本节需要用的函数
基本函数（step、sigmoid）
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import numpy as np
def identity_function(x):
return x
def step_function(x):
return np.array(x > 0, dtype=np.int)
def sigmoid(x):
return 1 / (1 + np.exp(-x))
def sigmoid_grad(x):
return (1.0 - sigmoid(x)) * sigmoid(x)
def relu(x):
return np.maximum(0, x)
def relu_grad(x):
grad = np.zeros(x)
grad[x>=0] = 1
return grad
def softmax(x):
if x.ndim == 2:
x = x.T
x = x - np.max(x, axis=0)
y = np.exp(x) / np.sum(np.exp(x), axis=0)
return y.T
x = x - np.max(x) # 溢出对策
return np.exp(x) / np.sum(np.exp(x))
def mean_squared_error(y, t):
return 0.5 * np.sum((y-t)**2)
def cross_entropy_error(y, t):
if y.ndim == 1:
t = t.reshape(1, t.size)
y = y.reshape(1, y.size)
# 监督数据是one-hot-vector的情况下，转换为正确解标签的索引
if t.size == y.size:
t = t.argmax(axis=1)
batch_size = y.shape[0]
return -np.sum(np.log(y[np.arange(batch_size), t] + 1e-7)) / batch_size
def softmax_loss(X, t):
y = softmax(X)
return cross_entropy_error(y, t)
数值微分
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def _numerical_gradient_1d(f, x):
h = 1e-4 # 0.0001
grad = np.zeros_like(x)
for idx in range(x.size):
tmp_val = x[idx]
x[idx] = float(tmp_val) + h
fxh1 = f(x) # f(x+h)
x[idx] = tmp_val - h
fxh2 = f(x) # f(x-h)
grad[idx] = (fxh1 - fxh2) / (2*h)
x[idx] = tmp_val # 还原值
return grad
def numerical_gradient_2d(f, X):
if X.ndim == 1:
return _numerical_gradient_1d(f, X)
else:
grad = np.zeros_like(X)
for idx, x in enumerate(X):
grad[idx] = _numerical_gradient_1d(f, x)
return grad
def numerical_gradient(f, x):
h = 1e-4 # 0.0001
grad = np.zeros_like(x)
it = np.nditer(x, flags=['multi_index'], op_flags=['readwrite'])
while not it.finished:
idx = it.multi_index
tmp_val = x[idx]
x[idx] = float(tmp_val) + h
fxh1 = f(x) # f(x+h)
x[idx] = tmp_val - h
fxh2 = f(x) # f(x-h)
grad[idx] = (fxh1 - fxh2) / (2*h)
x[idx] = tmp_val # 还原值
it.iternext()
return grad
基本层（ReLU、Sigmoid）
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class Relu:
def __init__(self):
self.mask = None
def forward(self, x):
self.mask = (x <= 0)
out = x.copy()
out[self.mask] = 0
return out
def backward(self, dout):
dout[self.mask] = 0
dx = dout
return dx
class Sigmoid:
def __init__(self):
self.out = None
def forward(self, x):
out = sigmoid(x)
self.out = out
return out
def backward(self, dout):
dx = dout * (1.0 - self.out) * self.out
return dx
class Affine:
def __init__(self, W, b):
self.W =W
self.b = b
self.x = None
self.original_x_shape = None
# 权重和偏置参数的导数
self.dW = None
self.db = None
def forward(self, x):
# 对应张量
self.original_x_shape = x.shape
x = x.reshape(x.shape[0], -1)
self.x = x
out = np.dot(self.x, self.W) + self.b
return out
def backward(self, dout):
dx = np.dot(dout, self.W.T)
self.dW = np.dot(self.x.T, dout)
self.db = np.sum(dout, axis=0)
dx = dx.reshape(*self.original_x_shape) # 还原输入数据的形状（对应张量）
return dx
class SoftmaxWithLoss:
def __init__(self):
self.loss = None
self.y = None # softmax的输出
self.t = None # 监督数据
def forward(self, x, t):
self.t = t
self.y = softmax(x)
self.loss = cross_entropy_error(self.y, self.t)
return self.loss
def backward(self, dout=1):
batch_size = self.t.shape[0]
if self.t.size == self.y.size: # 监督数据是one-hot-vector的情况
dx = (self.y - self.t) / batch_size
else:
dx = self.y.copy()
dx[np.arange(batch_size), self.t] -= 1
dx = dx / batch_size
return dx
class Dropout:
"""
http://arxiv.org/abs/1207.0580
"""
def __init__(self, dropout_ratio=0.5):
self.dropout_ratio = dropout_ratio
self.mask = None
def forward(self, x, train_flg=True):
if train_flg:
self.mask = np.random.rand(*x.shape) > self.dropout_ratio
return x * self.mask
else:
return x * (1.0 - self.dropout_ratio)
def backward(self, dout):
return dout * self.mask
class BatchNormalization:
"""
http://arxiv.org/abs/1502.03167
"""
def __init__(self, gamma, beta, momentum=0.9, running_mean=None, running_var=None):
self.gamma = gamma
self.beta = beta
self.momentum = momentum
self.input_shape = None # Conv层的情况下为4维，全连接层的情况下为2维
# 测试时使用的平均值和方差
self.running_mean = running_mean
self.running_var = running_var
# backward时使用的中间数据
self.batch_size = None
self.xc = None
self.std = None
self.dgamma = None
self.dbeta = None
def forward(self, x, train_flg=True):
self.input_shape = x.shape
if x.ndim != 2:
N, C, H, W = x.shape
x = x.reshape(N, -1)
out = self.__forward(x, train_flg)
return out.reshape(*self.input_shape)
def __forward(self, x, train_flg):
if self.running_mean is None:
N, D = x.shape
self.running_mean = np.zeros(D)
self.running_var = np.zeros(D)
if train_flg:
mu = x.mean(axis=0)
xc = x - mu
var = np.mean(xc**2, axis=0)
std = np.sqrt(var + 10e-7)
xn = xc / std
self.batch_size = x.shape[0]
self.xc = xc
self.xn = xn
self.std = std
self.running_mean = self.momentum * self.running_mean + (1-self.momentum) * mu
self.running_var = self.momentum * self.running_var + (1-self.momentum) * var
else:
xc = x - self.running_mean
xn = xc / ((np.sqrt(self.running_var + 10e-7)))
out = self.gamma * xn + self.beta
return out
def backward(self, dout):
if dout.ndim != 2:
N, C, H, W = dout.shape
dout = dout.reshape(N, -1)
dx = self.__backward(dout)
dx = dx.reshape(*self.input_shape)
return dx
def __backward(self, dout):
dbeta = dout.sum(axis=0)
dgamma = np.sum(self.xn * dout, axis=0)
dxn = self.gamma * dout
dxc = dxn / self.std
dstd = -np.sum((dxn * self.xc) / (self.std * self.std), axis=0)
dvar = 0.5 * dstd / self.std
dxc += (2.0 / self.batch_size) * self.xc * dvar
dmu = np.sum(dxc, axis=0)
dx = dxc - dmu / self.batch_size
self.dgamma = dgamma
self.dbeta = dbeta
return dx
下载MNIST
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try:
import urllib.request
except ImportError:
raise ImportError('You should use Python 3.x')
import os.path
from IPython.terminal.embed import InteractiveShellEmbed
import gzip
import pickle
import os
import numpy as np
url_base = 'http://yann.lecun.com/exdb/mnist/'
key_file = {
'train_img':'train-images-idx3-ubyte.gz',
'train_label':'train-labels-idx1-ubyte.gz',
'test_img':'t10k-images-idx3-ubyte.gz',
'test_label':'t10k-labels-idx1-ubyte.gz'
}
## if you run in terminal, run this
# dataset_dir = os.path.dirname(os.path.abspath(__file__))
## if you run in IPython, run this
ip_shell = InteractiveShellEmbed()
dataset_dir = ip_shell.magic("%pwd")
save_file = dataset_dir + "/mnist.pkl"
train_num = 60000
test_num = 10000
img_dim = (1, 28, 28)
img_size = 784
def _download(file_name):
file_path = dataset_dir + "/" + file_name
if os.path.exists(file_path):
return
print("Downloading " + file_name + " ... ")
urllib.request.urlretrieve(url_base + file_name, file_path)
print("Done")
def download_mnist():
for v in key_file.values():
_download(v)
def _load_label(file_name):
file_path = dataset_dir + "/" + file_name
print("Converting " + file_name + " to NumPy Array ...")
with gzip.open(file_path, 'rb') as f:
labels = np.frombuffer(f.read(), np.uint8, offset=8)
print("Done")
return labels
def _load_img(file_name):
file_path = dataset_dir + "/" + file_name
print("Converting " + file_name + " to NumPy Array ...")
with gzip.open(file_path, 'rb') as f:
data = np.frombuffer(f.read(), np.uint8, offset=16)
data = data.reshape(-1, img_size)
print("Done")
return data
def _convert_numpy():
dataset = {}
dataset['train_img'] = _load_img(key_file['train_img'])
dataset['train_label'] = _load_label(key_file['train_label'])
dataset['test_img'] = _load_img(key_file['test_img'])
dataset['test_label'] = _load_label(key_file['test_label'])
return dataset
def init_mnist():
download_mnist()
dataset = _convert_numpy()
print("Creating pickle file ...")
with open(save_file, 'wb') as f:
pickle.dump(dataset, f, -1)
print("Done!")
def _change_one_hot_label(X):
T = np.zeros((X.size, 10))
for idx, row in enumerate(T):
row[X[idx]] = 1
return T
def load_mnist(normalize=True, flatten=True, one_hot_label=False):
"""读入MNIST数据集
Parameters
----------
normalize : 将图像的像素值正规化为0.0~1.0
one_hot_label :
one_hot_label为True的情况下，标签作为one-hot数组返回
one-hot数组是指[0,0,1,0,0,0,0,0,0,0]这样的数组
flatten : 是否将图像展开为一维数组
Returns
-------
(训练图像, 训练标签), (测试图像, 测试标签)
"""
if not os.path.exists(save_file):
init_mnist()
with open(save_file, 'rb') as f:
dataset = pickle.load(f)
if normalize:
for key in ('train_img', 'test_img'):
dataset[key] = dataset[key].astype(np.float32)
dataset[key] /= 255.0
if one_hot_label:
dataset['train_label'] = _change_one_hot_label(dataset['train_label'])
dataset['test_label'] = _change_one_hot_label(dataset['test_label'])
if not flatten:
for key in ('train_img', 'test_img'):
dataset[key] = dataset[key].reshape(-1, 1, 28, 28)
return (dataset['train_img'], dataset['train_label']), (dataset['test_img'], dataset['test_label'])
if __name__ == '__main__':
init_mnist()
SGD
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import numpy as np
class SGD:
"""随机梯度下降法（Stochastic Gradient Descent）"""
def __init__(self, lr=0.01):
self.lr = lr
def update(self, params, grads):
for key in params.keys():
params[key] -= self.lr * grads[key]
class Momentum:
"""Momentum SGD"""
def __init__(self, lr=0.01, momentum=0.9):
self.lr = lr
self.momentum = momentum
self.v = None
def update(self, params, grads):
if self.v is None:
self.v = {}
for key, val in params.items():
self.v[key] = np.zeros_like(val)
for key in params.keys():
self.v[key] = self.momentum*self.v[key] - self.lr*grads[key]
params[key] += self.v[key]
class Nesterov:
"""Nesterov's Accelerated Gradient (http://arxiv.org/abs/1212.0901)"""
def __init__(self, lr=0.01, momentum=0.9):
self.lr = lr
self.momentum = momentum
self.v = None
def update(self, params, grads):
if self.v is None:
self.v = {}
for key, val in params.items():
self.v[key] = np.zeros_like(val)
for key in params.keys():
self.v[key] *= self.momentum
self.v[key] -= self.lr * grads[key]
params[key] += self.momentum * self.momentum * self.v[key]
params[key] -= (1 + self.momentum) * self.lr * grads[key]
class AdaGrad:
"""AdaGrad"""
def __init__(self, lr=0.01):
self.lr = lr
self.h = None
def update(self, params, grads):
if self.h is None:
self.h = {}
for key, val in params.items():
self.h[key] = np.zeros_like(val)
for key in params.keys():
self.h[key] += grads[key] * grads[key]
params[key] -= self.lr * grads[key] / (np.sqrt(self.h[key]) + 1e-7)
class RMSprop:
"""RMSprop"""
def __init__(self, lr=0.01, decay_rate = 0.99):
self.lr = lr
self.decay_rate = decay_rate
self.h = None
def update(self, params, grads):
if self.h is None:
self.h = {}
for key, val in params.items():
self.h[key] = np.zeros_like(val)
for key in params.keys():
self.h[key] *= self.decay_rate
self.h[key] += (1 - self.decay_rate) * grads[key] * grads[key]
params[key] -= self.lr * grads[key] / (np.sqrt(self.h[key]) + 1e-7)
class Adam:
"""Adam (http://arxiv.org/abs/1412.6980v8)"""
def __init__(self, lr=0.001, beta1=0.9, beta2=0.999):
self.lr = lr
self.beta1 = beta1
self.beta2 = beta2
self.iter = 0
self.m = None
self.v = None
def update(self, params, grads):
if self.m is None:
self.m, self.v = {}, {}
for key, val in params.items():
self.m[key] = np.zeros_like(val)
self.v[key] = np.zeros_like(val)
self.iter += 1
lr_t = self.lr * np.sqrt(1.0 - self.beta2**self.iter) / (1.0 - self.beta1**self.iter)
for key in params.keys():
#self.m[key] = self.beta1*self.m[key] + (1-self.beta1)*grads[key]
#self.v[key] = self.beta2*self.v[key] + (1-self.beta2)*(grads[key]**2)
self.m[key] += (1 - self.beta1) * (grads[key] - self.m[key])
self.v[key] += (1 - self.beta2) * (grads[key]**2 - self.v[key])
params[key] -= lr_t * self.m[key] / (np.sqrt(self.v[key]) + 1e-7)
#unbias_m += (1 - self.beta1) * (grads[key] - self.m[key]) # correct bias
#unbisa_b += (1 - self.beta2) * (grads[key]*grads[key] - self.v[key]) # correct bias
#params[key] += self.lr * unbias_m / (np.sqrt(unbisa_b) + 1e-7)
训练器
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import numpy as np
class Trainer:
"""进行神经网络的训练的类
"""
def __init__(self, network, x_train, t_train, x_test, t_test,
epochs=20, mini_batch_size=100,
optimizer='SGD', optimizer_param={'lr':0.01},
evaluate_sample_num_per_epoch=None, verbose=True):
self.network = network
self.verbose = verbose
self.x_train = x_train
self.t_train = t_train
self.x_test = x_test
self.t_test = t_test
self.epochs = epochs
self.batch_size = mini_batch_size
self.evaluate_sample_num_per_epoch = evaluate_sample_num_per_epoch
# optimzer
optimizer_class_dict = {'sgd':SGD, 'momentum':Momentum, 'nesterov':Nesterov,
'adagrad':AdaGrad, 'rmsprpo':RMSprop, 'adam':Adam}
self.optimizer = optimizer_class_dict[optimizer.lower()](**optimizer_param)
self.train_size = x_train.shape[0]
self.iter_per_epoch = max(self.train_size / mini_batch_size, 1)
self.max_iter = int(epochs * self.iter_per_epoch)
self.current_iter = 0
self.current_epoch = 0
self.train_loss_list = []
self.train_acc_list = []
self.test_acc_list = []
def train_step(self):
batch_mask = np.random.choice(self.train_size, self.batch_size)
x_batch = self.x_train[batch_mask]
t_batch = self.t_train[batch_mask]
grads = self.network.gradient(x_batch, t_batch)
self.optimizer.update(self.network.params, grads)
loss = self.network.loss(x_batch, t_batch)
self.train_loss_list.append(loss)
if self.verbose: print("train loss:" + str(loss))
if self.current_iter % self.iter_per_epoch == 0:
self.current_epoch += 1
x_train_sample, t_train_sample = self.x_train, self.t_train
x_test_sample, t_test_sample = self.x_test, self.t_test
if not self.evaluate_sample_num_per_epoch is None:
t = self.evaluate_sample_num_per_epoch
x_train_sample, t_train_sample = self.x_train[:t], self.t_train[:t]
x_test_sample, t_test_sample = self.x_test[:t], self.t_test[:t]
train_acc = self.network.accuracy(x_train_sample, t_train_sample)
test_acc = self.network.accuracy(x_test_sample, t_test_sample)
self.train_acc_list.append(train_acc)
self.test_acc_list.append(test_acc)
if self.verbose: print("=== epoch:" + str(self.current_epoch) + ", train acc:" + str(train_acc) + ", test acc:" + str(test_acc) + " ===")
self.current_iter += 1
def train(self):
for i in range(self.max_iter):
self.train_step()
test_acc = self.network.accuracy(self.x_test, self.t_test)
if self.verbose:
print("=============== Final Test Accuracy ===============")
print("test acc:" + str(test_acc))
卷积神经网络
卷积神经网络（Convolutional Neural Newtwork，CNN） 与之前介绍的神经网络的不同之处在于，之前的神经网络的所有神经元之间都有连接，称为 全连接（fully-connected），这是通过 Affine层（全连接层）实现的。比如下面的一个神经网络：
卷积网络中，增加了 Convolution层（卷积层） 和 Pooling层（池化层），将之前的 Affine-ReLU 改成了 Convolution-ReLU-Pooling
下面来介绍新增的这两个层。
卷积层
全连接层有个很大的问题，就是无论数据是几维的，都会拉平成一维数据。比如之前的MNIST数据集，本来是 $28\times 28$ 的图像，输入时却成了 $784\times 1$ 的数组。因此，图像中的很多空间信息都损失了。而卷积层则可以接受图像输入，并输出图像，我们将输入输出的数据称为 特征图（feature map）。
卷积层主要做的是卷积运算，运算过程如下：用一个 滤波器 对图像进行扫描，将图像的被扫部分与滤波器进行元素相乘再求和，从而得到结果中的一个子像素。
\[\newcommand{\blue}[1]{\color{Cyan}#1} \begin{bmatrix} 1 & 2 & 3 & 0\\ 0 & 1 & 2 & 3\\ 3 & 0 & 1 & 2\\ 2 & 3 & 0 & 1 \end{bmatrix} * \begin{bmatrix} 2 & 0 & 1\\ 0 & 1 & 2\\ 1 & 0 & 2 \end{bmatrix} =\begin{bmatrix} 15 & 16\\ 6 & 15 \end{bmatrix} \\ \begin{bmatrix} 15 & \\ & \end{bmatrix} =\begin{bmatrix} \blue{1} & \blue2 & \blue3 & 0\\ \blue0 & \blue1 & \blue2 & 3\\ \blue3 & \blue0 & \blue1 & 2\\ 2 & 3 & 0 & 1 \end{bmatrix} \odot \begin{bmatrix} 2 & 0 & 1\\ 0 & 1 & 2\\ 1 & 0 & 2 \end{bmatrix} \\ \begin{bmatrix} & 16 \\ & \end{bmatrix} =\begin{bmatrix} 1 & \blue2 & \blue3 & \blue0\\ 0 & \blue1 & \blue2 & \blue3\\ 3 & \blue0 & \blue1 & \blue2\\ 2 & 3 & 0 & 1 \end{bmatrix} \odot \begin{bmatrix} 2 & 0 & 1\\ 0 & 1 & 2\\ 1 & 0 & 2 \end{bmatrix} \\ \begin{bmatrix} & \\ 6 & \end{bmatrix} =\begin{bmatrix} 1 & 2 & 3 & 0\\ \blue0 & \blue1 & \blue2 & 3\\ \blue3 & \blue0 & \blue1 & 2\\ \blue2 & \blue3 & \blue0 & 1 \end{bmatrix} \odot \begin{bmatrix} 2 & 0 & 1\\ 0 & 1 & 2\\ 1 & 0 & 2 \end{bmatrix} \\ \begin{bmatrix} & \\ & 15 \end{bmatrix} =\begin{bmatrix} 1 & 2 & 3 & 0\\ 0 & \blue1 & \blue2 & \blue3\\ 3 & \blue0 & \blue1 & \blue2\\ 2 & \blue3 & \blue0 & \blue1 \end{bmatrix} \odot \begin{bmatrix} 2 & 0 & 1\\ 0 & 1 & 2\\ 1 & 0 & 2 \end{bmatrix}\]卷积层中也有偏置，偏置一般就是一个数字，加到卷积结果的所有元素上。这里就不附图了。
注意到上面 $4\times 4$ 矩阵经过卷积后变成了 $2\times 2$ 矩阵，说明卷积会缩小图像。如果经过多次卷积，那么图像最终会缩成 $1\times 1$。为了使得图像大小不改变，我们会对输入数据进行 填充（padding），就是在边上加一圈 0，这样卷积完还是一样大：
\[\newcommand{\grey}[1]{\color{Grey}#1} \begin{bmatrix} \grey0 & \grey0 & \grey0 & \grey0 & \grey0 & \grey0\\ \grey0 & 1 & 2 & 3 & 0 & \grey0\\ \grey0 & 0 & 1 & 2 & 3 & \grey0\\ \grey0 & 3 & 0 & 1 & 2 & \grey0\\ \grey0 & 2 & 3 & 0 & 1 & \grey0\\ \grey0 & \grey0 & \grey0 & \grey0 & \grey0 & \grey0 \end{bmatrix} * \begin{bmatrix} 2 & 0 & 1\\ 0 & 1 & 2\\ 1 & 0 & 2 \end{bmatrix} =\begin{bmatrix} 7 & 12 & 10 & 2\\ 4 & 15 & 16 & 10\\ 10 & 6 & 15 & 6\\ 8 & 10 & 4 & 3 \end{bmatrix}\]此外，还有一个重要参数就是 步幅（stride），即每次滤波器上下左右移动的距离。前面的步幅是 1，我们也可以改成 3：
\[\newcommand{\grey}[1]{\color{Grey}#1} \begin{bmatrix} \grey0 & \grey0 & \grey0 & \grey0 & \grey0 & \grey0\\ \grey0 & 1 & 2 & 3 & 0 & \grey0\\ \grey0 & 0 & 1 & 2 & 3 & \grey0\\ \grey0 & 3 & 0 & 1 & 2 & \grey0\\ \grey0 & 2 & 3 & 0 & 1 & \grey0\\ \grey0 & \grey0 & \grey0 & \grey0 & \grey0 & \grey0 \end{bmatrix} * \begin{bmatrix} 2 & 0 & 1\\ 0 & 1 & 2\\ 1 & 0 & 2 \end{bmatrix} =\begin{bmatrix} 7 & 2\\ 8 & 3 \end{bmatrix}\]假设输入大小为 $(H,W)$，滤波器大小为 $(FH,FW)$，填充为 $P$，步幅为 $S$，那么输出大小 $(OH,OW)$ 满足：
\[OH = \frac{H+2P-FH}{S}+1\\ OW = \frac{W+2P-FW}{S}+1\]如果除不尽的话，可以报错，也可以四舍五入。
对于高维数据，可以用高维滤波器进行卷积。下图展示了一个三维数据经过三维滤波器卷积后，得到了一个一维数据。
或者也可以用多个的滤波器进行卷积，这样会得到一个多维的数据：
对于有多个通道的图像（3维数据），我们一般按照（channel, height, width）的顺序写，滤波器也同理。我们可以用一个方块的厚度、长、宽来表示，如下图（或上图）：
卷积同样可以进行批处理，此时一个batch的数据写成（batch_size, channel, height, width）
池化层
池化层与卷积层类似，但它并不是相乘再求和，而是取最大值（Max池化）或平均值（Average池化）。比如：
\[\newcommand{\blue}[1]{\color{Cyan}#1} \newcommand{\red}[1]{\color{Magenta}#1} \newcommand{\orange}[1]{\color{Orange}#1} \newcommand{\green}[1]{\color{Lime}#1} \begin{bmatrix} \blue1 & \blue2 & \red3 & \red0\\ \blue0 & \blue1 & \red2 & \red3\\ \orange3 & \orange0 & \green1 & \green2\\ \orange2 & \orange3 & \green0 & \green1 \end{bmatrix} \Rightarrow \begin{bmatrix} \blue2 & \red3\\ \orange4 & \green5 \end{bmatrix}\]一般来说，池化的窗口大小与步幅会设定成相同的值。池化层与卷积的不同之处在于：
- 池化层没有要学习的参数，只是取最大或平均值
- 数据经过池化层后通道数不发生变换，各通道是独立进行的
- 池化层对微小的变化具有鲁棒性（这样图像就算有微小的位移也不会造成影响）
卷积层的实现
如果使用循环来实现卷积，那么就会十分复杂且速度很慢。这里我们利用 im2col 来实现。im2col 会将图像中每个滤波区域横向转化成一列（我也不知道为什么不说“转化为一行”），将滤波器纵向转化成一列，这样，就可以通过矩阵相乘一次性得到卷积结果，然后再 reshape 为输出数据的大小。
im2col
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import numpy as np
def conv_output_size(input_size, filter_size, stride=1, pad=0):
return (input_size + 2*pad - filter_size) / stride + 1
def im2col(input_data, filter_h, filter_w, stride=1, pad=0):
"""
Parameters
----------
input_data : 由(数据量, 通道, 高, 长)的4维数组构成的输入数据
filter_h : 滤波器的高
filter_w : 滤波器的长
stride : 步幅
pad : 填充
Returns
-------
col : 2维数组
"""
N, C, H, W = input_data.shape #获取数据的形状
out_h = (H + 2*pad - filter_h)//stride + 1 #输出数据的高度
out_w = (W + 2*pad - filter_w)//stride + 1 #输出数据的宽度
img = np.pad(input_data, [(0,0), (0,0), (pad, pad), (pad, pad)], 'constant') #填充
col = np.zeros((N, C, filter_h, filter_w, out_h, out_w))
for y in range(filter_h):
y_max = y + stride*out_h
for x in range(filter_w):
x_max = x + stride*out_w
col[:, :, y, x, :, :] = img[:, :, y:y_max:stride, x:x_max:stride]
col = col.transpose(0, 4, 5, 1, 2, 3).reshape(N*out_h*out_w, -1)
return col
def col2im(col, input_shape, filter_h, filter_w, stride=1, pad=0):
"""
Parameters
----------
col :
input_shape : 输入数据的形状（例：(10, 1, 28, 28)）
filter_h :
filter_w
stride
pad
Returns
-------
"""
N, C, H, W = input_shape
out_h = (H + 2*pad - filter_h)//stride + 1
out_w = (W + 2*pad - filter_w)//stride + 1
col = col.reshape(N, out_h, out_w, C, filter_h, filter_w).transpose(0, 3, 4, 5, 1, 2)
img = np.zeros((N, C, H + 2*pad + stride - 1, W + 2*pad + stride - 1))
for y in range(filter_h):
y_max = y + stride*out_h
for x in range(filter_w):
x_max = x + stride*out_w
img[:, :, y:y_max:stride, x:x_max:stride] += col[:, :, y, x, :, :]
return img[:, :, pad:H + pad, pad:W + pad]
x = np.random.rand(1,3,7,7)
col1 = im2col(x,5,5,stride=1,pad=0)
print(col1.shape)
OUTPUT:
(9, 75)
经过 im2col 后，卷积和Affine十分相似，其反向传播的过程也是类似的。实现方法如下：
class Convolution:
def __init__(self, W, b, stride=1, pad=0):
self.W = W
self.b = b
self.stride = stride
self.pad = pad
# 中间数据（backward时使用）
self.x = None
self.col = None
self.col_W = None
# 权重和偏置参数的梯度
self.dW = None
self.db = None
def forward(self, x):
FN, C, FH, FW = self.W.shape
N, C, H, W = x.shape
out_h = 1 + int((H + 2*self.pad - FH) / self.stride)
out_w = 1 + int((W + 2*self.pad - FW) / self.stride)
col = im2col(x, FH, FW, self.stride, self.pad)
col_W = self.W.reshape(FN, -1).T
out = np.dot(col, col_W) + self.b
out = out.reshape(N, out_h, out_w, -1).transpose(0, 3, 1, 2)
self.x = x
self.col = col
self.col_W = col_W
return out
def backward(self, dout):
FN, C, FH, FW = self.W.shape
dout = dout.transpose(0,2,3,1).reshape(-1, FN)
self.db = np.sum(dout, axis=0)
self.dW = np.dot(self.col.T, dout)
self.dW = self.dW.transpose(1, 0).reshape(FN, C, FH, FW)
dcol = np.dot(dout, self.col_W.T)
dx = col2im(dcol, self.x.shape, FH, FW, self.stride, self.pad)
return dx
池化层的实现
池化层也是利用 im2col 来实现。
class Pooling:
def __init__(self, pool_h, pool_w, stride=1, pad=0):
self.pool_h = pool_h
self.pool_w = pool_w
self.stride = stride
self.pad = pad
self.x = None
self.arg_max = None
def forward(self, x):
N, C, H, W = x.shape
out_h = int(1 + (H - self.pool_h) / self.stride)
out_w = int(1 + (W - self.pool_w) / self.stride)
col = im2col(x, self.pool_h, self.pool_w, self.stride, self.pad)
col = col.reshape(-1, self.pool_h*self.pool_w)
arg_max = np.argmax(col, axis=1)
out = np.max(col, axis=1)
out = out.reshape(N, out_h, out_w, C).transpose(0, 3, 1, 2)
self.x = x
self.arg_max = arg_max
return out
def backward(self, dout):
dout = dout.transpose(0, 2, 3, 1)
pool_size = self.pool_h * self.pool_w
dmax = np.zeros((dout.size, pool_size))
dmax[np.arange(self.arg_max.size), self.arg_max.flatten()] = dout.flatten()
dmax = dmax.reshape(dout.shape + (pool_size,))
dcol = dmax.reshape(dmax.shape[0] * dmax.shape[1] * dmax.shape[2], -1)
dx = col2im(dcol, self.x.shape, self.pool_h, self.pool_w, self.stride, self.pad)
return dx
构建CNN
下面我们来构建一个简单的CNN网络用于识别 MNIST，其构成为：
- Conv
- ReLU
- Pooling
- Affine
- ReLU
- Affine
- Softmax
import pickle
import numpy as np
from collections import OrderedDict
class SimpleConvNet:
"""简单的ConvNet
conv - relu - pool - affine - relu - affine - softmax
Parameters
----------
input_size : 输入大小（MNIST的情况下为784）
hidden_size_list : 隐藏层的神经元数量的列表（e.g. [100, 100, 100]）
output_size : 输出大小（MNIST的情况下为10）
activation : 'relu' or 'sigmoid'
weight_init_std : 指定权重的标准差（e.g. 0.01）
指定'relu'或'he'的情况下设定“He的初始值”
指定'sigmoid'或'xavier'的情况下设定“Xavier的初始值”
"""
def __init__(self, input_dim=(1, 28, 28),
conv_param={'filter_num':30, 'filter_size':5, 'pad':0, 'stride':1},
hidden_size=100, output_size=10, weight_init_std=0.01):
filter_num = conv_param['filter_num']
filter_size = conv_param['filter_size']
filter_pad = conv_param['pad']
filter_stride = conv_param['stride']
input_size = input_dim[1]
conv_output_size = (input_size - filter_size + 2*filter_pad) / filter_stride + 1
pool_output_size = int(filter_num * (conv_output_size/2) * (conv_output_size/2))
# 初始化权重
self.params = {}
self.params['W1'] = weight_init_std * \
np.random.randn(filter_num, input_dim[0], filter_size, filter_size)
self.params['b1'] = np.zeros(filter_num)
self.params['W2'] = weight_init_std * \
np.random.randn(pool_output_size, hidden_size)
self.params['b2'] = np.zeros(hidden_size)
self.params['W3'] = weight_init_std * \
np.random.randn(hidden_size, output_size)
self.params['b3'] = np.zeros(output_size)
# 生成层
self.layers = OrderedDict()
self.layers['Conv1'] = Convolution(self.params['W1'], self.params['b1'],
conv_param['stride'], conv_param['pad'])
self.layers['Relu1'] = Relu()
self.layers['Pool1'] = Pooling(pool_h=2, pool_w=2, stride=2)
self.layers['Affine1'] = Affine(self.params['W2'], self.params['b2'])
self.layers['Relu2'] = Relu()
self.layers['Affine2'] = Affine(self.params['W3'], self.params['b3'])
self.last_layer = SoftmaxWithLoss()
def predict(self, x):
for layer in self.layers.values():
x = layer.forward(x)
return x
def loss(self, x, t):
"""求损失函数
参数x是输入数据、t是教师标签
"""
y = self.predict(x)
return self.last_layer.forward(y, t)
def accuracy(self, x, t, batch_size=100):
if t.ndim != 1 : t = np.argmax(t, axis=1)
acc = 0.0
for i in range(int(x.shape[0] / batch_size)):
tx = x[i*batch_size:(i+1)*batch_size]
tt = t[i*batch_size:(i+1)*batch_size]
y = self.predict(tx)
y = np.argmax(y, axis=1)
acc += np.sum(y == tt)
return acc / x.shape[0]
def numerical_gradient(self, x, t):
"""求梯度（数值微分）
Parameters
----------
x : 输入数据
t : 教师标签
Returns
-------
具有各层的梯度的字典变量
grads['W1']、grads['W2']、...是各层的权重
grads['b1']、grads['b2']、...是各层的偏置
"""
loss_w = lambda w: self.loss(x, t)
grads = {}
for idx in (1, 2, 3):
grads['W' + str(idx)] = numerical_gradient(loss_w, self.params['W' + str(idx)])
grads['b' + str(idx)] = numerical_gradient(loss_w, self.params['b' + str(idx)])
return grads
def gradient(self, x, t):
"""求梯度（误差反向传播法）
Parameters
----------
x : 输入数据
t : 教师标签
Returns
-------
具有各层的梯度的字典变量
grads['W1']、grads['W2']、...是各层的权重
grads['b1']、grads['b2']、...是各层的偏置
"""
# forward
self.loss(x, t)
# backward
dout = 1
dout = self.last_layer.backward(dout)
layers = list(self.layers.values())
layers.reverse()
for layer in layers:
dout = layer.backward(dout)
# 设定
grads = {}
grads['W1'], grads['b1'] = self.layers['Conv1'].dW, self.layers['Conv1'].db
grads['W2'], grads['b2'] = self.layers['Affine1'].dW, self.layers['Affine1'].db
grads['W3'], grads['b3'] = self.layers['Affine2'].dW, self.layers['Affine2'].db
return grads
def save_params(self, file_name="params.pkl"):
params = {}
for key, val in self.params.items():
params[key] = val
with open(file_name, 'wb') as f:
pickle.dump(params, f)
def load_params(self, file_name="params.pkl"):
with open(file_name, 'rb') as f:
params = pickle.load(f)
for key, val in params.items():
self.params[key] = val
for i, key in enumerate(['Conv1', 'Affine1', 'Affine2']):
self.layers[key].W = self.params['W' + str(i+1)]
self.layers[key].b = self.params['b' + str(i+1)]
import numpy as np
import matplotlib.pyplot as plt
# 读入数据
(x_train, t_train), (x_test, t_test) = load_mnist(flatten=False)
# 处理花费时间较长的情况下减少数据
x_train, t_train = x_train[:2000], t_train[:2000]
x_test, t_test = x_test[:1000], t_test[:1000]
max_epochs = 20
network = SimpleConvNet(input_dim=(1,28,28),
conv_param = {'filter_num': 30, 'filter_size': 5, 'pad': 0, 'stride': 1},
hidden_size=100, output_size=10, weight_init_std=0.01)
trainer = Trainer(network, x_train, t_train, x_test, t_test,
epochs=max_epochs, mini_batch_size=100,
optimizer='Adam', optimizer_param={'lr': 0.001},
evaluate_sample_num_per_epoch=1000)
trainer.train()
# 保存参数
network.save_params("params.pkl")
print("Saved Network Parameters!")
# 绘制图形
markers = {'train': 'o', 'test': 's'}
x = np.arange(max_epochs)
plt.plot(x, trainer.train_acc_list, marker='o', label='train', markevery=2)
plt.plot(x, trainer.test_acc_list, marker='s', label='test', markevery=2)
plt.xlabel("epochs")
plt.ylabel("accuracy")
plt.ylim(0, 1.0)
plt.legend(loc='lower right')
plt.show()
我们可以对比一下训练前与训练后的卷积核的图像，可以发现，训练后，卷积核有了很强的“方向性”，白色部分大多是打横或打竖的。
import numpy as np
import matplotlib.pyplot as plt
def filter_show(filters, nx=8, margin=3, scale=10):
"""
c.f. https://gist.github.com/aidiary/07d530d5e08011832b12#file-draw_weight-py
"""
FN, C, FH, FW = filters.shape
ny = int(np.ceil(FN / nx))
fig = plt.figure()
fig.subplots_adjust(left=0, right=1, bottom=0, top=1, hspace=0.05, wspace=0.05)
for i in range(FN):
ax = fig.add_subplot(ny, nx, i+1, xticks=[], yticks=[])
ax.imshow(filters[i, 0], cmap=plt.cm.gray_r, interpolation='nearest')
plt.show()
network = SimpleConvNet()
# 随机进行初始化后的权重
filter_show(network.params['W1'])
# 学习后的权重
network.load_params("params.pkl")
filter_show(network.params['W1'])
