Seq2seq and attention model

#Generating Language

#Synthesis

  • Input: symbols as one-hot vectors
    • Dimensionality of the vector is the size of the 「vocabulary
    • Projected down to lower-dimensional “embeddings
  • The hidden units are (one or more layers of) LSTM units
  • Output at each time: A probability distribution that ideally assigns peak probability to the next word in the sequence
  • Divergence

$$ \operatorname{Div}(\mathbf{Y}_{\text {target}}(1 \ldots T), \mathbf{Y}(1 \ldots T))=\sum_{t}\operatorname{Xent}(\mathbf{Y}_{\text {target}}(t), \mathbf{Y}(t))=-\sum_{t} \log Y(t, w_{t+1}) $$

  • Feed the drawn word as the next word in the series
  • And draw the next word from the output probability distribution

#Beginnings and ends

  • A sequence of words by itself does not indicate if it is a complete sentence or not
  • To make it explicit, we will add two additional symbols (in addition to the words) to the base vocabulary
    • <sos>: Indicates start of a sentence
    • <eos> : Indicates end of a sentence
  • When do we stop?
    • Continue this process until we draw an <eos>
    • Or we decide to terminate generation based on some other criterion

#Delayed sequence to sequence

#Pseudocode

  • Problem: Each word that is output depends only on current hidden state, and not on previous outputs
  • The input sequence feeds into a recurrent structure
  • The input sequence is terminated by an explicit <eos> symbol
    • The hidden activation at the <eos> “stores” all information about the sentence
  • Subsequently a second RNN uses the hidden activation as initial state to produce a sequence of outputs
    • The output at each time becomes the input at the next time
    • Output production continues until an <eos> is produced

#Autoencoder

  • The recurrent structure that extracts the hidden representation from the input sequence is the encoder
  • The recurrent structure that utilizes this representation to produce the output sequence is the decoder

#Generating output

  • At each time the network produces a probability distribution over words, given the entire input and previous outputs
  • At each time a word is drawn from the output distribution

$$ P\left(O_{1}, \ldots, O_{L} \mid W_{1}^{i n}, \ldots, W_{N}^{i n}\right)=y_{1}^{O_{1}} y_{1}^{O_{2}} \ldots y_{1}^{O_{L}} $$

  • The objective of drawing: Produce the most likely output (that ends in an <eos>)

$$ \underset{O_{1}, \ldots, O_{L}}{\operatorname{argmax}} y_{1}^{O_{1}} y_{1}^{O_{2}} \ldots y_{1}^{O_{L}} $$

  • How to draw words?
    • Greedy answer
      • Select the most probable word at each time
      • Not good, making a poor choice at any time commits us to a poor future
    • Randomly draw a word at each time according to the output probability distribution
      • Not guaranteed to give you the most likely output
    • Beam search
      • Search multiple choices and prune
      • At each time, retain only the top K scoring forks
      • Terminate: When the current most likely path overall ends in <eos>

#Train

  • In practice, if we apply SGD, we may randomly sample words from the output to actually use for the backprop and update
    • Randomly select training instance: (input, output)
    • Forward pass
    • Randomly select a single output $y(t)$ and corresponding desired output $d(t)$ for backprop
  • Trick
    • The input sequence is fed in reverse order
      • This happens both for training and during actual decode
  • Problem
    • All the information about the input sequence is embedded into a single vector
    • In reality: All hidden values carry information

#Attention model

  • Compute a weighted combination of all the hidden outputs into a single vector
    • Weights vary by output time
  • Require a time-varying weight that specifies relationship of output time to input time
    • Weights are functions of current output state

$$ e_{i}(t)=g\left(\boldsymbol{h}_{i}, \boldsymbol{s}_{t-1}\right) $$

$$ w_{i}(t)=\frac{\exp \left(e_{i}(t)\right)}{\sum_{j} \exp \left(e_{j}(t)\right)} $$

#Attention weight

  • Typical option for $g()$
    • Inner product
      • $$g\left(\boldsymbol{h}_{i}, \boldsymbol{s}_{t-1}\right)=\boldsymbol{h}_{i}^{T} \boldsymbol{s}_{t-1}$$
    • Project to the same demension
      • $$g\left(\boldsymbol{h}_{i}, \boldsymbol{s}_{t-1}\right)=\boldsymbol{h}_{i}^{T} \boldsymbol{W}_{g} \boldsymbol{s}_{t-1}$$
    • Non-linear activation
      • $$g\left(\boldsymbol{h}_{i}, \boldsymbol{s}_{t-1}\right)=v_{g}^{T} \boldsymbol{t} \boldsymbol{a} \boldsymbol{n} \boldsymbol{h}\left(\boldsymbol{W}_{g}\left[\begin{array}{c}\boldsymbol{h}_{i} \\ \boldsymbol{s}_{t-1}\end{array}\right]\right)$$
    • MLP
      • $$g\left(\boldsymbol{h}_{i}, \boldsymbol{s}_{t-1}\right)=\operatorname{MLP}\left(\left[\boldsymbol{h}_{i}, \boldsymbol{s}_{t-1}\right]\right)$$

#Pseudocode

#Train

  • Back propagation also updates parameters of the “attention” function
  • Trick: Occasionally pass drawn output instead of ground truth, as input
    • Randomly select from output, force network to produce correct word even the prioir word is not correct

#variants

  • Bidirectional processing of input sequence
  • Local attention vs global attention
  • Multihead attention
    • Derive 「value」, and multiple 「keys」 from the encoder
      • $V_{i}, K_{i}^{l}, i=1 \ldots T, l=1 \ldots N_{\text {head}}$
    • Derive one or more 「queries」 from decoder
      • $Q_{j}^{l}, j=1 \ldots M, l=1 \ldots N_{\text {head}}$
    • Each query-key pair gives you one attention distribution
      • And one context vector
      • $a_{j, i}^{l}=$attention$\left(Q_{j}^{l}, K_{i}^{l}, i=1 \ldots T\right), \quad C_{j}^{l}=\sum_{i} a_{j, i}^{l} V_{i}$
    • Concatenate set of context vectors into one extended context vector
      • $C_{j}=\left[C_{j}^{1} C_{j}^{2} \ldots C_{j}^{N_{\text {head}}}\right]$
    • Each 「attender」 focuses on a different aspect of the input that’s important for the decode
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