如何为机器学习MNIST数据集正确实现反向传播？

Question

So, I'm using Michael Nielson's machine learning book as a reference for my code (it is basically identical): http://neuralnetworksanddeeplearning.com/chap1.html

The code in question:

    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)

My algorithm can only get to the first round backpropagation before this error occurs:

  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)

I get why it's an error. The number of columns in weights doesn't match the number of rows in the pixel image, so I can't do matrix multiplication. Here's where I'm confused -- there are 30 neurons used in the backpropagation, each with 50,000 images being evaluated. My understanding is that each of the 50,000 should have 784 weights attached, one for each pixel. But when I modify the code accordingly:

        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

I still get a similar error:

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

I'm just really confuzzled by all the linear algebra involved and I think I'm just missing something about the structure of the weight matrix. Any help at all would be greatly appreciated.

Answer 1

It looks like the issue is in your changes to the original code.

I'be downloaded example from the link you provided and it works without any errors:

Here is full source code I used:

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)

Additional info:

However, I would recommend using one of existing frameworks, for example - Keras to don't reinvent the wheel

Also, it was checked with python 3.6:

Answer 2

Kudos on digging into Nielsen's code. It's a great resource to develop thorough understanding of NN principles. Too many people leap ahead to Keras without knowing what goes on under the hood.

Each training example doesn't get its own weights. Each of the 784 features does. If each example got its own weights then each weight set would overfit to its corresponding training example. Also, if you later used your trained network to run inference on a single test example, what would it do with 50,000 sets of weights when presented with just one handwritten digit? Instead, each of the 30 neurons in your hidden layer learns a set of 784 weights, one for each pixel, that offers high predictive accuracy when generalized to any handwritten digit.

Import network.py and instantiate a Network class like this without modifying any code:

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

..which gives you a network with 784 input neurons, 30 hidden neurons and 10 output neurons. Your weight matrices will have dimensions [30, 784] and [10, 30] , respectively. When you feed the network an input array of dimensions [784, 1] the matrix multiplication that gave you an error is valid because dim 1 of the weight matrix equals dim 0 of the input array (both 784).

Your problem is not implementation of backprop but rather setting up a network architecture appropriate for the shape of your input data. If memory serves Nielsen leaves backprop as a black box in chapter 1 and doesn't dive into it until chapter 2. Keep at it, and good luck!