Machine Learning for Structured Prediction

Transcription

1 Machine Learning for Structured Prediction Grzegorz Chrupa la National Centre for Language Technology School of Computing Dublin City University NCLT Seminar Grzegorz Chrupa la (DCU) Machine Learning for Structured Prediction NCLT Seminar / 19

2 Structured vs Non-structured prediction In supervised learning we try to learn a function h : X Y where x X are inputs and y Y are outputs. Binary classification: Y = { 1, +1} Multiclass classification: Y = {1,...,K} (finite set of labels) Regression: Y = R In contrast, in structured prediction elements Y are complex The prediction is based on the feature function Φ : X F where usually F = R D (D-dimensional vector space) Grzegorz Chrupa la (DCU) Machine Learning for Structured Prediction NCLT Seminar / 19

3 Structured prediction tasks Sequence labeling: e.g. POS tagging Parsing: given an input sequence, build a tree whose yield (leaves) are the elements in the sequence and whose structure obeys some grammar. Collective classification: E.g. relation learning problems, such as labeling web pages given link information. Bipartite matching: given a bipartite graph, find the best possible matching. e.g. word alignment in NLP and protein structure prediction in computational biology. Grzegorz Chrupa la (DCU) Machine Learning for Structured Prediction NCLT Seminar / 19

4 Linear classifiers In binary classification, linear classifiers separate classes in a multidimensional input space with a hyperplane The classification function giving the output score is where y = f ( x w + b) = f ( d D w dx d + b) x is the input feature vector = Φ(x), w is a real vector of feature weights, b is the bias term, and f maps the dot product of the two vectors to output (e.g. f (x) = 1 if x < 0, +1 otherwise) The use of kernels permits to use linear classifiers for non-linearly separable problems such as the XOR function Grzegorz Chrupa la (DCU) Machine Learning for Structured Prediction NCLT Seminar / 19

5 Linear classifier hyperplanes Figure: Three separating hyperplanes in 2-dimensional space (Wikipedia) Grzegorz Chrupa la (DCU) Machine Learning for Structured Prediction NCLT Seminar / 19

6 Perceptron The Perceptron iterates I times through the 1...n training examples, updating the parameters (weights and bias) if the current example n is classified incorrectly: w w + y n x n b b + yn where y n is the output for the n th training example Averaged Perceptron, which generalizes better to unseen examples, does weight averaging, i.e. the final weights returned are the average of all weight vectors used during the run of the algorithm Grzegorz Chrupa la (DCU) Machine Learning for Structured Prediction NCLT Seminar / 19

7 Perceptron Algorithm Daume III (2006) Grzegorz Chrupa la (DCU) Machine Learning for Structured Prediction NCLT Seminar / 19

8 Structured Perceptron For structured prediction the feature function is Φ : X Y F (second argument is the hypothesized output) Structured prediction algorithms need Φ to admit tractable search The argmax problem: ŷ = argmax y Y w Φ(x,y) Structured Perceptron: for each example n wherever the predicted ŷ n for x n differs from y n the weights are updated: w w + Φ(x n, y n ) Φ(x n, ŷ n ) Similar to plain Perceptron key difference: ŷ is calculated using the arg max. Incremental Perceptron is a variant used in cased where arg max isn t available a beam search algorithm is used instead Grzegorz Chrupa la (DCU) Machine Learning for Structured Prediction NCLT Seminar / 19

9 Logistic regression / Maximum Entropy Conditional probability of class y is proportional to exp f (x) p(y x; w,b) = 1 Z x; w,b exp [y( w Φ(x) + b)] = exp [ 2y( w Φ(x) + b)] For training we try to find w and b which maximize the likelihood of training data Gaussian prior used to penalize large weights Generalized Iterative Scaling algorithm used to solve the maximization problem Grzegorz Chrupa la (DCU) Machine Learning for Structured Prediction NCLT Seminar / 19

10 Maximum Entropy Markov Models MEMMs are an extension of MaxEnt to sequence labeling Replace p(observation state) with p(state observation) For first-order MEMM, the conditional distribution on the label y n given full input x, the previous label y n 1, feature function Φ and weights w is: p(y n x,y n 1 ; w ) = 1 Z x;yn 1 w exp [ w Φ(x,y n,y n 1 )] When MEMM is trained true label y n 1 is used At prediction time, Viterbi is applied to solve arg max Predicted y n 1 label is used in classifying n th element of the sequence Grzegorz Chrupa la (DCU) Machine Learning for Structured Prediction NCLT Seminar / 19

11 Conditional Random Fields Alternative generalization of MaxEnt to structured prediction Main difference: while MEMM uses per state exponential models for the conditional probabilities of next states given the current state CRF has a single exponential model for the joint probability of the whole sequence of labels The formulation is: p(y x; w ) = 1 Z x; w exp [ w Φ(x,y)] Z x; w = y Y exp [ w Φ(x,y )] where elements y Y are incorrect outputs Grzegorz Chrupa la (DCU) Machine Learning for Structured Prediction NCLT Seminar / 19

12 CRF - experimental comparison to HMM and MEMMs J. Lafferty, A. McCallum, F. Pereira - Proc. 18th International Conf. on Machine Learning, 2001 Grzegorz Chrupa la (DCU) Machine Learning for Structured Prediction NCLT Seminar / 19

13 Large Margin Models Choose parameter settings which maximize the distance between the hyperplane separating classes and the nearest data points on either side. Support Vector Machines. For data separable with margin 1: subject to minimize w,b 1 2 w 2 n, y n [ w Φ(x n ) + b] 1 Grzegorz Chrupa la (DCU) Machine Learning for Structured Prediction NCLT Seminar / 19

14 SVM Soft-margin formulation: don t enforce perfect classification of training examples The slack variables ξ measures how far an example is from the correct hard margin constraint subject to minimize w,b 1 2 w 2 + C N n=1 ξ n n, y n [ w Φ(x n ) + b] 1 ξ n ξ n 0 The parameter C trades off fitting the training data and a small weight vector Grzegorz Chrupa la (DCU) Machine Learning for Structured Prediction NCLT Seminar / 19

15 Margins and Support Vectors Figure: SVM maximum-margin hyperplanes for binary classification. Examples along the hyperplanes are support vectors (from Wikipedia) Grzegorz Chrupa la (DCU) Machine Learning for Structured Prediction NCLT Seminar / 19

16 Maximum Margin models for Structured Prediction Maximum Margin Markov Networks (M 3 N) The difference in score (under loss function l) between true output y and any incorrect output ŷ is at least l(x,y,ŷ) M 3 N scales the margin to be proportional to loss. Support Vector Machines for Interdependent and Structured Outputs: SVM struct Instead of scaling the margin, scale the slack variables by the loss The two approaches differ in the optimization techniques used (SVM struct constrains the loss function less but is more difficult to optimize). Grzegorz Chrupa la (DCU) Machine Learning for Structured Prediction NCLT Seminar / 19

17 Summary of structured prediction algorithms Figure from Daume III (2006) Grzegorz Chrupa la (DCU) Machine Learning for Structured Prediction NCLT Seminar / 19

18 Other approaches to structured outputs Searn (Search + Learn) Analogical learning Miscellaneous hacks: Reranking Stacking Class n-grams Grzegorz Chrupa la (DCU) Machine Learning for Structured Prediction NCLT Seminar / 19

19 References M. Collins, 2002, Discriminative Training Methods for Hidden Markov Models: Theory and Experiments with Perceptron Algorithms, EMNLP-02 McCallum, Andrew, Dayne Freitag, and Fernando Pereira Maximum entropy Markov models for information extraction and segmentation. ICML J. Lafferty, A. McCallum, and F. Pereira Conditional random elds: Probabilistic models for segmenting and labeling sequence data. ICML B. Taskar, C. Guestrin, and D. Koller Max-margin Markov networks. In Advances in Neural Information Processing Systems (NIPS). I. Tsochantaridis, T. Hofmann, T. Joachims, and Y. Altun Large margin methods for structured and interdependent output variables. JMLR Hal Daumé III, 2006, Practical Structured Learning Techniques for Natural Language Processing, PhD thesis Grzegorz Chrupa la (DCU) Machine Learning for Structured Prediction NCLT Seminar / 19