python实现卷积层的前向后向传播过程

出自:网易云课堂视频课程《用Python做深度学习1——数学基础》
作者:黑板客
转载请注明出处,原文链接和作者。

conv_python

卷积层是深度学习基础中最重要的部分。其实卷积层和数学里的卷积公式没有直接关系,跟随我把卷积层从公式到python实现走一遍,你会发现其实它并不那么难理解。(代码基于standford的cs231n课程大作业)

Convolution Layer Forward

卷积层的前向激活过程,我们首先忽略激活层。认为f(x)=x,那么纯卷积层的前向激活公式如下:

$$ \ out_{n,f,ho,wo} = conv(XP,W,b,params) $$ $$ \ =\sum\limits_{c = 0,ho = 0,wo = 0}^{C - 1,Ho - 1,Wo - 1} {{XP_{n,c,ho * S + (1:HH),wo * S + (1:WW)}}*{W_{f,c,:,:}}} + b_{f} $$

n是输入的个数,比如输入100张图片,n=100.

C是input channel,比如输入的图片是RGB三通道的,C=3.

S是stride,stride为1,逐行扫描。stride为2,隔一行扫描一次。不理解stride的还要先查查其它文章。

XP是填0后的输入。若不填0,则XP=X. 不理解填0操作的还要先查查其它文章。

F是filter number,系数的高和宽分别是HH,WW。Ho和Wo是输出的高,宽。

根据这个公式我们可以写出最基础的前向过程,理解原理不用担心你的for循环有几层,那些是以后优化的工作。理解了上面的公式,你就可以理解下面的实现代码。


import numpy as np

%load_ext autoreload
%autoreload 2

def conv_forward_naive(x, w, b, conv_param):
  """
  A naive implementation of the forward pass for a convolutional layer.

  The input consists of N data points, each with C channels, height H and width
  W. We convolve each input with F different filters, where each filter spans
  all C channels and has height HH and width HH.

  Input:
  - x: Input data of shape (N, C, H, W)
  - w: Filter weights of shape (F, C, HH, WW)
  - b: Biases, of shape (F,)
  - conv_param: A dictionary with the following keys:
    - 'stride': The number of pixels between adjacent receptive fields in the
      horizontal and vertical directions.
    - 'pad': The number of pixels that will be used to zero-pad the input.

  Returns a tuple of:
  - out: Output data, of shape (N, F, H', W') where H' and W' are given by
    H' = 1 + (H + 2 * pad - HH) / stride
    W' = 1 + (W + 2 * pad - WW) / stride
  - cache: (x, w, b, conv_param)
  """
  out = None
  N,C,H,W = x.shape
  F,_,HH,WW = w.shape
  S = conv_param['stride']
  P = conv_param['pad']
  Ho = 1 + (H + 2 * P - HH) / S
  Wo = 1 + (W + 2 * P - WW) / S
  x_pad = np.zeros((N,C,H+2*P,W+2*P))
  x_pad[:,:,P:P+H,P:P+W]=x
  #x_pad = np.pad(x, ((0,), (0,), (P,), (P,)), 'constant')
  out = np.zeros((N,F,Ho,Wo))

  for f in xrange(F):
    for i in xrange(Ho):
      for j in xrange(Wo):
        # N*C*HH*WW, C*HH*WW = N*C*HH*WW, sum -> N*1
        out[:,f,i,j] = np.sum(x_pad[:, :, i*S : i*S+HH, j*S : j*S+WW] * w[f, :, :, :], axis=(1, 2, 3)) 

    out[:,f,:,:]+=b[f]
  cache = (x, w, b, conv_param)
  return out, cache

我们可以用几个例子试试它的输出


x_shape = (2, 3, 4, 4) #n,c,h,w
w_shape = (2, 3, 3, 3) #f,c,hw,ww
x = np.ones(x_shape)
w = np.ones(w_shape)
b = np.array([1,2])

conv_param = {'stride': 1, 'pad': 0}
out, _ = conv_forward_naive(x, w, b, conv_param)

print out
print out.shape  #n,f,ho,wo

结果如下:

[[[[ 28. 28.] [ 28. 28.]]

[[ 29. 29.] [ 29. 29.]]]

[[[ 28. 28.] [ 28. 28.]]

[[ 29. 29.] [ 29. 29.]]]]

(2, 2, 2, 2)

设置pad为1,自己再计算一下结果。尤其是结果的维数变化。 设置stride为3,pad为1呢? 还可以怎么设置呢?

Convolution Layer Backward

后向传播过程复杂一些,不过一旦你掌握了偏微分和链式法则,应该也难不倒你。

假设卷积层后直接跟了Loss层,那么

$$ \frac{{\partial L}}{{\partial w}} = \frac{{\partial L}}{{\partial out}}*\frac{{\partial out}}{{\partial w}} $$

$$ \frac{{\partial L}}{{\partial x}} = \frac{{\partial L}}{{\partial out}}*\frac{{\partial out}}{{\partial x}} $$

$$ \frac{{\partial L}}{{\partial b}} = \frac{{\partial L}}{{\partial out}}*\frac{{\partial out}}{{\partial b}} $$

而且

$$ \frac{{\partial L}}{{\partial out}} = dout $$

dout在卷积层的后向过程是已知的,所以公式看上去很简单,就是下标处理复杂了点。我们慢慢来继续推导它。

$$ \frac{{\partial L}}{{\partial {W_{f,c,:,:}}}} = \sum\limits_{n = 0,ho = 0,wo = 0}^{N - 1,Ho - 1,Wo - 1} {dou{t_{n,f,ho,wo}}} * \frac{{\partial ({XP_{n,c,h_{win},w_{win}}} * {W_{f,c,:,:}})}}{{\partial {W_{f,c,:,:}}}} $$ $$ \ = \sum\limits_{n = 0,ho = 0,wo = 0}^{N - 1,Ho - 1,Wo - 1} {dou{t_{n,f,ho,wo}} * {XP_{n,c,h_{win},w_{win}}}} $$

为了简化期间,我们把偏导一个w,改成偏导一个w的二维矩阵,这样dout的偏微分就更好理解一些。对x的偏导我们也同样处理。其中XP的下标h_win,w_win是前向过程公式里hos+(1:HH)和wos+(1:WW)的缩写。

$$ \frac{{\partial L}}{{\partial {XP_{n,c,h_{win},w_{win}}}}} = \sum\limits_{f = 0,ho = 0,wo = 0}^{F - 1,Ho - 1,Wo - 1} {dou{t_{n,f,ho,wo}}} * \frac{{\partial ({XP_{n,c,h_{win},w_{win}}} * {w_{f,c,:,:}})}}{{\partial {X_{n,c,h_{win},w_{win}}}}} $$ $$\ = \sum\limits_{f = 0,ho = 0,wo = 0}^{F - 1,Ho - 1,Wo - 1} {dou{t_{n,f,ho,wo}} * {W_{f,c,:,:}}} $$

$$ \frac{{\partial L}}{{\partial {b_f}}} = \sum\limits_{n = 0,ho = 0,wo = 0}^{N - 1,Ho - 1,Wo - 1} {dou{t_{n,f,ho,wo}}} * \frac{{\partial ({XP_{n,c,h_{win},w_{win}}} * {W_{f,c,:,:}}+b_f)}}{{\partial {b_f}}} $$ $$\ = \sum\limits_{n = 0,ho = 0,wo = 0}^{N - 1,Ho - 1,Wo - 1} {dou{t_{n,f,ho,wo}}} $$

理解了上面的公式,接下来我们再理解下面的实现代码就简单多了。


def conv_backward_naive(dout, cache):
  """
  A naive implementation of the backward pass for a convolutional layer.

  Inputs:
  - dout: Upstream derivatives.
  - cache: A tuple of (x, w, b, conv_param) as in conv_forward_naive

  Returns a tuple of:
  - dx: Gradient with respect to x
  - dw: Gradient with respect to w
  - db: Gradient with respect to b
  """
  dx, dw, db = None, None, None

  N, F, H1, W1 = dout.shape
  x, w, b, conv_param = cache
  N, C, H, W = x.shape
  HH = w.shape[2]
  WW = w.shape[3]
  S = conv_param['stride']
  P = conv_param['pad']


  dx, dw, db = np.zeros_like(x), np.zeros_like(w), np.zeros_like(b)
  x_pad = np.pad(x, [(0,0), (0,0), (P,P), (P,P)], 'constant')
  dx_pad = np.pad(dx, [(0,0), (0,0), (P,P), (P,P)], 'constant')
  db = np.sum(dout, axis=(0,2,3))

  for n in xrange(N):
    for i in xrange(H1):
      for j in xrange(W1):
        # Window we want to apply the respective f th filter over (C, HH, WW)
        x_window = x_pad[n, :, i * S : i * S + HH, j * S : j * S + WW]

        for f in xrange(F):
          dw[f] += x_window * dout[n, f, i, j] #F,C,HH,WW
          #C,HH,WW
          dx_pad[n, :, i * S : i * S + HH, j * S : j * S + WW] += w[f] * dout[n, f, i, j]

  dx = dx_pad[:, :, P:P+H, P:P+W]

  return dx, dw, db

上面的实现代码是最原始的。 matlab上为了加速,使用已有的conv函数实现上述过程,才有了很多博文上提到的翻转180度两次的过程,翻来翻去的反而不容易理解整个过程。其实卷积层的前向和后向传播,跟信号处理的卷积操作没有直接关系。就是相关和点乘操作。其它实现都是优化加速方法。

我们对反向传播也举个例子


x_shape = (2, 3, 4, 4)
w_shape = (2, 3, 3, 3)
x = np.ones(x_shape)
w = np.ones(w_shape)
b = np.array([1,2])

conv_param = {'stride': 1, 'pad': 0}

Ho = (x_shape[3]+2*conv_param['pad']-w_shape[3])/conv_param['stride']+1
Wo = Ho

dout = np.ones((x_shape[0], w_shape[0], Ho, Wo))

out, cache = conv_forward_naive(x, w, b, conv_param)
dx, dw, db = conv_backward_naive(dout, cache)

print "out shape",out.shape
print "dw=========================="
print dw
print "dx=========================="
print dx
print "db=========================="
print db

out shape (2, 2, 2, 2)

dw==========================

[[[[ 8. 8. 8.] [ 8. 8. 8.] [ 8. 8. 8.]]

[[ 8. 8. 8.] [ 8. 8. 8.] [ 8. 8. 8.]]

[[ 8. 8. 8.] [ 8. 8. 8.] [ 8. 8. 8.]]]

[[[ 8. 8. 8.] [ 8. 8. 8.] [ 8. 8. 8.]]

[[ 8. 8. 8.] [ 8. 8. 8.] [ 8. 8. 8.]]

[[ 8. 8. 8.] [ 8. 8. 8.] [ 8. 8. 8.]]]]

dx==========================

[[[[ 2. 4. 4. 2.] [ 4. 8. 8. 4.] [ 4. 8. 8. 4.] [ 2. 4. 4. 2.]]

[[ 2. 4. 4. 2.] [ 4. 8. 8. 4.] [ 4. 8. 8. 4.] [ 2. 4. 4. 2.]]

[[ 2. 4. 4. 2.] [ 4. 8. 8. 4.] [ 4. 8. 8. 4.] [ 2. 4. 4. 2.]]]

[[[ 2. 4. 4. 2.] [ 4. 8. 8. 4.] [ 4. 8. 8. 4.] [ 2. 4. 4. 2.]]

[[ 2. 4. 4. 2.] [ 4. 8. 8. 4.] [ 4. 8. 8. 4.] [ 2. 4. 4. 2.]]

[[ 2. 4. 4. 2.] [ 4. 8. 8. 4.] [ 4. 8. 8. 4.] [ 2. 4. 4. 2.]]]]

db==========================

[ 8. 8.]

汇聚层(Pool Layer)和卷积层的前向后向过程都可以通过公式来理解和实现。成功实现前向和后向传播过程是实现卷积神经网络的基础。

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