用numpy实现CNN卷积神经网络

为了加深对卷积神经网络底层原理的理解，本文通过使用numpy来搭建一个基础的包含卷积层、池化层、全连接层和Softmax层的卷积神经网络，并选择relu作为我们的激活函数，选择多分类交叉熵损失函数，最后使用了mnist数据集进行了训练和测试。

关于卷积网络的详细原理和实现可参考下列文章：

刘建平Pinard：卷积网络前向反向传播算法

卷积层的反向传播

手把手带你 Numpy实现CNN

1、卷积层

卷积层的前向传播输出由卷积核和特征图作卷积运算得到，反向传播时需要计算kernel和bias的梯度以及delta的反向传播误差，kernel的梯度由原特征图和delta作卷积得到，bias每个通道的梯度由对delta每个通道直接求和得到，delta的反向传播误差由delta和旋转180度的卷积核作卷积运算得到。其中卷积运算在实现时先将特征图的对应部分和卷积核展开成了向量的形式，再作向量乘法运算，这样可以通过并行运算加快速度，实现代码如下：

def img2col(x, ksize, stride):
    wx, hx, cx = x.shape                     # [width,height,channel]
    feature_w = (wx - ksize) // stride + 1   # 返回的特征图尺寸       
    image_col = np.zeros((feature_w*feature_w, ksize*ksize*cx))
    num = 0
    for i in range(feature_w):
        for j in range(feature_w): 
            image_col[num] =  x[i*stride:i*stride+ksize, j*stride:j*stride+ksize, :].reshape(-1)
            num += 1
    return image_col
    
class Conv(object):
    def __init__(self, kernel_shape, stride=1, pad=0):
        width, height, in_channel, out_channel = kernel_shape
        self.stride = stride
        self.pad = pad
        scale = np.sqrt(3*in_channel*width*height/out_channel)   #batch=3
        self.k = np.random.standard_normal(kernel_shape) / scale
        self.b = np.random.standard_normal(out_channel) / scale
        self.k_gradient = np.zeros(kernel_shape)
        self.b_gradient = np.zeros(out_channel)

    def forward(self, x):
        self.x = x
        if self.pad != 0:
            self.x = np.pad(self.x, ((0,0),(self.pad,self.pad),(self.pad,self.pad),(0,0)), 'constant')
        bx, wx, hx, cx = self.x.shape
        wk, hk, ck, nk = self.k.shape             # kernel的宽、高、通道数、个数
        feature_w = (wx - wk) // self.stride + 1  # 返回的特征图尺寸
        feature = np.zeros((bx, feature_w, feature_w, nk))
                       
        self.image_col = []
        kernel = self.k.reshape(-1, nk)
        for i in range(bx):
            image_col = img2col(self.x[i], wk, self.stride)                       
            feature[i] = (np.dot(image_col, kernel)+self.b).reshape(feature_w,feature_w,nk)
            self.image_col.append(image_col)
        return feature

    def backward(self, delta, learning_rate):
        bx, wx, hx, cx = self.x.shape # batch,14,14,inchannel
        wk, hk, ck, nk = self.k.shape # 5,5,inChannel,outChannel
        bd, wd, hd, cd = delta.shape  # batch,10,10,outChannel

        # 计算self.k_gradient,self.b_gradient
        delta_col = delta.reshape(bd, -1, cd)
        for i in range(bx):
            self.k_gradient += np.dot(self.image_col[i].T, delta_col[i]).reshape(self.k.shape)
        self.k_gradient /= bx
        self.b_gradient += np.sum(delta_col, axis=(0, 1))
        self.b_gradient /= bx    

        # 计算delta_backward
        delta_backward = np.zeros(self.x.shape)
        k_180 = np.rot90(self.k, 2, (0,1))      # numpy矩阵旋转180度
        k_180 = k_180.swapaxes(2, 3)
        k_180_col = k_180.reshape(-1,ck)

        if hd-hk+1 != hx:
            pad = (hx-hd+hk-1) // 2
            pad_delta = np.pad(delta, ((0,0),(pad,pad),(pad,pad),(0,0)), 'constant')
        else:
            pad_delta = delta

        for i in range(bx):
            pad_delta_col = img2col(pad_delta[i], wk, self.stride)
            delta_backward[i] = np.dot(pad_delta_col, k_180_col).reshape(wx,hx,ck)

        # 反向传播
        self.k -=  self.k_gradient * learning_rate
        self.b -=  self.b_gradient * learning_rate

        return delta_backward

在这里顺便给出relu的实现代码：

class Relu(object):        
    def forward(self, x):
        self.x = x
        return np.maximum(x, 0)
    
    def backward(self, delta):
        delta[self.x<0] = 0
        return delta

2、池化层

池化层实现了ksize=2、stride=2的最大池化，前向传播时取对应核的最大值作为输出，并记录最大值的位置，反向传播时先将特征图按原值扩充一次，再将非最大值位置置0即可。

class Pool(object):
    def forward(self, x):
        b, w, h, c = x.shape
        feature_w = w // 2
        feature = np.zeros((b, feature_w, feature_w, c))
        self.feature_mask = np.zeros((b, w, h, c))   # 记录最大池化时最大值的位置信息用于反向传播
        for bi in range(b):
            for ci in range(c):
                for i in range(feature_w):
                    for j in range(feature_w):
                        feature[bi, i, j, ci] = np.max(x[bi,i*2:i*2+2,j*2:j*2+2,ci])
                        index = np.argmax(x[bi,i*2:i*2+2,j*2:j*2+2,ci])
                        self.feature_mask[bi, i*2+index//2, j*2+index%2, ci] = 1                    
        return feature

    def backward(self, delta):
        return np.repeat(np.repeat(delta, 2, axis=1), 2, axis=2) * self.feature_mask

3、全连接层

全连接层的实现前文已经给出，这里给出了封装成单独的类后的形式，增强了复用性：

class Linear(object):
    def __init__(self, inChannel, outChannel):
        scale = np.sqrt(inChannel/2)
        self.W = np.random.standard_normal((inChannel, outChannel)) / scale
        self.b = np.random.standard_normal(outChannel) / scale
        self.W_gradient = np.zeros((inChannel, outChannel))
        self.b_gradient = np.zeros(outChannel)

    def forward(self, x):
        self.x = x
        x_forward = np.dot(self.x, self.W) + self.b
        return x_forward

    def backward(self, delta, learning_rate):
        ## 梯度计算
        batch_size = self.x.shape[0]
        self.W_gradient = np.dot(self.x.T, delta) / batch_size  # bxin bxout
        self.b_gradient = np.sum(delta, axis=0) / batch_size 
        delta_backward = np.dot(delta, self.W.T)                # bxout inxout
        ## 反向传播
        self.W -= self.W_gradient * learning_rate
        self.b -= self.b_gradient * learning_rate 

        return delta_backward

4、Softmax层

一般分类模型在全连接层给出每个类别的预测值后会再经过softmax层来得到最终的预测值，其前向传播公式如下：

\[a_j=\frac{e^{z_j}}{\sum\limits_k{e^{z_k}}} \]

在将标签onehot编码后，反向传播公式可给出向量形式如下：

\[\delta=a-y \]

对单个样本，其多分类交叉熵loss计算公式给出向量形式如下：

\[loss=-\sum y\log a \]

最后给出代码实现：

class Softmax(object):
    def cal_loss(self, predict, label):
        batchsize, classes = predict.shape
        self.predict(predict)
        loss = 0
        delta = np.zeros(predict.shape)
        for i in range(batchsize):
            delta[i] = self.softmax[i] - label[i]
            loss -= np.sum(np.log(self.softmax[i]) * label[i])
        loss /= batchsize
        return loss, delta

    def predict(self, predict):
        batchsize, classes = predict.shape
        self.softmax = np.zeros(predict.shape)
        for i in range(batchsize):
            predict_tmp = predict[i] - np.max(predict[i])
            predict_tmp = np.exp(predict_tmp)
            self.softmax[i] = predict_tmp / np.sum(predict_tmp)
        return self.softmax

5、训练和测试

训练和测试是直接使用的torchvision集成的mnist数据集，训练后将权重参数通过numpy提供的接口保存到本地文件中，测试时再从文件中读取权重参数，在只训练了两个epoch的情况下测试集的准确率达到了98.05%，相比使用全连接的神经网络提高了不少。训练和测试的代码如下：

def train():
    # Mnist手写数字集
    dataset_path = "D://datasets//mnist"
    train_data = torchvision.datasets.MNIST(root=dataset_path, train=True, download=True)
    train_data.data = train_data.data.numpy()  # [60000,28,28]
    train_data.targets = train_data.targets.numpy()  # [60000]
    train_data.data = train_data.data.reshape(60000, 28, 28, 1) / 255.   # 输入向量处理
    train_data.targets = onehot(train_data.targets, 60000) # 标签one-hot处理 (60000, 10) 
    
    conv1 = Conv(kernel_shape=(5,5,1,6))   # 24x24x6
    relu1 = Relu()
    pool1 = Pool()                         # 12x12x6
    conv2 = Conv(kernel_shape=(5,5,6,16))  # 8x8x16
    relu2 = Relu()
    pool2 = Pool()                         # 4x4x16
    nn = Linear(256, 10)
    softmax = Softmax()
   
    lr = 0.01
    batch = 3
    for epoch in range(10):        
        for i in range(0, 60000, batch):
            X = train_data.data[i:i+batch]
            Y = train_data.targets[i:i+batch]

            predict = conv1.forward(X)
            predict = relu1.forward(predict)
            predict = pool1.forward(predict)
            predict = conv2.forward(predict)
            predict = relu2.forward(predict)
            predict = pool2.forward(predict)
            predict = predict.reshape(batch, -1)
            predict = nn.forward(predict)

            loss, delta = softmax.cal_loss(predict, Y)

            delta = nn.backward(delta, lr)
            delta = delta.reshape(batch,4,4,16)
            delta = pool2.backward(delta)
            delta = relu2.backward(delta)
            delta = conv2.backward(delta, lr)
            delta = pool1.backward(delta)
            delta = relu1.backward(delta)
            conv1.backward(delta, lr)

            print("Epoch-{}-{:05d}".format(str(epoch), i), ":", "loss:{:.4f}".format(loss))

        lr *= 0.95**(epoch+1)
        np.savez("data2.npz", k1=conv1.k, b1=conv1.b, k2=conv2.k, b2=conv2.b, w3=nn.W, b3=nn.b)

def eval():
    r = np.load("data2.npz")

    # Mnist手写数字集
    dataset_path = "D://datasets//mnist"
    test_data = torchvision.datasets.MNIST(root=dataset_path, train=False)
    test_data.data = test_data.data.numpy()        # [10000,28,28]
    test_data.targets = test_data.targets.numpy()  # [10000]

    test_data.data = test_data.data.reshape(10000, 28, 28, 1) / 255.

    conv1 = Conv(kernel_shape=(5, 5, 1, 6))  # 24x24x6
    relu1 = Relu()
    pool1 = Pool()  # 12x12x6
    conv2 = Conv(kernel_shape=(5, 5, 6, 16))  # 8x8x16
    relu2 = Relu()
    pool2 = Pool()  # 4x4x16
    nn = Linear(256, 10)
    softmax = Softmax()

    conv1.k = r["k1"]
    conv1.b = r["b1"]
    conv2.k = r["k2"]
    conv2.b = r["b2"]
    nn.W = r["w3"]
    nn.n = r["b3"]

    num = 0
    for i in range(10000):
        X = test_data.data[i]
        X = X[np.newaxis, :]
        Y = test_data.targets[i]

        predict = conv1.forward(X)
        predict = relu1.forward(predict)
        predict = pool1.forward(predict)
        predict = conv2.forward(predict)
        predict = relu2.forward(predict)
        predict = pool2.forward(predict)
        predict = predict.reshape(1, -1)
        predict = nn.forward(predict)

        predict = softmax.predict(predict)

        if np.argmax(predict) == Y:
            num += 1

    print("TEST-ACC: ", num/10000*100, "%")

6、完整代码

import numpy as np
import torchvision
import time, functools
import logging
np.set_printoptions(threshold=np.inf)  


def onehot(targets, num):
    result = np.zeros((num, 10))
    for i in range(num):
        result[i][targets[i]] = 1
    return result


def img2col(x, ksize, stride):
    wx, hx, cx = x.shape                     # [width,height,channel]
    feature_w = (wx - ksize) // stride + 1   # 返回的特征图尺寸       
    image_col = np.zeros((feature_w*feature_w, ksize*ksize*cx))
    num = 0
    for i in range(feature_w):
        for j in range(feature_w): 
            image_col[num] =  x[i*stride:i*stride+ksize, j*stride:j*stride+ksize, :].reshape(-1)
            num += 1
    return image_col
    

## nn
class Linear(object):
    def __init__(self, inChannel, outChannel):
        scale = np.sqrt(inChannel/2)
        self.W = np.random.standard_normal((inChannel, outChannel)) / scale
        self.b = np.random.standard_normal(outChannel) / scale
        self.W_gradient = np.zeros((inChannel, outChannel))
        self.b_gradient = np.zeros(outChannel)

    def forward(self, x):
        self.x = x
        x_forward = np.dot(self.x, self.W) + self.b
        return x_forward

    def backward(self, delta, learning_rate):
        ## 梯度计算
        batch_size = self.x.shape[0]
        self.W_gradient = np.dot(self.x.T, delta) / batch_size  # bxin bxout
        self.b_gradient = np.sum(delta, axis=0) / batch_size 
        delta_backward = np.dot(delta, self.W.T)                # bxout inxout
        ## 反向传播
        self.W -= self.W_gradient * learning_rate
        self.b -= self.b_gradient * learning_rate 

        return delta_backward

## conv
class Conv(object):
    def __init__(self, kernel_shape, stride=1, pad=0):
        width, height, in_channel, out_channel = kernel_shape
        self.stride = stride
        self.pad = pad
        scale = np.sqrt(3*in_channel*width*height/out_channel)
        self.k = np.random.standard_normal(kernel_shape) / scale
        self.b = np.random.standard_normal(out_channel) / scale
        self.k_gradient = np.zeros(kernel_shape)
        self.b_gradient = np.zeros(out_channel)

    def forward(self, x):
        self.x = x
        if self.pad != 0:
            self.x = np.pad(self.x, ((0,0),(self.pad,self.pad),(self.pad,self.pad),(0,0)), 'constant')
        bx, wx, hx, cx = self.x.shape
        wk, hk, ck, nk = self.k.shape             # kernel的宽、高、通道数、个数
        feature_w = (wx - wk) // self.stride + 1  # 返回的特征图尺寸
        feature = np.zeros((bx, feature_w, feature_w, nk))
                       
        self.image_col = []
        kernel = self.k.reshape(-1, nk)
        for i in range(bx):
            image_col = img2col(self.x[i], wk, self.stride)                       
            feature[i] = (np.dot(image_col, kernel)+self.b).reshape(feature_w,feature_w,nk)
            self.image_col.append(image_col)
        return feature

    def backward(self, delta, learning_rate):
        bx, wx, hx, cx = self.x.shape # batch,14,14,inchannel
        wk, hk, ck, nk = self.k.shape # 5,5,inChannel,outChannel
        bd, wd, hd, cd = delta.shape  # batch,10,10,outChannel

        # 计算self.k_gradient,self.b_gradient
        delta_col = delta.reshape(bd, -1, cd)
        for i in range(bx):
            self.k_gradient += np.dot(self.image_col[i].T, delta_col[i]).reshape(self.k.shape)
        self.k_gradient /= bx
        self.b_gradient += np.sum(delta_col, axis=(0, 1))
        self.b_gradient /= bx    

        # 计算delta_backward
        delta_backward = np.zeros(self.x.shape)
        k_180 = np.rot90(self.k, 2, (0,1))      # numpy矩阵旋转180度
        k_180 = k_180.swapaxes(2, 3)
        k_180_col = k_180.reshape(-1,ck)

        if hd-hk+1 != hx:
            pad = (hx-hd+hk-1) // 2
            pad_delta = np.pad(delta, ((0,0),(pad,pad),(pad,pad),(0,0)), 'constant')
        else:
            pad_delta = delta

        for i in range(bx):
            pad_delta_col = img2col(pad_delta[i], wk, self.stride)
            delta_backward[i] = np.dot(pad_delta_col, k_180_col).reshape(wx,hx,ck)

        # 反向传播
        self.k -=  self.k_gradient * learning_rate
        self.b -=  self.b_gradient * learning_rate

        return delta_backward

## pool
class Pool(object):
    def forward(self, x):
        b, w, h, c = x.shape
        feature_w = w // 2
        feature = np.zeros((b, feature_w, feature_w, c))
        self.feature_mask = np.zeros((b, w, h, c))   # 记录最大池化时最大值的位置信息用于反向传播
        for bi in range(b):
            for ci in range(c):
                for i in range(feature_w):
                    for j in range(feature_w):
                        feature[bi, i, j, ci] = np.max(x[bi,i*2:i*2+2,j*2:j*2+2,ci])
                        index = np.argmax(x[bi,i*2:i*2+2,j*2:j*2+2,ci])
                        self.feature_mask[bi, i*2+index//2, j*2+index%2, ci] = 1                    
        return feature

    def backward(self, delta):
        return np.repeat(np.repeat(delta, 2, axis=1), 2, axis=2) * self.feature_mask

## Relu
class Relu(object):        
    def forward(self, x):
        self.x = x
        return np.maximum(x, 0)
    
    def backward(self, delta):
        delta[self.x<0] = 0
        return delta

## Softmax
class Softmax(object):
    def cal_loss(self, predict, label):
        batchsize, classes = predict.shape
        self.predict(predict)
        loss = 0
        delta = np.zeros(predict.shape)
        for i in range(batchsize):
            delta[i] = self.softmax[i] - label[i]
            loss -= np.sum(np.log(self.softmax[i]) * label[i])
        loss /= batchsize
        return loss, delta

    def predict(self, predict):
        batchsize, classes = predict.shape
        self.softmax = np.zeros(predict.shape)
        for i in range(batchsize):
            predict_tmp = predict[i] - np.max(predict[i])
            predict_tmp = np.exp(predict_tmp)
            self.softmax[i] = predict_tmp / np.sum(predict_tmp)
        return self.softmax

def train():
    # Mnist手写数字集
    dataset_path = "D://datasets//mnist"
    train_data = torchvision.datasets.MNIST(root=dataset_path, train=True, download=True)
    train_data.data = train_data.data.numpy()  # [60000,28,28]
    train_data.targets = train_data.targets.numpy()  # [60000]
    train_data.data = train_data.data.reshape(60000, 28, 28, 1) / 255.   # 输入向量处理
    train_data.targets = onehot(train_data.targets, 60000) # 标签one-hot处理 (60000, 10) 
    
    conv1 = Conv(kernel_shape=(5,5,1,6))   # 24x24x6
    relu1 = Relu()
    pool1 = Pool()                         # 12x12x6
    conv2 = Conv(kernel_shape=(5,5,6,16))  # 8x8x16
    relu2 = Relu()
    pool2 = Pool()                         # 4x4x16
    nn = Linear(256, 10)
    softmax = Softmax()
   
    lr = 0.01
    batch = 3
    for epoch in range(10):        
        for i in range(0, 60000, batch):
            X = train_data.data[i:i+batch]
            Y = train_data.targets[i:i+batch]

            predict = conv1.forward(X)
            predict = relu1.forward(predict)
            predict = pool1.forward(predict)
            predict = conv2.forward(predict)
            predict = relu2.forward(predict)
            predict = pool2.forward(predict)
            predict = predict.reshape(batch, -1)
            predict = nn.forward(predict)

            loss, delta = softmax.cal_loss(predict, Y)

            delta = nn.backward(delta, lr)
            delta = delta.reshape(batch,4,4,16)
            delta = pool2.backward(delta)
            delta = relu2.backward(delta)
            delta = conv2.backward(delta, lr)
            delta = pool1.backward(delta)
            delta = relu1.backward(delta)
            conv1.backward(delta, lr)

            print("Epoch-{}-{:05d}".format(str(epoch), i), ":", "loss:{:.4f}".format(loss))

        lr *= 0.95**(epoch+1)
        np.savez("data2.npz", k1=conv1.k, b1=conv1.b, k2=conv2.k, b2=conv2.b, w3=nn.W, b3=nn.b)

def eval():
    r = np.load("data2.npz")

    # Mnist手写数字集
    dataset_path = "D://datasets//mnist"
    test_data = torchvision.datasets.MNIST(root=dataset_path, train=False)
    test_data.data = test_data.data.numpy()        # [10000,28,28]
    test_data.targets = test_data.targets.numpy()  # [10000]

    test_data.data = test_data.data.reshape(10000, 28, 28, 1) / 255.

    conv1 = Conv(kernel_shape=(5, 5, 1, 6))  # 24x24x6
    relu1 = Relu()
    pool1 = Pool()  # 12x12x6
    conv2 = Conv(kernel_shape=(5, 5, 6, 16))  # 8x8x16
    relu2 = Relu()
    pool2 = Pool()  # 4x4x16
    nn = Linear(256, 10)
    softmax = Softmax()

    conv1.k = r["k1"]
    conv1.b = r["b1"]
    conv2.k = r["k2"]
    conv2.b = r["b2"]
    nn.W = r["w3"]
    nn.n = r["b3"]

    num = 0
    for i in range(10000):
        X = test_data.data[i]
        X = X[np.newaxis, :]
        Y = test_data.targets[i]

        predict = conv1.forward(X)
        predict = relu1.forward(predict)
        predict = pool1.forward(predict)
        predict = conv2.forward(predict)
        predict = relu2.forward(predict)
        predict = pool2.forward(predict)
        predict = predict.reshape(1, -1)
        predict = nn.forward(predict)

        predict = softmax.predict(predict)

        if np.argmax(predict) == Y:
            num += 1

    print("TEST-ACC: ", num/10000*100, "%")

if __name__ == '__main__':
    #train()
    eval()