disc12-regular

Q2. Backpropagation (a) Perform forward propagation on the neural network below for x = 1 by filling in the values in the table. Note that (i), . . . , (vii) are outputs after performing the appropriate operation as indicated in the node. (i) (ii) (iii) (iv) (v) (vi) (vii) x ∗ 2 ∗ 3 ∗ 4 ∑ max min max (i) (ii) (iii) (iv) (v) (vi) (vii) (b) Below is a neural network with weights a, b, c, d, e, f . The inputs are x 1 and x 2 . The first hidden layer computes r 1 = max( c · x 1 + e · x 2 , 0) and r 2 = max( d · x 1 + f · x 2 , 0). The second hidden layer computes s 1 = 1 1+exp( − a · r 1 ) and s 2 = 1 1+exp( − b · r 2 ) . The output layer computes y = s 1 + s 2 . Note that the weights a, b, c, d, e, f are indicated along the edges of the neural network here. Suppose the network has inputs x 1 = 1 , x 2 = − 1. The weight values are a = 1 , b = 1 , c = 4 , d = 1 , e = 2 , f = 2. Forward propagation then computes r 1 = 2 , r 2 = 0 , s 1 = 0 . 9 , s 2 = 0 . 5 , y = 1 . 4. Note: some values are rounded. x 1 x 2 r 1 r 2 s 1 s 2 y a b c d e f Using the values computed from forward propagation , use backpropagation to numerically calcu- late the following partial derivatives. Write your answers as a single number (not an expression). You do not need a calculator. Use scratch paper if needed. Hint: For g ( z ) = 1 1+exp( − z ) , the derivative is ∂g ∂z = g ( z )(1 − g ( z )). ∂y ∂a ∂y ∂b ∂y ∂c ∂y ∂d ∂y ∂e ∂y ∂f 2