Transcript Unsupervised Disambiguation

Bootstrapping
•
Goals:
–
–
•
Utilize a minimal amount of (initial) supervision
Obtain learning from many unlabeled examples (vs.
selective sampling)
General scheme:
1.
2.
3.
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Initial supervision – seed examples for training an
initial model or seed model itself (indicative features)
Classify corpus with seed model
Add most confident classifications to training and
iterate.
Exploits feature redundancy in examples
Bootstrapping Decision List for
Word Sense Disambiguation
• Yarowsky (1995) scheme
• Relies on “one sense per collocation”
1. Initialize seed model with all collocations in the sense
dictionary definition – an initial decision list for each
sense.
– Alternatively – train from seed examples
2. Classify all examples with current model
– Any example containing a feature for some sense
3. Compute odds ratio for each feature-sense combination; if
above threshold then add the feature to the decision list
4. Optional: add the “one sense per discourse constraint” to
add/filter examples
5. Iterate (2)
“One Sense per Collocation”
Results
• Applies the “one sense per discourse” constraint after final
classification – classify all word occurrences in a
document by the majority sense
• Evaluation on ~37,000 examples:
Co-training for Name Classification
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Collins and Singer, EMNLP 1999
A bootstrapping approach that relies on a
(natural) partition of the feature space to two
different sub-spaces
The bootstrapping process iterates by switching
sub-space in each iteration
Name classification task: person, organization,
location
Features for decision list:
1. Spelling features (full, contains, cap, non-alpha)
2. Context features (words, syntactic type)
Seed Features
• Initialized with 0.9999 score for the decision list
DL-CoTrain Algorithm
1.
2.
3.
4.
Initialize n=5 (max. #rules learned in an iteration)
Initialize spelling decision list with seed
Classify corpus by spelling rules (where applicable)
Induce contextual rule list from classified
examples; keep at most the n most frequent rules
whose score>θ for each class
5. Classify corpus by contextual rules
6. Induce spelling rules, as in step 4, and add to list
7. If n<2500 then n n+5, and return to step 3.
Otherwise, classify corpus by combined list and
induce a final classifier from these labeled examples
Co-Train vs. Yarowsky’s Algorithm
• Collins & Singer implemented a cautious
version of Yarowsky’s algorithm, where n
rules are added at each iteration
• Same accuracy (~91%) (as well as a
boosting-based bootstrapping algorithm)
• Abney (ACL 2002) provides some analysis
for both algorithm, showing that they are
based on different independence assmptions
Clustering and Unsupervised
Disambiguation
• Word sense disambiguation without labeled
training or other information sources
• Cannot label to predefined senses (there are
none), so try to find “natural” senses
• Use clustering methods to divide different
contexts of a word into “sensible” classes
• Other applications of clustering:
– Thesaurus construction, document clustering
– Forming word classes for generalization in
language modeling and disambiguation
Clustering
Clustering Techniques
• Hierarchical:
– Bottom-up (agglomerative)
– Top-down (divisive)
• Flat (non-hierarchical)
– Usually iterative
– Soft (vs. hard) clustering
• “Degree” of membership
Comparison
Hierarchical:
• Preferable for detailed
data analysis
• More information
provided
• No clearly preferred
algorithm
• Less efficient (at least
O(n2))
Non-hierarchical:
• Preferable if efficiency
is important or lots of
data
• K-means is the
simplest method and
often good enough
• If no Euclidean space,
can use EM instead
Hierarchical Clustering
and
but
in
on
with
for
at
from
of
to
as
Similarity Functions
• sim(c1,c2) = similarity between clusters
– Defined over pairs of individual elements, and
(possibly inductively) over cluster pairs
• Values typically between 0 and 1
For hierarchical clustering, require:
• Monotonicity:
For all possible clusters,
min[sim(c1,c2),sim(c1,c3)]  sim(c1,c2  c3)
Merging two clusters cannot increase similarity!
Bottom-Up Clustering
Input: Set X = {x1,…,xn} of objects
Similarity function sim: 2X  2X
for i 1 to n do:
ci {xi}
C {ci, ..., cn}
j n + 1
while |C| > 1 do:
(ca, cb)  arg max(cu,cv) sim(cu,cv)
cj ca  cb; Trace merge to construct cluster hierarchy
C  (C \ {ca,cb})  {cj}
j++
Types of Cluster Pair Similarity
single link
similarity of most similar
(nearest) members
complete link similarity of most dissimilar
(remote) members
- property of unified cluster
group average average pairwise similarity of
members in unified cluster
- property of unified cluster
Single Link Clustering (“chaining” risk)
Complete Link Clustering
• “Global view” yields tighter clusters – usually
more suitable in NLP
Efficiency of Merge Step
• Maintain pairwise similarities between
clusters, and maximal similarity per cluster:
• Single link update - O(n):
– sim(c1  c2, c3) = max(sim(c1,c3),sim(c2,c3))
• Complete link update - O(n log(n)):
– Have to compare all pairwise similarities with
all points in merged cluster to find minimum
Group Average Clustering
• A “mid-point” between single and complete link –
produces relatively tight clusters
• Efficiency depends on choice of similarity metric
• Computing average similarity for a newly formed
cluster from scratch is O(n2)
– Goal – incremental computation
• Represent objects as m-dimensional unit vectors
(length-normalized), and use cosine for similarity:
 
xy
 
 
 
x y

i i i
sim ( x , y )  cos( x , y )    
 x y
2
2
x  y
x
y
i i i i
Cosine Similarity
cos 1 > cos 2 > cos 3
3
2
1
Efficient Cosine Similarity
• Average cosine similarity within a cluster c (to be
maximized after merge):
 
1
A(c) 
sim ( x , y )


c ( c  1) xc x  yc


• Define sum vector for cluster: s (c)   x


s (c )  s (c ) 
A(c)

xc
 
 x  y


xc yc

 
c ( c  1) A(c)   x  x

c ( c  1) A(c)  c


s (c )  s (c )  c


xc
c ( c  1)
A(c) 
A(ca  cb ) 


( s ( c ))  ( s ( c ))  c
c ( c  1)




( s ( ca )  s ( cb ))  ( s ( ca )  s ( cb ))  ( ca  c2 )
( ca  cb )( ca  cb  1)
• On merge, update sum:




s (c j )  s (ca  cb )  s (ca )  s (cb )
• Then update A for all new pairs of clusters
• Complexity: O(n) per iteration (assuming constant
dimensionality); Overall for algorithm: O(n2)
Top-Down Clustering
Input: Set X = {x1,…,xn} of objects
Coherence measure
coh: 2X
Splitting function
split: 2X 2X  2X
C { X }
j 1
while C contains a cluster larger than 1 element do:
ca  arg mincu coh(cu)
(cj+1,cj+2)  split(ca)
C  (C \ ca)  {cj+1,cj+2}
j += 2
Top-Down Clustering (cont.)
• Coherence measure – can be any type of cluster
quality function, including those used for
agglomerative clustering
– Single link – maximal similarity in cluster
– Complete link – minimal similarity in cluster
– Group average – average pairwise similarity
• Split – can be handled as a clustering problem on
its own, aiming to form two clusters
– Can use hierarchical clustering, but often more natural
to use non-hierarchical clustering (see next)
• May provide a more “global” view of the data (vs.
the locally greedy bottom-up procedure)
Non-Hierarchical Clustering
• Iterative clustering:
– Start with initial (random) set of clusters
– Assign each object to a cluster (or clusters)
– Re-compute cluster parameters
• E.g. centroids:


s (c ) 1
 (c ) 

x

| c | | c | xc

– Stop when clustering is “good”
• Q: How many clusters?
K-means Algorithm
Input: Set X = {x1,…,xn} of objects
Distance measure d: X  X 
Mean function : 2XX
Select k initial cluster centers f1, ..., fk
while not finished do:
for all clusters cj do:
cj  { xi | fj = arg minf d(xi, f) }
for all means fj do:
fj  (cj)
Complexity: O(n), assuming constant number of iterations
K-means Clustering
Example
Clustering of words from NY Times using
cooccurring words
1.
2.
3.
4.
5.
ballot, polls, Gov, seats
profit, finance, payments
NFL, Reds, Sox, inning, quarterback, score
researchers, science
Scott, Mary, Barbara, Edward
Buckshot Algorithm
• Often, random initialization for K-means works
well
• If not:
– Randomly choose n points
– Run hierarchical group average clustering:
O(( n)2)=O(n)
– Use those cluster means as starting points for K-means
– O(n) total complexity
• Scatter/Gather (Cutting, Karger, Pedersen, 1993)
– An interactive clustering-based browsing scheme for
text collections and search results (constant time
improvements)
The EM Algorithm
• Soft clustering method to solve
* = arg max P(X | m())
– Soft clustering – probabilistic association between
clusters and objects ( - the model parameters)
Note: Any occurrence of the data consists of:
– Observable variables: The objects we see
• Words in each context
• Word sequence in POS tagging
– Hidden variables: Which cluster generated which
object
• Which sense generates each context
• Underlying POS sequence
• Soft clustering can capture ambiguity (words) and
multiple topics (documents)
Two Principles
Expectation: If we knew  we could
compute the expectation of the hidden
variables (e.g, probability of x belonging to some
cluster)
Maximization: If we knew the hidden
structure, we could compute the maximum
likelihood value of 
EM for Word-sense “Discrimination”
Cluster the contexts (occurrences) of an ambiguous word,
where each cluster constitutes a “sense”:
• Initialize randomly model parameters P(vj | sk) and P(sk)
• E-step: Compute P(sk | ci) for each context ci and sense sk
• M-step: Re-estimate the model parameters P(vj | sk) and
P(sk), for context words and senses
• Continue as long as log-likelihood of all corpus contexts
1 ≤ i ≤ I increases (EM – guaranteed to increase in each
step till convergence to a maximum, possibly local):
log  P(ci | sk ) P( sk )  log  P(ci | sk ) P( sk )
i
k
i
k
E-Step
• To compute P(ci | sk), use naive Bayes assumption:
P(ci | sk ) 
 P (v
j
| sk )
v j sk
• For each context i & each sense sk, estimate the
posterior probability hik = P(sk | ci) (an expected
“count” of the sense for ci), using Bayes rule:
P(ci | sk ) P( sk )
hik 
k ' P(ci | sk ' ) P(sk ' )
M-Step
Re-estimate parameters using maximumlikelihood estimation:
P (v j
 h
|s )
 h
ci :v j ci
ik
k
j ci :v j ci
h

P( s ) 
 h
i
ik
k
ik
k
i
ik
h


i
I
ik
Decision Procedure
Assign senses by (same as Naïve Bayes):


s( wi )  arg max s log P( sk )   log P(v j | sk )
v c


• Can adjust the pre-determined number of senses k
to get finer or coarser distinctions
k
j
i
– Bank as physical location vs. abstract corporation
• If adding more senses doesn’t increase loglikelihood much, then stop
Results (Schütze 1998)
Word
suit
motion
train
Sense
lawsuit
suit you wear
physical movement
proposal for action
line of railroad cars
to teach
Accuracy
95  0
96  0
85  1
88  13
79  19
55  33
• Works better for topic-related senses (given the
broad-context features used)
• Improved IR performance by 14% - representing
both query and document as senses, and
combining results with word-based retrieval