 1

我仍然会遇到类似的错误：

ValueError: shapes (50000,) and (784,1) not aligned: 50000 (dim 0) != 784 (dim 0)

---

我真的被所有涉及的线性代数弄糊涂了，我想我只是缺少了一些关于权重矩阵结构的东西。任何帮助都将不胜感激。

问题似乎在于您对原始代码的更改

我将从您提供的链接下载示例，它可以正常工作，没有任何错误：

以下是我使用的完整源代码：

import cPickle
import gzip
import numpy as np
import random

def load_data():
    """Return the MNIST data as a tuple containing the training data,
    the validation data, and the test data.
    The ``training_data`` is returned as a tuple with two entries.
    The first entry contains the actual training images.  This is a
    numpy ndarray with 50,000 entries.  Each entry is, in turn, a
    numpy ndarray with 784 values, representing the 28 * 28 = 784
    pixels in a single MNIST image.
    The second entry in the ``training_data`` tuple is a numpy ndarray
    containing 50,000 entries.  Those entries are just the digit
    values (0...9) for the corresponding images contained in the first
    entry of the tuple.
    The ``validation_data`` and ``test_data`` are similar, except
    each contains only 10,000 images.
    This is a nice data format, but for use in neural networks it's
    helpful to modify the format of the ``training_data`` a little.
    That's done in the wrapper function ``load_data_wrapper()``, see
    below.
    """
    f = gzip.open('../data/mnist.pkl.gz', 'rb')
    training_data, validation_data, test_data = cPickle.load(f)
    f.close()
    return (training_data, validation_data, test_data)

def load_data_wrapper():
    """Return a tuple containing ``(training_data, validation_data,
    test_data)``. Based on ``load_data``, but the format is more
    convenient for use in our implementation of neural networks.
    In particular, ``training_data`` is a list containing 50,000
    2-tuples ``(x, y)``.  ``x`` is a 784-dimensional numpy.ndarray
    containing the input image.  ``y`` is a 10-dimensional
    numpy.ndarray representing the unit vector corresponding to the
    correct digit for ``x``.
    ``validation_data`` and ``test_data`` are lists containing 10,000
    2-tuples ``(x, y)``.  In each case, ``x`` is a 784-dimensional
    numpy.ndarry containing the input image, and ``y`` is the
    corresponding classification, i.e., the digit values (integers)
    corresponding to ``x``.
    Obviously, this means we're using slightly different formats for
    the training data and the validation / test data.  These formats
    turn out to be the most convenient for use in our neural network
    code."""
    tr_d, va_d, te_d = load_data()
    training_inputs = [np.reshape(x, (784, 1)) for x in tr_d[0]]
    training_results = [vectorized_result(y) for y in tr_d[1]]
    training_data = zip(training_inputs, training_results)
    validation_inputs = [np.reshape(x, (784, 1)) for x in va_d[0]]
    validation_data = zip(validation_inputs, va_d[1])
    test_inputs = [np.reshape(x, (784, 1)) for x in te_d[0]]
    test_data = zip(test_inputs, te_d[1])
    return (training_data, validation_data, test_data)

def vectorized_result(j):
    """Return a 10-dimensional unit vector with a 1.0 in the jth
    position and zeroes elsewhere.  This is used to convert a digit
    (0...9) into a corresponding desired output from the neural
    network."""
    e = np.zeros((10, 1))
    e[j] = 1.0
    return e

class Network(object):

    def __init__(self, sizes):
        """The list ``sizes`` contains the number of neurons in the
        respective layers of the network.  For example, if the list
        was [2, 3, 1] then it would be a three-layer network, with the
        first layer containing 2 neurons, the second layer 3 neurons,
        and the third layer 1 neuron.  The biases and weights for the
        network are initialized randomly, using a Gaussian
        distribution with mean 0, and variance 1.  Note that the first
        layer is assumed to be an input layer, and by convention we
        won't set any biases for those neurons, since biases are only
        ever used in computing the outputs from later layers."""
        self.num_layers = len(sizes)
        self.sizes = sizes
        self.biases = [np.random.randn(y, 1) for y in sizes[1:]]
        self.weights = [np.random.randn(y, x)
                        for x, y in zip(sizes[:-1], sizes[1:])]

    def feedforward(self, a):
        """Return the output of the network if ``a`` is input."""
        for b, w in zip(self.biases, self.weights):
            a = sigmoid(np.dot(w, a)+b)
        return a

    def SGD(self, training_data, epochs, mini_batch_size, eta,
            test_data=None):
        """Train the neural network using mini-batch stochastic
        gradient descent.  The ``training_data`` is a list of tuples
        ``(x, y)`` representing the training inputs and the desired
        outputs.  The other non-optional parameters are
        self-explanatory.  If ``test_data`` is provided then the
        network will be evaluated against the test data after each
        epoch, and partial progress printed out.  This is useful for
        tracking progress, but slows things down substantially."""
        if test_data: n_test = len(test_data)
        n = len(training_data)
        for j in xrange(epochs):
            random.shuffle(training_data)
            mini_batches = [
                training_data[k:k+mini_batch_size]
                for k in xrange(0, n, mini_batch_size)]
            for mini_batch in mini_batches:
                self.update_mini_batch(mini_batch, eta)
            if test_data:
                print "Epoch {0}: {1} / {2}".format(
                    j, self.evaluate(test_data), n_test)
            else:
                print "Epoch {0} complete".format(j)

    def update_mini_batch(self, mini_batch, eta):
        """Update the network's weights and biases by applying
        gradient descent using backpropagation to a single mini batch.
        The ``mini_batch`` is a list of tuples ``(x, y)``, and ``eta``
        is the learning rate."""
        nabla_b = [np.zeros(b.shape) for b in self.biases]
        nabla_w = [np.zeros(w.shape) for w in self.weights]
        for x, y in mini_batch:
            delta_nabla_b, delta_nabla_w = self.backprop(x, y)
            nabla_b = [nb+dnb for nb, dnb in zip(nabla_b, delta_nabla_b)]
            nabla_w = [nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)]
        self.weights = [w-(eta/len(mini_batch))*nw
                        for w, nw in zip(self.weights, nabla_w)]
        self.biases = [b-(eta/len(mini_batch))*nb
                       for b, nb in zip(self.biases, nabla_b)]

    def backprop(self, x, y):
        """Return a tuple ``(nabla_b, nabla_w)`` representing the
        gradient for the cost function C_x.  ``nabla_b`` and
        ``nabla_w`` are layer-by-layer lists of numpy arrays, similar
        to ``self.biases`` and ``self.weights``."""
        nabla_b = [np.zeros(b.shape) for b in self.biases]
        nabla_w = [np.zeros(w.shape) for w in self.weights]
        # feedforward
        activation = x
        activations = [x] # list to store all the activations, layer by layer
        zs = [] # list to store all the z vectors, layer by layer
        for b, w in zip(self.biases, self.weights):
            z = np.dot(w, activation)+b
            zs.append(z)
            activation = sigmoid(z)
            activations.append(activation)
        # backward pass
        delta = self.cost_derivative(activations[-1], y) * \
            sigmoid_prime(zs[-1])
        nabla_b[-1] = delta
        nabla_w[-1] = np.dot(delta, activations[-2].transpose())
        # Note that the variable l in the loop below is used a little
        # differently to the notation in Chapter 2 of the book.  Here,
        # l = 1 means the last layer of neurons, l = 2 is the
        # second-last layer, and so on.  It's a renumbering of the
        # scheme in the book, used here to take advantage of the fact
        # that Python can use negative indices in lists.
        for l in xrange(2, self.num_layers):
            z = zs[-l]
            sp = sigmoid_prime(z)
            delta = np.dot(self.weights[-l+1].transpose(), delta) * sp
            nabla_b[-l] = delta
            nabla_w[-l] = np.dot(delta, activations[-l-1].transpose())
        return (nabla_b, nabla_w)

    def evaluate(self, test_data):
        """Return the number of test inputs for which the neural
        network outputs the correct result. Note that the neural
        network's output is assumed to be the index of whichever
        neuron in the final layer has the highest activation."""
        test_results = [(np.argmax(self.feedforward(x)), y)
                        for (x, y) in test_data]
        return sum(int(x == y) for (x, y) in test_results)

    def cost_derivative(self, output_activations, y):
        """Return the vector of partial derivatives \partial C_x /
        \partial a for the output activations."""
        return (output_activations-y)

#### Miscellaneous functions
def sigmoid(z):
    """The sigmoid function."""
    return 1.0/(1.0+np.exp(-z))

def sigmoid_prime(z):
    """Derivative of the sigmoid function."""
    return sigmoid(z)*(1-sigmoid(z))

training_data, validation_data, test_data = load_data_wrapper()
net = Network([784, 30, 10])
net.SGD(training_data, 30, 10, 3.0, test_data=test_data)

其他信息：

然而，我建议使用一个现有的框架，例如Keras，不要重新发明轮子

此外，还使用python 3.6对其进行了检查：

---

挖掘尼尔森代码的功劳。这是深入理解NN原理的一个很好的资源。太多的人在不知道引擎盖下发生了什么的情况下跳到了Keras

每个训练示例都没有自己的权重。784功能中的每一项都可以。如果每个示例都有自己的权重，那么每个权重集都会过度适合其相应的训练示例。此外，如果您以后使用经过培训的网络对单个测试示例进行推理，那么当仅显示一个手写数字时，它将如何处理50000组权重？相反，隐藏层中的30个神经元中的每一个都学习一组784个权重，每个像素一个，当广义化到任何手写数字时，该权重提供了高预测精度

导入

network.py

并像这样实例化一个网络类，而不修改任何代码：

net = network.Network([784, 30, 10])

…这个网络有784个输入神经元，30个隐藏神经元和10个输出神经元。您的权重矩阵将分别具有维度

[30784]

和

[10,30]

。当您向网络馈送维度为

[784，1]

的输入数组时，产生错误的矩阵乘法是有效的，因为权重矩阵的

dim 1

等于输入数组的

dim 0

（两者均为784）

---

您的问题不是backprop的实现，而是设置适合输入数据形状的网络体系结构。如果记忆起作用，尼尔森在第一章中把backprop作为一个黑匣子留下，直到第二章才投入其中。坚持下去，祝你好运

谢谢。另外，我试图让这段代码工作的唯一原因是，我试图理解神经网络背后的实际机制，并了解它为什么会这样工作。我还必须能够向我所在的数据科学俱乐部解释这一点。在我完成这项工作之后，我将继续讨论现有的框架，可能是tensorflow。好的，我再次查看了代码，除了更改了变量名和更新了Python 3的兼容性（您的代码是Python 2）之外，它几乎是一样的。仍然出现此错误：\n delta=np.dot（self.weights[-l+1].transpose（），delta）*sp ValueError:操作数无法与形状（30,1）（10,1）一起广播。您是否愿意查看我的代码，看看我是否遗漏了一些不太明显的内容？Repo:@Eli：我检查了链接中的代码，它工作正常，至少在我使用python 2.7的环境中是这样。之后，我用Python3.6检查了代码（请参见添加到我答案中的屏幕截图）——效果也很好。我不确定是什么原因导致了您在环境中提到的错误，可能是某些软件包的错误版本或配置错误。你能尝试升级或重新安装你的jumpy软件包吗？如果没有帮助的话，我建议重新安装python环境和所有的软件包！谢谢你这么做，Stepan。我最后只是重做了这个项目（不难，主要是复制粘贴），它工作得非常好。如果我做了与第一次不同的事情，那么我没有注意到。非常奇怪。@StepanNovikov此代码不能在我的机器上使用Python3.5。