Predicting Sequences: Structured Perceptron. CS 6355: Structured Prediction

Transcription

1 Predicting Sequences: Structured Perceptron CS 6355: Structured Prediction 1

2 Conditional Random Fields summary An undirected graphical model Decompose the score over the structure into a collection of factors Each factor assigns a score to assignment of the random variables it is connected to Training and prediction Final prediction via argmax w T Á(x, y) Train by maximum (regularized) likelihood Connections to other models Effectively a linear classifier A generalization of logistic regression to structures An instance of Markov Random Field, with some random variables observed We will see this soon 2

3 Global features The feature function decomposes over the sequence y 0 y 1 y 2 y 3 x w T Á(x, y 0, y 1 ) w T Á(x, y 1, y 2 ) w T Á(x, y 2, y 3 ) 3

4 Outline Sequence models Hidden Markov models Inference with HMM Learning Conditional Models and Local Classifiers Global models Conditional Random Fields Structured Perceptron for sequences 4

5 HMM is also a linear classifier Consider the HMM 5

6 HMM is also a linear classifier Consider the HMM Log probabilities = transition scores + emission scores 6

7 HMM is also a linear classifier Consider the HMM Define indicators: I z = 1 if z is true; else 0 7

8 HMM is also a linear classifier Consider the HMM Define indicators: I z = 1 if z is true; else 0 8

9 HMM is also a linear classifier Consider the HMM Define indicators: I z = 1 if z is true; else 0 9

10 HMM is also a linear classifier Consider the HMM Indicators: I z = 1 if z is true; else 0 10

11 HMM is also a linear classifier Consider the HMM Indicators: I z = 1 if z is true; else 0 Or equivalently This is a linear function log P terms are the weights; counts and indicators are features Can be written as w T Á(x, y) and add more features 11

12 HMM is also a linear classifier Consider the HMM Indicators: I z = 1 if z is true; else 0 Or equivalently This is a linear function log P terms are the weights; counts and indicators are features Can be written as w T Á(x, y) and add more features 12

13 HMM is a linear classifier: An example Det Noun Verb Det The dog ate the Noun homework 13

14 HMM is a linear classifier: An example Det Noun Verb Det Noun Consider The dog ate the homework 14

15 HMM is a linear classifier: An example Det Noun Verb Det Noun Consider The dog ate the homework Transition scores + Emission scores 15

16 HMM is a linear classifier: An example Det Noun Verb Det Noun Consider The dog ate the homework log P(Det! Noun) 2 + log P(Noun! Verb) 1 + Emission scores + log P(Verb! Det) 1 16

17 HMM is a linear classifier: An example Det Noun Verb Det Noun Consider The dog ate the homework log P(The Det) 1 log P(Det! Noun) 2 + log P(Noun! Verb) 1 + log P(Verb! Det) log P(dog Noun) 1 + log P(ate Verb) 1 + log P(the Det) 1 + log P(homework Noun) 1 17

18 HMM is a linear classifier: An example Det Noun Verb Det Noun Consider The dog ate the homework log P(The Det) 1 log P(Det! Noun) 2 + log P(Noun! Verb) 1 + log P(Verb! Det) log P(dog Noun) 1 + log P(ate Verb) 1 + log P(the Det) 1 w: Parameters of the model + log P(homework Noun) 1 18

19 HMM is a linear classifier: An example Det Noun Verb Det Noun Consider The dog ate the homework log P(The Det) 1 log P(Det! Noun) 2 + log P(Noun! Verb) log P(dog Noun) 1 + log P(ate Verb) 1 Á(x, y): Properties of this output and the input + log P(Verb! Det) 1 + log P(the Det) 1 + log P(homework Noun) 1 19

20 HMM is a linear classifier: An example Det Noun Verb Det Noun Consider The dog ate the homework log P(Det Noun) log P(Noun Verb) log P(Verb Det) log P The Det) log P dog Noun) log P ate Verb) log P the Det log P homework Noun) w: Parameters of the model Á(x, y): Properties of this output and the input 20

21 HMM is a linear classifier: An example Det Noun Verb Det Noun Consider The dog ate the homework log P(Det Noun) log P(Noun Verb) log P(Verb Det) log P The Det) log P dog Noun) log P ate Verb) log P the Det log P homework Noun) log P(x, y) = A linear scoring function = w T Á(x,y) w: Parameters of the model Á(x, y): Properties of this output and the input 21

22 Towards structured Perceptron 1. HMM is a linear classifier Can we treat it as any linear classifier for training? If so, we could add additional features that are global properties As long as the output can be decomposed for easy inference 2. The Viterbi algorithm calculates max w T Á(x, y) Viterbi only cares about scores to structures (not necessarily normalized) 3. We could push the learning algorithm to train for un-normalized scores If we need normalization, we could always normalize by computing exponentiating and dividing by Z That is, the learning algorithm can effectively just focus on the score of y for a particular x Train a discriminative model! 22

23 Towards structured Perceptron 1. HMM is a linear classifier Can we treat it as any linear classifier for training? If so, we could add additional features that are global properties As long as the output can be decomposed for easy inference 2. The Viterbi algorithm calculates max w T Á(x, y) Viterbi only cares about scores to structures (not necessarily normalized) 3. We could push the learning algorithm to train for un-normalized scores If we need normalization, we could always normalize by computing exponentiating and dividing by Z That is, the learning algorithm can effectively just focus on the score of y for a particular x Train a discriminative model! 23

24 Towards structured Perceptron 1. HMM is a linear classifier Can we treat it as any linear classifier for training? If so, we could add additional features that are global properties As long as the output can be decomposed for easy inference 2. The Viterbi algorithm calculates max w T Á(x, y) Viterbi only cares about scores to structures (not necessarily normalized) 3. We could push the learning algorithm to train for un-normalized scores If we need normalization, we could always normalize by exponentiating and dividing by Z That is, the learning algorithm can effectively just focus on the score of y for a particular x Train a discriminative model instead of the generative one! 24

25 Structured Perceptron algorithm Given a training set D = {(x,y)} 1. Initialize w = 0 2 < n 2. For epoch = 1 T: 1. For each training example (x, y) 2 D: 1. Predict Structured y = argmax Perceptron y w T Á(x, update y ) 2. If y y, update w à w + learningrate (Á(x, y) - Á(x, y )) 3. Return w Prediction: argmax y w T Á(x, y) 25

26 Structured Perceptron algorithm Given a training set D = {(x,y)} 1. Initialize w = 0 2 < n 2. For epoch = 1 T: 1. For each training example (x, y) 2 D: 1. Predict y = argmax y w T Á(x, y ) 2. If y y, update w à w + learningrate (Á(x, y) - Á(x, y )) 3. Return w Prediction: argmax y w T Á(x, y) 26

27 Structured Perceptron algorithm Given a training set D = {(x,y)} 1. Initialize w = 0 2 < n 2. For epoch = 1 T: 1. For each training example (x, y) 2 D: 1. Predict y = argmax y w T Á(x, y ) 2. If y y, update w à w + learningrate (Á(x, y) - Á(x, y )) 3. Return w Prediction: argmax y w T Á(x, y) Update only on an error. Perceptron is an mistakedriven algorithm. If there is a mistake, promote y and demote y 27

28 Structured Perceptron algorithm Given a training set D = {(x,y)} 1. Initialize w = 0 2 < n 2. For epoch = 1 T: 1. For each training example (x, y) 2 D: 1. Predict y = argmax y w T Á(x, y ) 2. If y y, update w à w + learningrate (Á(x, y) - Á(x, y )) 3. Return w Prediction: argmax y w T Á(x, y) T is a hyperparameter to the algorithm Update only on an error. Perceptron is an mistakedriven algorithm. If there is a mistake, promote y and demote y 28

29 Structured Perceptron algorithm Given a training set D = {(x,y)} 1. Initialize w = 0 2 < n 2. For epoch = 1 T: 1. For each training example (x, y) 2 D: 1. Predict y = argmax y w T Á(x, y ) 2. If y y, update w à w + learningrate (Á(x, y) - Á(x, y )) 3. Return w Prediction: argmax y w T Á(x, y) In practice, good to shuffle D before the inner loop T is a hyperparameter to the algorithm Update only on an error. Perceptron is an mistakedriven algorithm. If there is a mistake, promote y and demote y 29

30 Structured Perceptron algorithm Given a training set D = {(x,y)} 1. Initialize w = 0 2 < n 2. For epoch = 1 T: 1. For each training example (x, y) 2 D: 1. Predict y = argmax y w T Á(x, y ) 2. If y y, update w à w + learningrate (Á(x, y) - Á(x, y )) 3. Return w Prediction: argmax y w T Á(x, y) In practice, good to shuffle D before the inner loop T is a hyperparameter to the algorithm Inference in training loop! Update only on an error. Perceptron is an mistakedriven algorithm. If there is a mistake, promote y and demote y 30

31 Notes on structured perceptron Mistake bound for separable data, just like perceptron In practice, use averaging for better generalization Initialize a = 0 After each step, whether there is an update or not, a à w + a Note, we still check for mistake using w not a Return a at the end instead of w Exercise: Optimize this for performance modify a only on errors Global update One weight vector for entire sequence Not for each position Same algorithm can be derived via constraint classification Create a binary classification data set and run perceptron 31

32 Structured Perceptron with averaging Given a training set D = {(x,y)} 1. Initialize w = 0 2 < n, a = 0 2 < n 2. For epoch = 1 T: 1. For each training example (x, y) 2 D: 1. Predict y = argmax y w T Á(x, y ) 2. If y y, update w à w + learningrate (Á(x, y) - Á(x, y )) 3. Set a à a + w 3. Return a 32

33 CRF vs. structured perceptron Stochastic gradient descent update for CRF For a training example (x i, y i ) Structured perceptron For a training example (x i, y i ) Expectation vs max Caveat: Adding regularization will change the CRF update, averaging changes the perceptron update 33

34 The lay of the land HMM: A generative model, assigns probabilities to sequences 34

35 The lay of the land HMM: A generative model, assigns probabilities to sequences Two roads diverge 35

36 The lay of the land HMM: A generative model, assigns probabilities to sequences Hidden Markov Models are actually just linear classifiers Don t really care whether we are predicting probabilities. We are assigning scores to a full output for a given input (like multiclass) Generalize algorithms for linear classifiers. Sophisticated models that can use arbitrary features Structured Perceptron Structured SVM Two roads diverge 36

37 The lay of the land HMM: A generative model, assigns probabilities to sequences Two roads diverge Hidden Markov Models are actually just linear classifiers Don t really care whether we are predicting probabilities. We are assigning scores to a full output for a given input (like multiclass) Generalize algorithms for linear classifiers. Sophisticated models that can use arbitrary features Structured Perceptron Structured SVM Model probabilities via exponential functions. Gives us the log-linear representation Log-probabilities for sequences for a given input Learn by maximizing likelihood. Sophisticated models that can use arbitrary features Conditional Random field 37

38 The lay of the land HMM: A generative model, assigns probabilities to sequences Hidden Markov Models are actually just linear classifiers Don t really care whether we are predicting probabilities. We are assigning scores to a full output for a given input (like multiclass) Generalize algorithms for linear classifiers. Sophisticated models that can use arbitrary features Two roads diverge Discriminative/Conditional models Model probabilities via exponential functions. Gives us the log-linear representation Log-probabilities for sequences for a given input Learn by maximizing likelihood. Sophisticated models that can use arbitrary features Structured Perceptron Structured SVM Conditional Random field 38

39 The lay of the land HMM: A generative model, assigns probabilities to sequences Hidden Markov Models are actually just linear classifiers Don t really care whether we are predicting probabilities. We are assigning scores to a full output for a given input (like multiclass) Generalize algorithms for linear classifiers. Sophisticated models that can use arbitrary features Two roads diverge Discriminative/Conditional models Model probabilities via exponential functions. Gives us the log-linear representation Log-probabilities for sequences for a given input Learn by maximizing likelihood. Sophisticated models that can use arbitrary features Structured Perceptron Structured SVM Conditional Random field Applicable beyond sequences Eventually, similar objective minimized with different loss functions 39

40 The lay of the land HMM: A generative model, assigns probabilities to sequences Hidden Markov Models are actually just linear classifiers Don t really care whether we are predicting probabilities. We are assigning scores to a full output for a given input (like multiclass) Generalize algorithms for linear classifiers. Sophisticated models that can use arbitrary features Two roads diverge Discriminative/Conditional models Model probabilities via exponential functions. Gives us the log-linear representation Log-probabilities for sequences for a given input Learn by maximizing likelihood. Sophisticated models that can use arbitrary features Structured Perceptron Structured SVM Conditional Random field Coming soon Applicable beyond sequences Eventually, similar objective minimized with different loss functions 40

41 Sequence models: Summary Goal: Predict an output sequence given input sequence Hidden Markov Model Inference Predict via Viterbi algorithm Conditional models/discriminative models Local approaches (no inference during training) MEMM, conditional Markov model Global approaches (inference during training) CRF, structured perceptron Prediction is not always tractable for general structures Same dichotomy for more general structures To think What are the parts in a sequence model? How is each model scoring these parts? 41