This question provides a simple two-layer neural network with a hidden layer of three neurons an...

 

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This question provides a simple two-layer neural network with a hidden layer of three neurons an output layer of one neuron, and two input neurons, In both the hidden layer and the output layer, the sigmoid function is considered as an activation function. X= [x1, x2] is a vector that the network receives as input. A parameter vector called W (¹)= [w1,w2, w3, w4, w5, w6] connects the input layer to the hidden layer. A parameter vector W(2) [w7, w8, w9] is the parameter that connects the hidden layer to the output layer (examples shown in the figure below). X1 Input W₁ W7 W₂ Output (a) Write the mathematical expression for the output of the hidden layer, including the bias term. (10%) (b) Write the mathematical expression for the output of the output layer, including the bias term. (10 %) (c) The true label for this input is y = 0. The cost function used in this network is the mean squared error (MSE). Write the mathematical expression for the cost for this input. (10%) (d) Use backpropagation to calculate the gradient descent of the cost with respect to the weights connecting the input layer to the hidden layer [w1,w2, w3, w4, w5, w6] and the weights connecting the hidden layer to the output layer [w7, w8, w9]. (10%) This question provides a simple two-layer neural network with a hidden layer of three neurons an output layer of one neuron, and two input neurons, In both the hidden layer and the output layer, the sigmoid function is considered as an activation function. X= [x1, x2] is a vector that the network receives as input. A parameter vector called W (¹)= [w1,w2, w3, w4, w5, w6] connects the input layer to the hidden layer. A parameter vector W(2) [w7, w8, w9] is the parameter that connects the hidden layer to the output layer (examples shown in the figure below). X1 Input W₁ W7 W₂ Output (a) Write the mathematical expression for the output of the hidden layer, including the bias term. (10%) (b) Write the mathematical expression for the output of the output layer, including the bias term. (10 %) (c) The true label for this input is y = 0. The cost function used in this network is the mean squared error (MSE). Write the mathematical expression for the cost for this input. (10%) (d) Use backpropagation to calculate the gradient descent of the cost with respect to the weights connecting the input layer to the hidden layer [w1,w2, w3, w4, w5, w6] and the weights connecting the hidden layer to the output layer [w7, w8, w9]. (10%)

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a Output of the hidden layer h W1X b1 where h is a vector of size 3 representing the output of the hidden layer neurons is the sigmoid function z 1 1 expz W1 is the weight matrix of size 3x2 connectin... View the full answer