#### Long Short Term Memory (LSTM) ## Quiz

#### Problem 1

How many bits of information can be modeled by the hidden state (at some specific time) of a Hidden Markov Model with 16 hidden units?

• $2$
• $4$
• $16$
• $>16$

$n$比特的信息可以表示$2^n$个二进制数，也就是$2^n$种状态，所以$m$种状态对应了$\text{log}_2(m)$个比特的信息。对于HMM来说，$16$个隐藏单元对应了$16$种状态，所以需要$\text{log}_2(16)=4$个比特的信息，答案为4。

#### Problem 2

This question is about speech recognition. To accurately recognize what phoneme is being spoken at a particular time, one needs to know the sound data from 100ms before that time to 100ms after that time, i.e. a total of 200ms of sound data. Which of the following setups have access to enough sound data to recognize what phoneme was being spoken 100ms into the past?

• A Recurrent Neural Network (RNN) with 200ms of input
• A Recurrent Neural Network (RNN) with 30ms of input
• A feed forward Neural Network with 30ms of input
• A feed forward Neural Network with 200ms of input

#### Problem 3

The figure below shows a Recurrent Neural Network (RNN) with one input unit xx, one logistic hidden unit hh, and one linear output unit $y$. The RNN is unrolled in time for $T=0,1$, and $2$. The network parameters are: $W_{xh}=0.5,W_{hh}=-1.0, W_{hy}=-0.7 , h_{bias}=-1.0$, and $y_{bias}=0.0$. Remember, $\sigma(k) = \frac{1}{1+\exp(-k)}$.

If the input xx takes the values $9 , 4 , -2$ at time steps $0, 1 ,2$ respectively, what is the value of the output $y$ at $T=1$? Give your answer to at least two digits after the decimal point.

#### Problem 4

The figure below shows a Recurrent Neural Network (RNN) with one input unit xx, one logistic hidden unit hh, and one linear output unit $y$. The RNN is unrolled in time for $T=0,1$, and $2$. The network parameters are: $W_{xh}=-0.1,W_{hh}=0.5, W_{hy}=0.25 , h_{bias}=0.4$, and $y_{bias}=0.0$.

If the input $x$ takes the values $18 , 9 , -8$ at time steps $0, 1 ,2$ respectively, the hidden unit values will be $0.2 , 0.4 , 0.8$ and the output unit values will be $0.05 , 0.1 , 0.2$ (you can check these values as an exercise). A variable $z$ is defined as the total input to the hidden unit before the logistic nonlinearity.

If we are using the squared loss, with targets $t_0,t_1,t_2$, then the sequence of calculations required to compute the total error $E$ is as follows:

If the target output values are $t_0 = 0.1 , t_1=-0.1 , t_2=-0.2​$ and the squared error loss is used, what is the value of the error derivative just before the hidden unit nonlinearity at $T=1​$ (i.e. $\frac{\partial E}{\partial z_1}​$)? Write your answer up to at least the fourth decimal place.

#### Problem 5

Consider a Recurrent Neural Network with one input unit, one logistic hidden unit, and one linear output unit. This RNN is for modeling sequences of length $4$ only, and the output unit exists only at the last time step, i.e. $T=3$. This diagram shows the RNN unrolled in time: Suppose that the model has learned the following parameter values:

• $w_{xh}=1$
• $w_{hh}=2$
• $w_{hy}=1$
• All biases are 0

For one specific training case, the input is $1$ at $T=0$ and $0$ at $T=1, T=2,$ and $T=3$. The target output (at $T=3$) is $0.5$, and we’re using the squared error loss function.

We’re interested in the gradient for $w_{xh}$, i.e. $\frac{\partial E}{\partial w_{xh}}$. Because it’s only at $T=0$ that the input is not zero, and it’s only at $T=3$ that there’s an output, the error needs to be backpropagated from $T=3$ to $T=0$, and that’s the kind of situations where RNN’s often get either vanishing gradients or exploding

gradients. Which of those two occurs in this situation?

You can either do the calculations, and find the answer that way, or you can find the answer with more thinking and less math, by thinking about the slope $\frac{\partial y}{\partial z}$ of the logistic function, and what role that plays in the backpropagation process.

#### Problem 6

Consider the following Recurrent Neural Network (RNN): As you can see, the RNN has two input units, two hidden units, and one output unit.

For this question, every arrow denotes the effect of a variable at time $t$ on a variable at time $t+1$.

Which feed forward Neural Network is equivalent to this network unrolled in time?   