$(2)$ Problem $4$

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Final due Sep$9,202407$:$59E^{nT}$

Recurrent neural networks iriviv$/$ vai ve useu as ciassimamui iumeis iui mile seiles data. Here we have a

simple RNN as shown in the figure above, where

$s_t=f_1(W^{s,s}{s}_{t-1}+W^{s,x_2}{x}_t),t=1,2,$dots,$T$

and

$y=f_2(W^{{}_1{y}_s}{s}_T+W_0)$

We assume all offsets are $0$ except $W_0$ for the final output layer and we decide the two activation functions to

be:

$f_1(z)=$RELU$(z)=$max$(0,z)$

and

$f_2(z)=$sign$(z)=\{\begin{array}{ccc}1,& if& z0\\ 0,& if& z<mn>0</mn>\end{array}$

Note that the RELU $(z)$ can be applied elementwise if $z$ is a vector.

Suppose we want to apply this model to classify sentences into different categories $($e$.$g$.$ positive$/$negative

sentiment$),$ we need to encode each word in a sentence into a vector as the input $x_t$ to the model. One way to

do this is to represent the $t$ th word as a column vector of length $|V|,$ where $V$ is the set of the entire

vocabulary. The $i$ th element of $x_t$ is $1$ if the word is the $i$ th word in the vocabulary and all other elements are

zero.

$3$ points possible $($graded$,$ results hidden$)$

Now compute the final hidden state $s_T^{(1)},s_T^{(2)},s_T^{(3)}$ for each of the three senteces $AA,$ABB,BAA in this

RNN$.$

$($Enter $,0$ for $S_T=[0,0{]}^T{?}^T.)$

$s_T^{(1)}=$

$s_T^{(2)}=$

$s_T^{(3)}=$