所以,我使用迈克尔·尼尔森的机器学习书作为我的代码的参考(它基本相同)：
http://neuralnetworksanddeeplearning.com/chap1.html
有问题的代码：
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)
我的算法只能在发生此错误之前进入第一轮反向传播：
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)
我明白为什么这是一个错误.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.这就是我困惑的地方 – 在反向传播中使用了30个神经元,每个神经元都有50,000个图像被评估.我的理解是,50,000个中的每一个都应该附加784个权重,每个像素一个.但是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
我仍然得到一个类似的错误：
ValueError: shapes (50000,) and (784,1) not aligned: 50000 (dim 0) != 784 (dim 0)
我只是被所涉及的所有线性代数所困扰,我想我只是缺少一些关于权重矩阵结构的东西.任何帮助都将非常感激.
最佳答案 看起来问题出在您对原始代码的更改中.
我从你提供的链接下载示例,它没有任何错误：
这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)
附加信息：
但是,我建议使用现有的框架之一,例如 – Keras不要重新发明轮子
另外,用python 3.6检查了它：
