1. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Part 4: Conditional Random Fields
Sebastian Nowozin and Christoph H. Lampert
Colorado Springs, 25th June 2011
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2. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Problem (Probabilistic Learning)
Let d(y|x) be the (unknown) true conditional distribution.
Let D = {(x1 , y 1 ), . . . , (xN , y N )} be i.i.d. samples from d(x, y).
Find a distribution p(y|x) that we can use as a proxy for d(y|x).
or
Given a parametrized family of distributions, p(y|x, w), find the
parameter w∗ making p(y|x, w) closest to d(y|x).
Open questions:
What do we mean by closest?
What’s a good candidate for p(y|x, w)?
How to actually find w∗ ?
conceptually, and
numerically
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3. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Principle of Parsimony (Parsimoney, aka Occam’s razor)
“Pluralitas non est ponenda sine neccesitate.”
William of Ockham
“We are to admit no more causes of natural things than such as are both
true and sufficient to explain their appearances.”
Isaac Newton
“Make everything as simple as possible, but not simpler.”
Albert Einstein
“Use the simplest explanation that covers all the facts.”
what we’ll use
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4. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
1) Define what aspects we consider relevant facts about the data.
2) Pick the simplest distribution reflecting that.
Definition (Simplicity ≡ Entropy)
The simplicity of a distribution p is given by its entropy:
H(p) = − p(z) log p(z)
z∈Z
Definition (Relevant Facts ≡ Feature Functions)
By φi : Z → R for i = 1, . . . , D we denote a set of feature functions that
express everything we want to be able to model about our data.
For example: the grayvalue of a pixel,
a bag-of-words histogram of an image,
the time of day an image was taken,
a flag if a pixel is darker than half of its neighbors.
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5. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Principle (Maximum Entropy Principle)
Let z 1 , . . . , z N be samples from a distribution d(z). Let φ1 , . . . , φD be
1
feature functions, and denote by µi := N n φi (z n ) their means over the
sample set.
The maximum entropy distribution, p, is the solution to
max H(p) subject to Ez∼p(z) {φi (z)} = µi .
p is a prob.distr.
be faithful to what we know
be as simple as possible
Theorem (Exponential Family Distribution)
Under some very reasonable conditions, the maximum entropy distribution
has the form
1
p(z) = exp wi φi (z)
Z i
for some parameter vector w = (w1 , . . . , wD ) and constant Z.
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6. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Example:
Let Z = R, φ1 (z) = z, φ2 (z) = z 2 .
The exponential family distribution is
1
p(z) = exp( w1 z + w2 z 2 )
Z(w)
b2 a 2 w1
= exp( a z − b ) for a = w2 , b = − .
Z(a, b) w2
It’s a Gaussian!
Given examples z 1 , . . . , z N , we can compute a and b, and derive w.
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7. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Example:
Let Z = {1, . . . , K}, φk (z) = z = k , for k = 1, . . . , K.
The exponential family distribution is
1
p(z) = exp( wk φk (z) )
Z(w)
k
exp(w1 )/Z for z = 1,
exp(w )/Z for z = 2,
2
=
. . .
exp(w )/Z for z = K.
K
with Z = exp(w1 ) + · · · + exp(wK ).
It’s a Multinomial!
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8. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Example:
Let Z = {0, 1}N ×M image grid,
φi (y) := yi for each pixel i,
φN M (y) = i∼j yi = yj (summing over all 4-neighbor pairs)
The exponential family distribution is
1
p(z) = exp( w, φ(y) + w
˜ yi = yj )
Z(w)
i,j
It’s a (binary) Markov Random Field!
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9. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Conditional Random Field Learning
Assume:
a set of i.i.d. samples D = {(xn , y n )}n=1,...,N , (xn , y n ) ∼ d(x, y)
feature functions ( φ1 (x, y), . . . , φD (x, y) ) ≡: φ(x, y)
1
parametrized family p(y|x, w) = Z(x,w) exp( w, φ(x, y) )
Task:
adjust w of p(y|x, w) based on D.
Many possible technique to do so:
Expectation Matching
Maximum Likelihood
Best Approximation
MAP estimation of w
Punchline: they all turn out to be (almost) the same!
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10. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Maximum Likelihood Parameter Estimation Idea: maximize conditional
likelihood of observing outputs y 1 , . . . , y N for inputs x1 , . . . , xN
w∗ = argmax p(y 1 , . . . , y N |x1 , . . . , xN , w)
w∈RD
N
i.i.d.
= argmax p(y n |xn , w)
w∈RD n=1
N
− log(·)
= argmin − log p(y n |xn , w)
w∈RD n=1
negative conditional log-likelihood (of D)
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11. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Best Approximation
Idea: find p(y|x, w) that is closest to d(y|x)
Definition (Similarity between conditional distributions)
For fixed x ∈ X : KL-divergence measure similarity
d(y|x)
KLcond (p||d)(x) := d(y|x) log
p(y|x, w)
y∈Y
For x ∼ d(x), compute expectation:
KLtot (p||d) : = Ex∼d(x) KLcond (p||d)(x)
d(y|x)
= d(x, y) log
p(y|x, w)
x∈X y∈Y
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12. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Best Approximation
Idea: find p(y|x, w) of minimal KLtot -distance to d(y|x)
d(y|x)
w∗ = argmin d(x, y) log
w∈RD x∈X y∈Y p(y|x, w)
drop const.
= argmin − d(x, y) log p(y|x, w)
w∈RD (x,y)∈X ×Y
N
(xn ,y n )∼d(x,y)
≈ argmin − log p(y n |xn , w)
w∈RD n=1
negative conditional log-likelihood (of D)
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13. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
MAP Estimation of w Idea: Treat w as random variable; maximize
posterior probability p(w|D)
N
p(x1 , y 1 , . . . , xn , y n |w)p(w) i.i.d.
Bayes p(y n |xn , w)
p(w|D) = = p(w)
p(D) p(y n |xn )
n=1
p(w): prior belief on w (cannot be estimated from data).
w∗ = argmax p(w|D) = argmin − log p(w|D)
w∈RD w∈RD
N
= argmin − log p(w) − log p(y n |xn , w) + log p(y n |xn )
w∈RD n=1
indep. of w
N
= argmin − log p(w) − log p(y n |xn , w)
w∈RD n=1
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14. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
N
w∗ = argmin − log p(w) − log p(y n |xn , w)
w∈RD n=1
Choices for p(w):
p(w) :≡ const. (uniform; in RD not really a distribution)
N
w∗ = argmin − log p(y n |xn , w) + const.
w∈RD n=1
negative conditional log-likelihood
1 2
p(w) := const. · e− 2σ2 w
(Gaussian)
N
1
w∗ = argmin − w 2
+ log p(y n |xn , w) + const.
w∈RD 2σ 2
n=1
regularized negative conditional log-likelihood
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15. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Probabilistic Models for Structured Prediction - Summary
Negative (Regularized) Conditional Log-Likelihood (of D)
N
1 2 w,φ(xn ,y)
L(w) = 2 w − w, φ(xn , y n ) − log e
2σ
n=1 y∈Y
(σ 2 → ∞ makes it unregularized)
Probabilistic parameter estimation or training means solving
w∗ = argmin L(w).
w∈RD
Same optimization problem as for multi-class logistic regression.
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17. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Steepest Descent Minimization – minimize L(w)
input tolerance > 0
1: wcur ← 0
2: repeat
3: v ← w L(wcur )
4: η ← argminη∈R L(wcur − ηv)
5: wcur ← wcur − ηv
6: until v <
output wcur
Alternatives:
L-BFGS (second-order descent without explicit Hessian)
Conjugate Gradient
We always need (at least) the gradient of L.
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18. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
N
1 2 w,φ(xn ,y)
L(w) = w − w, φ(xn , y n ) + log e
2σ 2
n=1 y∈Y
N
e w,φ(xn ,y) φ(xn , y)
1 n n y∈Y
w L(w) = 2 w − φ(x , y ) − w,φ(xn ,¯)
y
σ y ∈Y
¯ e
n=1
N
1
= w− φ(xn , y n ) − p(y|xn , w)φ(xn , y)
σ2
n=1 y∈Y
N
1
= w− φ(xn , y n ) − Ey∼p(y|xn ,w) φ(xn , y)
σ2
n=1
N
1
∆L(w) = IdD×D + [Ey∼p(y|xn ,w) φ(xn , y)][Ey∼p(y|xn ,w) φ(xn , y)]
σ2
n=1
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19. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
N
1 2 w,φ(xn ,y)
L(w) = w − w, φ(xn , y n ) + log e
2σ 2
n=1 y∈Y
C ∞ -differentiable on all RD .
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20. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
N
1
w L(w) = w− φ(xn , y n ) − Ey∼p(y|xn ,w) φ(xn , y)
σ2
n=1
For σ → ∞:
Ey∼p(y|xn ,w) φ(xn , y) = φ(xn , y n ) ⇒ w L(w) = 0,
criticial point of L (local minimum/maximum/saddle point).
Interpretation:
We aim for expectation matching: Ey∼p φ(x, y) = φ(x, y obs )
but discriminatively: only for x ∈ {x1 , . . . , xn }.
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21. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
N
1
∆L(w) = IdD×D + [Ey∼p(y|xn ,w) φ(xn , y)][Ey∼p(y|xn ,w) φ(xn , y)]
σ2
n=1
positive definite Hessian matrix → L(w) is convex
→ w L(w) = 0 implies global minimum.
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22. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Milestone I: Probabilistic Training (Conditional Random Fields)
p(y|x, w) log-linear in w ∈ RD .
Training: many probabilistic derivations lead to same optimization
problem → minimize negative conditional log-likelihood, L(w)
L(w) is differentiable and convex,
→ gradient descent will find global optimum with w L(w) =0
Same structure as multi-class logistic regression.
For logistic regression: this is where the textbook ends. we’re done.
For conditional random fields: we’re not in safe waters, yet!
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23. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Task: Compute v = w L(wcur ), evaluate L(wcur + ηv):
N
1 2 w,φ(xn ,y)
L(w) = w − w, φ(xn , y n ) + log e
2σ 2
n=1 y∈Y
N
1
w L(w) = w− φ(xn , y n ) − p(y|xn , w)φ(xn , y)
σ2
n=1 y∈Y
Problem: Y typically is very (exponentially) large:
binary image segmentation: |Y| = 2640×480 ≈ 1092475
ranking N images: |Y| = N !, e.g. N = 1000: |Y| ≈ 102568 .
We must use the structure in Y, or we’re lost.
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24. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Solving the Training Optimization Problem Numerically
N
1
w L(w) = w− φ(xn , y n ) − Ey∼p(y|xn ,w) φ(xn , y)
σ2
n=1
Computing the Gradient (naive): O(K M N D)
N
1 2
L(w) = 2 w − w, φ(xn , y n ) + log Z(xn , w)
2σ
n=1
Line Search (naive): O(K M N D) per evaluation of L
N : number of samples
D: dimension of feature space
M : number of output nodes ≈ 100s to 1,000,000s
K: number of possible labels of each output nodes ≈ 2 to 100s
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25. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Probabilistic Inference to the Rescue Remember: in a graphical model with
factors F, the features decompose:
φ(x, y) = φF (x, yF )
F ∈F
Ey∼p(y|x,w) φ(x, y) = Ey∼p(y|x,w) φF (x, yF )
F ∈F
= EyF ∼p(yF |x,w) φF (x, yF ) F ∈F
EyF ∼p(yF |x,w) φF (x, yF ) = p(yF |x, w) φF (x, yF )
yF ∈YF
factor marginals
K |F | terms
Factor marginals µF = p(yF |x, w)
are much smaller than complete joint distribution p(y|x, w),
can be computed/approximated, e.g., with (loopy) belief propagation.
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26. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Solving the Training Optimization Problem Numerically
N
1
w L(w) = w− φ(xn , y n ) − Ey∼p(y|xn ,w) φ(xn , y)
σ2
n=1
Computing the Gradient: X O(M K |Fmax | N D):
O(K MX X
XXX
nd),
N
1 2 w,φ(xn ,y)
L(w) = w − w, φ(xn , y n ) + log e
2σ 2
n=1 y∈Y
Line Search: XX O(M K |Fmax | N D) per evaluation of L
O(K MX X
X X
nd),
N : number of samples ≈ 10s to 1,000,000s
D: dimension of feature space
M : number of output nodes
K: number of possible labels of each output nodes
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27. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
What, if the training set D is too large (e.g. millions of examples)?
Simplify Model
Train simpler model (smaller factors)
Less expressive power ⇒ results might get worse rather than better
Subsampling
Create random subset D ⊂ D. Train model using D
Ignores all information in D D
Parallelize
Train multiple models in parallel. Merge the models.
Follows the multi-core/GPU trend
How to actually merge? (or ?)
Doesn’t really save computation.
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28. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
What, if the training set D is too large (e.g. millions of examples)?
Stochastic Gradient Descent (SGD)
Keep maximizing p(w|D).
In each gradient descent step:
Create random subset D ⊂ D, ← often just 1–3 elements!
Follow approximate gradient
˜w L(w) = 1 w − φ(xn , y n ) − Ey∼p(y|xn ,w) φ(xn , y)
σ2
(xn ,y n )∈D
Line search no longer possible. Extra parameter: stepsize η
SGD converges to argminw L(w)! (if η chosen right)
SGD needs more iterations, but each one is much faster
more: see L. Bottou, O. Bousquet: ”The Tradeoffs of Large Scale Learning”, NIPS 2008.
also: http://leon.bottou.org/research/largescale
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29. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Solving the Training Optimization Problem Numerically
N
1
w L(w) = w− φ(xn , y n ) − Ey∼p(y|xn ,w) φ(xn , y)
σ2
n=1
Computing the Gradient: X O(M K 2 N D) (if BP is possible):
O(K MX X
XXX
nd),
N
1 2 w,φ(xn ,y)
L(w) = w − w, φ(xn , y n ) + log e
2σ 2
n=1 y∈Y
Line Search: XX O(M K 2 N D) per evaluation of L
O(K MX X
X X
nd),
N : number of samples
D: dimension of feature space: ≈ φi,j 1–10s, φi : 100s to 10000s
M : number of output nodes
K: number of possible labels of each output nodes
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30. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Typical feature functions in image segmentation:
φi (yi , x) ∈ R≈1000 : local image features, e.g. bag-of-words
→ wi , φi (yi , x) : local classifier (like logistic-regression)
φi,j (yi , yj ) = yi = yj ∈ R1 : test for same label
→ wij , φij (yi , yj ) : penalizer for label changes (if wij 0)
combined: argmaxy p(y|x) is smoothed version of local cues
original local classification local + smoothness
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31. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Typical feature functions in handwriting recognition:
φi (yi , x) ∈ R≈1000 : image representation (pixels, gradients)
→ wi , φi (yi , x) : local classifier if xi is letter yi
φi,j (yi , yj ) = eyi ⊗ eyj ∈ R26·26 : letter/letter indicator
→ wij , φij (yi , yj ) : encourage/suppress letter combinations
combined: argmaxy p(y|x) is ”corrected” version of local cues
local classification local + ”correction”
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32. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Typical feature functions in pose estimation:
φi (yi , x) ∈ R≈1000 : local image representation, e.g. HoG
→ wi , φi (yi , x) : local confidence map
φi,j (yi , yj ) = good fit(yi , yj ) ∈ R1 : test for geometric fit
→ wij , φij (yi , yj ) : penalizer for unrealistic poses
together: argmaxy p(y|x) is sanitized version of local cues
original local classification local + geometry
[V. Ferrari, M. Marin-Jimenez, A. Zisserman: ”Progressive Search Space Reduction for Human Pose Estimation”, CVPR 2008.]
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33. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Typical feature functions for CRFs in Computer Vision:
φi (yi , x): local representation, high-dimensional
→ wi , φi (yi , x) : local classifier
φi,j (yi , yj ): prior knowledge, low-dimensional
→ wij , φij (yi , yj ) : penalize outliers
learning adjusts parameters:
unary wi : learn local classifiers and their importance
binary wij : learn importance of smoothing/penalization
argmaxy p(y|x) is cleaned up version of local prediction
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34. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Solving the Training Optimization Problem Numerically Idea: split
learning of unary potentials into two parts:
local classifiers,
their importance.
Two-Stage Training
pre-train fiy (x) = log p(yi |x)
ˆ
use φ˜i (yi , x) := f y (x) ∈ RK (low-dimensional)
i
keep φij (yi , yj ) are before
˜
perform CRF learning with φi and φij
Advantage:
lower dimensional feature space during inference → faster
fiy (x) can be stronger classifiers, e.g. non-linear SVMs
Disadvantage:
if local classifiers are bad, CRF training cannot fix that.
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35. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Solving the Training Optimization Problem Numerically CRF training
methods is based on gradient-descent optimization.
The faster we can do it, the better (more realistic) models we can use:
N
˜w L(w) = w − φ(xn , y n ) − p(y|xn , w) φ(xn , y) ∈ RD
σ2 n=1 y∈Y
A lot of research on accelerating CRF training:
problem ”solution” method(s)
|Y| too large exploit structure (loopy) belief propagation
smart sampling contrastive divergence
use approximate L e.g. pseudo-likelihood
N too large mini-batches stochastic gradient descent
D too large trained φunary two-stage training
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36. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Summary – CRF Learning Given:
i.i.d.
training set {(x1 , y 1 ), . . . , (xn , y n )} ⊂ X × Y, (xn , y n ) ∼ d(x, y)
feature function φ : X × RD .
1
Task: find parameter vector w such that Z exp( w, φ(x, y) ) ≈ d(y|x).
CRF solution derived by minimizing negative conditional log-likelihood:
N
∗ 1 2 w,φ(xn ,y)
w = argmin w − w, φ(xn , y n ) − log e
w 2σ 2
n=1 y∈Y
convex optimization problem → gradient descent works
training needs repeated runs of probabilistic inference
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37. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Extra I: Beyond Fully Supervised Learning So far, training was fully
supervised, all variables were observed.
In real life, some variables can be unobserved even during training.
missing labels in training data latent variables, e.g. part location
latent variables, e.g. part occlusion latent variables, e.g. viewpoint
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38. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Graphical model: three types of variables
x ∈ X always observed,
y ∈ Y observed only in training,
z ∈ Z never observed (latent).
Marginalization over Latent Variables
Construct conditional likelihood as usual:
1
p(y, z|x, w) = exp( w, φ(x, y, z) )
Z(x, w)
Derive p(y|x, w) by marginalizing over z:
1
p(y|x, w) = p(y, z|x, w)
Z(x, w)
z∈Z
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39. Sebastian Nowozin and Christoph Lampert – Structured Models in Computer Vision – Part 4. Conditional Random Fields
Negative regularized conditional log-likelihood:
N
2
L(w) = λ w − log p(y n |xn , w)
n=1
N
2
=λ w − log p(y n , z|xn , w)
n=1 z∈Z
N
2
=λ w − log exp( w, φ(xn , y n , z) )
n=1 z∈Z
N
+ log exp( w, φ(xn , y, z) )
n=1 z∈Z
y∈Y
L is not convex in w → can have local minima
no agreed on best way for treating latent variables
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