GRU(Gated Recurrent Unit) 更新过程推导及简单代码实现

GRU(Gated Recurrent Unit) 更新过程推导及简单代码实现

RNN

GRU

matlab codes

RNN网络考虑到了具有时间数列的样本数据，但是RNN仍存在着一些问题，比如随着时间的推移，RNN单元就失去了对很久之前信息的保存和处理的能力，而且存在着gradient vanishing问题。

所以有些特殊类型的RNN网络相继被提出，比如LSTM(long short term memory)和GRU(gated recurrent unit)(Chao,et al. 2014).这里我主要推导一下GRU参数的迭代过程

GRU单元结构如下图所示

1479126283494.jpg

数据流过程如下

其中表示Hadamard积，即对应元素乘积；下标表示节点的index，上标表示时刻；表示隐层到输出层的参数矩阵,分别是隐层和输出层的节点个数；分别表示输入和上一时刻隐层到更新门z的连接矩阵，表示输入数据的维度；分别表示输入和上一时刻隐层到重置门r的连接矩阵；分别表示输入和上一时刻的隐层到待选状态的连接矩阵。

针对于时刻t，使用链式求导法则，计算参数矩阵的梯度,其中E是代价函数，首先计算对隐层输出的梯度，因为隐层输出牵涉到多个时刻

所以

其中分别是对应激活函数的线性和部分

现在对参数计算梯度

令

则

将上面的式子矢量化(行向量)表示：

那接下来使用matlab来实现一个小例子，看看GRU的效果，同样是二进制相加的问题

1. function error= GRUtest( ) 
   

   ---

2. % 初始化训练数据 
   

   ---

3. uNum=16;%单元个数 
   

   ---

4. maxInt=2^uNum; 
   

   ---

5. % 初始化网络结构 
   

   ---

6. xdim=2; 
   

   ---

7. ydim=1; 
   

   ---

8. hdim=16; 
   

   ---

9. eta=0.1; 
   

   ---

10. %初始化网络参数 
    

    ---

11. Wy=rand(hdim,ydim)*2-1; 
    

    ---

12. Wr=rand(xdim,hdim)*2-1; 
    

    ---

13. Ur=rand(hdim,hdim)*2-1; 
    

    ---

14. W =rand(xdim,hdim)*2-1; 
    

    ---

15. U =rand(hdim,hdim)*2-1; 
    

    ---

16. Wz=rand(xdim,hdim)*2-1; 
    

    ---

17. Uz=rand(hdim,hdim)*2-1; 
    

    ---

18. ---

19. rvalues=zeros(uNum+1,hdim); 
    

    ---

20. zvalues=zeros(uNum+1,hdim); 
    

    ---

21. hbarvalues=zeros(uNum,hdim); 
    

    ---

22. hvalues = zeros(uNum,hdim); 
    

    ---

23. yvalues=zeros(uNum,ydim); 
    

    ---

24. ---

25. for p=1:10000 
    

    ---

26. aInt=randi(maxInt/2); 
    

    ---

27. bInt=randi(maxInt/2); 
    

    ---

28. cInt=aInt+bInt; 
    

    ---

29. at=dec2bin(aInt)-'0'; 
    

    ---

30. bt=dec2bin(bInt)-'0'; 
    

    ---

31. ct=dec2bin(cInt)-'0'; 
    

    ---

32. a=zeros(1,uNum); 
    

    ---

33. b=zeros(1,uNum); 
    

    ---

34. c=zeros(1,uNum); 
    

    ---

35. a(1:size(at,2))=at(end:-1:1); 
    

    ---

36. b(1:size(bt,2))=bt(end:-1:1); 
    

    ---

37. c(1:size(ct,2))=ct(end:-1:1); 
    

    ---

38. xvalues=[a;b]'; 
    

    ---

39. d=c'; 
    

    ---

40. ---

41. % 前向计算 
    

    ---

42. rvalues(1,:)=sigmoid(xvalues(1,:)*Wr); 
    

    ---

43. hbarvalues(1,:)=outTanh(xvalues(1,:)*W); 
    

    ---

44. zvalues(1,:)=sigmoid(xvalues(1,:)*Wz); 
    

    ---

45. hvalues(1,:)=zvalues(1,:).*hbarvalues(1,:); 
    

    ---

46. yvalues(1,:)=sigmoid(hvalues(1,:)*Wy); 
    

    ---

47. for t=2:uNum 
    

    ---

48. rvalues(t,:)=sigmoid(xvalues(t,:)*Wr+hvalues(t-1,:)*Ur); 
    

    ---

49. hbarvalues(t,:)=outTanh(xvalues(t,:)*W+(rvalues(t,:).*hvalues(t-1,:))*U); 
    

    ---

50. zvalues(t,:)=sigmoid(xvalues(t,:)*Wz+hvalues(t-1,:)*Uz); 
    

    ---

51. hvalues(t,:)=(1-zvalues(t,:)).*hvalues(t-1,:)+zvalues(t,:).*hbarvalues(t,:); 
    

    ---

52. yvalues(t,:)=sigmoid(hvalues(t,:)*Wy);  
    

    ---

53. end 
    

    ---

54. ---

55. % 误差反向传播 
    

    ---

56. delta_r_next=zeros(1,hdim); 
    

    ---

57. delta_z_next=zeros(1,hdim); 
    

    ---

58. delta_h_next=zeros(1,hdim); 
    

    ---

59. delta_next=zeros(1,hdim); 
    

    ---

60. ---

61. dWy=zeros(hdim,ydim); 
    

    ---

62. dWr=zeros(xdim,hdim); 
    

    ---

63. dUr=zeros(hdim,hdim); 
    

    ---

64. dW=zeros(xdim,hdim); 
    

    ---

65. dU=zeros(hdim,hdim); 
    

    ---

66. dWz=zeros(xdim,hdim); 
    

    ---

67. dUz=zeros(hdim,hdim); 
    

    ---

68. ---

69. for t=uNum:-1:2 
    

    ---

70. delta_y=(yvalues(t,:)-d(t,:)).*diffsigmoid(yvalues(t,:)); 
    

    ---

71. delta_h=delta_y*Wy'+delta_z_next*Uz'+delta_next*U'.*rvalues(t+1,:)+delta_r_next*Ur'+delta_h_next.*(1-zvalues(t+1,:)); 
    

    ---

72. delta_z=delta_h.*(hbarvalues(t,:)-hvalues(t-1,:)).*diffsigmoid(zvalues(t,:)); 
    

    ---

73. delta =delta_h.*zvalues(t,:).*diffoutTanh(hbarvalues(t,:)); 
    

    ---

74. delta_r=hvalues(t-1,:).*((delta_h.*zvalues(t,:).*diffoutTanh(hbarvalues(t,:)))*U').*diffsigmoid(rvalues(t,:)); 
    

    ---

75. ---

76. dWy=dWy+hvalues(t,:)'*delta_y; 
    

    ---

77. dWz=dWz+xvalues(t,:)'*delta_z; 
    

    ---

78. dUz=dUz+hvalues(t-1,:)'*delta_z; 
    

    ---

79. dW =dW+xvalues(t,:)'*delta; 
    

    ---

80. dU =dU+(rvalues(t,:).*hvalues(t-1,:))'*delta ; 
    

    ---

81. dWr=dWr+xvalues(t,:)'*delta_r; 
    

    ---

82. dUr=dUr+hvalues(t-1,:)'*delta_r; 
    

    ---

83. ---

84. delta_r_next=delta_r; 
    

    ---

85. delta_z_next=delta_z; 
    

    ---

86. delta_h_next=delta_h; 
    

    ---

87. delta_next =delta; 
    

    ---

88. ---

89. end 
    

    ---

90. ---

91. t=1; 
    

    ---

92. delta_y=(yvalues(t,:)-d(t,:)).*diffsigmoid(yvalues(t,:)); 
    

    ---

93. delta_h=delta_y*Wy'+delta_z_next*Uz'+delta_next*U'.*rvalues(t+1,:)+delta_r_next*Ur'+delta_h_next.*(1-zvalues(t+1,:)); 
    

    ---

94. delta_z=delta_h.*(hbarvalues(t,:)-0).*diffsigmoid(zvalues(t,:)); 
    

    ---

95. delta =delta_h.*zvalues(t,:).*diffoutTanh(hbarvalues(t,:)); 
    

    ---

96. delta_r=0.*((delta_h.*zvalues(t,:).*diffoutTanh(hbarvalues(t,:)))*U').*diffsigmoid(rvalues(t,:)); 
    

    ---

97. ---

98. dWy=dWy+hvalues(t,:)'*delta_y; 
    

    ---

99. dWz=dWz+xvalues(t,:)'*delta_z; 
    

    ---

100. dW =dW+xvalues(t,:)'*delta; 
     

     ---

101. dWr=dWr+xvalues(t,:)'*delta_r; 
     

     ---

102. ---

103. Wy = Wy-eta*dWy; 
     

     ---

104. Wr = Wr-eta*dWr; 
     

     ---

105. Ur = Ur-eta*dUr; 
     

     ---

106. W = W -eta*dW; 
     

     ---

107. U = U-eta*dU; 
     

     ---

108. Wz = Wz-eta*dWz; 
     

     ---

109. Uz = Uz-eta*dUz; 
     

     ---

110. error = (norm(yvalues-d,2))/2.0; 
     

     ---

111. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
     

     ---

112. if mod(p,500)==0 
     

     ---

113. fprintf('******************第%s次迭代****************\n',int2str(p)); 
     

     ---

114. yvalues=round(yvalues(end:-1:1)); 
     

     ---

115. y=bin2dec(int2str(yvalues')); 
     

     ---

116. fprintf('y=%d\n',y); 
     

     ---

117. fprintf('c=%d\n',cInt); 
     

     ---

118. fprintf('样本误差:e=%f\n',error); 
     

     ---

119. end 
     

     ---

120. end 
     

     ---

121. end 
     

     ---

122. ---

123. function f=sigmoid(x) 
     

     ---

124. f=1./(1+exp(-x)); 
     

     ---

125. end 
     

     ---

126. ---

127. function fd = diffsigmoid(f) 
     

     ---

128. fd=f.*(1-f); 
     

     ---

129. end 
     

     ---

130. ---

131. function g=outTanh(x) 
     

     ---

132. g=1-2./(1+exp(2*x)); 
     

     ---

133. end 
     

     ---

134. ---

135. function gd=diffoutTanh(g) 
     

     ---

136. gd=1-g.^2; 
     

     ---

137. end 
     

     ---

部分实验结果

1479392393541.jpg