$6.[$Neural Networks$](5$ pts$)$

a$.(1$ pts$)$ True or False: For any neural network, the validation loss will always decrease monotonically with the number of iterations of gradient descent, provided the step size is sufficiently small.

$-$ True

$-$ False

b$.$ Let $\backslash ($ f $\backslash )$ be a fully$-$connected neural network with input $\backslash ($ x $\backslash$in $\backslash$mathbb$\{$R$\}^\{$M$\},$ P $\backslash )$ hidden layers with $\backslash ($ K $\backslash )$ nodes per layer and logistic activation functions, and a single logistic output. Let $\backslash ($ g $\backslash )$ be the same network as $\backslash ($ f $\backslash ),$ except we insert another hidden layer with $\backslash ($ K $\backslash )$ nodes that have no activation function $($or equivalently, the identity activation function$),$ so that $\backslash ($ g $\backslash )$ has $\backslash ($ P$+1\backslash )$ hidden layers. Denote this new layer $\backslash ($ L$^\{\backslash$text $\{$new $\}\}\backslash ).$ Assume that there are no bias terms for any layer nor for the input. $($Please select one option for all the following questions.$)$

i$.(1$ pts$)\backslash ($ f $\backslash )$ can learn the same decision boundary as $\backslash ($ g $\backslash )$ if the additional linear layer is placed

$-$ immediately after the input.

$-$ immediately before the last sigmoid activation function.

$-$ anywhere in between the above two choices.

$-$ none of the above.

ii$.(1$ pts$)$ Assume that $\backslash ($ L$^\{\backslash$text $\{$new $\}\}\backslash )$ is placed in between two other hidden layers in $\backslash ($ g $\backslash ).$ How many more parameters does $\backslash ($ g $\backslash )$ learn compared to $\backslash ($ f $\backslash )?$

$-\backslash ($ K $\backslash )$

$-\backslash ($ K$^\{2\}\backslash )$

$-\backslash ($ K P $\backslash )$

$-\backslash ($ K M $\backslash )$

$-\backslash (2$ K$^\{2\}\backslash )$

iii. $(1$ pts$)$ True or False: After training $\backslash ($ f $\backslash )$ and $\backslash ($ g $\backslash )$ to convergence, $\backslash ($ g $\backslash )$ can have a lower training loss than $\backslash ($ f $\backslash ).($Use the same assumption as in ii$).$

$-$ True

$-$ False

iv$.(1$ pts$)$ True or False: After training $\backslash ($ f $\backslash )$ and $\backslash ($ g $\backslash )$ to convergence, $\backslash ($ f $\backslash )$ can have a lower training loss than $\backslash ($ g $\backslash ).($Use the same assumption as in ii$).$

$-$ True

$-$ False