Transcript Homotopy-based Semi-Supervised Hidden Markov Models for

Homotopy-based SemiSupervised Hidden Markov
Models for Sequence Labeling
Gholamreza Haffari
Anoop Sarkar
Presenter: Milan Tofiloski
Natural Language Lab
Simon Fraser university
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Outline
• Motivation & Contributions
• Experiments
• Homotopy method
• More experiments
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Maximum Likelihood Principle
• Parameter setting for the joint probability of
input-output which maximizes probability of
the given data:
• L : labeled data
• U : unlabeled data
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Deficiency of MLE
• Usually |U| >> |L|, then
• Which means the relationship of input-output
is ignored when estimating the parameters !
– MLE focuses on modeling the input distribution P(x)
– But we are interested in modeling the joint distribution
P(x,y)
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Remedy for the Deficiency
• Balance the effect of lab and unlab data:
• Find
which maximally take
advantage of lab and unlab data
•
MLE
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An experiment with HMM
Lower is Better
MLE Performance
• MLE can hurt the performance
• Balancing lab and unlab data related terms is beneficial
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Our Contributions
1. Introducing a principled way to
choose  for HMM in sequence
labeling (tagging) tasks
2. Introducing an efficient dynamic
programming algorithm to compute
second order statistics in HMM
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Outline
• Motivation & Contributions
• Experiments
• Homotopy method
• More experiments
8
Task
• Field segmentation in information extraction
• 13 tag fields: AUTHOR, TITLE, …
EDITOR
EDITOR
A
.
TITLE
Models
for
-
EDITOR
Elmagarmid
TITLE
PUB
EDITOR
TITLE
,
EDITOR
EDITOR
editor
TITLE
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Transaction
TITLE
Advanced Database Applications
PUB
Kaufmann
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DATE
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1992
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TITLE
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Morgan
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Experimental Setup
• Use an HMM with 13 states
– Freeze the transition (state->state) probabilities to what
has been observed in the lab data
– Use the Homotopy method to just learn the emission
(state->alphabet) probabilities
– Do add- smoothing for the initial values of emission
and transition probabilities
• Data statistics:
– Average seq. length : 36.7
– Average number of segments in a seq: 5.4
– Size of Lab/Unlab data is 300/700
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Baselines
• Held-out: put aside part of the lab data as
a held-out set, and use it t choose 
• Oracle: choose  based on test data using
per position accuracy
• Supervised: forgetting about unlab data,
and just using lab data
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Homotopy vs Baselines
Higher is Better
Even very small values of  can be useful.
In homotopy =.004, and in supervised  = 0
• Sequence of most probable states decoding
See paper for more results
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Outline
• Motivation & Contributions
• Experiments
• Homotopy method
• More experiments
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Path of Solutions
• Look at  as  changes from 0 to 1
• Choose the best  based on the path





Discontinuity

Bifurcation
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EMfor HMM
•
Let
•
To find best parameter values  which (locally) maximizes the
objective function for a fixed :
be a state->state or state->observation event in our HMM
Repeat until convergence
EM()
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Fixed Points of EM
• Useful fact
• At the fixed points
, then
• This is similar to using Homotopy for root finding
– Same numerical techniques should be applicable here
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Homotopy for Root Finding
• To find a root of G()
– start from a root of a simple problem F()
– trace the roots of intermediate problems while
morphing F to G
• To find  which satisfy the above:
– Set the derivative to zero: gives differential equation
– Numerically solve the resulting differential eqn.
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Solving the Differential Eqn
Jaccobian of EM1
M
.
v=0
Repeat until
–
Update
in a proper direction parallel to v=Kernel(M)
– Update M
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Jaccobian of EM1
See the paper for details
• So, we need to compute the covariance
matrix of the events
• The entry in the row
and column
of
the covariance matrix is
Challenging for HMM
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Forward-Backward
Expected Quadratic Counts for HMM
• Dynamic programming algorithm to efficiently
compute
• Pre-compute a table Zx for each sequence
…
k2
k1
… xi
xi+1
…
xj
…
…
• Having table Zx, the EQC can be computed efficiently
– The time complexity is
where K is the number of states
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in the HMM (see paper for more details)
How to Choose  based on Path
• monotone: the first point at which the
monotonocity of  changes
• MaxEnt: choose  for which the model has
maximum entropy on the unlab data
• minEig: when solving the diff eqn, consider
the minimum singular value of the matrix M.
Across rounds, choose  for which the
minimum singular value is the smallest
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Outline
• Motivation & Contributions
• Experiments
• Homotopy method
• More experiments
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Varying the Size of Unlab Data
• The three Homotopy-based methods outperform EM
• maxEnt outperforms minEig and monotone
• minEig and monotone have similar performances
Size of the labeled data: 100
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Picked  Values
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Picked  Values
• EM gives higher weight to
unlabeled data compared to
Homotopy-based method
•  selected by
− maxEnt are much smaller than those
selected by minEig and monotone
− minEig and monotone are close
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Conclusion and Future Work
• Using EM can hurt performance in the
case |L| << |U|
• Proposed a method to alleviate this
problem for HMMs for seq. labeling tasks
• To speed up the method
– Using sampling to find approximation to
covariance matrix
– Using faster methods in recovering the
solution path, e.g. predictor-corrector
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Questions?
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Is Oracle outperformed by Homotopy?
• No!
- The performance measure used in selecting 
in oracle method may be different from that
used in comparing homotopy and oracle
- The decoding alg used in oracle may be
different from that used in comparing
homotopy and oracle
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Why not set
?
• This adhoc way of setting  has two
drawbacks:
- It still may hurt the performance. The proper  may be
much smaller than that.
- In some situations, the right choice of  may be a big
value. Setting
is very conservative and
dose not fully take advantage of the available
unlabeled data.
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Homotopy vs Baselines
Our method
(see the paper
for more results)
Higher is Better
– Viterbi Decoding: most probable seq of states decoding
– SMS Decoding: seq of most probable states decoding 30