Natural Language Processinganoopsarkar.github.io/nlp-class/assets/slides/nlm.pdf · Anoop Sarkar...

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SFUNatLangLab

Natural Language Processing

Anoop Sarkaranoopsarkar.github.io/nlp-class

Simon Fraser University

November 6, 2019

http://anoopsarkar.github.io/nlp-class

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Part 1: Long distance dependencies


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Long distance dependencies

Example

I He doesn’t have very much confidence in himself

I She doesn’t have very much confidence in herself

n-gram Language Models: P(wi | w i−1i−n+1)

P(himself | confidence, in)

P(herself | confidence, in)

What we want: P(wi | w<i)

P(himself | He, . . . , confidence)

P(herself | She, . . . , confidence)

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Long distance dependenciesOther examples

I Selectional preferences: I ate lunch with a fork vs. I atelunch with a backpack

I Topic: Babe Ruth was able to touch the home plate yet againvs. Lucy was able to touch the home audiences with herhumour

I Register: Consistency of register in the entire sentence, e.g.informal (Twitter) vs. formal (scientific articles)

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Language Models

Chain Rule and ignore some history: the trigram model

p(w1, . . . ,wn)

≈ p(w1)p(w2 | w1)p(w3 | w1,w2) . . . p(wn | wn−2,wn−1)

≈∏t

p(wt+1 | wt−1,wt)

How can we address the long-distance issues?

I Skip n-gram models. Skip an arbitrary distance for n-gramcontext.

I Variable n in n-gram models that is adaptive

I Problems: Still ”all or nothing”. Categorical rather than soft.

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Part 2: Neural Language Models


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Neural Language Models

Use Chain rule and approximate using a neural network

p(w1, . . . ,wn) ≈∏t

p(wt+1 | φ(w1, . . . ,wt)︸︷︷︸capture history with vector s(t)

)

Recurrent Neural NetworkI Let y be the output wt+1 for current word wt and history

w1, . . . ,wt

I s(t) = f (Uxh ·w(t) +Whh · s(t− 1)) where f is sigmoid / tanh

I s(t) encapsulates history using single vector of size h

I Output word at time step wt+1 is provided by y(t)

I y(t) = g(Vhy · s(t)) where g is softmax

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Neural Language ModelsRecurrent Neural Network

Single time step in RNN:I Input layer is a one hot vector and

output layer y have the samedimensionality as vocabulary(10K-200K).

I One hot vector is used to look up wordembedding w

I “Hidden” layer s is orders ofmagnitude smaller (50-1K neurons)

I U is the matrix of weights betweeninput and hidden layer

I V is the matrix of weights betweenhidden and output layer

I Without recurrent weights W , this isequivalent to a bigram feedforwardlanguage model

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y(1) y(2) y(3) y(4) y(5) y(6)

s(1) s(2) s(3) s(4) s(5) s(6)

w(1) w(2) w(3) w(4) w(5) w(6)

Uxh

Vhy

Uxh

Vhy

Uxh

Vhy

Uxh

Vhy

Uxh

Vhy

Uxh

Vhy

Whh Whh Whh Whh Whh

What is stored and what is computed:

I Model parameters: w ∈ Rx (word embeddings);Uxh ∈ Rx×h;Whh ∈ Rh×h;Vhy ∈ Rh×y where y = |V|.

I Vectors computed during forward pass: s(t) ∈ Rh; y(t) ∈ Ry

and each y(t) is a probability over vocabulary V.

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Part 3: Training RNN Language Models


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Computational Graph for an RNN Language Model

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Training of RNNLM

I The training is performed using Stochastic Gradient Descent(SGD)

I We go through all the training data iteratively, and update theweight matrices U, W and V (after processing every word)

I Training is performed in several “epochs” (usually 5-10)

I An epoch is one pass through the training data

I As with feedforward networks we have two passes:

Forward pass : collect the values to make a prediction (foreach time step)

Backward pass : back-propagate the error gradients (througheach time step)

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Training of RNNLMForward pass

I In the forward pass we compute a hidden state s(t) based onprevious states 1, . . . , t − 1I s(t) = f (Uxh · w(t) + Whh · s(t − 1))I s(t) = f (Uxh · w(t) + Whh · f (Uxh · w(t) + Whh · s(t − 2)))I s(t) = f (Uxh · w(t) + Whh ·

f (Uxh · w(t) + Whh · f (Uxh · w(t) + Whh · s(t − 3))))I etc.

I Let us assume f is linear, e.g. f (x) = x .

I Notice how we have to compute Whh ·Whh · . . . =∏

i Whh

I By examining this repeated matrix multiplication we can showthat the norm of Whh →∞ (explodes)

I This is why f is set to a function that returns a boundedvalue (sigmoid / tanh)

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Training of RNNLMBackward pass

I Gradient of the error vector in the output layer eo(t) iscomputed using a cross entropy criterion:

eo(t) = d(t)− y(t)

I d(t) is a target vector that represents the word w(t + 1)represented as a one-hot (1-of-V) vector

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I Weights V between the hidden layer s(t) and the output layery(t) are updated as

V (t+1) = V (t) + s(t) · eo(t) · α

I where α is the learning rate

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I Next, gradients of errors are propagated from the output layerto the hidden layer

eh(t) = dh(eo · V , t)

I where the error vector is obtained using function dh() that isapplied element-wise:

dhj(x , t) = x · sj(t)(1− sj(t))

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I Weights U between the input layer w(t) and the hidden layers(t) are then updated as

U(t+1) = U(t) + w(t) · eh(t) · α

I Similarly the word embeddings w can also be updated usingthe error gradient.

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Training of RNNLM: Backpropagation through timeBackward pass

I The recurrent weights W are updated by unfolding them intime and training the network as a deep feedforward neuralnetwork.

I The process of propagating errors back through the recurrentweights is called Backpropagation Through Time (BPTT).

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Training of RNNLM: Backpropagation through timeFig. from [1]: RNN unfolded as a deep feedforward network 3 time steps back in time

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I Error propagation is done recursively as follows (it requires thestates of the hidden layer from the previous time steps τ to bestored):

e(t − τ − 1) = dh(eh(t − τ) ·W , t − τ − 1)

I The error gradients quickly vanish as they get backpropagatedin time (less likely if we use sigmoid / tanh)

I We use gated RNNs to stop gradients from vanishing orexploding.

I Popular gated RNNs are long short-term memory RNNs akaLSTMs and gated recurrent units aka GRUs.

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I The recurrent weights W are updated as:

W (t+1) = W (t) +T∑

z=0

s(t − z − 1) · eh(t − z) · α

I Note that the matrix W is changed in one update at once,not during backpropagation of errors.

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Part 4: Gated Recurrent Units


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Interpolation for hidden unitsu: use history or forget history

I For RNN state s(t) ∈ Rh create a binary vector u ∈ {0, 1}h

ui =

{1 use the new hidden state (standard RNN update)0 copy previous hidden state and ignore RNN update

I Create an intermediate hidden state s(t) where f is tanh:

s(t) = f (Uxh · w(t) + Whh · s(t − 1))

I Use the binary vector u to interpolate between copying priorstate s(t − 1) and using new state s(t):

s(t) = (1− u) �

� is elementwise multiplication

s(t − 1) + u�s(t)

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Interpolation for hidden unitsr : reset or retain each element of hidden state vector

I For RNN state s(t−1) ∈ Rh create a binary vector r ∈ {0, 1}h

ri =

{1 if si (t − 1) should be used0 if si (t − 1) should be ignored

I Modify intermediate hidden state s(t) where f is tanh:

s(t) = f (Uxh · w(t) + Whh · (r � s(t − 1)))

I Use the binary vector u to interpolate between s(t − 1) ands(t):

s(t) = (1− u)� s(t − 1) + u � s(t)

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Interpolation for hidden unitsLearning u and r

I Instead of binary vectors u ∈ {0, 1}h and r ∈ {0, 1}h we wantto learn u and r

I Let u ∈ [0, 1]h and r ∈ [0, 1]h

I Learn these two h dimensional vectors using equations similarto the RNN hidden state equation:

u(t) = σ (Uuxh · w(t) + W u

hh · s(t − 1))

r(t) = σ (U rxh · w(t) + W r

hh · s(t − 1))

I The sigmoid function σ ensures that each element of u and ris between [0, 1]

I The use history u and reset element r vectors use differentparameters Uu,W u and U r ,W r

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Interpolation for hidden unitsGated Recurrent Unit (GRU)

I Putting it all together:

u(t) = σ (Uuxh · w(t) + W u

hh · s(t − 1))

r(t) = σ (U rxh · w(t) + W r

hh · s(t − 1))

s(t) = tanh(Uxh · w(t) + Whh · (r(t)� s(t − 1)))

s(t) = (1− u(t))� s(t − 1) + u(t)� s(t)

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Interpolation for hidden unitsLong Short-term Memory (LSTM)

I Split up u(t) into two different gates i(t) and f (t):

i(t) = σ(U ixh · w(t) + W i

hh · s(t − 1))

f (t) = σ(U fxh · w(t) + W f

hh · s(t − 1))

r(t) = σ (U rxh · w(t) + W r

hh · s(t − 1))

s(t) = tanh(Uxh · w(t) + Whh · s(t − 1)︸︷︷︸GRU:r(t)�s(t−1)

)

s(t) = f (t)� s(t − 1) + i(t)� s(t)︸︷︷︸GRU:(1−u(t))�s(t−1)+u(t)�s(t)

s(t) = r(t)� tanh (s(t))

I So LSTM is a GRU plus an extra Uxh, Whh and tanh.

I Q: what happens if f (t) is set to 1− i(t)? A: read [3]

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Part 5: Sequence prediction using RNNs


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Representation: finding the right parameters

Problem: Predict ?? using context, P(?? | context)

Profits/N soared/V at/P Boeing/?? Co. , easily topping forecastson Wall Street , as their CEO Alan Mulally announced first quarterresults .

Representation: history

I The input is a tuple: (x[1:n], i) [ignoring y−1 for now]

I x[1:n] are the n words in the input

I i is the index of the word being tagged

I For example, for x4 = BoeingI We can use an RNN to summarize the entire context at i = 4

I x[1:i−1] = (Profits, soared, at)I x[i+1:n] = (Co., easily, ..., results, .)

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Locally normalized RNN taggers

Log-linear model over history, tag pair (h, t)

log Pr(y | h) = w · f(h, y)− log∑y ′

exp(w · f(h, y ′)

)f(h, y) is a vector of feature functions

RNN for tagging

I Replace f(h, y) with RNN hidden state s(t)

I Define the output logprob: log Pr(y | h) = log y(t)

I y(t) = g(V · s(t)) where g is softmax

I In neural LMs the output y ∈ V (vocabulary)

I In sequence tagging using RNNs the output y ∈ T (tagset)

log Pr(y[1:n] | x[1:n]) =n∑

i=1

log Pr(yi | hi )

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Bidirectional RNNsFig. from [2]

sb(1) sb(2) sb(3) sb(4) sb(5) sb(6)

sf (1) sf (2) sf (3) sf (4) sf (5) sf (6)

x(1) x(2) x(3) x(4) x(5) x(6)

Bidirectional RNN

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Bidirectional RNNs can be StackedFig. from [2]

sf2(1) sf2(2) sf2(3) sf2(4) sf2(5) sf2(6)

sb2(1) sb2(2) sb2(3) sb2(4) sb2(5) sb2(6)

sf1(1) sf1(2) sf1(3) sf1(4) sf1(5) sf1(6)

sb1(1) sb1(2) sb1(3) sb1(4) sb1(5) sb1(6)

x(1) x(2) x(3) x(4) x(5) x(6)

Two Bidirectional RNNs stacked on top of each other

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Part 6: Training RNNs on GPUs


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Parallelizing RNN computationsFig. from [2]

Apply RNNs to batches of sequencesPresent the data as a 3D tensor of (T × B × F ). Each dynamicupdate will now be a matrix multiplication.

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Binary MasksFig. from [2]

A mask matrix may be used to aid with computations that ignorethe padded zeros.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0

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Binary MasksFig. from [2]

It may be necessary to (partially) sort your data.

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[1] Tomas MikolovRecurrent Neural Networks for Language Models. GoogleTalk.2010.

[2] Philemon BrakelMLIA-IQIA Summer School notes on RNNs2015.

[3] Klaus Greff, Rupesh Kumar Srivastava, Jan Koutnık, Bas R.Steunebrink, Jurgen SchmidhuberLSTM: A Search Space Odyssey2017.

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Acknowledgements

Many slides borrowed or inspired from lecture notes by MichaelCollins, Chris Dyer, Kevin Knight, Chris Manning, Philipp Koehn,Adam Lopez, Graham Neubig, Richard Socher and LukeZettlemoyer from their NLP course materials.

All mistakes are my own.

A big thank you to all the students who read through these notesand helped me improve them.

Natural Language Processinganoopsarkar.github.io/nlp-class/assets/slides/nlm.pdf · Anoop Sarkar...

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