![Machine Learning Srihari
Discriminative Classifier: Logistic Regression
Logistic Sigmoid
• Feature vector x
• Two-class classification: class variable σ(a)
y has values C1 and C2
a
• A posteriori probability p(C1|x) written Properties:
as A. Symmetry
p(C1|x) =f(x) = σ (wTx) where σ(-a)=1-σ(a)
1 B. Inverse
σ (a) = a=ln(σ /1-σ)
1 + exp(−a )
known as logit.
• It is known as logistic regression in Also known as
statistics log odds since
it is the ratio
– Although it is a model for classification
ln[p(C1|x)/p(C2|x)]
rather than for regression
C. Derivative 20
dσ/da=σ(1-σ)](https://image.slidesharecdn.com/machine-learning-generative-and-discriminative-models501/85/Machine-Learning-Generative-and-Discriminative-Models-20-320.jpg)
Machine Learning: Generative and Discriminative Models
- 1. Machine Learning: Generative and Discriminative Models Sargur N. Srihari srihari@cedar.buffalo.edu Machine Learning Course: http://www.cedar.buffalo.edu/~srihari/CSE574/index.html
- 2. Machine Learning Srihari Outline of Presentation 1. What is Machine Learning? ML applications, ML as Search 2. Generative and Discriminative Taxonomy 3. Generative-Discriminative Pairs Classifiers: Naïve Bayes and Logistic Regression Sequential Data: HMMs and CRFs 4. Performance Comparison in Sequential Applications NLP: Table extraction, POS tagging, Shallow parsing, Handwritten word recognition, Document analysis 5. Advantages, disadvantages 6. Summary 7. References 2
- 3. Machine Learning Srihari 1. Machine Learning • Programming computers to use example data or past experience • Well-Posed Learning Problems – A computer program is said to learn from experience E – with respect to class of tasks T and performance measure P, – if its performance at tasks T, as measured by P, improves with experience E. 3
- 4. Machine Learning Srihari Problems Too Difficult To Program by Hand • Learning to drive an autonomous vehicle – Train computer-controlled vehicles to steer correctly – Drive at 70 mph for 90 miles on public highways – Associate steering commands with image sequences Task T: driving on public, 4-lane highway using vision sensors Perform measure P: average distance traveled before error (as judged by human overseer) Training E: sequence of images and steering commands recorded 4 while observing a human driver
- 5. Machine Learning Srihari Example Problem: Handwritten Digit Recognition • Handcrafted rules will result in large no of rules and exceptions • Better to have a machine that learns from a large training Wide variability of same numeral set 5
- 6. Machine Learning Srihari Other Applications of Machine Learning • Recognizing spoken words – Speaker-specific strategies for recognizing phonemes and words from speech – Neural networks and methods for learning HMMs for customizing to individual speakers, vocabularies and microphone characteristics • Search engines – Information extraction from text • Data mining – Very large databases to learn general regularities implicit in data – Classify celestial objects from image data – Decision tree for objects in sky survey: 3 terabytes 6
- 7. Machine Learning Srihari ML as Searching Hypotheses Space • Very large space of possible hypotheses to fit: – observed data and – any prior knowledge held by the observer Method Hypothesis Space Concept Learning Boolean Expressions Decision Trees All Possible Trees Neural Networks Weight Space 7
- 8. Machine Learning Srihari ML Methodologies are increasingly statistical • Rule-based expert systems being replaced by probabilistic generative models • Example: Autonomous agents in AI – ELIZA : natural language rules to emulate therapy session – Manual specification of models, theories are increasingly difficult • Greater availability of data and computational power to migrate away from rule-based and manually specified models to probabilistic data-driven modes 8
- 9. Machine Learning Srihari The Statistical ML Approach 1. Data Collection Large sample of data of how humans perform the task 2. Model Selection Settle on a parametric statistical model of the process 3. Parameter Estimation Calculate parameter values by inspecting the data Using learned model perform: 4. Search Find optimal solution to given problem 9
- 10. Machine Learning Srihari 2. Generative and Discriminative Models: An analogy • The task is to determine the language that someone is speaking • Generative approach: – is to learn each language and determine as to which language the speech belongs to • Discriminative approach: – is determine the linguistic differences without learning any language– a much easier task! 10
- 11. Machine Learning Srihari Taxonomy of ML Models • Generative Methods – Model class-conditional pdfs and prior probabilities – “Generative” since sampling can generate synthetic data points – Popular models • Gaussians, Naïve Bayes, Mixtures of multinomials • Mixtures of Gaussians, Mixtures of experts, Hidden Markov Models (HMM) • Sigmoidal belief networks, Bayesian networks, Markov random fields • Discriminative Methods – Directly estimate posterior probabilities – No attempt to model underlying probability distributions – Focus computational resources on given task– better performance – Popular models • Logistic regression, SVMs • Traditional neural networks, Nearest neighbor • Conditional Random Fields (CRF) 11
- 12. Generative Models (graphical) Parent node Quick Medical Markov Random selects between Reference -DT Field components Diagnosing Diseases from Symptoms
- 13. Machine Learning Srihari Successes of Generative Methods • NLP – Traditional rule-based or Boolean logic systems (eg Dialog and Lexis-Nexis) are giving way to statistical approaches (Markov models and stochastic context free grammars) • Medical Diagnosis – QMR knowledge base, initially a heuristic expert systems for reasoning about diseases and symptoms has been augmented with decision theoretic formulation • Genomics and Bioinformatics – Sequences represented as generative HMMs 13
- 14. Machine Learning Srihari Discriminative Classifier: SVM (x1, x2) (x1, x2, x1x2) Nonlinear decision boundary Linear boundary in high-dimensional space 14
- 15. Machine Learning Srihari Support Vector Machines Three support vectors are • Support vectors are those shown as solid dots nearest patterns at distance b from hyperplane • SVM finds hyperplane with maximum distance from nearest training patterns • For full description of SVMs see http://www.cedar.buffalo.edu/ ~srihari/CSE555/SVMs.pdf 15
- 16. Machine Learning Srihari 3. Generative-Discriminative Pairs • Naïve Bayes and Logistic Regression form a generative-discriminative pair for classification • Their relationship mirrors that between HMMs and linear-chain CRFs for sequential data 16
- 17. Machine Learning Srihari Graphical Model Relationship Hidden Markov Model GENERATIVE Naïve Bayes Classifier y Y y1 yN SEQUENCE x X p(Y,X) p(y,x) x1 xM x1 xN CONDITION CONDITION p(y/x) p(Y/X) DISCRIMINATIVE SEQUENCE Logistic Regression Conditional Random Field 17
- 18. Machine Learning Srihari Generative Classifier: Bayes • Given variables x =(x1,..,xM) and class variable y • Joint pdf is p(x,y) – Called generative model since we can generate more samples artificially • Given a full joint pdf we can – Marginalize p( y ) = ∑ p(x, y ) x p (x, y ) p ( y | x) = – Condition p (x) – By conditioning the joint pdf we form a classifier • Computational problem: – If x is binary then we need 2M values – If 100 samples are needed to estimate a given probability, M=10, and there are two classes then we need 2048 samples 18
- 19. Machine Learning Srihari Naïve Bayes Classifier • Goal is to predict single class variable y given a vector of features x=(x1,..,xM) • Assume that once class labels are known the features are independent • Joint probability model has the form M p ( y, x) = p ( y )∏ p( xm | y ) m =1 – Need to estimate only M probabilities • Factor graph obtained by defining factors ψ(y)=p(y), ψm(y,xm)=p(xm,y) 19
- 20. Machine Learning Srihari Discriminative Classifier: Logistic Regression Logistic Sigmoid • Feature vector x • Two-class classification: class variable σ(a) y has values C1 and C2 a • A posteriori probability p(C1|x) written Properties: as A. Symmetry p(C1|x) =f(x) = σ (wTx) where σ(-a)=1-σ(a) 1 B. Inverse σ (a) = a=ln(σ /1-σ) 1 + exp(−a ) known as logit. • It is known as logistic regression in Also known as statistics log odds since it is the ratio – Although it is a model for classification ln[p(C1|x)/p(C2|x)] rather than for regression C. Derivative 20 dσ/da=σ(1-σ)
- 21. Machine Learning Srihari Logistic Regression versus Generative Bayes Classifier • Posterior probability of class variable y is p (x | C1 ) p (C1 ) p (C1 | x) = p (x | C1 ) p (C1 ) + p (x | C2 ) p (C2 ) 1 p (x | C1 ) p (C1 ) = = σ (a ) where a = ln 1 + exp(−a ) p (x | C2 ) p(C2 ) • In a generative model we estimate the class- conditionals (which are used to determine a) • In the discriminative approach we directly estimate a as a linear function of x i.e., a = wTx 21
- 22. Machine Learning Srihari Logistic Regression Parameters • For M-dimensional feature space logistic regression has M parameters w=(w1,..,wM) • By contrast, generative approach – by fitting Gaussian class-conditional densities will result in 2M parameters for means, M(M+1)/2 parameters for shared covariance matrix, and one for class prior p(C1) – Which can be reduced to O(M) parameters by assuming independence via Naïve Bayes 22
- 23. Machine Learning Srihari Multi-class Logistic Regression p ( x | Ck ) p(Ck ) p(Ck | x) = • Case of K>2 classes ∑ p( x | C j ) p(C j ) j exp(ak ) = ∑ exp(a j ) j • Known as normalized exponential where ak=ln p(x|Ck)p(Ck) • Normalized exponential also known as softmax since if ak>>aj then p(Ck|x)=1 and p(Cj|x)=0 • In logistic regression we assume activations given by ak=wkTx 23
- 24. Machine Learning Srihari Graphical Model for Logistic Regression • Multiclass logistic regression can be written as p( y | x) = 1 exp ⎧λ + ∑ λ x ⎫ where ⎨ ⎬ y K yj j Z (x) ⎩ j =1 ⎭ ⎧ K ⎫ Z (x) = ∑ y exp ⎨λ y + ∑ λ yj x j ⎬ ⎩ j =1 ⎭ • Rather than using one weight per class we can define feature functions that are nonzero only for a single class 1 ⎧K ⎫ p ( y | x) = exp ⎨∑ λk f k ( y, x) ⎬ Z (x) ⎩ k =1 ⎭ • This notation mirrors the usual notation for CRFs 24
- 25. Machine Learning Srihari 4. Sequence Models • Classifiers predict only a single class variable • Graphical Models are best to model many variables that are interdependent • Given sequence of observations X={xn}n=1N • Underlying sequence of states Y={yn}n=1N 25
- 26. Machine Learning Srihari Generative Model: HMM • X is observed data sequence to be labeled, y1 y2 yn yN Y is the random variable over the label sequences • HMM is a distribution that models x1 x2 xn xN p(Y, X) N • Joint distribution is p( Y,X ) = ∏ p ( yn | yn −1 ) p(x n | yn ) n =1 • Highly structured network indicates conditional independences, – past states independent of future states – Conditional independence of observed given its state. 26
- 27. Machine Learning Srihari Discriminative Model for Sequential Data • CRF models the conditional distribution p(Y/X) • CRF is a random field globally y1 y2 yn yN conditioned on the observation X • The conditional distribution X p(Y|X) that follows from the joint distribution p(Y,X) can be rewritten as a Markov Random Field 27
- 28. Machine Learning Srihari Markov Random Field (MRF) • Also called undirected graphical model • Joint distribution of set of variables x is defined by an undirected graph as 1 p (x) = Z ∏ψ C C (x C ) where C is a maximal clique (each node connected to every other node), xC is the set of variables in that clique, ψC is a potential function (or local or compatibility function) such that ψC(xC) > 0, typically ψC(xC) = exp{-E(xC)}, and Z = ∑ ∏ψ C (x C ) is the partition function for normalization x C • Model refers to a family of distributions and Field refers to a specific one 28
- 29. Machine Learning Srihari MRF with Input-Output Variables • X is a set of input variables that are observed – Element of X is denoted x • Y is a set of output variables that we predict – Element of Y is denoted y • A are subsets of X U Y – Elements of A that are in A ^ X are denoted xA – Element of A that are in A ^ Y are denoted yA • Then undirected graphical model has the form 1 p (x,y) = ∏ Ψ A (x A , y A ) where Z= ∑ ∏ Ψ A (x A , y A ) Z A x,y A 29
- 30. Machine Learning Srihari MRF Local Function • Assume each local function has the form ⎧ ⎫ Ψ A (x A , y A ) = exp ⎨∑ θ Am f Am (x A , y A ) ⎬ ⎩m ⎭ where θA is a parameter vector, fA are feature functions and m=1,..M are feature subscripts 30
- 31. Machine Learning Srihari From HMM to CRF • In an HMM Indicator function: N 1{x = x’} takes value 1when p( Y,X ) = ∏ p ( yn | yn −1 ) p(x n | yn ) n =1 x=x’ and 0 otherwise • Can be rewritten as Parameters of 1 ⎧ ⎫ the distribution: p ( Y, X) = exp ⎨∑ ∑ λij 1{ yn =i}1{ yn−1 = j} + ∑∑∑ μoi 1{ yn =i}1{ xn =o} ⎬ θ ={λij,μoi} Z ⎩ n i , j∈S n i∈S o∈O ⎭ • Further rewritten as Feature Functions have 1 ⎧ M ⎫ p (Y, X) = exp ⎨∑ λm f m ( yn , yn −1 , x n ) ⎬ the form fm(yn,yn-1,xn): Z ⎩ m =1 ⎭ Need one feature for each state transition (i,j) • Which gives us fij(y,y’,x)=1{y=i}1{y’=j} and ⎧M ⎫ exp ⎨∑ λm f m ( yn , yn −1 , x n ) ⎬ one for each state- p ( Y | X) = p( y, x) = ⎩ m =1 ⎭ observation pair ∑ y ' p( y ', x) ∑ exp ⎧∑ λm f m ( yn , yn−1 , xn ) ⎫ ⎨ M ⎬ fio(y,y’,x)=1{y=i}1{x=o} ⎩ m =1 ⎭ y' 31 • Note that Z cancels out
- 32. Machine Learning Srihari CRF definition • A linear chain CRF is a distribution p(Y|X) that takes the form 1 ⎧M ⎫ p ( Y | X) = exp ⎨∑ λm f m ( yn , yn −1 , x n ) ⎬ Z (X) ⎩ m =1 ⎭ • Where Z(X) is an instance specific normalization function ⎧M ⎫ Z (X) = ∑ exp ⎨∑ λm f m ( yn , yn −1 , x n ) ⎬ y ⎩ m =1 ⎭ 32
- 33. Machine Learning Srihari Functional Models Naïve Bayes Classifier Hidden Markov Model y Y y1 yn yN GENERATIVE x X x1 xM x1 xn xN M p(y, x ) = p(y)∏ p ( xm | y ) N p( Y,X ) = ∏ p( yn | yn −1 ) p(x n | yn ) m =1 n =1 ⎧M ⎫ ⎧ M ⎫ exp ⎨∑ λm f m ( yn , yn −1 , x n ) ⎬ exp ⎨∑ λm f m ( y, x) ⎬ p ( Y | X) = ⎩ m =1 ⎭ p ( y | x) = ⎩ m =1 ⎭ ⎧ M ⎫ ⎧ M ⎫ ∑ y ' exp ⎨∑1 λm f m ( yn ', yn−1 ', xn ) ⎬ ∑ y ' exp ⎨∑1 λm f m ( y ', x) ⎬ ⎩ m= ⎭ DISCRIMINATIVE ⎩ m= ⎭ Logistic Regression Conditional Random Field 33
- 34. Machine Learning Srihari NLP: Part Of Speech Tagging For a sequence of words w = {w1,w2,..wn} find syntactic labels s for each word: w = The quick brown fox jumped over the lazy dog s = DET VERB ADJ NOUN-S VERB-P PREP DET ADJ NOUN-S Baseline is already 90% Model Error • Tag every word with its most frequent tag HMM 5.69% • Tag unknown words as nouns CRF 5.55% Per-word error rates for POS tagging on the Penn treebank 34
- 35. Machine Learning Srihari Table Extraction To label lines of text document: Whether part of table and its role in table. Finding tables and extracting information is necessary component of data mining, question-answering and IR tasks. HMM CRF 89.7% 99.9% 35
- 36. Machine Learning Srihari Shallow Parsing • Precursor to full parsing or information extraction – Identifies non-recursive cores of various phrase types in text • Input: words in a sentence annotated automatically with POS tags • Task: label each word with a label indicating – word is outside a chunk (O), starts a chunk (B), continues a chunk (I) NP chunks CRFs beat all reported single-model NP chunking results on standard evaluation dataset 36
- 37. Machine Learning Srihari Handwritten Word Recognition Given word image and lexicon, find most probable lexical entry Algorithm Outline • Oversegment image segment combinations are potential characters • Given y = a word in lexicon, s = grouping of segments, x = input word image features • Find word in lexicon and segment grouping that maximizes P(y,s | x), CRF Model eψ ( y , x ;θ ) m ⎛ ⎞ P( y | x,θ ) = ψ ( y, x;θ ) = ∑ ⎜ A( j, y j , x;θ s + ∑ I ( j , k , y j , yk , x,θ t ) ⎟ ∑y' eψ ( y ', x ;θ ) ⎜ j =1 ⎝ ( j , k )∈E ⎟ ⎠ where yi ε (a-z,A-Z,0-9}, θ : model parameters Association Potential (state term) 1 0.98 SDP CRF A( j , y j , x;θ s ) = ∑ ( f i s ( j , y j , x ) ⋅ θ ijs ) CRF 0.96 0.94 Precision i 0.92 Segment-DP Precision 0.9 Interaction Potential 0.88 0.86 I ( j , k , y j , y k , x; θ ) = ∑ ( f i ( j , k , y j , y k , x ) ⋅ θ ) 0.84 t t t ijk 0.82 0.8 37 0 20 40 60 80 100 120 Word Recognition Rank i WR Rank
- 38. Machine Learning Srihari Document Analysis (labeling regions) error rates CRF Neural Naive Network Bayes Machine 1.64% 2.35% 11.54% Printed Text Handwritten 5.19% 20.90% 25.04% Text Noise 10.20% 15.00% 12.23% Total 4.25% 7.04% 12.58% 38
- 39. Machine Learning Srihari 5. Advantage of CRF over Other Models • Other Generative Models – Relax assuming conditional independence of observed data given the labels – Can contain arbitrary feature functions • Each feature function can use entire input data sequence. Probability of label at observed data segment may depend on any past or future data segments. • Other Discriminative Models – Avoid limitation of other discriminative Markov models biased towards states with few successor states. – Single exponential model for joint probability of entire sequence of labels given observed sequence. – Each factor depends only on previous label, and not future labels. P(y | x) = product of factors, one for each label. 39
- 40. Machine Learning Srihari Disadvantages of Discriminative Classifiers • Lack elegance of generative – Priors, structure, uncertainty • Alternative notions of penalty functions, regularization, kernel functions • Feel like black-boxes – Relationships between variables are not explicit and visualizable 40
- 41. Machine Learning Srihari Bridging Generative and Discriminative • Can performance of SVMs be combined elegantly with flexible Bayesian statistics? • Maximum Entropy Discrimination marries both methods – Solve over a distribution of parameters (a distribution over solutions) 41
- 42. Machine Learning Srihari 6. Summary • Machine learning algorithms have great practical value in a variety of application domains – A well-defined learning problem requires a well-specified task, performance metric, and source of experience • Generative and Discriminative methods are two-broad approaches: – former involve modeling, latter directly solve classification • Generative and Discriminative Method Pairs – Naïve Bayes and Logistic Regression are a corresponding pair for classification – HMM and CRF are a corresponding pair for sequential data • CRF performs better in language related tasks • Generative models are more elegant, have explanatory power 42
- 43. Machine Learning Srihari 7. References 1. T. Mitchell, Machine Learning, McGraw-Hill, 1997 2. C. Bishop, Pattern Recognition and Machine Learning, Springer, 2006 3. T. Jebarra, Machine Learning: Discriminative and Generative, Kluwer, 2004 4. R.O. Duda, P.E. Hart and D. Stork, Pattern Classification, 2nd Ed, Wiley 2002 5. C. Sutton and A. McCallum, An Introduction to Conditional Random Fields for Relational Learning 6. S. Shetty, H. Srinivasan and S. N. Srihari, Handwritten Word Recognition using CRFs, ICDAR 2007 7. S. Shetty, H.Srinivasan and S. N. Srihari, Segmentation and Labeling of Documents using CRFs, SPIE-DRR 2007 43