A TUTORIAL ON CONDITIONAL RANDOM FIELDS WITH APPLICATIONS TO MUSIC ANALYSIS
Slim Essid
Telecom ParisTech
November 2013
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 1 / 122 Slides available online
http://www.telecom-paristech.fr/~essid/resources.htm
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 2 / 122 Contents
I Introduction
I References to applications in MIR-related tasks
I The logistic regression model
I Conditional Random Fields (for linear-chain data)
I Improvements and extensions to original CRFs
I Conclusion
I References
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 3 / 122 I Appendix Introduction Preliminaries The supervised classification problem
Goal: predict labels y (aka classes or outputs) for some observations o (aka data points, inputs). Examples: • Predict genre, mood, user tag... for a music excerpt. • Predict instrument, chord, notes played... for a music segment. Supervised classification: • Each observation o is supposed to pertain to a predefined class Ck : the k-th (discrete) class of a classification problem; k = 1, ··· , K. • This is represented using a label y for each o; y ∈ Y, e.g. Y = {0, 1}, Y = {1, 2, 3, ..., K}. Examples of classes: Tags: Jazz, Bebop, Fast, Exciting, Sax, Drums, ... Chords: C7, Gmaj7, F min7, ... Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 4 / 122 Introduction Preliminaries Features
• Classification relies on features x: descriptors of some qualities/ attributes of the inputs o. Two types of features: Continuous features Discrete/categorical
−7 −7 −7 −7 o1 o2 o3 o4 C C F F
x1 x2 x3 x4 CCFF real-valued: e.g. MFCC, chroma, symbolic: e.g. note/chord played, tempo... key... • Usually assembled as feature vectors x.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 5 / 122 Introduction Preliminaries Notations
• o: an input (observation) to be classified; e.g.: a music excerpt, a music symbol, an audio frame/segment...
T D • x = (x1, ··· , xD ) : a D-dimensional column vector (usually in R ); xT is a row vector.
• xn is a feature vector among a collection of N examples x1, ··· , xN .
• xjn is the j-th feature coefficient of xn ; 1 ≤ j ≤ D.
• D ={x1, ..., xN } : the set of all training feature-vector examples.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 6 / 122 Introduction Preliminaries Feature functions Different from features!
Definition A feature function is a real-valued function of both the input space O (observations) and the output space Y (target labels), fj : O × Y → R, that can be used to compute characteristics of the observations.
• An alternative way to express the characteristics of the observations, in a more flexible manner:
− using output-specific features; − describing the context.
( 1 if oi = C, oi+1 = E and y = Cmaj Example: fj (oi , yi ) = 0 otherwise
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 7 / 122 Introduction Preliminaries Feature functions
I Remarks: − Different attributes may thus be considered for different classes.
− Feature functions are more general than features: one can define
∆ I fj (o, y) = xj ; or ∆ I fj (o, y) = x.
− In the following:
I Feature-function notations will be used only when needed.
I Otherwise, feature-vectors will be preferred.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 8 / 122 Introduction Preliminaries Probabilistic classification
Take decisions based on the MAP rule: yˆ = argmax p(y|x) y∈Y
in order to minimize the error rate (here the expected 0-1 loss). MAP: Maximum A Posteriori probability
→ this is the Bayes decision rule (for the 0-1 loss.)
How to get there?
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 9 / 122 Introduction Preliminaries Generative model based classification
• Objective: yˆ = argmaxy p(y|x). p(y,x) p(y)p(x|y) • By the Bayes rule p(y|x) = p(x) = p(x) ,
p(y)p(x|y) yˆ = argmax = argmax p(y)p(x|y). y p(x) y
• 1 Assuming a fixed prior p(y) (possibly uninformative: p(y) = K ), one is left with: yˆ = argmax p(x|y). y → Our decision criterion becomes a maximum-likelihood criterion. → This is a generative approach to classification: a probabilistic model of “how to generate x given a class y” is targeted.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 10 / 122 Introduction Preliminaries Discriminative model based classification
Directly models p(y|x) without wasting efforts on modeling the observations, which is not needed for the goal yˆ = argmaxy p(y|x).
I Pros: − The class posteriors p(y = c|x) are potentially simpler than the class-conditional densities.
5 1.2
1 4 p(y = 1|x) p(y = 2|x) p(x|y = 2) 0.8 3 p(x|y = 1) 0.6 2
Class posteriors 0.4
1 0.2 Class conditional densities
0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 xx Generated using pmtk3 (Dunham and Murphy, 2010)
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 11 / 122 Introduction Preliminaries Discriminative model based classification
Directly models p(y|x) without wasting efforts on modeling the observations, which is not needed for the goal yˆ = argmaxy p(y|x). I Pros: − The class posteriors p(y = k|x) are potentially simpler than the class-conditional densities. − Avoids making unwarranted assumptions about the features which may be highly dependent (especially with structured data). − Improved robustness to model imperfections, as independence assumptions will be made only among the labels, not the observations.
I Cons: − Classes need to be learned jointly and data should be available for all classes. − Models do not allow for generating observations.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 12 / 122 Introduction Preliminaries Predicting structured-output data
• In many MIR tasks the outputs are structured, e.g.: Harmonic analysis tasks: chord progressions follow predefined patterns (cadences)
Linear-chain structure
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 13 / 122 Introduction Preliminaries Predicting structured-output data
• In many MIR tasks the outputs are structured, e.g.:
Musical instrument recognition
o1 o2 o3 o4
Dr BsDr BsDrPn BsDr
Linear-chain structure
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 13 / 122 Introduction Preliminaries Predicting structured-output data
• In many MIR tasks the outputs are structured, e.g.:
Autotagging tasks: target tags are correlated (e.g. bebop, Jazz, fast tempo)
Tree structure
→ Need for predictors able to take advantage of this structure.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 13 / 122 Introduction Preliminaries Predicting structured-output data
• In many MIR tasks the outputs are structured, e.g.:
Autotagging tasks: target tags are correlated (e.g. bebop, Jazz, fast tempo)
Tree structure
→ Need for predictors able to take advantage of this structure.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 13 / 122 Introduction Preliminaries Predicting sequential data
• In this tutorial, we focus on linear-chain data
o1 o2 o3 o4 o1 o2 o3 o4
Dr BsDr BsDrPn BsDr C−7 C−7 F−7 F−7
• Specialized inference algorithms can then be used (forward-backward method), which are easier to apprehend. • More general methods can be used for more general structure (belief propagation and extensions), see e.g. (Jensen and Nielsen, 2007).
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 14 / 122 Introduction Preliminaries More notations
• x is a sequence of observations: x = (x1, ··· , xn).
• y is the corresponding sequence of labels: y = (y1, ··· , yn). • We assume we have a training dataset D of N (i.i.d) such sequences: D = {(x(1), y (1)), ··· , (x(N), y (N))}.
• Remarks:
− Observations are no longer assumed to be i.i.d within each sequence.
(q) − Sequences x do not necessarily have the same length, when needed nq will denote the length of x(q).
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 15 / 122 Introduction A first look at CRF The CRF model A discriminative model for structured-output data
CRF model definition
D 1 X p(y|x; θ) = exp θ F (x, y) Z(x, θ) j j j=1 1 = Ψ(x, y; θ); θ = {θ , ··· , θ }. Z(x, θ) 1 D
• P P Z(x, θ) = y exp j θj Fj (x, y) is called a partition function.
• PD Ψ(x, y; θ) = exp j=1 θj Fj (x, y) is called a potential function.
• Remark: feature functions Fj (x, y) depend on the whole sequence of observations x and labels y.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 16 / 122 Introduction A first look at CRF Applications of CRFs
CRF models have proven to be superior to competitors in a variety of application fields.
• They are the state-of-the-art techniques in many natural language processing (NLP) tasks (Taskar et al., 2002; Settles, 2004; Lavergne et al., 2010) part-of-speech tagging (POS), named-entity recognition (NER)...
• They have been successfully used for various computer vision tasks (He et al., 2004; Quattoni et al., 2007; Wang et al., 2006; Morency et al., 2007; Rudovic et al., 2012) image labeling, object and gesture recognition, facial expressions...
• Also for speech analysis tasks (Gunawardana et al., 2005; Reiter et al., 2007; Morris and Fosler-Lussier, 2008; Hong, 2010) speech recognition, speech segmentation, speaker identification...
• To date rarely used for music analysis, despite a great potential...
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 17 / 122 References to applications in MIR-related tasks
I Introduction
I References to applications in MIR-related tasks
I The logistic regression model
I Conditional Random Fields (for linear-chain data)
I Improvements and extensions to original CRFs
I Conclusion
I References
I Appendix
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 18 / 122 References to applications in MIR-related tasks Overview
• Chord recognition (Corey and Fujinaga, 2007). • Autotagging: exploiting correlations between tags for multi-label classification:
− considering pairs of tags (Duan et al., 2008); − attempting to account for correlations between multiple labels and inputs (Li, 2011). • Musical emotion analysis: modeling temporal dynamics of musical emotion (in the V-A space):
− targeting discretized V-A values and using discretized features (Schmidt and Kim, 2011); − targeting continuous V-A values based on CCRF for regression (Imbrasaite et al., 2013).
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 19 / 122 References to applications in MIR-related tasks Audio-to-score alignment based on CRF
• A sophisticated alignment system
− exploiting context of local feature observations depending on (discrete) tempo variables (Joder et al., 2010).
− carefully modeling the temporal structure using a semi-Markov CRF and a tempo-variation model (Joder et al., 2011).
− learning optimal score-audio affinity features (Joder et al., 2013).
• Substantial increase in performance compared to HMM-based approach: on pop music, 97.8% (vs 65.4%) of note onsets correctly associated, using 100-ms tolerance-window.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 20 / 122 References to applications in MIR-related tasks Beat detection based on CRF To appear...
• System developed by Thomas Fillon and Cyril Joder.
• Evaluation by Simon Durand, on a dataset of Classical, Jazz, Blues and Electro. music (Durand, 2013).
• Tolerance on temporal precision: 10% of the tempo value.
System (Davies and Plumbley, 2007) (Ellis, 2007) (Klapuri et al., 2006) CRF F-measure 68,4 55,6 69,6 72,7
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 21 / 122 The logistic regression model
I Introduction
I References to applications in MIR-related tasks
I The logistic regression model – Model specification – Maximum Entropy Modeling – Parameter estimation – Improvements to the logistic regression model
I Conditional Random Fields (for linear-chain data)
I Improvements and extensions to original CRFs
I Conclusion
ISlimReferences Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 22 / 122
I Appendix The logistic regression model Model specification The logistic regression model
Approach: model the posterior probabilities of the K classes using linear functions of the inputs x, according to:
P(C1|x) T log = w10 + w1 x P(CK |x)
P(C2|x) T log = w20 + w2 x P(CK |x) . . P(CK−1|x) T log = w(K−1)0 + wK−1x P(CK |x)
Defines a log-linear model specified in terms of K − 1 log-odds: log P(Ck |x) . P(CK |x)
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 23 / 122 The logistic regression model Model specification The logistic regression model
P(Ck |x) T • From log = wk0 + w x ; k = 1, ··· , K − 1; it is easy to deduce P(CK |x) k that:
Multiclass logistic regression model exp(w + wT x) P(C |x) = k0 k ; k = 1, ··· , K − 1, k PK−1 T 1 + l=1 exp(wl0 + wl x) 1 P(C |x) = K PK−1 T 1 + l=1 exp(wl0 + wl x)
• Remarks
− The model is a classification model (not a regression model!) − It is a discriminative model as it targets P(Ck |x) (as opposed to modeling p(x|Ck ) in generative models.)
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 24 / 122 The logistic regression model Model specification Binary classification case
• When K = 2
1 P(C1|x) = p = T 1 + exp −(w10 + w1 x) P(C2|x) = 1 − p 1 • p = ; a = w + wT x 1 + exp −a 10 1
Logistic sigmoid function 1 σ(a) =∆ 1 + exp −a
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 25 / 122 The logistic regression model Model specification The logistic sigmoid function
1 ∆ 1 0.9 σ(a) = 1+exp −a 0.8 0.7
0.6
0.5 • Properties: 0.4 0.3
0.2
Symmetry: σ(−a) = 1 − σ(a) 0.1
0 −8 −6 −4 −2 0 2 4 6 8 σ a Inverse: a = log 1−σ : logit function
• p p The odds 1−p ∈ [0, +∞] hence the log-odds log 1−p ∈ [−∞, +∞]
• Logistic regression models the log-odds as linear functions of the inputs... why is this a good idea? → Study the link to maximum entropy models.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 26 / 122 The logistic regression model Maximum Entropy Modeling Maximum Entropy: an introductory example Inspired by (Berger et al., 1996)
Goal: perform chord transcription using notes played as input (in the symbolic domain). Method: Use a training dataset to estimate p(y|o): the probability to assign the chord y to the observed note o; to be used for MAP decision.
• The structure of both the ground-truth labels y and the observations o reflect a set of facts about the data: rules of harmony. → Our model should capture these facts to perform accurate predictions.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 27 / 122 The logistic regression model Maximum Entropy Modeling Using facts about the data
• Let’s assume we observe the note C. → The “matching” chord is among {Cmaj, Cmin, A[maj, Amin, Fmaj, Fmin}.
In terms of statistics P(Cmaj) + P(Cmin) + P(A[maj) + P(Amin) + P(Fmaj) + P(Fmin) = 1.
• How to choose P(Cmaj), ··· , P(Fmin)?
• Safe choice:
In terms of statistics 1 P(Cmaj) = P(Cmin) = ··· = P(Fmin) = 6
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 28 / 122 The logistic regression model Maximum Entropy Modeling Why “uniform”?
• Intuitively: the most uniform model according to our knowledge, the only unbiased assumption
• Ancient wisdom:
− Occam’s razor (William of Ockham, 1287-1347): principle of parsimony: “Nunquam ponenda est pluralitas sine necesitate.” [Plurality must never be posited without necessity.]
− Laplace: “when one has no information to distinguish between the probability of two events, the best strategy is to consider them equally likely.” (Principle of Insufficient Reason)
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 29 / 122 The logistic regression model Maximum Entropy Modeling More facts
• The matching chord is Cmaj or Fmaj 30% of the time:
P(Cmaj) + P(Fmaj) = 3/10 P(Cmaj) + P(Cmin) + ··· + P(Fmaj) + P(Fmin) = 1
• Again many solutions... and a resonable choice is:
P(Cmaj) = P(Fmaj) = 3/20 P(Cmin) = P(A[maj) = P(Amin) = P(Fmin) = 7/40
• How to generalize this? How to determine the “most uniform” model subject to the constraints at hand?
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 30 / 122 The logistic regression model Maximum Entropy Modeling Using feature functions
• Need to express the facts about the observations in a flexible way, to make sure the model will match them:
− make use of statistics of the observations: e.g. if C is played, the matching chord is Cmaj or Fmaj with frequency 3/10. − allow for using the context: e.g. if C is followed by E then the chord is Cmaj with frequency 1/2.
→ define feature functions to capture these statistics and use them to impose constraints to the model. ( 1 if oi = C, oi+1 = E and y = Cmaj Example: fj (oi , yi ) = 0 otherwise
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 31 / 122 The logistic regression model Maximum Entropy Modeling Defining constraints through feature functions
• The training sample can be described in terms of its empirical probability distribution p˜(o, y): 1 p˜(o, y) =∆ × number of times that (o, y) occurs in the sample N
∆ • ˜ P E(fj ) = o,y p˜(o, y)fj (o, y): expected value of fj w.r.t p˜(o, y).
∆ • P E(fj ) = o,y p(o)p(y|o)fj (o, y): expected v. of fj w.r.t the model p(o, y).
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 32 / 122 The logistic regression model Maximum Entropy Modeling Defining constraints through feature functions
• The observed statistics (facts) are captured by enforcing:
Constraint equation ˜ E(fj ) = E(fj ), i.e.
X X p(o)p(y|o)fj (o, y) = p˜(o, y)fj (o, y) o,y o,y
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 33 / 122 The logistic regression model Maximum Entropy Modeling Maximum entropy principle
• Now how to implement the idea of uniform modeling?
• Among the set M of probability distributions that satisfy the ˜ constraints, E(fj ) = E(fj ), choose:
Maximum entropy criterion
p∗(y|o) = argmax H(y|o); p(y|o)∈M
∆ X H(y|o) = − p(o)p(y|o) log p(y|o): the conditional entropy o,y
• Hint from information theory: the discrete distribution with maximum entropy is the uniform distribution.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 34 / 122 The logistic regression model Maximum Entropy Modeling Solving the problem
∗ Primal: p (y|o) = argmaxp(y|o)∈M H(y|o) ˜ P Constraints: E(fj ) = E(fj ) and y p(y|o) = 1 ∆ P Lagrangian: L(p, λ) = H(y|o) + λ0 y p(y|o) − 1 + P ˜ j λj E(fj ) − E(fj )
Equating the derivative of the Lagrangian with 0: 1 X pλ(y|o) = exp λj fj (o, y); Zλ(o) j P P Zλ(x) = y exp j λj fj (o, y)
∗ The solution is given by the dual optimal: λ = argmaxλ L(p, λ).
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 35 / 122 The logistic regression model Maximum Entropy Modeling Compare to the LR model
I Maxent model: 1 P p(y = k|o) = exp λjk fj (o, y) ; Zλ(o) j P P Zλ(o) = y exp j λjk fj (o, y) .
I Logistic regression model:
exp(w + wT x) p(y = k|x) = k0 k PK−1 T 1 + l=1 exp(wl0 + wl x) exp(w 0 + w0T x) = k0 k PK 0 0T l=1 exp(wl0 + wl x) 1 0 0T = exp(wk0 + wk x). Zw(x)
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 36 / 122 The logistic regression model Maximum Entropy Modeling Compare to the LR model
I Maxent model:
1 X p(y = k|o) = exp λkj fj (o, y) Zλ(o) j
I Logistic regression model:
1 0 0T p(y = k|x) = exp(wk0 + wk x) Zw(x)
• Using: T − feature-function: fj (o, y) = xj ; f0(o, y) = 1 and x = (x1, ··· , xj , ··· , xD ) ;
0 D 0 − 0 T P wk0 + wk x = j=0 wkj fj (o, y);
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 37 / 122 The logistic regression model Maximum Entropy Modeling Compare to the CRF model
I Maxent model:
1 X p(y = k|o) = exp λkj fj (o, y) Zλ(o) j
I Logistic regression model:
1 X 0 p(y = k|o) = exp wkj fj (o, y) Zw(o) j
I CRF model: 1 X p(y|x; θ) = exp θj Fj (x, y) Zθ(x) j
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 38 / 122 The logistic regression model Maximum Entropy Modeling Conclusion
The solution to the maximum entropy models has the same parametric form as logistic regression and CRF models.
• It is easily shown that the optimal solution is the maximum-likelihood 1 P solution in the parametric family pλ(y|x) = exp( λj xj ). Zλ(x) j
• We’ve only considered discrete inputs, what about continuous inputs? − It is found that if the class-conditional densities p(x|y) are members of the exponential family of distributions, then the posterior probabilities are again given by logistic sigmoids of a linear function.
− In particular, the model is optimal with Gaussian densities (with a shared covariance matrix).
The logistic regression model is quite well justified in a variety of situations.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 39 / 122 The logistic regression model Parameter estimation
I Introduction
I References to applications in MIR-related tasks
I The logistic regression model – Model specification – Maximum Entropy Modeling – Parameter estimation – Improvements to the logistic regression model
I Conditional Random Fields (for linear-chain data)
I Improvements and extensions to original CRFs
I Conclusion
ISlimReferences Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 40 / 122
I Appendix The logistic regression model Parameter estimation Fitting the LR models
• Done by maximum likelihood estimation; in practice minimizing the Negative Log-Likelihood (NLL).
• Let θ denote the set of all parameters: θ = {w10, w1, ··· , w(K−1)0, wK−1}.
• The log-likelihood for the N (i.i.d) feature-vector observations is: N ∆ X L(D; θ) = − log p(yi |xi ; θ) i=1
• To simplify, we focus on the bi-class case...
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 41 / 122 The logistic regression model Parameter estimation NLL for bi-class LR
• Let yi = 1 for C1 observations and yi = 0 for C2 observations. ∆ • Let p(x; θ) = p(yi = 1|xi ; θ); hence p(yi = 0|xi ; θ) = 1 − p(x; θ).
• We can write: p(y|x; θ) = p(x; θ)y (1 − p(x; θ))1−y .
Negative Log-Likelihood
N X L(D; θ) = L(w˜ ) = − {yi log p(xi ; w˜ ) + (1 − yi ) log (1 − p(xi ; w˜ ))} i=1 N X n T T o = − yi w˜ x˜i − log 1 + exp(w˜ x˜i ) i=1
T T where w˜ = (w0, w) and x˜i = (1, xi ) so that w˜ x˜i = w0 + w xi .
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 42 / 122 The logistic regression model Parameter estimation Gradient and Hessian of the NLL
PN Gradient: ∇L(D; w˜ ) = − i=1 x˜i (yi − p(xi ; w˜ ))
∂2L(D;w˜) PN T Hessian: ∂w˜∂w˜ T = i=1 x˜i x˜i p(xi ; w˜ )(1 − p(xi ; w˜ ))
→ so the Hessian is positive semi-definite, → the NLL is convex and it has a global minimum.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 43 / 122 The logistic regression model Parameter estimation Minimizing the NLL
• By setting the derivatives to zero: N ∂L(D; w˜ ) X = − x˜ji (yi − p(xi ; w˜ )) = 0 ; 0 ≤ j ≤ D. ∂wj i=1
Optimization problem Solve for w˜ the D + 1 non-linear equations:
N N X X yi x˜ji = x˜ji p(xi ; w˜ ); 0 ≤ j ≤ D i=1 i=1
• For j = 0, since the first coefficient of x˜i is 1, that is x˜0i = 1, we get: N N X X yi = p(xi ; w˜ ). i=1 i=1
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 44 / 122 The logistic regression model Parameter estimation Optimization methods
PN PN Objective: Solve i=1 yi x˜ji = i=1 x˜ji p(xi ; w˜ ) Problem: No closed-form solution in general (system of D + 1 non-linear equations). Solution: use descent methods.
Among the many available descent algorithms, two are widely used:
− the Newton-Raphson method: fast... but complex (efficient variants exist);
− the stochastic gradient descent method: easy to implement, adapted to large scale problems.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 45 / 122 The logistic regression model Parameter estimation Optimization with the Newton-Raphson method
• To minimize g(θ), consider its second-order Taylor series approximation around θn: 1 g(θ) ≈ g(θ ) + ∇g(θ )T (θ − θ ) + (θ − θ )T H(θ )(θ − θ ); n n n 2 n n n
∇g(θn) and H(θn) are resp. the gradient and Hessian of g(θ) at θn.
• This approximation is a quadratic function which is minimized by solving:
∇g(θn) + H(θn)(θ − θn) = 0.
Hence the Newton-Raphson step
−1 θn+1 = θn − H(θn) ∇(θ).
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 46 / 122 The logistic regression model Parameter estimation Optimization with the Newton-Raphson method Illustration
g(θ) g quad (θ) θn
θn θn + sn
Generated using pmtk3 (Dunham and Murphy, 2010)
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 47 / 122 The logistic regression model Parameter estimation Optimization with the Newton-Raphson method Illustration
g(θ) g quad (θ) θn+1
θn θn+1 θn+1 + sn+1
Generated using pmtk3 (Dunham and Murphy, 2010)
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 47 / 122 The logistic regression model Parameter estimation Optimization with the Newton-Raphson method Discussion
• Typically the algorithm converges (though overshooting may occur), and convergence speed is quadratic.
• D has to be small enough so that it is not too costly to recompute and store the inverse Hessian matrix at each iteration. • Otherwise use Quasi-Newton methods:
− BFGS (Broyden, Fletcher, Goldfarb and Shanno) method: approximates the inverse Hessian using successive gradient values. − L-BFGS (limited memory BFGS) method: stores only a few vectors used to approximate the inverse Hessian.
• Alternatively, use stochastic gradient learning (see Appendix):
− Makes gradient updates based on one training example at a time. − In practice: simple approach, slow convergence, less accurate than L-BFGS.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 48 / 122 The logistic regression model Improvements to the logistic regression model
I Introduction
I References to applications in MIR-related tasks
I The logistic regression model – Model specification – Maximum Entropy Modeling – Parameter estimation – Improvements to the logistic regression model
I Conditional Random Fields (for linear-chain data)
I Improvements and extensions to original CRFs
I Conclusion
ISlimReferences Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 49 / 122
I Appendix The logistic regression model Improvements to the logistic regression model
`2-regularization
• To avoid overfitting the complexity of the model should be penalized.
• Similarly to ridge regression (Hastie et al., 2009), a quadratic regularization term can be added to the NLL:
Regularized logistic regression problem
γ wˆ = argmin L(D; w˜ ) + ||w||2 w˜ 2 N D X h i γ X = argmin − y w˜ T x˜ − log 1 + exp w˜ T x˜ + w 2 i i i 2 j w˜ i=1 j=1
γ ≥ 0 : complexity parameter controlling the amount of shrinkage; usually tuned by cross-validation.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 50 / 122 The logistic regression model Improvements to the logistic regression model
`2-regularization Discussion
Recall that: γ wˆ = argmin L(D; w˜ ) + ||w||2 w˜ 2 is equivalent to: ( wˆ = argminw˜ L(D; w˜ ) subject to ||w||2 ≤ t
for some t which has a correspondence to γ.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 51 / 122 The logistic regression model Improvements to the logistic regression model
`2-regularization Gradient and Hessian
Gradient: ∇L2(D; w˜ ) = ∇L2(D; w˜ ) + γw
Hessian: H2(w˜ ) = H(w˜ ) + γID+1
→ So the Hessian becomes positive definite, the NLL is now strictly convex and it has a unique global minimum. → The previous optimization methods can be straightforwardly adapted by modifying the expressions of the gradient and Hessian.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 52 / 122 The logistic regression model Improvements to the logistic regression model
`1-regularization
• Proceed as in the LASSO (Hastie et al., 2009), using a `1-regularization.
`1-regularized logistic regression problem
wˆ = argmin L(D; w˜ ) + γ||w||1 w˜ D X = argmin L(D; w˜ ) + γ |wj | ; γ ≥ 0. w˜ j=1
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 53 / 122 The logistic regression model Improvements to the logistic regression model
`1-regularization Discussion
• `1-regularization achieves feature selection.
Illustration by Alexandre Gramfort, Telecom ParisTech
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 54 / 122 The logistic regression model Improvements to the logistic regression model
`1-regularization Discussion
• `1-regularization achieves feature selection.
• Difficulties:
− The regularizer is not differentiable at zero yielding non-smooth optimization problem. → specific optimization techniques needed (Yuan et al., 2010).
− In configurations with groups of highly correlated features:
I `1-regularization tends to select randomly one feature in each group; I `2-regularization tends to yield better prediction performance. → Consider the elastic net model (Hastie et al., 2009):
2 L(D; w˜) + γ2||w||2 + γ1||w||1
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 54 / 122 The logistic regression model Improvements to the logistic regression model Kernel logistic regression (KLR)
• Let K: positive definite kernel and HK: the RKHS generated by K.
• Let φ ∈ HK, a feature mapping to HK.
KLR model
1 T p(yi |xi ) = ; g(x) = w0 + w φ(x) 1 + exp −g(xi )
KLR model estimation problem: N X min L(w˜ ) = − [yi g(xi ) − log(1 + exp g(xi ))] w˜ i=1
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 55 / 122 The logistic regression model Improvements to the logistic regression model Regularized KLR
Regularized KLR model estimation problem: N X γ 2 min L(w˜ ) = − [yi g(xi ) − log(1 + exp g(xi ))] + ||g||H w˜ 2 K i=1
• PN By the representer theorem: g(x) = w0 + i=1 αi K(xi , x)
• The problem is strictly convex and can be solved using classic solvers.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 56 / 122 The logistic regression model Improvements to the logistic regression model KLR vs SVM
• It can be shown that KLR and SVM are quite related (see Appendix). • Very similar prediction performance and optimal margin properties. • Same refinements are possible: SMO, MKL...
+ Provides well-calibrated class probabilities. + Naturally generalizes to multi-class problems.
- No support vectors! → Import Vector Machines (Zhu and Hastie, 2002).
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 57 / 122 Conditional Random Fields (for linear-chain data)
I Introduction
I References to applications in MIR-related tasks
I The logistic regression model
I Conditional Random Fields (for linear-chain data) – Introduction – Inference – Parameter estimation
I Improvements and extensions to original CRFs
I Conclusion
I References Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 58 / 122 I Appendix Conditional Random Fields (for linear-chain data) Introduction Structured-output data
o1 o2 o3 o4 o1 o2 o3 o4
Dr BsDr BsDrPn BsDr C−7 C−7 F−7 F−7
Musical instrument classification Chord transcription
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 59 / 122 Conditional Random Fields (for linear-chain data) Introduction Recalling the notations
• x is a sequence of observations: x = (x1, ··· , xn).
• y is the corresponding sequence of labels: y = (y1, ··· , yn). • We assume we have a training dataset D of N (i.i.d) such sequences: D = {(x(1), y (1)), ··· , (x(N), y (N))}.
• Remarks:
− Observations are no longer assumed to be i.i.d within each sequence.
(q) − Sequences x do not necessarily have the same length, when needed nq will denote the length of x(q).
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 60 / 122 Conditional Random Fields (for linear-chain data) Introduction The CRF model
CRF model definition
D 1 X p(y|x; θ) = exp θ F (x, y) Z(x, θ) j j j=1 1 = Ψ(x, y; θ); θ = {θ , ··· , θ }. Z(x, θ) 1 D
• P P Z(x, θ) = y exp j θj Fj (x, y) is called a partition function. • PD Ψ(x, y; θ) = exp j=1 θj Fj (x, y) is called a potential function.
• Remarks: − CRFs appear to be an extension of logistic regression to structured data. − Feature functions Fj (x, y) depend on the whole sequence of observations x and labels y.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 61 / 122 Conditional Random Fields (for linear-chain data) Introduction Defining label constraints
• Without any further assumptions on the structure of y the model is hardly usable:
one needs to enumerate all possible sequences y for:
− P P Z(x, θ) = y exp j θj Fj (x, y);
− yˆ = argmaxy p(y|x; θ).
with |Y|n possible assignments !
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 62 / 122 Conditional Random Fields (for linear-chain data) Introduction Defining label constraints Using feature functions
• Consider feature functions Fj (x, y) such that: n X Fj (x, y) = fj (yi−1, yi , x, i); where n is the length of x. i=1
→ defines linear-chain CRFs: at each position i, 1 ≤ i ≤ n,
− each fj depends on the whole observation sequence,
− but only on the current and previous labels.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 63 / 122 Conditional Random Fields (for linear-chain data) Introduction Defining label constraints Valid feature functions
n X Fj (x, y) = fj (yi−1, yi , x, i) i=1
Examples of such feature functions (for discrete observations):
− The current observation is G, the current label is Cmin7 and the previous is G7;
− The past 4 observations..., the current label is...
− The next observation is...
− The current label is...
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 64 / 122 Conditional Random Fields (for linear-chain data) Introduction Defining label constraints Observation and transition feature functions
• For convenience, one can define two types of feature functions:
− Observation (aka state) feature functions: bj (yi , x, i);
− Transition feature functions: tj (yi−1, yi , x, i).
• Hence: n D n D 1 X Xo X Xt p(y|x; θ) = exp θ b (y , x, i) + θ t (y y , x, i) Z(x, θ) j j i j j i−1, i i=1 j=1 i=1 j=1
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 65 / 122 Conditional Random Fields (for linear-chain data) Introduction Connection to HMM
The Hidden Markov Model ∆ Qn ∆ phmm(y, x) = i=1 p(yi |yi−1)p(xi |yi ) ; where p(y1|y0) = p(y1).
One can write: ( n n ) X X phmm(y, x) = exp log p(yi |yi−1) + log p(xi |yi ) i=1 i=1 n X X = exp λlqI(yi = l)I(yi−1 = q) i=1 l,q∈Y
n X X + µol I(y = l)I(xi = o) ; i=1 l∈Y,o∈X
where λlq = log p(yi = l|yi−1 = q) and µol = log p(xi = o|yi = l).
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 66 / 122 Conditional Random Fields (for linear-chain data) Introduction Connection to HMM
• Using the feature functions:
− bj (yi , x, i)= I(y = l)I(xi = o), where each j indexes a different “l, o configuration”; − tj (yi−1, yi , x, i)= I(yi = l)I(yi−1 = q), where j indexes a different “l, q configuration”;
p(y,x) P 0 • also using p(y|x) = P 0 and letting Z(x) = 0 p(y , x), one gets: y0 p(y ,x) y
n n 1 X X X X p (y|x) = exp θ b (y , x, i)+ θ t (y , y , x, i) ; hmm Z(x) j j i j j i−1 i i=1 j i=1 j
→ HMMs are a particular type of linear-chain CRFs.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 67 / 122 Conditional Random Fields (for linear-chain data) Introduction Connection to HMM Discussion
CRFs have a number of advantages over HMMs, as a consequence of two major differences:
• CRFs are discriminative models. • CRFs are undirected models.
yi−1 yi yi+1 yi−1 yi yi+1
xi−1 xi xi+1 xi−1 xi xi+1
HMM CRF
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 68 / 122 Conditional Random Fields (for linear-chain data) Introduction Connection to HMM Advantage of the discriminative nature of CRF
HMM: observation xi is independent of all other variables given its parent state yi . CRF: no assumptions on the dependencies among the observations: only p(y|x) is modeled.
→ CRFs can safely:
− exploit overlapping features; − account for long-term dependencies, considering the whole sequence of observations x at each location i (i 7→ bj (yi , x, i));
− use transition feature-functions tj (yi−1,yi , x, i).
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 69 / 122 Conditional Random Fields (for linear-chain data) Introduction Using linear-chain CRFs
Problems to be solved:
• Inference: given a model θ, how to compute:
− yˆ = argmaxy p(y|x; θ) ?
− P P Z(x, θ) = y exp j Fj (x, y) to deduce 1 PD p(y|x; θ) = Z(x,θ) exp j=1 θj Fj (x, y) ?
• Parameter estimation: given a training dataset D = {(x(1), y (1)), ··· , (x(N), y (N))}, how to estimate the optimal θ?
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 70 / 122 Conditional Random Fields (for linear-chain data) Inference Decoding the optimal sequence
n • Problem: solve yˆ = argmaxy∈Yn p(y|x; θ), with |Y| possible assignments!
• Solution: use the Viterbi algorithm.
Exploit the linear-chain structure:
D 1 X yˆ = argmax p(y|x; θ) = argmax exp θj Fj (x, y) y y Z(x, θ) j=1 D X = argmax θj Fj (x, y) y j=1 n D X X = argmax θj fj (yi−1, yi , x, i) y i=1 j=1
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 71 / 122 Conditional Random Fields (for linear-chain data) Inference Decoding the optimal sequence
∆ PD Let: gi (yi−1, yi ) = j=1 θj fj (yi−1, yi , x, i); then:
n D n X X X yˆ = argmax θj fj (yi−1, yi , x, i) = argmax gi (yi−1, yi ). y y i=1 j=1 i=1
Let δm(s) be the optimal “intermediate score” such that at time step m the label value is s:
"m−1 # ∆ X δm(s) = max gi (yi−1, yi ) + gm(ym−1, s) {y ,··· ,y } 1 m−1 i=1
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 72 / 122 Conditional Random Fields (for linear-chain data) Inference Decoding the optimal sequence Trellis representation
"m−1 # ∆ X δm(s) = max gi (yi−1, yi ) + gm(ym−1, s) {y ,··· ,y } 1 m−1 i=1
y1
y2 s
y3
yK m − 1 mn ...
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 73 / 122 Conditional Random Fields (for linear-chain data) Inference Decoding the optimal sequence
1 • The intermediate scores δm(s) can be efficiently computed using :
Viterbi recursion
δm(s) = max [δm−1(ym−1) + gm(ym−1, s)] ; 1 ≤ m ≤ n ym−1∈Y
• As we proceed we need to keep track of the selected predecessor of s, at each time step m.
• We use ψm(s) for this purpose.
1See Appendix for more details. Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 74 / 122 Conditional Random Fields (for linear-chain data) Inference Decoding the optimal sequence Viterbi algorithm Initialization: δ1(s) = g1(y0, s); ∀s ∈ Y; y0 = start
ψ1(s) = start Recursion: ∀s ∈ Y; 1 ≤ m ≤ n
δm(s) = max [δm−1(y) + gm(y, s)] y∈Y
ψm(s) = argmax [δm−1(y) + gm(y, s)] y∈Y
Termination: n ∗ X δn(yn ) = max δn(y) = max gi (yi−1, yi ). y∈Y y i=1 ∗ yn = argmax δn(y) y∈Y Path backtracking: ∗ ∗ ym = ψm+1(ym+1); m = n − 1, n − 2, ··· , 1. Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 75 / 122 Conditional Random Fields (for linear-chain data) Inference Decoding the optimal sequence Backtracking
∗ ∗ ym = ψm+1(ym+1); m = n − 1, n − 2, ··· , 1.
y1
y2
y3
yK m − 1 mn ... Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 76 / 122 Conditional Random Fields (for linear-chain data) Inference Complexity of Viterbi decoding
Remarks on the computational cost:
• O(K 2n) in the worst case; K = |Y|.
• In practice: O(T Kn), where T : average number of possible “transitions” between labels y.
• Can be reduced using beam search: exploring a subset of possible labels at each time position (the “most promising” ones) (Ortmanns et al., 1996).
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 77 / 122 Conditional Random Fields (for linear-chain data) Inference Using linear-chain CRFs
Problems to be solved:
• Inference: given a model θ, how to compute:
− yˆ = argmaxy p(y|x; θ) X
− P P Z(x, θ) = y exp j Fj (x, y) to deduce 1 PD p(y|x; θ) = Z(x,θ) exp j=1 θj Fj (x, y) ?
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 78 / 122 Conditional Random Fields (for linear-chain data) Inference Computing the partition function Z(x, θ) The sum-product problem
Recall the CRF model: 1 Qn p(y|x; θ) = Z(x,θ) i=1 Mi (yi−1, yi , x) ; PD Mi (yi−1, yi , x)= exp j=1 θj fj (yi−1, yi , x, i) ;
P Qn Z(x, θ) = y∈Yn i=1 Mi (yi−1, yi , x): intractable as is...
→ use the forward-backward method: reduces complexity from O(K n) to O(nK 2).
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 79 / 122 Conditional Random Fields (for linear-chain data) Inference The forward-backward method
• Defining α (y ) = P M (y , y )α (y ); 2 ≤ m ≤ n, it is m m ym−1 m m−1 m m−1 m−1 easily shown2 that:
At the end of the sequence X Z(x, θ) = αn(yn).
yn∈Y
• Alternatively, defining β (y ) = P M (y , y )β (y ); 1 ≤ m ≤ n − 1 and m m ym+1 m+1 m m+1 m+1 m+1 βn(yn) = 1, one gets: At the beginning of the sequence X Z(x, θ) = M1(y0, y1)β1(y1). y1∈Y 2See Appendix for more details Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 80 / 122 Conditional Random Fields (for linear-chain data) Inference Marginal probability
X p(ym−1, ym|x) = p(y|x)
y\{ym−1,ym}
Marginal probability by forward-backward 1 p(y , y |x) = α (y )M (y , y , x)β (y ). m−1 m Z(x) m−1 m−1 m m−1 m m m
More details in the appendix.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 81 / 122 Conditional Random Fields (for linear-chain data) Parameter estimation
I Introduction
I References to applications in MIR-related tasks
I The logistic regression model
I Conditional Random Fields (for linear-chain data) – Introduction – Inference – Parameter estimation
I Improvements and extensions to original CRFs
I Conclusion
I References Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 82 / 122 I Appendix Conditional Random Fields (for linear-chain data) Parameter estimation Negative log-likelihood (NLL)
• Given training data D = {(x(1), y (1)), ··· , (x(N), y (N))}, the NLL is:
N ∆ X L(D; θ) = − log p(y (q)|x(q); θ) q=1 N nq D X (q) X X (q) (q) (q) = log Z(x ; θ) − θj fj (yi−1, yi , x , i) q=1 i=1 j=1
N D X (q) X (q) (q) = log Z(x ; θ) − θj Fj (x , y ) . q=1 j=1
• L(D; θ) is convex → gradient-descent will converge to global minimum.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 83 / 122 Conditional Random Fields (for linear-chain data) Parameter estimation NLL gradient
N ∂L(D; θ) X ∂ Gradient: = log Z(x(q); θ) − F (x(q), y (q)) . ∂θ ∂θ k k q=1 k D ∂ 1 X ∂ X log Z(x; θ)= exp θ F (x, y) ∂θ Z(x; θ) ∂θ j j k y∈Yn k j=1 D 1 X X = F (x, y) exp θ F (x, y) Z(x; θ) k j j y∈Yn j=1 P X exp j θj Fj (x, y) = F (x, y) k Z(x; θ) y∈Yn X = Fk (x, y)p(y|x; θ) y∈Yn = Ep(y|x;θ) Fk (x, y) .
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 84 / 122 Conditional Random Fields (for linear-chain data) Parameter estimation NLL gradient
N ∂L(D; θ) X ∂ Gradient: = log Z(x(q); θ) − F (x(q), y (q)) . ∂θ ∂θ k k q=1 k
∂ log Z(x; θ)= Fk (x, y) : conditional expectation given x. ∂θk Ep(y|x;θ)
N ∂L(D; θ) X n h (q) i (q) (q) o = (q) F (x , y) − F (x , y ) . ∂θ Ep(y|x ;θ) k k k q=1
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 84 / 122 Conditional Random Fields (for linear-chain data) Parameter estimation Optimality condition
∂L(D;θ) • Setting the derivatives to 0, i.e. = 0, yields: ∂θk
N N X h (q) i X (q) (q) Ep(y|x(q);θ) Fk (x , y) = Fk (x , y ); 1 ≤ k ≤ D q=1 q=1
• No closed-form solution: numerical optimization is again needed.
• (q) Need to compute Ep(y|x(q);θ) Fk (x , y) efficiently.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 85 / 122 Conditional Random Fields (for linear-chain data) Parameter estimation Optimality condition
∂L(D;θ) • Setting the derivatives to 0, i.e. = 0, yields: ∂θk
N N 1 X h (q) i 1 X (q) (q) (q) F (x , y) = F (x , y ); 1 ≤ k ≤ D N Ep(y|x ;θ) k N k q=1 q=1
• Average expectation under the model = empirical mean.
• No closed-form solution: numerical optimization is again needed.
• (q) Need to compute Ep(y|x(q);θ) Fk (x , y) efficiently.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 85 / 122 Conditional Random Fields (for linear-chain data) Parameter estimation Efficient gradient computation
X Ep(y|x;θ) Fk (x, y) = Fk (x, y)p(y|x; θ) y∈Yn n X X = fk (yi−1, yi , x)p(y|x; θ) i=1 y∈Yn n X X = fk (yi−1, yi , x)p(yi−1, yi |x; θ) 2 i=1 yi−1,yi ∈Y
p(yi−1, yi |x; θ) is the marginal probability which thanks to the forward-backward algorithm is obtained by: 1 p(y , y |x) = α (y )M (y , y , x)β (y ). i−1 i Z(x) i−1 i−1 i i−1 i i i
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 86 / 122 Conditional Random Fields (for linear-chain data) Parameter estimation Optimization
Now that we are able to compute the gradient, we can use a descent method to solve for θ.
Many algorithms are available (see Sokolovska, 2010; Lavergne et al., 2010):
• Generalized iterative scaling (Lafferty et al., 2001): original algorithm, slow convergence, suboptimal.
• Conjugate gradient (Wallach, 2002): faster convergence, better quality.
• L-BFGS (McCallum, 2002): fast convergence, scalable; a good option, most used.
• Stochastic gradient: suboptimal, simple, online, large-scale applications.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 87 / 122 Improvements and extensions to original CRFs
I Introduction
I References to applications in MIR-related tasks
I The logistic regression model
I Conditional Random Fields (for linear-chain data)
I Improvements and extensions to original CRFs – Regularization – Hidden-state CRF model – Other extensions
I Conclusion
I References Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 88 / 122 I Appendix Improvements and extensions to original CRFs Regularization
Using `2-norm
||θ||2 • 2 Redefine the objective function as: L(D; θ) = L(D; θ) + 2σ2 ; σ2: a free parameter penalizing large weights (as in ridge regression).
• The gradient coefficients become: n o ∂L(D;θ) PN (q) (q) (q) θk = (q) Fk (x , y) − Fk (x , y ) + 2 . ∂θk q=1 Ep(y|x ;θ) σ • Advantages:
− The objective becomes strictly convex. − Shrinkage of θ coefficients is achieved avoiding overfitting and numerical problems.
• σ2 needs to be tuned (usually by cross-validation).
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 89 / 122 Improvements and extensions to original CRFs Regularization Regularization
Using `1-norm to perform feature selection
• Redefine the objective function as: PD L(D; θ) = L(D; θ) + ρ||θ||1 = L(D; θ) + ρ j=1 |θj | (as in the LASSO).
• Advantage: performs feature selection in some NLP apps: up to 95% of the features can be discarded without affecting performance! (see Sokolovska, 2010).
• Difficulties:
− The regularizer is not differentiable at zero: specific optimization techniques needed (Sokolovska, 2010).
− In configurations with groups of highly correlated features, tend to select randomly one feature in each group.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 90 / 122 Improvements and extensions to original CRFs Hidden-state CRF model
I Introduction
I References to applications in MIR-related tasks
I The logistic regression model
I Conditional Random Fields (for linear-chain data)
I Improvements and extensions to original CRFs – Regularization – Hidden-state CRF model – Other extensions
I Conclusion
I References Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 91 / 122 I Appendix Improvements and extensions to original CRFs Hidden-state CRF model Motivation
Problem: the CRF model does not support hidden states.
CRF ... (i) (i) Input: x1 x2 x3 x4 D = {(x , y )}i Output: y1 y2 y3 y4 Am Dm7 G7 CMaj7
Hidden-state CRF Input: x1 x2 x3 x4 ... (i) (i) D = {(x , y )}i Hidden: h1 h2 h3 h4 Output: y = ’TRUMPET’
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 92 / 122 Improvements and extensions to original CRFs Hidden-state CRF model Motivation
Problem: the CRF model does not support hidden states.
yi−1 yi yi+1 CRF
(i) (i) D = {(x , y )}i xi−1 xi xi+1
y
hi−1 hi hi+1 Hidden-state CRF (i) (i) D = {(x , y )}i
xi−1 xi xi+1
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 92 / 122 Improvements and extensions to original CRFs Hidden-state CRF model The HCRF model (Quattoni et al., 2007)
• Each sequence of observations x = (x1, ··· , xn) is associated with: − a unique label y; − a sequence of latent variables h = (h1, ··· , hn), where hi ∈ H.
HCRF model definition D 1 X p(y, h|x; θ) = exp θ F (x, y, h) Z(x, θ) j j j=1 P PD Z(x, θ) = y,h exp j=1 θj Fj (x, y, h); θ = {θ1, ··· , θD }.
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• 1 PD Using the HCRF model: p(y, h|x; θ) = Z(x,θ) exp j=1 θj Fj (x, y, h); entails being able to compute:
− ∗ yˆ = argmaxy∈Y p(y|x; θ ), to classify new test cases; − the partition function Z(x, θ), to evaluate posterior probabilities.
∆ • 0 P PD Let Z (y, x, θ) = h∈Hn exp j=1 θj Fj (x, y, h): marginalization wrt h.
• We have: Z 0(y,x,θ) − P p(y|x; θ) = n p(y, h|x; θ) = P 0 ; h∈H y Z (y,x,θ)
− P 0 Z(x, θ) = y Z (y, x, θ). • Z 0(y, x, θ) can be easily computed using forward/backward recursions (as done in CRF).
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 94 / 122 Improvements and extensions to original CRFs Hidden-state CRF model Negative log-likelihood
N ∆ X L(D; θ) = − log p(y (q)|x(q); θ) q=1 N ( ! ) X X = log Z 0(y, x(q), θ) − log Z 0(y (q), x(q), θ) ; q=1 y
D 0 ∆ X X Z (y, x, θ) = exp θj Fj (x, y, h) h∈Hn j=1
L(D; θ) is no longer convex → convergence to a local minimum.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 95 / 122 Improvements and extensions to original CRFs Hidden-state CRF model NLL gradient
N ( ∂L(D; θ) X X (q) = Fk (x , y, h)p(y, h|x; θ) ∂θk q=1 y,h ) X (q) (q) (q) (q) − Fk (x , y , h)p(h|y , x ; θ) h
which can be again computed using the forward-backward method.
A gradient descent method (L-BFGS) can be again used to solve for θ.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 96 / 122 Improvements and extensions to original CRFs Hidden-state CRF model Application to musical instrument classification
Input: x1 x2 x3 x4 ...
Hidden: h1 h2 h3 h4
Output: y = ’TRUMPET’
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 97 / 122 Improvements and extensions to original CRFs Hidden-state CRF model Feature functions used Following (Quattoni et al., 2007)
N N N X X X Ψ(x, y, h, θ) = < θ(hi ), xi > + θ(y, hi ) + θ(y, hi−1, hi ) i=1 i=1 i=1
• < θ(hi ), xi >: compatibility between observation xi and hidden state hi ∈ H;
• θ(y, hi ) : compatibility between hidden state hi and label y;
• θ(y, hi−1, hi ) : compatibility between transition hi−1 ↔ hi and label y.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 98 / 122 Improvements and extensions to original CRFs Hidden-state CRF model Evaluation
• Classifying 1-second long segments of solo excerpts of Cello, Guitar, Piano, Bassoon and Oboe. • Data: − training set: 2505 segments (i.e. 42’); − testing set: 2505 segments. • Classifiers:
− `2-regularized HCRF with 3 hidden states; − Linear SVM. • Features: 47 cepstral, perceptual and temporal features.
• Results Classifier SVM HCRF Average accuracy 75% 76%
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 99 / 122 Improvements and extensions to original CRFs Other extensions
I Introduction
I References to applications in MIR-related tasks
I The logistic regression model
I Conditional Random Fields (for linear-chain data)
I Improvements and extensions to original CRFs – Regularization – Hidden-state CRF model – Other extensions
I Conclusion
I References Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 100 / 122 I Appendix Improvements and extensions to original CRFs Other extensions Other extensions
• LDCRF: Latent-Dynamic Conditional Random Field (Sung and Jurafsky, 2009) modeling both hidden-states and structured-outputs.
y yi−1 yi yi+1
yi−1 yi yi+1 hi−1 hi hi+1 hi−1 hi hi+1
xi−1 xi xi+1 xi−1 xi xi+1 xi−1 xi xi+1
CRF HCRF LDCRF
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 101 / 122 Improvements and extensions to original CRFs Other extensions Other extensions
• LDCRF: Latent-Dynamic Conditional Random Field (Sung and Jurafsky, 2009) modeling both hidden-states and structured-outputs.
• Kernel-CRF (Laferty et al., 2004; Altun et al., 2004) introducing implicit features to account for (non-linear) interactions between original features.
• Semi-Markov CRF (Sarawagi and Cohen, 2005) modeling segment-level labels.
• CCRF: Continuous CRF (Qin and Liu, 2008) modeling continuous labels in a regression setting.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 101 / 122 Conclusion Take-home messagesI
• CRFs are powerful structured-data prediction models (more flexible than HMMs and other more general Bayesian networks) as they are:
− discriminative models: focus on modeling the target labels;
→ can handle a high number of feature functions, including transition features, and account for long-range dependencies.
− undirected models: no need to normalize potentials locally.
→ allow for incorporating prior knowledge about constraints and label dependencies in an intuitive way.
• Easily extendable with key mechanisms: regularization, sparsity, latent variables, kernels...
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 102 / 122 Conclusion Take-home messagesII
• Great potential for various MIR-related tasks (both symbolic and numerically-continuous data):
− transcription tasks: notes, chords, harmonic analysis...
− “classification” (autotagging) tasks:
I accounting for label-correlations in multi-label problems;
I accounting for temporal dynamics using HCRF...
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 103 / 122 References CRF software packages
Package Language Main features Reference
CRF++ C++ Linear-chain CRF, NLP, L-BFGS optimization (Taku-ku, 2003) Matlab, Linear-chain CRF, categorical features, L-BFGS crfChain Schmidt (2008) C mex optimization Linear-chain CRF, NLP, various regularization C++, CRFsuite and optimization methods (L-BFGS), designed (Okazaki, 2007) Python for fast training C++, HCRF CRF, HCRF, LDCRF, continuous inputs, Matlab, (Morency, 2010) library L-BFGS optimization Python CRF, maxent, HMM, NLP, text feature
Mallet Java extraction routines, various optimization (McCallum, 2002) methods (L-BFGS) Linear-chain CRF, NLP, large label and feature
Wapiti C99 sets, various regularization and optimization (Lavergne et al., 2010) methods (L-BFGS, SGD), multi-threaded
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 104 / 122 References CRF tutorials
• Charles Sutton and Andrew McCallum. An Introduction to Conditional Random Fields for Relational Learning. In Introduction to Statistical Relational Learning. Edited by Lise Getoor and Ben Taskar. MIT Press, 2006. • Charles Elkan. Log-linear Models and Conditional Random Fields. Notes for a tutorial at CIKM’08, October 2008. • Roman Klinger and Katrin Tomanek. Classical Probabilistic Models and Conditional Random Fields. Algorithm Engineering Report TR07-2-2013, December 2007. • Hanna M. Wallach. Conditional Random Fields: An Introduction. Technical Report MS-CIS-04-21, Department of Computer and Information Science, University of Pennsylvania, 2004. • Roland Memisevic. An Introduction to Structured Discriminative Learning. Technical Report, University of Toronto, 2006. • Rahul Gupta. Conditional Random Fields. Unpublished report, IIT Bombay, 2006.
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Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 110 / 122 I Introduction
I References to applications in MIR-related tasks
I The logistic regression model Appendix I Conditional Random Fields (for linear-chain data)
I Improvements and extensions to original CRFs
I Conclusion
I References
I Appendix – Optimization with stochastic gradient learning – Comparing KLR and SVM – Derivation of the Viterbi algorithm – The forward-backward method
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 111 / 122 Appendix Optimization with stochastic gradient learning LR model learning with stochastic gradient descent (SGL)
• Idea: make gradient updates based on one training example at a time ∂L(D;w˜) • Use: = (yi − p(xi ; w˜ )) xji ∂wj
Algorithm - Initialise w˜ - Repeat (until convergence)
- Randomly permute training examples xi - For i = 1 : N
wj ← wj + t (yσi − pσi ) xjσi ; j = 1, ··· , D
• t : step size, to be tuned • Complexity of SGL: O(NFD) per epoch; with F the average number of non-zero feature coefficients per example; an epoch is a “complete” update using all training examples.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 112 / 122 Appendix Comparing KLR and SVM Support Vector Machines Recalling SVM as a regularized function fitting problem
T • The SVM solution, g(x) = w0 + w φ(x), can be found by solving:
N X 0 γ 2 0 min 1 − yi g(xi ) + ||g||H ; yi ∈ {−1, 1} w˜ + 2 K i=1
3 Hinge
2.5
2 Hinge loss 1.5 0 0 loss [1 − yi g(xi )]+ = max(0, 1 − yi g(xi )) 1
0.5
0
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 y'g(x)
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 113 / 122 Appendix Comparing KLR and SVM KLR vs SVM
( 1 if yi = 1 • 0 Let yi = −1 if yi = 0
• The negative log-likelihood of the KLR model can then be written as PN 0 L(D; w˜ ) = i=1 log (1 + exp −yi g(xi )) .
• Both KLR and SVM solve:
N X 0 λ 2 min l(yi g(xi )) + ||g||H ; w˜ 2 K i=1 KLR SVM
0 0 0 0 l(yi g(xi )) = log 1 + exp −yi f (xi ) l(yi g(xi )) = 1 − yi f (xi ) +
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 114 / 122 Appendix Comparing KLR and SVM KLR vs SVM Hinge vs negative binomial log-likelihood
3 Hinge NLL 2.5
2
1.5 loss
1
0.5
0
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 y'g(x)
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 115 / 122 Appendix Derivation of the Viterbi algorithm Decoding the optimal sequence
D ∆ X Let: gi (yi−1, yi ) = θj fj (yi−1, yi , x, i); then: j=1
n D n X X X yˆ = argmax θj fj (yi−1, yi , x, i) = argmax gi (yi−1, yi ). y y i=1 j=1 i=1
"m−1 # ∆ X Let δm(s) = max gi (yi−1, yi ) + gm(ym−1, s) {y ,··· ,y } 1 m−1 i=1 m−1 X = max gi (yi−1, yi ) + max gm(ym−1, s). {y ,··· ,y } ym−1 1 m−1 i=1
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 116 / 122 Appendix Derivation of the Viterbi algorithm Viterbi decoding
m−1 X δ(s) = max gi (yi−1, yi ) + max gm(ym−1, s) {y ,··· ,y } ym−1 1 m−1 i=1
"m−2 # X So: δm−1(ym−1) = max gi (yi−1, yi ) + gm−1(ym−2, y m−1) {y ,··· ,y } 1 m−2 i=1 m−1 X = max gi (yi−1, yi ). {y ,··· ,y } 1 m−2 i=1
δm(s) = max [δm−1(ym−1) + gm(ym−1, s)]. ym−1∈Y
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 117 / 122 Appendix Derivation of the Viterbi algorithm Viterbi decoding The algorithm Initialization: δ1(s) = g1(y0, s); ∀s ∈ Y; y0 = start
ψ1(s) = start Recursion: ∀s ∈ Y; 1 ≤ m ≤ n
δm(s) = max [δm−1(y) + gm(y, s)] y∈Y
ψm(s) = argmax [δm−1(y) + gm(y, s)] y∈Y Termination: ∗ δn(yn ) = max δn(y) y∈Y ∗ yn = argmax δn(y) y∈Y Path backtracking: ∗ ∗ ym = ψm+1(ym+1); m = n − 1, n − 2, ··· , 1.
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 118 / 122 Appendix The forward-backward method The forward recursion
Define α scores as:
α1(y1) = M1(y0, y1) X α2(y2) = M2(y1, y2)α1(y1)
y1∈Y X α3(y3) = M3(y2, y3)α2(y2)
y2∈Y . . X αm(ym)= Mm(ym−1, ym)αm−1(ym−1); 2 ≤ m ≤ n ym−1
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 119 / 122 Appendix The forward-backward method The forward recursion
Define α scores as:
α1(y1) = M1(y0, y1) X α2(y2) = M2(y1, y2)α1(y1)
y1∈Y X X α3(y3) = M3(y2, y3)α2(y2) = M3(y2, y3)M2(y1, y2)M1(y0, y1)
y2∈Y y1,y2 . . X αm(ym) = Mm(ym−1, ym)αm−1(ym−1); 2 ≤ m ≤ n ym−1
At the end of the sequence n X X Y αn(yn) = Mi (yi−1, yi , x) = Z(x, θ). n yn∈Y y∈Y i=1
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 119 / 122 Appendix The forward-backward method The forward recursion
Define α scores as: X αm(ym) = Mm(ym−1, ym)αm−1(ym−1); 2 ≤ m ≤ n. ym−1
At the end of the sequence n X X Y αn(yn) = Mi (yi−1, yi , x) = Z(x, θ). n yn∈Y y∈Y i=1
Complexity: reduced from O(K n) to O(nK 2).
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 119 / 122 Appendix The forward-backward method The backward recursion
X βm(ym) = Mm+1(ym, ym+1)βm+1(ym+1); 1 ≤ m ≤ n − 1 ym+1∈Y
βn(yn) = 1
At the beginning of the sequence X Z(x, θ)= M1(y0, y1)β1(y1). y1∈Y
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 120 / 122 Appendix The forward-backward method Marginal probability
p(y , y |x) = P p(y|x); m−1 m y\{ym−1,ym} ∆ y\{ym−1, ym} = {y1, ··· , ym−2, ym+1, ··· , yn}.
n 1 X Y p(y , y |x) = M (y , y , x) m−1 m Z(x) i i−1 i y\{ym−1,ym} i=1 m−1 1 X Y = M (y , y , x) × M (y , y , x) Z(x) i i−1 i m m−1 m y\{ym−1,ym} i=1 n Y × Mi (yi−1, yi , x) i=m+1
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 121 / 122 Appendix The forward-backward method Marginal probability
m−1 1 X Y p(y , y |x) = M (y , y , x) × M (y , y , x) m−1 m Z(x) m m−1 m i i−1 i {y1,···,ym−2} i=1 n X Y × Mi (yi−1, yi , x)
{ym+1,···,yn} i=m+1
1 p(y , y |x) = α (y )M (y , y , x)β (y ). m−1 m Z(x) m−1 m−1 m m−1 m m m
Slim Essid (Telecom ParisTech) A tutorial on CRF - ISMIR 2013 November 2013 122 / 122