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Conditional Random Fields

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Conditional Random Fields

- Given a sequence (in NLP, words), assign appropriate labels to each word.
- For example, POS tagging:

DT

NN

VBD

IN

DT

NN

.

The cat sat on the mat .

- Given a sequence (in NLP, words), assign appropriate labels to each word.
- Another example, partial parsing (aka chunking):

B-NP

I-NP

B-VP

B-PP

B-NP

I-NP

The cat sat on the mat

- Given a sequence (in NLP, words), assign appropriate labels to each word.
- Another example, relation extraction:

B-Arg

I-Arg

B-Rel

I-Rel

B-Arg

I-Arg

The cat sat on the mat

- A CRF model consists of
- F = <f1, …, fk>, a vector of “feature functions”
- θ = <θ1, …, θk>, a vector of weights for each feature function.

- Let O = < o1, …, oT> be an observed sentence
- Let X = <x1, …, xT> be the latent variables.
- This is the same as the Maximum Entropy equation!

- Note that the denominator depends on O, but not on y (it’s marginalizing over y).
- Typically, we write
- where

Making Structured Predictions

Recall: max. ent. for text classification:

CRFs for sequence labeling:

What’s the difference?

Two (related) differences, both for the sake of efficiency:

- Feature functions in CRFs are restricted to graph parts (described later)
- We can’t do brute force to compute the argmax. Instead, we do Viterbi.

Best sequence is

Recall from HMM discussion:

If there are

K possible states for each xi variable,

and N total xi variables,

Then there are KN possible settings for x

So brute force can’t find the best sequence.

Instead, we resort to a Viterbi-like dynamic program.

X1

Xt-1

Xt=hj

o1

ot-1

ot

ot+1

oT

The state sequence which maximizes the score of seeing the observations to time t-1, landing in state hj at time t, and seeing the observation at time t

o1

ot-1

ot

ot+1

oT

x1

xt-1

xt

xt+1

xT

Compute the most likely state sequence by working backwards

X1

Xt-1

Xt=hj

Xt+1

o1

ot-1

ot

ot+1

oT

??!

Recursive Computation

??!

To make efficient computation (dynamic programs) possible, we restrict the feature functions to:

Graph parts (or just parts): A feature function that counts how often a particular configuration occurs for a clique in the CRF graph.

Clique: a set of completely connected nodes in a graph. That is, each node in the clique has an edge connecting it to every other node in the clique.

X1

X2

X3

X4

X5

X6

o1

o2

o3

o4

o5

o6

The cliques in a linear chain CRF are the set of individual nodes, and the set of pairs of consecutive nodes.

CRF

X1

X2

X3

X4

X5

X6

o1

o2

o3

o4

o5

o6

The cliques in a linear chain CRF are the set of individual nodes, and the set of pairs of consecutive nodes.

Individual node cliques

CRF

X1

X2

X3

X4

X5

X6

o1

o2

o3

o4

o5

o6

The cliques in a linear chain CRF are the set of individual nodes, and the set of pairs of consecutive nodes.

Pair-of-node cliques

CRF

X1

X2

X3

X4

X5

X6

o1

o2

o3

o4

o5

o6

For non-linear-chain CRFs (something we won’t normally consider in this class), you can get larger cliques:

X5’

CRF

Larger cliques

x1=D

x2=N

x3=V

x4=D

x5=A

x6=N

o1

o2

o3

o4

o5

o6

Graph parts are feature functions f(x,o)that count how many cliques have a particular configuration.

For example, f(x,o) = count of [xi = Noun].

Here, x2 and x6 are both Nouns, so f(x,o) = 2.

CRF

x1=D

x2=N

x3=V

x4=D

x5=A

x6=N

o1

o2

o3

o4

o5

o6

For a pair-of-nodes example,

f(x,o) = count of [xi = Noun,xi+1=Verb]

Here, x2 is a Noun and x3 is a Verb, so f(x,o) = 1.

CRF

X1

X1

X2

X2

X3

X3

X4

X4

X5

X5

X6

X6

o1

o1

o2

o2

o3

o3

o4

o4

o5

o5

o6

o6

In a CRF, each feature function can depend on o, in addition to a clique in x

Normally, we draw a CRF like this:

HMM

CRF

X1

X2

X3

X4

X5

X6

o1

o2

o3

o4

o5

o6

X1

X2

X3

X4

X5

X6

o1

o2

o3

o4

o5

o6

In a CRF, each feature function can depend on o, in addition to a clique in x

But really, it’s more like this:

This would cause problems for a generative model, but in a conditional model, ois always a fixed constant. So we can still calculate relevant algorithms like Viterbi efficiently.

HMM

CRF

x1=D

x2=N

x3=V

x4=D

x5=A

x6=N

The

cat

chased

the

tiny

fly

An example part including x and o:

f(x,o) = count of [xi = A or D,xi+1=N,o2=cat]

Here, x1 is a D and x2 is a N, plus x5 is a A and x6 is a N, plus o2=cat, so f(x,o) = 2.

Notice that the clique x5-x6 is allowed to depend on o2.

CRF

x1=D

x2=N

x3=V

x4=D

x5=A

x6=N

The

cat

chased

the

tiny

fly

An more usual example including x and o:

f(x,o) = count of [xi = A or D,xi+1=N,oi+1=cat]

Here, x1 is a D and x2 is a N, plus o2=cat, so f(x,o)=1.

CRF

- A CRF model consists of
- P = <p1, …, pk>, a vector of parts
- θ = <θ1, …, θk>, a vector of weights for each part.

- Let O = < o1, …, oT> be an observed sentence
- Let X = <x1, …, xT> be the latent variables.

X1

Xt-1

Xt=hj

Xt+1

o1

ot-1

ot

ot+1

oT

Recursive Computation

Supervised Parameter Estimation

- Given a set of observations o and the correct labels xfor each, determine the best θ:
- Because the CRF equation is just a special form of the maximum entropy equation, we can train it exactly the same way:
- Determine the gradient
- Step in the direction of the gradient
- Repeat until convergence

Training is an optimization problem:

find the value for λ that maximizes the conditional log-likelihood of the training data:

Optimization is normally performed using some form of gradient descent:

0) Initialize λ0 to 0

1) Compute the gradient: ∇CLL

2) Take a step in the direction of the gradient:

λi+1 = λi + α∇CLL

3) Repeat until CLL doesn’t improve:

stop when |CLL(λi+1) – CLL(λi)| < ε

Computing the gradient:

Computing the gradient:

Involves a sum over all possible classes

- In ME models, each document d is classified independently.
- The sum involves as many terms as there are classes c’.
- Very doable.

The hard part for CRFs

- For CRFs, the term
involves an exponential sum.

- The solution again involves dynamic programming, very similar to the Forward algorithm for HMMs.

CRFs vs. HMMs

- HMMs are generative models: That is, they can compute the joint probability
P(sentence, hidden-states)

- From a generative model, one can compute
- Two conditional models:
- P(sentence | hidden-states) and
- P(hidden-states| sentence)

- Marginal models P(sentence) and P(hidden-states)

- Two conditional models:
- For sequence labeling, we want
P(hidden-states | sentence)

- Most often, people are most interested in the conditional probability
P(hidden-states | sentence)

For example, this is the distribution needed for sequence labeling.

- Discriminative(also called conditional) models directly represent the conditional distribution
P(hidden-states | sentence)

- These models cannot tell you the joint distribution, marginals, or other conditionals.
- But they’re quite good at this particular conditional distribution.

It’s possible to convert an HMM into a CRF:

Set pprior,state(x,o) = count[x1=state]

Set θprior,state = log PHMM(x1=state) = log state

Set ptrans,state1,state2(x,o)= count[xi=state1,xi+1=state2]

Set θtrans,state1,state2 = log PHMM(xi+1=state2|xi=state1)

= log Astate1,state2

Set pobs,state,word(x,o)= count[xi=state,oi=word]

Set θobs,state,word = log PHMM(oi=word|xi=state)

= log Bstate,word

If we convert an HMM to a CRF, all of the CRF parameters θ will be logs of probabilities.

- Therefore, they will all be between –∞ and 0
Notice: CRF parameters can be between –∞ and +∞.

So, how do HMMs and CRFs compare in terms of bias and variance (as sequence labelers)?

- HMMs have more bias
- CRFs have more variance

The biggest advantage of CRFs over HMMs is that they can handle overlapping features.

For example, for POS tagging, using words as a features (like xi=“the” or xj=“jogging”) is quite useful.

However, it’s often also useful to use “orthographic” features, like “the word ends in –ing” or “the word starts with a capital letter.”

These features overlap: some words end in “ing”, some don’t.

- Generative models have trouble handling overlapping features correctly
- Discriminative models don’t: they can simply use the features.

CRF Example

A CRF POS Tagger for English

We need to determine the set of possible word types V.

Let V =

{all types in 1 million tokens of Wall Street Journal text, which we’ll use for training}

U

{UNKNOWN} (for word types we haven’t seen)

Standard Penn Treebank tagset