Domain Adaptation with Multiple Sources

1
Domain Adaptation with Multiple Sources

• Yishay Mansour, Tel Aviv Univ. Google
• Mehryar Mohri, NYU Google
• Afshin Rostami, NYU

2
Adaptation
3
Adaptation motivation

• High level
• The ability to generalize from one domain to
  another
• Significance
• Basic human property
• Essential in most learning environments
• Implicit in many applications.

4
Adaptation - examples

• Sentiment analysis
• Users leave reviews
• products, sellers, movies,
• Goal score reviews as positive or negative.
• Adaptation example
• Learn for restaurants and airlines
• Generalize to hotels

5
Adaptation - examples

• Speech recognition
• Adaptation
• Learn a few accents
• Generalize to new accents
• think foreign accents.

6
Adaptation and generalization

• Machine Learning prediction
• Learn from examples drawn from distribution D
• predict the label of unseen examples
• drawn from the same distribution D
• generalization within a distribution
• Adaptation
• predict the label of unseen examples
• drawn from a different distribution D
• Generalization across distributions

7
Adaptation Related Work

• Learn from D and test on D
• relating the increase in error to dist(D,D)
• Ben-David et al. (2006), Blitzer et al. (2007),
• Single distribution varying label quality
• Cramer et al. (2005, 2006)

8
Our Model
9
Our Model - input
Typical loss function L(a,b)a-b and L(D,h,f)
ExD f(x)-h(x)
10
Our Model target distribution
?1
target distribution D?
?k
11
Our model Combination Rule

• Combine h1, , hk to a hypothesis h
• Low expected loss
• hopefully at most e
• combining rules
• let z S zi 1 and zi 0
• linear h(x) S zi hi(x)
• distribution weighted

. . .
hk
h1
combining rule
12
Combining Rules Pros

• Alternative Build a dataset for the mixture.
• Learning the mixture parameters is non-trivial
• Combined data set might be huge size
• Domain dependent data unavailable
• Combined data might be huge
• Sometimes only classifiers are given/exist
• privacy
• MOST IMPORTANT
• FUNDAMENTAL THEORY QUESTION

13
Our Results

• Linear Combining rule
• Seems like the first thing to try
• Can be very bad
• Simple settings where any linear combining rule
  performs badly.

14
Our Results

• Distribution weighted combining rules
• Given the mixture parameter ?
• there is a good distribution weighted combining
  rule.
• expected loss at most e
• For any target function f,
• there is a good distribution combining rule hz
• expected loss at most e
• Extension for multiple consistent target
  functions
• expected loss at most 3e
• OUTCOME This is the right hypothesis class

15
Known Distribution
16
Linear combining rules
h0 h1 f X
0 1 1 a
0 1 0 b
Original Loss e0 !!!
Db Da D
0 1 ½ a
1 0 ½ b
Any linear combining rule h has expected absolute
loss ½
17
Distribution weighted combining rule

• Target distribution a mixture
• D?(x)S ?i Di(x)
• Set z?
• Claim L(D?,h?,f) e

18
Distribution weighted combining rule
PROOF
19
Back to the bad example
h0 h1 f X
0 1 1 a
0 1 0 b
Original Loss e0 !!!
Db Da D
0 1 ½ a
1 0 ½ b
h(x) xa h(x)h1(x)1 xb h(x)h0(x)0
20
Unknown Distribution
21
Unknown mixture distribution

• Zero-sum game
• NATURE selects a distribution Di
• LEARNER selects a z
• hypothesis hz
• Payoff L(Di,hz,f)
• Restating to previous result
• For any mixed action ? of NATURE
• LEARNER has a pure action z ?
• such that the expected loss is at most e

22
Unknown mixture distribution

• Consequence
• LEARNER has a mixed action (over zs)
• for any mixed action ? of NATURE
• a mixture distribution D?
• The loss is at most e
• Challenge
• show a specific hypothesis hz
• pure, not mixed, action

23
Searching for a good hypothesis

• Uniformly good hypothesis hz
• for any Di we have L(Di, hz,f) e
• Assume all the hi are identical
• Extremely lucky and unlikely case
• If we have a good hypothesis we are done!
• L(D?,hz,f) S ?i L(Di,hz,f) S ?i e e
• We need to show in general a good hz !

24
Proof Outline

• Balancing the losses
• Show that some hz has identical loss on any Di
• uses Brouwer Fixed Point Theorem
• holds very generally
• Bounding the losses
• Show this hz has low loss for some mixture
• specifically Dz

25
Brouwer Fixed Point Theorem For any convex and
compact set A and any continuous mapping f
A?A, there exists a point x in A such that f(x)x
A compact and convex set
f A?A continuous mapping
26
Balancing Losses
Problem 1 Need to get f continuous
27
Balancing Losses
Fixed point zf(z)
Problem 2 Needs that zi ?0
28
Bounding the losses

• We can guarantee balanced losses even for linear
  combining rule !

h0 h1 f X
0 1 1 a
0 1 0 b
For z(½, ½) we have L(Da,hz,f)½ L(Db,hz,f)½
Db Da D
0 1 ½ a
1 0 ½ b
29
Bounding Losses

• Consider the previous z
• from Brouwer fixed point theorem
• Consider the mixture Dz
• Expected loss is at most e
• Also L(Dz,hz,f) SzjL(Dj,hz,f)?
• Conclusion
• For any mixture expected loss at most ? e

30
Solving the problems

• Redefine the distribution weighted rule
• Claim For any distribution D,
• is continuous in z.

31
Main Theorem

• For any target function f and any dgt0,
• there exists ?gt0 and z such that
• for any ? we have

32
Balancing Losses

• The set A S zi 1 and zi 0
• The simplex
• The mapping f with parameters ? and ?
• f(z)i (zi Li,z?/k)/ (SzjLj,z?)
• where Li,zL(Di,hz,?,f)
• For some z in A we have f(z)z
• zi (zi Li,z?/k)/ (SzjLj,z?) gt0
• Li,z (SzjLj,z)? - ?/(zi k) lt (SzjLj,z) ?

33
Bounding Losses

• Consider the previous z
• from Brouwer fixed point theorem
• Consider the mixture Dz
• Expected loss is at most e?
• By definition SzjLj,z L(Dz,hz,?,f)
• Conclusion ?SzjLj,z e?

34
Putting it together

• There exists (z,?) such that
• Expected loss of hz,? approximately balanced
• L(Di,hz,?,f) ??
• Bounding ? using Dz
• ? L(Dz,hz,?,f) e?
• For any mixture D?
• L(D?,hz,?,f) e? ?

35
A more general model

• So far NATURE first fixes target function f
• consistent target functions f
• the expected loss w.r.t. Di is at most e
• for any of the k distributions
• Function class F f is consistent
• New Model
• LEARNER picks a hypothesis h
• NATURE picks f in F and mixture D?
• Loss L(D?,h,f)
• RESULT L(D?,h,f) 3e.

36
Simple Algorithms
37
Uniform Algorithm

• Hypothesis sets z(1/k , , 1/k)
• Performance
• For any mixture, expected error ke
• There exists mixture with expected error O(ke)
• For k2, there exists a mixture with 2e-e2

38
Open Problem

• Find a uniformly good hypothesis
• efficiently !!!
• algorithmic issues
• Search over the zs
• Multiple local minima.

39
Empirical Results
40
Empirical Results

• Data-set of sentiment analysis
• good product takes a little time to start
  operating very good for the price a little
  trouble using it inside ca
• it rocks man this is the rockinest think i've
  ever seen or buyed dudes check it ou
• does not retract agree with the prior reviewers i
  can not get it to retract any longer and that was
  only after 3 uses
• dont buy not worth a cent got it at walmart can't
  even remove a scuff i give it 100 good thing i
  could return it
• flash drive excelent hard drive good price and
  good time for seller thanks

41
Empirical analysis

• Multiple domains
• dvd, books, electronics, kitchen appliance.
• Language model
• build a model for each domain
• unlike the theory, this is an additional error
  source
• Tested on mixture distribution
• known mixture parameters
• Target score (1-5)
• error Mean Square Error (MSE)

42
Distribution weighted
kitchen
dvd
linear
books
electronics
43
(No Transcript)
44
(No Transcript)
45
Summary
46
Summary

• Adaptation model
• combining rules
• linear
• distribution weighted
• Theoretical analysis
• mixture distribution
• Future research
• algorithms for combining rules
• beyond mixtures

47
Thank You!
48
Adaptation Our Model

• Input
• target function f
• k distributions D1, , Dk
• k hypothesis h1, , hk
• For every i L(Di,hi,f) e
• where L(D,h,f) defines the expected loss
• think L(D,h,f) ExD f(x)-h(x)