Conditional Random Fields

1. Conditional Random Fields

2. Sequence Labeling: The Problem • 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 .

3. Sequence Labeling: The Problem • 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

4. Sequence Labeling: The Problem • 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

5. The CRF Equation • 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!

6. CRF Equation, standard format • Note that the denominator depends on O, but not on y (it’s marginalizing over y). • Typically, we write • where

7. Making Structured Predictions

8. Structured prediction vs. Text Classification Recall: max. ent. for text classification: CRFs for sequence labeling: What’s the difference?

9. Structured prediction vs. Text Classification 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.

10. Finding the Best Sequence 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.

11. Viterbi Algorithm 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

12. o1 ot-1 ot ot+1 oT Viterbi Algorithm x1 xt-1 xt xt+1 xT Compute the most likely state sequence by working backwards

13. Viterbi Algorithm X1 Xt-1 Xt=hj Xt+1 o1 ot-1 ot ot+1 oT ??! Recursive Computation ??!

14. Feature functions and Graph parts 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.

15. X1 X2 X3 X4 X5 X6 o1 o2 o3 o4 o5 o6 Clique Example The cliques in a linear chain CRF are the set of individual nodes, and the set of pairs of consecutive nodes. CRF

16. X1 X2 X3 X4 X5 X6 o1 o2 o3 o4 o5 o6 Clique Example 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

17. X1 X2 X3 X4 X5 X6 o1 o2 o3 o4 o5 o6 Clique Example 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

18. X1 X2 X3 X4 X5 X6 o1 o2 o3 o4 o5 o6 Clique Example For non-linear-chain CRFs (something we won’t normally consider in this class), you can get larger cliques: X5’ CRF Larger cliques

19. x1=D x2=N x3=V x4=D x5=A x6=N o1 o2 o3 o4 o5 o6 Graph part as Feature Function Example 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

20. x1=D x2=N x3=V x4=D x5=A x6=N o1 o2 o3 o4 o5 o6 Graph part as Feature Function Example 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

21. X1 X1 X2 X2 X3 X3 X4 X4 X5 X5 X6 X6 o1 o1 o2 o2 o3 o3 o4 o4 o5 o5 o6 o6 Features can depend on the whole observation 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

22. X1 X2 X3 X4 X5 X6 o1 o2 o3 o4 o5 o6 X 1 X2 X3 X4 X5 X6 o1 o2 o3 o4 o5 o6 Features can depend on the whole observation 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

23. x1=D x2=N x3=V x4=D x5=A x6=N The cat chased the tiny fly Graph part as Feature Function Example 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

24. x1=D x2=N x3=V x4=D x5=A x6=N The cat chased the tiny fly Graph part as Feature Function Example 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

25. The CRF Equation, with Parts • 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.

26. Viterbi Algorithm – 2nd Try X1 Xt-1 Xt=hj Xt+1 o1 ot-1 ot ot+1 oT Recursive Computation

27. Supervised Parameter Estimation

28. Conditional Training • 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

29. Recall: Training a ME model Training is an optimization problem: find the value for λ that maximizes the conditional log-likelihood of the training data:

30. Recall: Training a ME model 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

31. Recall: Training a ME model Computing the gradient:

32. Recall: Training a ME model Computing the gradient: Involves a sum over all possible classes

33. Recall: Training a ME model:Expected feature counts • In ME models, each document d is classified independently. • The sum involves as many terms as there are classes c’. • Very doable.

34. Training a CRF The hard part for CRFs

35. Training a CRF: Expected feature counts • For CRFs, the term involves an exponential sum. • The solution again involves dynamic programming, very similar to the Forward algorithm for HMMs.

36. CRFs vs. HMMs

37. Generative (Joint Probability) Models • 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) • For sequence labeling, we want P(hidden-states | sentence)

38. Discriminative (Conditional) Models • 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.

39. Discriminative vs. Generative

40. CRFs vs. HMMs, a closer look 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

41. CRF vs. HMM, a closer look 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

42. Comparing feature functions 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.

43. CRF Example A CRF POS Tagger for English

44. Vocabulary 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)

45. L = Label Set Standard Penn Treebank tagset

46. L = Label Set

47. CRF Features