Answered Essay: The multi-layer network below is being trained using backpropagation. The current input/output pair is x_

The multi-layer network below is being trained using backpropagation. The current input/output pair s Xp(k) = (1.0, 1.0, 1.0)t and dp(k) = (0.0, 0.5, 1.0. The weights and node outputs are given in the table below (note that, for ease of making the table, Im using the letter w” to represent all weights since Ive numbered the nodes sequentially). Assume the sigmoid activation function (logistic function) for each node with a=1. Using a learning rate of 0.1 and a momentum term of 0.4, compute δ6 (k), δ4(k) and w64(k+1) (assume that the previous weight change was 0.08) 6 4 0.2 0.8 0.668 0.525 0.406 0.659 0.438 4 0.5 -2.0 -2.0 0.2 0.8 First, write out the backpropagation formula

The multi-layer network below is being trained using backpropagation. The current input/output pair is x_p(k) = (1.0, 1.0, 1.0)^t and d_p(k) = (0.0, 0.5, 1.0)^t. The weights and node outputs are given in the table below (note that, for ease of making the table, I’m using the letter “w” to represent all weights since I’ve numbered the nodes sequentially). Assume the sigmoid activation function (logistic function) for each node with a = 1. Using a learning rate of 0.1 and a momentum term of 0.4, compute delta_6 (k), delta_4(k) and w_64(k + 1) (assume that the previous weight change was 0.08). First, write out the backpropagation formula

Expert Answer

 

t is the target
units[l] is the number of units in layer l
n[l][i] is unit i in layer l
n[l][i].output is the output
n[l][i].delta is the delta
n[l][i].weight[j] is weight j
ek is the learning constant
adapt() {
int i,j,k,l;

  for(l=layers-1;l>=0;l--) 
    for(i=0;i<units[l];i++) 
      if(l==layers-1)
        n[l][i].delta=
          ek*n[l][i].output*
          (1.0-n[l][i].output)*
          (t[i]-n[l][i].output);
      else {
        n[l][i].delta=0.0;
        for(k=0;k<units[l];k++)
          n[l][i].delta+=
            n[l+1][k].delta*
            n[l+1][k].weight[i];
        n[l][i].delta=n[l][i].delta*
          ek*n[l][i].output*
          (1.0-n[l][i].output);
       }  
    
  
  for(l=layers-1;l>=1;l--) 
    for(i=0;i<units[l];i++) 
      for(j=0;j<weights;j++)
        n[l][i].weight[j]+=
           n[l-1][j].output*
           n[l][i].delta;
      
    
   for(i=0;i<units[0];i++)
     for(j=0;j<weights;j++)
       n[0][i].weight[j]+=
         input[j]*n[0][i].delta;
}

When this algorithm is applied to the XOR we get the following output.

iteration no 0, inputs 0 1, target 1, output 0.477995 
iteration no 20, inputs 0 0, target 1, output 0.447816 
iteration no 40, inputs 1 0, target 0, output 0.450292 
iteration no 60, inputs 0 0, target 1, output 0.549096 
iteration no 80, inputs 1 0, target 0, output 0.460706 
iteration no 100, inputs 0 0, target 1, output 0.507636 
iteration no 120, inputs 0 1, target 1, output 0.571619 
iteration no 140, inputs 1 0, target 0, output 0.451493 
iteration no 160, inputs 0 1, target 1, output 0.570574 
iteration no 180, inputs 0 0, target 1, output 0.575979 
iteration no 200, inputs 0 1, target 1, output 0.744079 
iteration no 220, inputs 1 0, target 0, output 0.233541 
iteration no 240, inputs 0 1, target 1, output 0.755600 
iteration no 260, inputs 1 1, target 0, output 0.185273 
iteration no 280, inputs 0 1, target 1, output 0.788309 
iteration no 300, inputs 1 1, target 0, output 0.167068 
iteration no 320, inputs 1 0, target 0, output 0.123461 
iteration no 340, inputs 1 1, target 0, output 0.132892 
iteration no 360, inputs 1 1, target 0, output 0.133583 
iteration no 380, inputs 1 1, target 0, output 0.116641 
iteration no 400, inputs 1 0, target 0, output 0.088269 
iteration no 420, inputs 0 0, target 1, output 0.861810 
iteration no 440, inputs 1 1, target 0, output 0.102406 
iteration no 460, inputs 1 0, target 0, output 0.080179 
iteration no 480, inputs 1 0, target 0, output 0.075584 
iteration no 500, inputs 0 0, target 1, output 0.884442 
iteration no 520, inputs 0 0, target 1, output 0.892789 
iteration no 540, inputs 0 1, target 1, output 0.923969 
iteration no 560, inputs 1 0, target 0, output 0.064146 
iteration no 580, inputs 1 1, target 0, output 0.071938 
iteration no 600, inputs 1 1, target 0, output 0.075764 
iteration no 620, inputs 1 1, target 0, output 0.074536 
iteration no 640, inputs 1 1, target 0, output 0.069014 
iteration no 660, inputs 1 1, target 0, output 0.066534 
iteration no 680, inputs 0 0, target 1, output 0.918422 
iteration no 700, inputs 0 0, target 1, output 0.924860 
iteration no 720, inputs 1 1, target 0, output 0.065864 
iteration no 740, inputs 1 0, target 0, output 0.052634 
iteration no 760, inputs 0 0, target 1, output 0.927081 
iteration no 780, inputs 1 0, target 0, output 0.050964 
iteration no 800, inputs 0 1, target 1, output 0.948869 
iteration no 820, inputs 1 0, target 0, output 0.049082 
iteration no 840, inputs 1 0, target 0, output 0.048074 
iteration no 860, inputs 1 1, target 0, output 0.057916 
iteration no 880, inputs 1 1, target 0, output 0.056088 
iteration no 900, inputs 0 1, target 1, output 0.954659 
iteration no 920, inputs 1 1, target 0, output 0.057337 
iteration no 940, inputs 0 0, target 1, output 0.944243 
iteration no 960, inputs 1 0, target 0, output 0.045653 
iteration no 980, inputs 0 0, target 1, output 0.946199
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