lecture 03: machine learning for language technology - linear classifiers
DESCRIPTION
We will talk about the following machine learning methods: perceptron, SVMs, MIRA, logistic regression/maximum entropyTRANSCRIPT
Machine Learning for Language Technology
Uppsala UniversityDepartment of Linguistics and Philology
Slides borrowed from previous courses.Thanks to Ryan McDonald (Google Research)
and Prof. Joakim Nivre
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Introduction
Linear Classifiers
I Classifiers covered so far:I Decision treesI Nearest neighbors
I Next two lectures: Linear classifiersI Statistics from Google Scholar (Sept 2013):
I “Maximum Entropy”&“NLP”; 11,400 hits (2013), 2,660(2009), 141 before 2000
I “SVM”&“NLP”: 10,900 hits (2013), 2,210 (2009), 16 before2000
I “Perceptron”&“NLP”: 3,160 hits (2013), 947 (2009), 118before 2000
I All are linear classifiers that have become important tools inLanguage Technology in the past 10 years or so.
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Introduction
Outline
I Today:I Preliminaries: input/output, features, etc.I Linear classifiers
I PerceptronI Large-margin classifiers (SVMs, MIRA)I Logistic regression
I Next time:I Structured prediction with linear classifiers
I Structured perceptronI Structured large-margin classifiers (SVMs, MIRA)I Conditional random fields
I Case study: Dependency parsing
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Preliminaries
Inputs and Outputs
I Input: x ∈ XI e.g., document or sentence with some words x = w1 . . .wn, or
a series of previous actions
I Output: y ∈ YI e.g., parse tree, document class, part-of-speech tags,
word-sense
I Input/output pair: (x,y) ∈ X × YI e.g., a document x and its label yI Sometimes x is explicit in y, e.g., a parse tree y will contain
the sentence x
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Preliminaries
Feature Representations
I We assume a mapping from input-output pairs (x,y) to ahigh dimensional feature vector
I f(x,y) : X × Y → Rm
I For some cases, i.e., binary classification Y = {−1,+1}, wecan map only from the input to the feature space
I f(x) : X → Rm
I However, most problems in NLP require more than twoclasses, so we focus on the multi-class case
I For any vector v ∈ Rm, let vj be the j th value
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Preliminaries
Features and Classes
I All features must be numericalI Numerical features are represented directly as fi (x,y) ∈ RI Binary (boolean) features are represented as fi (x,y) ∈ {0, 1}
I Multinomial (categorical) features must be binarizedI Instead of: fi (x,y) ∈ {v0, . . . , vp}I We have: fi+0(x,y) ∈ {0, 1}, . . . , fi+p(x,y) ∈ {0, 1}I Such that: fi+j(x,y) = 1 iff fi (x,y) = vj
I We need distinct features for distinct output classesI Instead of: fi (x) (1 ≤ i ≤ m)I We have: fi+0m(x,y), . . . , fi+Nm(x,y) for Y = {0, . . . ,N}I Such that: fi+jm(x,y) = fi (x) iff y = yj
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Preliminaries
Examples
I x is a document and y is a label
fj(x,y) =
1 if x contains the word“interest”
and y =“financial”0 otherwise
fj(x,y) = % of words in x with punctuation and y =“scientific”
I x is a word and y is a part-of-speech tag
fj(x,y) =
{1 if x = “bank”and y = Verb0 otherwise
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Preliminaries
Examples
I x is a name, y is a label classifying the name
f0(x, y) =
8<: 1 if x contains “George”and y =“Person”
0 otherwise
f1(x, y) =
8<: 1 if x contains “Washington”and y =“Person”
0 otherwise
f2(x, y) =
8<: 1 if x contains “Bridge”and y =“Person”
0 otherwise
f3(x, y) =
8<: 1 if x contains “General”and y =“Person”
0 otherwise
f4(x, y) =
8<: 1 if x contains “George”and y =“Object”
0 otherwise
f5(x, y) =
8<: 1 if x contains “Washington”and y =“Object”
0 otherwise
f6(x, y) =
8<: 1 if x contains “Bridge”and y =“Object”
0 otherwise
f7(x, y) =
8<: 1 if x contains “General”and y =“Object”
0 otherwise
I x=General George Washington, y=Person → f(x, y) = [1 1 0 1 0 0 0 0]
I x=George Washington Bridge, y=Object → f(x, y) = [0 0 0 0 1 1 1 0]
I x=George Washington George, y=Object → f(x, y) = [0 0 0 0 1 1 0 0]
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Preliminaries
Block Feature Vectors
I x=General George Washington, y=Person → f(x, y) = [1 1 0 1 0 0 0 0]
I x=George Washington Bridge, y=Object → f(x, y) = [0 0 0 0 1 1 1 0]
I x=George Washington George, y=Object → f(x, y) = [0 0 0 0 1 1 0 0]
I One equal-size block of the feature vector for each label
I Input features duplicated in each block
I Non-zero values allowed only in one block
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Linear Classifiers
Linear Classifiers
I Linear classifier: score (or probability) of a particularclassification is based on a linear combination of features andtheir weights
I Let w ∈ Rm be a high dimensional weight vectorI If we assume that w is known, then we define our classifier as
I Multiclass Classification: Y = {0, 1, . . . ,N}
y = arg maxy
w · f(x,y)
= arg maxy
m∑j=0
wj × fj(x,y)
I Binary Classification just a special case of multiclass
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Linear Classifiers
Linear Classifiers - Bias Terms
I Often linear classifiers presented as
y = arg maxy
m∑j=0
wj × fj(x,y) + by
I Where b is a bias or offset term
I But this can be folded into f
x=General George Washington, y=Person → f(x, y) = [1 1 0 1 1 0 0 0 0 0]
x=General George Washington, y=Object → f(x, y) = [0 0 0 0 0 1 1 0 1 1]
f4(x, y) =
1 y =“Person”0 otherwise
f9(x, y) =
1 y =“Object”0 otherwise
I w4 and w9 are now the bias terms for the labels
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Linear Classifiers
Binary Linear Classifier
Divides all points:
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Linear Classifiers
Multiclass Linear Classifier
Defines regions of space:
I i.e., + are all points (x,y) where + = arg maxy w · f(x,y)
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Linear Classifiers
Separability
I A set of points is separable, if there exists a w such thatclassification is perfect
Separable Not Separable
I This can also be defined mathematically (and we will shortly)
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Linear Classifiers
Supervised Learning – how to find w
I Input: training examples T = {(xt ,yt)}|T |t=1
I Input: feature representation fI Output: w that maximizes/minimizes some important
function on the training setI minimize error (Perceptron, SVMs, Boosting)I maximize likelihood of data (Logistic Regression)
I Assumption: The training data is separableI Not necessary, just makes life easierI There is a lot of good work in machine learning to tackle the
non-separable case
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Linear Classifiers
Perceptron
... The resulting hyperplane is called -perceptron- ...(Witten and Frank, 2005:126)
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Linear Classifiers
Perceptron
I Choose a w that minimizes error
w = arg minw
∑t
1− 1[yt = arg maxy
w · f(xt ,y)]
1[p] =
{1 p is true0 otherwise
I This is a 0-1 loss functionI Aside: when minimizing error people tend to use hinge-loss or
other smoother loss functions
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Linear Classifiers
Perceptron Learning Algorithm
Training data: T = {(xt ,yt)}|T |t=1
1. w(0) = 0; i = 02. for n : 1..N3. for t : 1..T (random order: shuffle the training set beforehand!)
4. Let y′ = arg maxy w(i) · f(xt ,y)5. if y′ 6= yt
6. w(i+1) = w(i) + f(xt ,yt)− f(xt ,y′)
7. i = i + 18. return wi
I See also Daume’ (2012:38-41).
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Linear Classifiers
Perceptron: Separability and Margin
I Given an training instance (xt ,yt), define:I Yt = Y − {yt}I i.e., Yt is the set of incorrect labels for xt
I A training set T is separable with margin γ > 0 if there existsa vector w with ‖w‖ = 1 such that:
w · f(xt ,yt)−w · f(xt ,y′) ≥ γ
for all y′ ∈ Yt and ||w|| =√∑
j w2j (Euclidean or L2 norm)
I Assumption: the training set is separable with margin γ
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Linear Classifiers
Perceptron: Main Theorem
I Theorem: For any training set separable with a margin of γ,the following holds for the perceptron algorithm:
mistakes made during training ≤ R2
γ2
where R ≥ ||f(xt ,yt)− f(xt ,y′)|| for all (xt ,yt) ∈ T and
y′ ∈ Yt
I Thus, after a finite number of training iterations, the error onthe training set will converge to zero
I For proof, see the Appendix to these slides
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Linear Classifiers
Perceptron Summary
I Learns a linear classifier that minimizes error
I Guaranteed to find a w in a finite amount of timeI Perceptron is an example of an online learning algorithm
I w is updated based on a single training instance in isolation
w(i+1) = w(i) + f(xt ,yt)− f(xt ,y′)
I Compare decision trees that perform batch learningI All training instances are used to find best split
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Linear Classifiers
Margin
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Linear Classifiers
Margin
Training Testing
Denote thevalue of themargin by γ
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Linear Classifiers
Maximizing Margin (i)
I For a training set T , the margin of a weight vector w is thesmallest γ such that
w · f(xt ,yt)−w · f(xt ,y′) ≥ γ
for every training instance (xt ,yt) ∈ T , y′ ∈ Yt
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Linear Classifiers
Maximizing Margin (ii)
I Intuitively maximizing margin makes sense
I More importantly, generalization error to unseen test data isproportional to the inverse of the margin (for the proof, seeDaume’, 2012: 45-46)
ε ∝ R2
γ2 × |T |
I Perceptron: we have shown that:I If a training set is separable by some margin, the perceptron
will find a w that separates the dataI However, the perceptron does not pick w to maximize the
margin!
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Linear Classifiers
Maximizing Margin (iii)
Let γ > 0max||w||≤1
γ
such that:w · f(xt ,yt)−w · f(xt ,y
′) ≥ γ
∀(xt ,yt) ∈ T
and y′ ∈ Yt
I Note: algorithm still minimizes error
I ||w|| is bound since scaling trivially produces larger margin
β(w · f(xt ,yt)−w · f(xt ,y′)) ≥ βγ, for some β ≥ 1
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Linear Classifiers
Max Margin = Min Norm
Let γ > 0
Max Margin:
max||w||≤1
γ
such that:
w·f(xt ,yt)−w·f(xt ,y′) ≥ γ
∀(xt ,yt) ∈ T
and y′ ∈ Yt
=
Min Norm:
minw
1
2||w||2
such that:
w·f(xt ,yt)−w·f(xt ,y′) ≥ 1
∀(xt ,yt) ∈ T
and y′ ∈ Yt
I Instead of fixing ||w|| we fix the margin γ = 1
I Technically γ ∝ 1/||w||
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Linear Classifiers
Support Vector Machines
a.k.a. SVM(s)
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Linear Classifiers
Support Vector Machines (i)
min1
2||w||2
such that:w · f(xt ,yt)−w · f(xt ,y
′) ≥ 1
∀(xt ,yt) ∈ T
and y′ ∈ Yt
I Quadratic programming problem – a well known convexoptimization problem
I Can be solved with out-of-the-box algorithms
I Batch learning algorithm – w set w.r.t. all training points
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Linear Classifiers
Support Vector Machines (ii)
I Problem: Sometimes |T | is far too large
I Thus the number of constraints might make solving thequadratic programming problem very difficult
I Common technique: Sequential Minimal Optimization (SMO)
I Sparse: solution depends only on features in support vectors
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Linear Classifiers
MIRA
Margin Infused Relaxed Algorithm
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Linear Classifiers
Margin Infused Relaxed Algorithm (MIRA)
I Another option – maximize margin using an online algorithmI Batch vs. Online
I Batch – update parameters based on entire training set (SVM)I Online – update parameters based on a single training instance
at a time (Perceptron)
I MIRA can be thought of as a max-margin perceptron or anonline SVM
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Linear Classifiers
MIRA
Batch (SVMs):
min1
2||w||2
such that:
w·f(xt ,yt)−w·f(xt ,y′) ≥ 1
∀(xt ,yt) ∈ T and y′ ∈ Yt
Online (MIRA):
Training data: T = {(xt ,yt)}|T |t=1
1. w(0) = 0; i = 02. for n : 1..N3. for t : 1..T4. w(i+1) = arg minw*
∥∥w*−w(i)∥∥
such that:w · f(xt ,yt)−w · f(xt ,y
′) ≥ 1∀y′ ∈ Yt
5. i = i + 16. return wi
I MIRA has much smaller optimizations with only |Yt |constraints
I Cost: sub-optimal optimization
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Linear Classifiers
Interim Summary
What we have coveredI Linear classifiers:
I PerceptronI SVMsI MIRA
I All are trained to minimize errorI With or without maximizing marginI Online or batch
What is nextI Logistic Regression
I Train linear classifiers to maximize likelihood
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Linear Classifiers
Logistic Regression
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Linear Classifiers
Logistic Regression (i)
Define a conditional probability:
P(y|x) =ew·f(x,y)
Zx, where Zx =
∑y′∈Y
ew·f(x,y′)
Note: still a linear classifier
arg maxy
P(y|x) = arg maxy
ew·f(x,y)
Zx
= arg maxy
ew·f(x,y)
= arg maxy
w · f(x,y)
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Linear Classifiers
Logistic Regression (ii)
P(y|x) =ew·f(x,y)
Zx
I Q: How do we learn weights w
I A: Set weights to maximize log-likelihood of training data:
w = arg maxw
∏t
P(yt |xt) = arg maxw
∑t
log P(yt |xt)
I In a nut shell we set the weights w so that we assign as muchprobability to the correct label y for each x in the training set
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Linear Classifiers
Logistic Regression
P(y|x) =ew·f(x,y)
Zx, where Zx =
∑y′∈Y
ew·f(x,y′)
w = arg maxw
∑t
log P(yt |xt) (*)
I The objective function (*) is concave
I Therefore there is a global maximumI No closed form solution, but lots of numerical techniques
I Gradient methods (gradient ascent, iterative scaling)I Newton methods (limited-memory quasi-newton)
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Linear Classifiers
Logistic Regression Summary
I Define conditional probability
P(y|x) =ew·f(x,y)
Zx
I Set weights to maximize log-likelihood of training data:
w = arg maxw
∑t
log P(yt |xt)
I Can find the gradient and run gradient ascent (or anygradient-based optimization algorithm)
OF (w) = (∂
∂w0F (w),
∂
∂w1F (w), . . . ,
∂
∂wmF (w))
∂
∂wiF (w) =
∑t
fi (xt ,yt)−∑
t
∑y′∈Y
P(y′|xt)fi (xt ,y′)
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Linear Classifiers
Linear Classification: Summary
I Basic form of (multiclass) classifier:
y = arg maxy
w · f(x,y)
I Different learning methods:I Perceptron – separate data (0-1 loss, online)I Support vector machine – maximize margin (hinge loss, batch)I Logistic regression – maximize likelihood (log loss, batch)
I All three methods are widely used in NLP
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Linear Classifiers
Aside: Min error versus max log-likelihood (i)
I Highly related but not identical
I Example: consider a training set T with 1001 points
1000× (xi ,y = 0) = [−1, 1, 0, 0] for i = 1 . . . 1000
1× (x1001,y = 1) = [0, 0, 3, 1]
I Now consider w = [−1, 0, 1, 0]
I Error in this case is 0 – so w minimizes error
[−1, 0, 1, 0] · [−1, 1, 0, 0] = 1 > [−1, 0, 1, 0] · [0, 0,−1, 1] = −1
[−1, 0, 1, 0] · [0, 0, 3, 1] = 3 > [−1, 0, 1, 0] · [3, 1, 0, 0] = −3
I However, log-likelihood = −126.9 (omit calculation)
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Linear Classifiers
Aside: Min error versus max log-likelihood (ii)
I Highly related but not identical
I Example: consider a training set T with 1001 points
1000× (xi ,y = 0) = [−1, 1, 0, 0] for i = 1 . . . 1000
1× (x1001,y = 1) = [0, 0, 3, 1]
I Now consider w = [−1, 7, 1, 0]
I Error in this case is 1 – so w does not minimizes error
[−1, 7, 1, 0] · [−1, 1, 0, 0] = 8 > [−1, 7, 1, 0] · [0, 0,−1, 1] = −1
[−1, 7, 1, 0] · [0, 0, 3, 1] = 3 < [−1, 7, 1, 0] · [3, 1, 0, 0] = 4
I However, log-likelihood = -1.4
I Better log-likelihood and worse error
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Linear Classifiers
Aside: Min error versus max log-likelihood (iii)
I Max likelihood 6= min errorI Max likelihood pushes as much probability on correct labeling
of training instanceI Even at the cost of mislabeling a few examples
I Min error forces all training instances to be correctly classified
I SVMs with slack variables – allows some examples to beclassified wrong if resulting margin is improved on otherexamples
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Linear Classifiers
Aside: Max margin versus max log-likelihood
I Let’s re-write the max likelihood objective function
w = arg maxw
∑t
log P(yt |xt)
= arg maxw
∑t
logew·f(xt ,yt)∑
y′∈Y ew·f(x,y′)
= arg maxw
∑t
w · f(xt ,yt)− log∑y′∈Y
ew·f(x,y′)
I Pick w to maximize score difference between correct labelingand every possible labeling
I Margin: maximize difference between correct and all incorrect
I The above formulation is often referred to as the soft-margin
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Linear Classifiers
Aside: Logistic Regression = MaximumEntropy
I Well known equivalenceI Max Ent: maximize entropy subject to constraints on features
I Empirical feature counts must equal expected counts
I Quick intuitionI Partial derivative in logistic regression
∂
∂wiF (w) =
∑t
fi (xt ,yt)−∑
t
∑y′∈Y
P(y′|xt)fi (xt ,y′)
I First term is empirical feature counts and second term isexpected counts
I Derivative set to zero maximizes functionI Therefore when both counts are equivalent, we optimize the
logistic regression objective!
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Appendix
Proofs and Derivations
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Convergence Proof for Perceptron
Perceptron Learning AlgorithmTraining data: T = {(xt , yt )}
|T |t=1
1. w(0) = 0; i = 02. for n : 1..N3. for t : 1..T
4. Let y′ = arg maxy w(i) · f(xt , y)
5. if y′ 6= yt
6. w(i+1) = w(i) + f(xt , yt ) − f(xt , y′)7. i = i + 1
8. return wi
I w (k−1) are the weights before kth
mistake
I Suppose kth mistake made at thetth example, (xt , yt)
I y′ = arg maxy w(k−1) · f(xt , y)
I y′ 6= yt
I w (k) = w(k−1) + f(xt , yt)− f(xt , y′)
I Now: u · w(k) = u · w(k−1) + u · (f(xt , yt)− f(xt , y′)) ≥ u · w(k−1) + γ
I Now: w(0) = 0 and u · w(0) = 0, by induction on k, u · w(k) ≥ kγ
I Now: since u · w(k) ≤ ||u|| × ||w(k)|| and ||u|| = 1 then ||w(k)|| ≥ kγ
I Now:
||w(k)||2 = ||w(k−1)||2 + ||f(xt , yt)− f(xt , y′)||2 + 2w(k−1) · (f(xt , yt)− f(xt , y
′))
||w(k)||2 ≤ ||w(k−1)||2 + R2
(since R ≥ ||f(xt , yt)− f(xt , y′)||
and w(k−1) · f(xt , yt)− w(k−1) · f(xt , y′) ≤ 0)
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Convergence Proof for Perceptron
Perceptron Learning Algorithm
I We have just shown that ||w(k)|| ≥ kγ and||w(k)||2 ≤ ||w(k−1)||2 + R2
I By induction on k and since w(0) = 0 and ||w(0)||2 = 0
||w(k)||2 ≤ kR2
I Therefore,k2γ2 ≤ ||w(k)||2 ≤ kR2
I and solving for k
k ≤ R2
γ2
I Therefore the number of errors is bounded!
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Gradient Ascent for Logistic Regression
Gradient Ascent
I Let F (w) =∑
t log ew·f(xt ,yt )
Zx
I Want to find arg maxw F (w)I Set w0 = Om
I Iterate until convergence
wi = wi−1 + αOF (wi−1)
I α > 0 and set so that F (wi ) > F (wi−1)I OF (w) is gradient of F w.r.t. w
I A gradient is all partial derivatives over variables wi
I i.e., OF (w) = ( ∂∂w0
F (w), ∂∂w1
F (w), . . . , ∂∂wm
F (w))
I Gradient ascent will always find w to maximize F
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Gradient Ascent for Logistic Regression
The partial derivatives
I Need to find all partial derivatives ∂∂wi
F (w)
F (w) =∑
t
log P(yt |xt)
=∑
t
logew·f(xt ,yt)∑
y′∈Y ew·f(xt ,y′)
=∑
t
loge
Pj wj×fj (xt ,yt)∑
y′∈Y eP
j wj×fj (xt ,y′)
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Gradient Ascent for Logistic Regression
Partial derivatives - some reminders
1. ∂∂x log F = 1
F∂∂x F
I We always assume log is the natural logarithm loge
2. ∂∂x eF = eF ∂
∂x F
3. ∂∂x
∑t Ft =
∑t
∂∂x Ft
4. ∂∂x
FG =
G ∂∂x
F−F ∂∂x
G
G2
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Gradient Ascent for Logistic Regression
The partial derivatives
∂
∂wiF (w) =
∂
∂wi
∑t
loge
Pj wj×fj (xt ,yt)∑
y′∈Y eP
j wj×fj (xt ,y′)
=∑
t
∂
∂wilog
eP
j wj×fj (xt ,yt)∑y′∈Y e
Pj wj×fj (xt ,y′)
=∑
t
(
∑y′∈Y e
Pj wj×fj (xt ,y
′)
eP
j wj×fj (xt ,yt))(
∂
∂wi
eP
j wj×fj (xt ,yt)∑y′∈Y e
Pwj
wj×fj (xt ,y′))
=∑
t
(Zxt
eP
j wj×fj (xt ,yt))(
∂
∂wi
eP
j wj×fj (xt ,yt)
Zxt
)
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Gradient Ascent for Logistic Regression
The partial derivativesNow,
∂
∂wi
eP
j wj×fj (xt ,yt )
Zxt
=Zxt
∂∂wi
eP
j wj×fj (xt ,yt ) − eP
j wj×fj (xt ,yt ) ∂∂wi
Zxt
Z 2xt
=Zxt e
Pj wj×fj (xt ,yt )fi (xt , yt) − e
Pj wj×fj (xt ,yt ) ∂
∂wiZxt
Z 2xt
=e
Pj wj×fj (xt ,yt )
Z 2xt
(Zxt fi (xt , yt) −∂
∂wiZxt )
=e
Pj wj×fj (xt ,yt )
Z 2xt
(Zxt fi (xt , yt)
−X
y′∈Y
eP
j wj×fj (xt ,y′)fi (xt , y
′))
because
∂
∂wiZxt =
∂
∂wi
Xy′∈Y
eP
j wj×fj (xt ,y′) =
Xy′∈Y
eP
j wj×fj (xt ,y′)fi (xt , y
′)
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Gradient Ascent for Logistic Regression
The partial derivativesFrom before,
∂
∂wi
eP
j wj×fj (xt ,yt )
Zxt
=e
Pj wj×fj (xt ,yt )
Z 2xt
(Zxt fi (xt , yt)
−X
y′∈Y
eP
j wj×fj (xt ,y′)fi (xt , y
′))
Sub this in,
∂
∂wiF (w) =
Xt
(Zxt
eP
j wj×fj (xt ,yt ))(
∂
∂wi
eP
j wj×fj (xt ,yt )
Zxt
)
=X
t
1
Zxt
(Zxt fi (xt , yt) −X
y′∈Y
eP
j wj×fj (xt ,y′)fi (xt , y
′)))
=X
t
fi (xt , yt) −X
t
Xy′∈Y
eP
j wj×fj (xt ,y′)
Zxt
fi (xt , y′)
=X
t
fi (xt , yt) −X
t
Xy′∈Y
P(y′|xt)fi (xt , y′)
Machine Learning for Language Technology 54(55)
Gradient Ascent for Logistic Regression
FINALLY!!!
I After all that,
∂
∂wiF (w) =
∑t
fi (xt ,yt)−∑
t
∑y′∈Y
P(y′|xt)fi (xt ,y′)
I And the gradient is:
OF (w) = (∂
∂w0F (w),
∂
∂w1F (w), . . . ,
∂
∂wmF (w))
I So we can now use gradient assent to find w!!
Machine Learning for Language Technology 55(55)