On Learning Invariant Representations for
Domain Adaptation
Han Zhao, Remi Tachet des Combes, Kun Zhang, and Geoﬀrey J. Gordon
Presented by: Han Zhao
han.zhao@cs.cmu.edu
Machine Learning Department, Carnegie Mellon University
June 11th, 20191

Background
Generalization: Source (Train) = Target (Test)

SourceTarget2

Background
Domain adaptation: Source 6= Target<latexit sha1_base64="DdKPZFrLlnuhD+vGJXEoK8bQvW4=">AAACHHicbVDLSgMxFE3qq9ZXq0s3wSK4KjMi6LLoxmUF2wrtUDJpphOax5hkhDL0F9zqyq9xJ24F/8ZMOwv7OBA4nHMu9+aECWfGet4vLG1sbm3vlHcre/sHh0fV2nHHqFQT2iaKK/0UYkM5k7RtmeX0KdEUi5DTbji+y/3uC9WGKfloJwkNBB5JFjGCbS71JX0eVOtew5sBrRK/IHVQoDWowVJ/qEgqqLSEY2N6vpfYIMPaMsLptNJPDU0wGeMR7TkqsaAmyGbHTtG5U4YoUto9adFM/T+RYWHMRIQuKbCNzbKXi+u8XmqjmyBjMkktlWS+KEo5sgrlP0dDpimxfOIIJpq5WxGJscbEun4Wt/CRcoFYTNfLjEwrrjV/uaNV0rls+F7Df7iqN2+L/srgFJyBC+CDa9AE96AF2oCAGLyCN/AOP+An/ILf82gJFjMnYAHw5w8BAaGx</latexit>

Source (with Labels)

Target (No Labels)

3

Background
Many applications…

4

Background
Many applications…
•

Face recognition (Sohn et al.’ 17)•

Object recognition (Ghifary et al.’ 15)•

Sentiment analysis (Glorot et al.’ 11)•

Accented speech recognition (Sun et al.’ 17)•

Indoor WiFi localization (Pan et al.’ 08)•

Relational reasoning (Davis et al.’ 09)

•

……

4

Background
Learning domain-invariant representations:

SourceTarget5

Background
Learning domain-invariant representations:g(·)<latexit sha1_base64="xTFpQDckv9/JC8ue01iyn2qvijU=">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</latexit>

Sourceg(·)<latexit sha1_base64="xTFpQDckv9/JC8ue01iyn2qvijU=">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</latexit>

Target5

Background
Learning domain-invariant representations:g(·)<latexit sha1_base64="xTFpQDckv9/JC8ue01iyn2qvijU=">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</latexit>

Sourceg(·)<latexit sha1_base64="xTFpQDckv9/JC8ue01iyn2qvijU=">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</latexit>

TargetFeature transformation
5

Background
Learning domain-invariant representations:g(·)<latexit sha1_base64="xTFpQDckv9/JC8ue01iyn2qvijU=">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</latexit>

Sourceg(·)<latexit sha1_base64="xTFpQDckv9/JC8ue01iyn2qvijU=">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</latexit>

TargetMatching feature distribution
5

Background
Learning domain-invariant representations:g(·)<latexit sha1_base64="xTFpQDckv9/JC8ue01iyn2qvijU=">AAACwnicbZFNb9NAEIY35qMlfLVw5GIRVSqXyG6R2mNVOMCtiKaNFEfVeD1OluyHu7tuEhb/Ca5w5E/xb1inRqQOI630ap53dnZ20oIzY6Podye4d//Bw63tR93HT54+e76z++LCqFJTHFDFlR6mYJAziQPLLMdhoRFEyvEynb2r+eUNasOUPLfLAscCJpLljIL1qeFkP6GZsm+udnpRP1pFuCniRvRIE2dXu51fSaZoKVBaysGYURwVduxAW0Y5Vt2kNFgAncEER15KEGjGbvXgKtzzmSzMlfZH2nCVXa9wIIxZitQ7BdipabM6+T82Km1+PHZMFqVFSW8b5SUPrQrr6cOMaaSWL70Aqpl/a0inoIFa/0fdvXC9T317AV/V3VkEzFArJfwsIClyTyXOqRICZOYSJrPKJXVpmrqPVYvOkU2m9q8hd/O2YbH4BxdtaArZUArcfd7EViOuOc6ryu81bm9xU1wc9OPD/sGnt72T02bD2+QVeU32SUyOyAn5QM7IgFDCyXfyg/wM3gdfguvA3FqDTlPzktyJ4NsfkdLj8Q==</latexit>h(·)<latexit sha1_base64="EDqE3aR0NEfzYm6sPUlnsGMAyYw=">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</latexit>`(h(g(x)), y)<latexit sha1_base64="m/58EYhT+smiOW3FgAnImjwCshI=">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</latexit>

Sourceg(·)<latexit sha1_base64="xTFpQDckv9/JC8ue01iyn2qvijU=">AAACwnicbZFNb9NAEIY35qMlfLVw5GIRVSqXyG6R2mNVOMCtiKaNFEfVeD1OluyHu7tuEhb/Ca5w5E/xb1inRqQOI630ap53dnZ20oIzY6Podye4d//Bw63tR93HT54+e76z++LCqFJTHFDFlR6mYJAziQPLLMdhoRFEyvEynb2r+eUNasOUPLfLAscCJpLljIL1qeFkP6GZsm+udnpRP1pFuCniRvRIE2dXu51fSaZoKVBaysGYURwVduxAW0Y5Vt2kNFgAncEER15KEGjGbvXgKtzzmSzMlfZH2nCVXa9wIIxZitQ7BdipabM6+T82Km1+PHZMFqVFSW8b5SUPrQrr6cOMaaSWL70Aqpl/a0inoIFa/0fdvXC9T317AV/V3VkEzFArJfwsIClyTyXOqRICZOYSJrPKJXVpmrqPVYvOkU2m9q8hd/O2YbH4BxdtaArZUArcfd7EViOuOc6ryu81bm9xU1wc9OPD/sGnt72T02bD2+QVeU32SUyOyAn5QM7IgFDCyXfyg/wM3gdfguvA3FqDTlPzktyJ4NsfkdLj8Q==</latexit>

Supervised training

Target6

Background
Learning domain-invariant representations:g(·)<latexit sha1_base64="xTFpQDckv9/JC8ue01iyn2qvijU=">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</latexit>h(·)<latexit sha1_base64="EDqE3aR0NEfzYm6sPUlnsGMAyYw=">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</latexit>`(h(g(x)), y)<latexit sha1_base64="m/58EYhT+smiOW3FgAnImjwCshI=">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</latexit>

Sourceg(·)<latexit sha1_base64="xTFpQDckv9/JC8ue01iyn2qvijU=">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</latexit>

Supervised training

TargetGoal: generalization on target domain
6

Background
Learning domain-invariant representations:g(·)<latexit sha1_base64="xTFpQDckv9/JC8ue01iyn2qvijU=">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</latexit>

Sourceg(·)<latexit sha1_base64="xTFpQDckv9/JC8ue01iyn2qvijU=">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</latexit>

Target7

Background
Learning domain-invariant representations:•

Adversarial discriminator (DANN, Ganin et al.’ 15)g(·)<latexit sha1_base64="xTFpQDckv9/JC8ue01iyn2qvijU=">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</latexit>

Sourceg(·)<latexit sha1_base64="xTFpQDckv9/JC8ue01iyn2qvijU=">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</latexit>

Target7

Background
Learning domain-invariant representations:•

Adversarial discriminator (DANN, Ganin et al.’ 15)•

Maximum mean discrepancy (DAN, Long et al.’ 15)g(·)<latexit sha1_base64="xTFpQDckv9/JC8ue01iyn2qvijU=">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</latexit>

Sourceg(·)<latexit sha1_base64="xTFpQDckv9/JC8ue01iyn2qvijU=">AAACwnicbZFNb9NAEIY35qMlfLVw5GIRVSqXyG6R2mNVOMCtiKaNFEfVeD1OluyHu7tuEhb/Ca5w5E/xb1inRqQOI630ap53dnZ20oIzY6Podye4d//Bw63tR93HT54+e76z++LCqFJTHFDFlR6mYJAziQPLLMdhoRFEyvEynb2r+eUNasOUPLfLAscCJpLljIL1qeFkP6GZsm+udnpRP1pFuCniRvRIE2dXu51fSaZoKVBaysGYURwVduxAW0Y5Vt2kNFgAncEER15KEGjGbvXgKtzzmSzMlfZH2nCVXa9wIIxZitQ7BdipabM6+T82Km1+PHZMFqVFSW8b5SUPrQrr6cOMaaSWL70Aqpl/a0inoIFa/0fdvXC9T317AV/V3VkEzFArJfwsIClyTyXOqRICZOYSJrPKJXVpmrqPVYvOkU2m9q8hd/O2YbH4BxdtaArZUArcfd7EViOuOc6ryu81bm9xU1wc9OPD/sGnt72T02bD2+QVeU32SUyOyAn5QM7IgFDCyXfyg/wM3gdfguvA3FqDTlPzktyJ4NsfkdLj8Q==</latexit>

Target7

Background
Learning domain-invariant representations:•

Adversarial discriminator (DANN, Ganin et al.’ 15)•

Maximum mean discrepancy (DAN, Long et al.’ 15)•

Wasserstein distance (Shen et al.’ 18)

g(·)<latexit sha1_base64="xTFpQDckv9/JC8ue01iyn2qvijU=">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</latexit>

Sourceg(·)<latexit sha1_base64="xTFpQDckv9/JC8ue01iyn2qvijU=">AAACwnicbZFNb9NAEIY35qMlfLVw5GIRVSqXyG6R2mNVOMCtiKaNFEfVeD1OluyHu7tuEhb/Ca5w5E/xb1inRqQOI630ap53dnZ20oIzY6Podye4d//Bw63tR93HT54+e76z++LCqFJTHFDFlR6mYJAziQPLLMdhoRFEyvEynb2r+eUNasOUPLfLAscCJpLljIL1qeFkP6GZsm+udnpRP1pFuCniRvRIE2dXu51fSaZoKVBaysGYURwVduxAW0Y5Vt2kNFgAncEER15KEGjGbvXgKtzzmSzMlfZH2nCVXa9wIIxZitQ7BdipabM6+T82Km1+PHZMFqVFSW8b5SUPrQrr6cOMaaSWL70Aqpl/a0inoIFa/0fdvXC9T317AV/V3VkEzFArJfwsIClyTyXOqRICZOYSJrPKJXVpmrqPVYvOkU2m9q8hd/O2YbH4BxdtaArZUArcfd7EViOuOc6ryu81bm9xU1wc9OPD/sGnt72T02bD2+QVeU32SUyOyAn5QM7IgFDCyXfyg/wM3gdfguvA3FqDTlPzktyJ4NsfkdLj8Q==</latexit>

Target7

Background
Learning domain-invariant representations:•

Adversarial discriminator (DANN, Ganin et al.’ 15)•

Maximum mean discrepancy (DAN, Long et al.’ 15)•

Wasserstein distance (Shen et al.’ 18)

•

……

g(·)
<latexit sha1_base64="xTFpQDckv9/JC8ue01iyn2qvijU=">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</latexit>

Sourceg(·)<latexit sha1_base64="xTFpQDckv9/JC8ue01iyn2qvijU=">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</latexit>

Target7

Background
Justification of learning domain-invariant representations:
Theorem (Ben-David et al.’ 10):1
"T (h)  "S (h) + dH H (T ; S) +2⇤

<latexit sha1_base64="61AbZdnkqFmEF6WmYw0WEedc8bY=">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</latexit>"T (h)/"S (h): true target/source errors•<latexit sha1_base64="OV+fChg/DB0JN+OhlBFqBPFxRqM=">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</latexit>dH H (T ; S): divergence between target/source distributions•<latexit sha1_base64="G6sdacGNmUQnTWJzdHsBXFKS37U=">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</latexit>⇤•<latexit sha1_base64="hR8cZye/m+6lsmMEJDbLNVq1nGc=">AAAD93icdVNdaxNBFJ1m/ajxK9VHXwZDaVUoSRUUpFDUh/hWadIWsjHMzt7NDpmP7cxsk7jsbxFBxFd/ha/66L9xdrvSZIMXFu6cc+69zNyzQcKZsZ3On42Gd+36jZubt5q379y9d7+19eDEqFRTGFDFlT4LiAHOJAwssxzOEg1EBBxOg+nbgj+9AG2Ykn27SGAkyESyiFFiHTRuvfa5E4fk41PshxDBOfYFk+Ms3vGZ9Hu93AetlR4f78Y7T/AzXB37xXHcanf2OmXg9aRbJW1UxdF4q/HVDxVNBUhLOTFm2O0kdpQRbRnlkDf91EBC6JRMYOhSSQSYUVbeMsfbDglxpLT7pMUlulyREWHMQgROKYiNTZ0rwP9xNhZ5HSummByXscI5sVWKm9WKou86PExt9GqUMZmkFiS9vEWUcmwVLvaBQ6aBWr5wCaGauYfANCaaUOu21tzG9REJ+aRWJwgyBa2UcA9FJAXuWAkzqoQgMszcGsM884vSIMje5zV2BmwS23+CKJvVBfP5FTmvkyaRFUsJz47XaasBlhT9usIZ7Irt1dnSa05wQTQkhnEla4LQ/SBLDd7VG5SOLhrQwnHOMPKgPuLcafLsYEmSO1936y5eT07297rP9/Y/vGgfvqkcvokeocdoF3XRS3SIeugIDRBFX9BP9Av99hbeZ++b9/1S2tioah6ilfB+/AU7xF1y</latexit>

00:= min
"
(h
)
+
"
(h
): optimal joint error
S
T
0
h 2H

8

Background
Justification of learning domain-invariant representations:
Theorem (Ben-David et al.’ 10):1
"T (h)  "S (h) + dH H (T ; S) +2⇤

<latexit sha1_base64="61AbZdnkqFmEF6WmYw0WEedc8bY=">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</latexit>"T (h)/"S (h): true target/source errors•<latexit sha1_base64="OV+fChg/DB0JN+OhlBFqBPFxRqM=">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</latexit>dH H (T ; S): divergence between target/source distributions•<latexit sha1_base64="G6sdacGNmUQnTWJzdHsBXFKS37U=">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</latexit>⇤•<latexit sha1_base64="hR8cZye/m+6lsmMEJDbLNVq1nGc=">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</latexit>

00:= min
"
(h
)
+
"
(h
): optimal joint error
S
T
0
h 2H

Bound-minimizing algorithm:

min<latexit sha1_base64="l5UOtk2LD8ug2mfzYzf0s8046+A=">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</latexit>

1"S (h) + dH H (T ; S)
28

Motivation
Question:
Is finding invariant representations while at the same timeachieving a small source error suﬃcient to guarantee asmall target error?

9Motivation
Question:
Is finding invariant representations while at the same timeachieving a small source error suﬃcient to guarantee asmall target error?
•

If not, under what conditions is it?

•

Is there any necessary condition for this family of methods?

9Overview
Question:
Is finding invariant representations while at the same timeachieving a small source error suﬃcient to guarantee asmall target error?
If not, under what conditions is it?
Suﬃcient condition: only when the conditional distributions match
Y |X
Y |X
D S ⇡ DT•<latexit sha1_base64="JRixQMo6MOB92ydbuyuVqJ9YgDk=">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</latexit>

Is there any necessary condition for this family of methods?
Necessary condition: only when marginal label distributions are close•YDS<latexit sha1_base64="9oLoCRbM8T/40qLIL65zo+7m3+k=">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</latexit>

Y
⇡ DT
10

A Simple Example
Consider a 1D adaptation problem:
target domain
y=1

y=0
x

1

0.5
0
y=0
y=1

0.5

1

1.5

2

source domain

Source: DS = U ( 1, 0)
Target: DT = U (1, 2)<latexit sha1_base64="B4uYRkZP0jnlIJMhvH2YBBZC274=">AAADznicdZJbaxNBFMenWS81XprWR18GQ6GCht0q6EugqA/xLdKmLWRDmJ09SYbMZTszm4vL4qufwidBv5Lfxtk0YjLBAwuH/+9/zpmdOUnGmbFh+HuvFty5e+/+/oP6w0ePnxw0Do8ujco1hR5VXOnrhBjgTELPMsvhOtNARMLhKpl+qPjVDLRhSl7YZQYDQcaSjRgl1knDxlGcuiHDc9zGvZNX0Uscvhg2mmErXAXeTaJ10kTr6A4Paz/iVNFcgLSUE2P6UZjZQUG0ZZRDWY9zAxmhUzKGvkslEWAGxerwJT52SopHSrtPWrxSNysKIoxZisQ5BbET47NK/B+zE1H6WjXFlHgVW8yZrVLcbFdUfXflfm5H7wYFk1luQdLbvxjlHFuFq2vGKdNALV+6hFDN3EVgOiGaUOseo36M/REZ+aK2JwgyBa2UcBdFJAXuqIQ5VUIQmRYxk2lZxFVpkhSfSo/OgY0n9q9hVMx9w2LxDy58aDK5ppTw4nwXWw2w4bjwHZ3OBu34FLR2D1/EM6IhM4wr6Rmqldxo8NFvkMIIbqoGtNo4tzCy7Y+4cZ6yaG9YSrfXkb/Fu8nlaSt63Tr9/KZ59n694fvoGXqOTlCE3qIz1EFd1EMULdB39BP9CrrBLCiDr7fW2t665inaiuDbH1CdTHA=</latexit><latexit sha1_base64="6ZM1hZpXB37ObrE4xzwLnLx14wc=">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</latexit>

11

A Simple Example
Consider a 1D adaptation problem:
target domain

1

(

)

y=1

0.5
0
y=0
y=1

0.5

<latexit sha1_base64="CD/+W6kdrKs/7eziwmMLrC6r9+w=">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</latexit>

1

y=0

1.5

x
2

<latexit sha1_base64="/CzBYNA9tf2BmD+v0JEp81UzyRI=">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</latexit>

source domain

Source: DS = U ( 1, 0)
Target: DT = U (1, 2)<latexit sha1_base64="B4uYRkZP0jnlIJMhvH2YBBZC274=">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</latexit>⇤

h (x) = 1 i↵ x 2 ( 1/2, 3/2)
<latexit sha1_base64="3wYd7tH2o6N+aqyfauy3LU+Qzjs=">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</latexit><latexit sha1_base64="6ZM1hZpXB37ObrE4xzwLnLx14wc=">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</latexit>

11

A Simple Example
Consider a 1D adaptation problem:
target domain

1

(

)

y=1

0.5
0
y=0
y=1

0.5

1

<latexit sha1_base64="CD/+W6kdrKs/7eziwmMLrC6r9+w=">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</latexit>

y=0

1.5

x
2

<latexit sha1_base64="/CzBYNA9tf2BmD+v0JEp81UzyRI=">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</latexit>

source domain

Source: DS = U ( 1, 0)
Target: DT = U (1, 2)⇤<latexit sha1_base64="B4uYRkZP0jnlIJMhvH2YBBZC274=">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</latexit>

h (x) = 1 i↵ x 2 ( 1/2, 3/2)
⇤
0
0
= min
"
(h
)
+
"
(h
)=0
S
T
0
<latexit sha1_base64="3wYd7tH2o6N+aqyfauy3LU+Qzjs=">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</latexit><latexit sha1_base64="6ZM1hZpXB37ObrE4xzwLnLx14wc=">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</latexit><latexit sha1_base64="IzxZKNuEc/Exe4o5W56dWzNATWI=">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</latexit>h11

A Simple Example
Consider a 1D adaptation problem: Adaptationg(x) = Ix0 (x) · (x + 1)

target domain+ Ix>0 (x) · (x<latexit sha1_base64="Hk7gK9ZvHZgvlaE6B/70QQwbEHM=">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</latexit>

y=1

y=0

target domain

1)

y=1

y=0

x
1

0.5
0
y=0
y=1

0.5

1

1.5

x

2

1

0.5

0

0.5
y=0

source domain

1

1.5

2

y=1

source domain

Bound-minimizing algorithm: min<latexit sha1_base64="l5UOtk2LD8ug2mfzYzf0s8046+A=">AAAD9XicdVNdaxNBFJ02ftT4leqjL4OhkCKEJBVUpFC0D/GtkqQtZEOYnb2bDJmPzcxskzjsTxFfiq/+Cl/12X/jbBppssELC4dzzr2XmTkbJpwZ22j82dkt3bl77/7eg/LDR4+fPK3sPzs3KtUUelRxpS9DYoAzCT3LLIfLRAMRIYeLcPIx1y+uQBumZNcuEhgIMpIsZpRYTw0r7wLBZDBNSYQD0FrpYac2PsSvcBBrQl0zc60sGrqg3Q5OgVviQVbrvsedw2Gl2qg3loW3QXMFqmhVZ8P93esgUjQVIC3lxJh+s5HYgSPaMsohKwepgYTQCRlB30NJBJiBW54xwweeiXCstP+kxUt2vcMRYcxChN4piB2bopaT/9PsWGRFLt9iMrysDc2brVLcbHbkc7fpfmrjtwPHZJJakPTmFHHKsVU4fw0cMQ3U8oUHhGrmLwLTMfE3b/2blQ9wcUVCvqjNDYJMQCsl/EURSYF7VcKMKiGIjFzAZJS5IG8NQ/cpK6gzYKOx/WeI3axomM9vxXlRNIlcqZRw19mWrQZYc3SLDp+lW7VdVJdx9IYroiExjCtZMET+91gbcFocEEEM03wAzRPnAyOPiyum3pO54zVL5nPdLKZ4G5y36s2jeuvz6+rJh1XC99AL9BLVUBO9QSeojc5QD1H0Df1Ev9Dv0qz0tXRd+n5j3d1Z9TxHG1X68Rfx5FzR</latexit>

1"S (h) + dH H (T ; S)
2

12

A Simple Example
Consider a 1D adaptation problem: Adaptationg(x) = Ix0 (x) · (x + 1)

target domain+ Ix>0 (x) · (x<latexit sha1_base64="Hk7gK9ZvHZgvlaE6B/70QQwbEHM=">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</latexit>

y=1

y=0

target domain

1)

y=1

y=0

x
1

0.5
0
y=0
y=1

0.5

1

1.5

x

2

1

0

0.5
y=0

source domain

1

1.5

2

y=1

source domain

1"S (h) + dH H (T ; S)
2

Bound-minimizing algorithm: min<latexit sha1_base64="l5UOtk2LD8ug2mfzYzf0s8046+A=">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</latexit>

Source: DS0 = U (0, 1)<latexit sha1_base64="TnmOc77dRtBXpwLBcUaVYLBOmxI=">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</latexit>

Target:0.5

DT0 = U (0, 1)

dH H (S, T ) = 0<latexit sha1_base64="yU9UWaTi87vcGT1ozmjrJz9YmDQ=">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</latexit><latexit sha1_base64="ev9y89oZU911nrIb1pfQtZ8tD3c=">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</latexit>

12

A Simple Example
Consider a 1D adaptation problem: Adaptationg(x) = Ix0 (x) · (x + 1)

target domain+ Ix>0 (x) · (x<latexit sha1_base64="Hk7gK9ZvHZgvlaE6B/70QQwbEHM=">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</latexit>

y=1

y=0

target domain

1)

y=1

y=0

x
1

0.5
0
y=0
y=1

0.5

1

1.5

x

2

1

0

0.5
y=0

source domain

1

1.5

2

y=1

source domain

1"S (h) + dH H (T ; S)
2

Bound-minimizing algorithm: min<latexit sha1_base64="l5UOtk2LD8ug2mfzYzf0s8046+A=">AAAD9XicdVNdaxNBFJ02ftT4leqjL4OhkCKEJBVUpFC0D/GtkqQtZEOYnb2bDJmPzcxskzjsTxFfiq/+Cl/12X/jbBppssELC4dzzr2XmTkbJpwZ22j82dkt3bl77/7eg/LDR4+fPK3sPzs3KtUUelRxpS9DYoAzCT3LLIfLRAMRIYeLcPIx1y+uQBumZNcuEhgIMpIsZpRYTw0r7wLBZDBNSYQD0FrpYac2PsSvcBBrQl0zc60sGrqg3Q5OgVviQVbrvsedw2Gl2qg3loW3QXMFqmhVZ8P93esgUjQVIC3lxJh+s5HYgSPaMsohKwepgYTQCRlB30NJBJiBW54xwweeiXCstP+kxUt2vcMRYcxChN4piB2bopaT/9PsWGRFLt9iMrysDc2brVLcbHbkc7fpfmrjtwPHZJJakPTmFHHKsVU4fw0cMQ3U8oUHhGrmLwLTMfE3b/2blQ9wcUVCvqjNDYJMQCsl/EURSYF7VcKMKiGIjFzAZJS5IG8NQ/cpK6gzYKOx/WeI3axomM9vxXlRNIlcqZRw19mWrQZYc3SLDp+lW7VdVJdx9IYroiExjCtZMET+91gbcFocEEEM03wAzRPnAyOPiyum3pO54zVL5nPdLKZ4G5y36s2jeuvz6+rJh1XC99AL9BLVUBO9QSeojc5QD1H0Df1Ev9Dv0qz0tXRd+n5j3d1Z9TxHG1X68Rfx5FzR</latexit>

Source: DS0 = U (0, 1)<latexit sha1_base64="TnmOc77dRtBXpwLBcUaVYLBOmxI=">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</latexit>

Target:0.5

DT0 = U (0, 1)

dH H (S, T ) = 0<latexit sha1_base64="yU9UWaTi87vcGT1ozmjrJz9YmDQ=">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</latexit><latexit sha1_base64="ev9y89oZU911nrIb1pfQtZ8tD3c=">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</latexit>

If "S (h) = 0 , then "T (h) = 1 !
<latexit sha1_base64="DbdtTuEiyXfd1WfkWW/nqh6vJb4=">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</latexit><latexit sha1_base64="vOhxmKPf21jmz+DidbSVu7dAWco=">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</latexit>

12

A Simple Exampleg(x) = Ix0 (x) · (x + 1)

target domain+ Ix>0 (x) · (x<latexit sha1_base64="Hk7gK9ZvHZgvlaE6B/70QQwbEHM=">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</latexit>

y=1

y=0

target domain

1)

y=1

y=0

x
1

0.5
0
y=0
y=1

0.5

1

1.5

x

2

1

0.5

0

0.5
y=0

source domain

1

1.5

2

y=1

source domain

Bound-minimizing algorithm: min<latexit sha1_base64="l5UOtk2LD8ug2mfzYzf0s8046+A=">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</latexit>

What’s wrong with this example?

1"S (h) + dH H (T ; S)
2

1
"T (h)  "S (h) + dH H (T ; S) +2⇤

<latexit sha1_base64="61AbZdnkqFmEF6WmYw0WEedc8bY=">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</latexit>||
0<latexit sha1_base64="cOLlPwZYWKmOBSc61BiW1c43xPU=">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</latexit><latexit sha1_base64="WZrYTG83/jkP4dur1KsWUXEZ+0k=">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</latexit>||
0<latexit sha1_base64="cOLlPwZYWKmOBSc61BiW1c43xPU=">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</latexit><latexit sha1_base64="WZrYTG83/jkP4dur1KsWUXEZ+0k=">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</latexit>

13

A Simple Exampleg(x) = Ix0 (x) · (x + 1)

target domain+ Ix>0 (x) · (x<latexit sha1_base64="Hk7gK9ZvHZgvlaE6B/70QQwbEHM=">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</latexit>

y=1

y=0

target domain

1)

y=1

y=0

x
1

0.5
0
y=0
y=1

0.5

1

1.5

x

2

1

0.5

0

0.5
y=0

source domain

1

1.5

2

y=1

source domain

Bound-minimizing algorithm: min<latexit sha1_base64="l5UOtk2LD8ug2mfzYzf0s8046+A=">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</latexit>

What’s wrong with this example?

1"S (h) + dH H (T ; S)
2

1
"T (h)  "S (h) + dH H (T ; S) +2⇤

<latexit sha1_base64="61AbZdnkqFmEF6WmYw0WEedc8bY=">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</latexit>||
0<latexit sha1_base64="cOLlPwZYWKmOBSc61BiW1c43xPU=">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</latexit><latexit sha1_base64="WZrYTG83/jkP4dur1KsWUXEZ+0k=">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</latexit>

In fact, 8h 2 2[0,1] ,<latexit sha1_base64="KXOs3WhbtJMqyvIrd3C6b3lih1M=">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</latexit>"S (h) + "T (h) = 1||0<latexit sha1_base64="cOLlPwZYWKmOBSc61BiW1c43xPU=">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</latexit><latexit sha1_base64="WZrYTG83/jkP4dur1KsWUXEZ+0k=">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</latexit><latexit sha1_base64="DKy1j5fzUbocOfAWTh6nPu7pmBE=">AAADx3icdZJLbxMxEMfdLI8SXi0cuVhElRCHKNsiwSVSBRzSWxFNWykbKq93NrHix8b25oG1Bz4FV5D4UnwbvGkQiSNGWmk0v//MeO1/WnBmbKfze68R3bl77/7+g+bDR4+fPD04fHZpVKkp9KniSl+nxABnEvqWWQ7XhQYiUg5X6eRDza9moA1T8sIuCxgKMpIsZ5RYXxom3Esz8uU17uL45qDVaXdWgXeTeJ200DrObw4bv5JM0VKAtJQTYwZxp7BDR7RllEPVTEoDBaETMoKBTyURYIZudeoKH/lKhnOl/SctXlU3OxwRxixF6pWC2LEJWV38H7NjUYW1eoup8Cq2mBdbpbjZ7qjn7pYHpc3fDR2TRWlB0tu/yEuOrcL1/eKMaaCWL31CqGb+IjAdE02o9a/QPMLhioJ8VdsbBJmAVkr4iyKSAvdUwpwqIYjMXMJkVrmkbk1Td1YFdA5sNLZ/Bbmbh4LF4h9chNAUck0p4e7zLrYaYENxESp6vQ3aCylo7R/eJTOioTCMKxkIMm/4jQEfwwEZ5DCtB9Dacd4wshuumHpN5bobksr7Og5dvJtcHrfjk/bxpzet0/drh++jF+gleoVi9Badoh46R31E0RR9Rz/Qz+gsUtEsWtxKG3vrnudoK6JvfwAfdUtB</latexit>

⇤

=1
13

A Simple Exampleg(x) = Ix0 (x) · (x + 1)

target domain+ Ix>0 (x) · (x<latexit sha1_base64="Hk7gK9ZvHZgvlaE6B/70QQwbEHM=">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</latexit>

y=1

y=0

target domain

1)

y=1

y=0

x
1

0.5
0
y=0
y=1

0.5

1

1.5

x

2

1

0.5

0

0.5
y=0

source domain

1

1.5

2

y=1

source domain

Bound-minimizing algorithm: min<latexit sha1_base64="l5UOtk2LD8ug2mfzYzf0s8046+A=">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</latexit>

What’s wrong with this example?

1"S (h) + dH H (T ; S)
2

1
"T (h)  "S (h) + dH H (T ; S) +2⇤

<latexit sha1_base64="61AbZdnkqFmEF6WmYw0WEedc8bY=">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</latexit>

||
0

||
0

<latexit sha1_base64="cOLlPwZYWKmOBSc61BiW1c43xPU=">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</latexit><latexit sha1_base64="cOLlPwZYWKmOBSc61BiW1c43xPU=">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</latexit><latexit sha1_base64="WZrYTG83/jkP4dur1KsWUXEZ+0k=">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</latexit>

In fact, 8h 2 2[0,1] ,<latexit sha1_base64="KXOs3WhbtJMqyvIrd3C6b3lih1M=">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</latexit><latexit sha1_base64="WZrYTG83/jkP4dur1KsWUXEZ+0k=">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</latexit>"S (h) + "T (h) = 1<latexit sha1_base64="DKy1j5fzUbocOfAWTh6nPu7pmBE=">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</latexit>⇤=1

Conditional distributions are maximally diﬀerent after adaptation:⇥

Y |X
Ex⇠U (0,1) |DS<latexit sha1_base64="tyND50zZby5aESwKlSFj2zC3Xgo=">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</latexit>

Y |X ⇤
DT | = 1

13

A Generalization Upper Bound
Failure mode: conditional shift increases during adaptation
Could we also take into account the conditional shift?

14

A Generalization Upper Bound
Failure mode: conditional shift increases during adaptation
Could we also take into account the conditional shift?
Theorem (deterministic setting): Let hDS , fS i and hDT , fT i be the source
and target domains. For any function class H ✓ [0, 1]X , and 8h 2 H ,the following inequality holds:<latexit sha1_base64="oeO3Id9qqDVNC1pMN4uQhYTi3T8=">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</latexit><latexit sha1_base64="1QdPM51J2GUl9zBOHrGaCFANtgQ=">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</latexit><latexit sha1_base64="CdTIWttOC609bflsrMwTJ7N9sfE=">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</latexit><latexit sha1_base64="7eNAKYOU0HRqWCL76F3YCdWzZ1A=">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</latexit>"S (h)<latexit sha1_base64="7DBUCBE0/4XbImNI3BbAdTJUWv8=">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</latexit>"T (h)  dH̃ (DS , DT ) + min{EDS [|fS

• H̃ := {sgn(|h(x)<latexit sha1_base64="4gxC8BZTbVHah9Q/teZcTxSucEo=">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</latexit>

h0 (x)|

fT |], EDT [|fS

fT |]}t) | h, h0 2 H, 0  t  1} , a hypothesis classinduced by H
• fS /fT labeling function in source/target domains<latexit sha1_base64="SM1q2BE3fqdv6Zb7lcRNP5QRJs0=">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</latexit><latexit sha1_base64="CgUVVB7kl+cZsFJd9QGgcFRBjpg=">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</latexit>

14

A Generalization Upper Bound
Failure mode: conditional shift increases during adaptation
Could we also take into account the conditional shift?
Theorem (deterministic setting): Let hDS , fS i and hDT , fT i be the source
and target domains. For any function class H ✓ [0, 1]X , and 8h 2 H ,the following inequality holds:<latexit sha1_base64="oeO3Id9qqDVNC1pMN4uQhYTi3T8=">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</latexit><latexit sha1_base64="1QdPM51J2GUl9zBOHrGaCFANtgQ=">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</latexit><latexit sha1_base64="CdTIWttOC609bflsrMwTJ7N9sfE=">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</latexit><latexit sha1_base64="7eNAKYOU0HRqWCL76F3YCdWzZ1A=">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</latexit>"S (h)<latexit sha1_base64="7DBUCBE0/4XbImNI3BbAdTJUWv8=">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</latexit>"T (h)  dH̃ (DS , DT ) + min{EDS [|fS

fT |], EDT [|fS

fT |]}

Shift between data distributions• H̃ := {sgn(|h(x)<latexit sha1_base64="4gxC8BZTbVHah9Q/teZcTxSucEo=">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</latexit>

h0 (x)|t) | h, h0 2 H, 0  t  1} , a hypothesis classinduced by H
• fS /fT labeling function in source/target domains<latexit sha1_base64="SM1q2BE3fqdv6Zb7lcRNP5QRJs0=">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</latexit><latexit sha1_base64="CgUVVB7kl+cZsFJd9QGgcFRBjpg=">AAAD33icdZJNb9NAEIa3MR8lfLVw5GIRVeIU7IIEl0oVUCncipo0leIoWq/HySr74e6um4SVz9wQHIF/w7/g37BOg5psxEiWRu/zzow9nrRgVJso+rPTCG7dvnN3917z/oOHjx7v7T8517JUBHpEMqkuUqyBUQE9Qw2Di0IB5imDfjp9X/P+FShNpeiaRQFDjseC5pRg46R+Pjp7mY+6o71W1I6WEW4n8SppoVWcjvYbv5NMkpKDMIRhrQdxVJihxcpQwqBqJqWGApMpHsPApQJz0EO7fN8qPHBKFuZSuUeYcKmuV1jMtV7w1Dk5NhPts1r8HzMTXvlaPUVX4TI2mDMbKZnerKj7bsuD0uRvh5aKojQgyPVX5CULjQzrzYYZVUAMW7gEE0XdIkIywQoT4/bfPAj9EQX+LDcncDwFJSV3i8KCAHNUwIxIzrHIbEJFVtmkLk1T+7Hy6AzoeGL+GXI78w3z+Q2c+1AXYkUJZvZsGxsFsObo+o5OZ412fApKuR9vkyusoNCUSeEZMnfqaw0++A0yyOGybkDqi3MHI478EZfOU9mjNYvnOJkXN/s7qSp39bF/49vJ+WE7ftU+/PS6dfxudf+76Bl6jl6gGL1Bx6iDTlEPETRF39FP9CvAwZfga/Dt2trYWdU8RRsR/PgLqA1WNQ==</latexit>

14

A Generalization Upper Bound
Failure mode: conditional shift increases during adaptation
Could we also take into account the conditional shift?
Theorem (deterministic setting): Let hDS , fS i and hDT , fT i be the source
and target domains. For any function class H ✓ [0, 1]X , and 8h 2 H ,the following inequality holds:<latexit sha1_base64="oeO3Id9qqDVNC1pMN4uQhYTi3T8=">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</latexit><latexit sha1_base64="1QdPM51J2GUl9zBOHrGaCFANtgQ=">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</latexit><latexit sha1_base64="CdTIWttOC609bflsrMwTJ7N9sfE=">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</latexit><latexit sha1_base64="7eNAKYOU0HRqWCL76F3YCdWzZ1A=">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</latexit>"S (h)<latexit sha1_base64="7DBUCBE0/4XbImNI3BbAdTJUWv8=">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</latexit>"T (h)  dH̃ (DS , DT ) + min{EDS [|fS

Shift between data distributions• H̃ := {sgn(|h(x)<latexit sha1_base64="4gxC8BZTbVHah9Q/teZcTxSucEo=">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</latexit>

h0 (x)|

fT |], EDT [|fS

fT |]}

Shift between conditional distributionst) | h, h0 2 H, 0  t  1} , a hypothesis classinduced by H
• fS /fT labeling function in source/target domains<latexit sha1_base64="SM1q2BE3fqdv6Zb7lcRNP5QRJs0=">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</latexit><latexit sha1_base64="CgUVVB7kl+cZsFJd9QGgcFRBjpg=">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</latexit>

14

A Generalization Upper Bound
Theorem (deterministic setting): Let hDS , fS i and hDT , fT i be the source
and target domains. For any function class H ✓ [0, 1]X , and 8h 2 H ,the following inequality holds:<latexit sha1_base64="oeO3Id9qqDVNC1pMN4uQhYTi3T8=">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</latexit><latexit sha1_base64="1QdPM51J2GUl9zBOHrGaCFANtgQ=">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</latexit><latexit sha1_base64="CdTIWttOC609bflsrMwTJ7N9sfE=">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</latexit><latexit sha1_base64="7eNAKYOU0HRqWCL76F3YCdWzZ1A=">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</latexit>"S (h)<latexit sha1_base64="7DBUCBE0/4XbImNI3BbAdTJUWv8=">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</latexit>"T (h)  dH̃ (DS , DT ) + min{EDS [|fS

Shift between data distributions

fT |], EDT [|fS

fT |]}

Shift between conditional distributions

15

A Generalization Upper Bound
Theorem (deterministic setting): Let hDS , fS i and hDT , fT i be the source
and target domains. For any function class H ✓ [0, 1]X , and 8h 2 H ,the following inequality holds:<latexit sha1_base64="oeO3Id9qqDVNC1pMN4uQhYTi3T8=">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</latexit><latexit sha1_base64="1QdPM51J2GUl9zBOHrGaCFANtgQ=">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</latexit><latexit sha1_base64="CdTIWttOC609bflsrMwTJ7N9sfE=">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</latexit><latexit sha1_base64="7eNAKYOU0HRqWCL76F3YCdWzZ1A=">AAAD+3icdVNNj9MwEPU2fCzlqwtHLoZqJQ6oShckuKy0AlYKt0Vsdys1pXKcSWvVsVPb2bZEOfNDuCE4Anf+Bf8Gpy1q64qRIo3ee56xZ16ijDNtfP/PXs27dv3Gzf1b9dt37t673zh4cKFlrih0qORSdSOigTMBHcMMh26mgKQRh8to/KbiL69AaSbFuZln0E/JULCEUWIsNGg8DoMg1HmkwcAE9/xnuN3/WIQpMSNKeNEty0Gj6bf8ReDdpL1KmmgVZ4OD2u8wljRPQRjKida9tp+ZfkGUYZRDWQ9zDRmhYzKEnk0FSUH3i8VbSnxokRgnUtlPGLxAN08UJNV6nkZWWV1Su1wF/o8zo7R0saqLLvEitjgrNlJyvX2iqrsL93KTvOoXTGS5AUGXr0hyjo3E1dRxzBRQw+c2IVQxOwhMR0QRauxu6ofYbZGRT3K7Q0rGoKRM7aCIoMAtK2BKZZoSERchE3G5XFsUFe9Kh50CG47MP0FSTF3BbLYmZy6pM1GuLfFhlzYKYENx7iqCYIMNXBaUsosvwiuiINOMS+EIYvsbbBR46xaIIYFJVYBWjrOGEcdui4nVlMXxhsRRnM6y9fxOF65vux7fTS6OWu3nraP3L5onr1f+30eP0BP0FLXRS3SCAnSGOoiiz+g7+ol+eaX3xfvqfVtKa3urMw/RVng//gKW72Ef</latexit>"T (h)  dH̃ (DS , DT ) + min{EDS [|fS

"S (h)<latexit sha1_base64="7DBUCBE0/4XbImNI3BbAdTJUWv8=">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</latexit>

Shift between data distributions•

Free of<latexit sha1_base64="slVYf49G6Yu+rcwWP4w4CcIA6YE=">AAAD4XicdZLNbtNAEMe3MR8lfLVw5GIRVUIcIqdFgkulCqgUbkU0bVEcqvV6nKyyH+7uuklY+QG4ITgCL8Nb8DasU6M6GzGSpdH8fzNjj/9Jzqg2UfRnoxXcuHnr9uad9t179x883Np+dKJloQgMiGRSnSVYA6MCBoYaBme5AswTBqfJ9E2ln16C0lSKY7PIYcTxWNCMEmxc6WPMHJriT8/PtzpRN1pGuJ706qSD6jg63279jlNJCg7CEIa1Hvai3IwsVoYSBmU7LjTkmEzxGIYuFZiDHtnlG5fhjqukYSaVe4QJl9Vmh8Vc6wVPHMmxmWhfq4r/08yEl36t2qLLcBkrmoONlEyvdlRz18vDwmSvRpaKvDAgyNVXZAULjQyr24YpVUAMW7gEE0XdIUIywQoT4/5Aeyf0V+T4s1zdwPEUlJTcHQoLAsypAmZEco5FamMq0tLGVWuS2Help86AjifmH5DZmQ/M59fi3Bd1LmqVYGY/rMtGATSIY5/o9xtq31dBKffjbXyJFeSaMik8IHVmbwx46w9IIYOLagCpHOcMI/b9FReOKe1+A/GIw3l+fb/DsnSu7/keX09Odru9ve7u+xedg9e1/zfRE/QUPUM99BIdoD46QgNEEEff0U/0KyDBl+Br8O0KbW3UPY/RSgQ//gIUglcu</latexit>⇤fT |], EDT [|fS

fT |]}

Shift between conditional distributions, the optimal joint error

15

A Generalization Upper Bound
Theorem (deterministic setting): Let hDS , fS i and hDT , fT i be the source
and target domains. For any function class H ✓ [0, 1]X , and 8h 2 H ,the following inequality holds:<latexit sha1_base64="oeO3Id9qqDVNC1pMN4uQhYTi3T8=">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</latexit><latexit sha1_base64="1QdPM51J2GUl9zBOHrGaCFANtgQ=">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</latexit><latexit sha1_base64="CdTIWttOC609bflsrMwTJ7N9sfE=">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</latexit><latexit sha1_base64="7eNAKYOU0HRqWCL76F3YCdWzZ1A=">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</latexit>"S (h)<latexit sha1_base64="7DBUCBE0/4XbImNI3BbAdTJUWv8=">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</latexit>"T (h)  dH̃ (DS , DT ) + min{EDS [|fS

Shift between data distributions••

fT |], EDT [|fS

fT |]}

Shift between conditional distributions

Free of ⇤, the optimal joint error
Two more noise terms if we extend the bound to stochastic setting<latexit sha1_base64="slVYf49G6Yu+rcwWP4w4CcIA6YE=">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</latexit>

15

A Generalization Upper Bound
Theorem (deterministic setting): Let hDS , fS i and hDT , fT i be the source
and target domains. For any function class H ✓ [0, 1]X , and 8h 2 H ,the following inequality holds:<latexit sha1_base64="oeO3Id9qqDVNC1pMN4uQhYTi3T8=">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</latexit><latexit sha1_base64="1QdPM51J2GUl9zBOHrGaCFANtgQ=">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</latexit><latexit sha1_base64="CdTIWttOC609bflsrMwTJ7N9sfE=">AAAD6XicdVLbbhMxEHW7XEq4NIVHXiyiSjxFSakEL5UqoNLyVkTTVspGkdc7m1jxZWt7mwRrP4I3BI/Aj/AX/A3eNKiJI0ZaaXTOmRnvzEkLzoztdP5sbUd37t67v/Og8fDR4ye7zb2n50aVmkKPKq70ZUoMcCahZ5nlcFloICLlcJFO3tX8xTVow5Q8s/MCBoKMJMsZJdZDw+ZukitNOMfjhMkkjofNVqfdWQTeTLrLpIWWcTrc2/6dZIqWAqSlnBjT73YKO3BEW0Y5VI2kNFAQOiEj6PtUEgFm4BYvr/C+RzLsn+A/afECXa1wRBgzF6lXCmLHJuRq8H+cHYsqxOoppsKLWOO82CrFzXpF3XcT7pc2fzNwTBalBUlv/iIvObYK1zvGGdNALZ/7hFDN/CIwHRNNqPWXaOzjcERBPqv1CYJMQCsl/KKIpMA9K2FKlRBEZs6fKqtcUpemqftQBewU2Ghs/wlyNw0Fs9ktOQtJU8glSwl3nzZpqwFWFGehIo5X2DhkQWt/eJdcEw2FYVzJQJB50680eB82yCCHq7oBrR3nDSOPwhFXXlO5oxVJoDiZFbf7O6kq7/pu6PHN5Pyg3X3VPvh42Dp+u/T/DnqOXqCXqIteo2MUo1PUQxSV6Dv6iX5Fk+hL9DX6diPd3lrWPENrEf34C40AWdk=</latexit><latexit sha1_base64="7eNAKYOU0HRqWCL76F3YCdWzZ1A=">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</latexit>"T (h)  dH̃ (DS , DT ) + min{EDS [|fS

"S (h)<latexit sha1_base64="7DBUCBE0/4XbImNI3BbAdTJUWv8=">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</latexit>

Shift between data distributions•••

fT |], EDT [|fS

fT |]}

Shift between conditional distributions

Free of ⇤, the optimal joint error
Two more noise terms if we extend the bound to stochastic setting
Can use standard concentration argument to get high probability bound of"S (h) and dH̃ (DS , DT )<latexit sha1_base64="slVYf49G6Yu+rcwWP4w4CcIA6YE=">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</latexit><latexit sha1_base64="cdZZM8f5t8VJPgUQIsHBgxgQY9s=">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</latexit><latexit sha1_base64="YvP5PmUuJGrClKe+1n8qDNuGc94=">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</latexit>

15

An Information-Theoretic Lower Bound
Can we extend the counter-example to more general cases?
Consider the general adaptation scenario:gh

X ! Z ! Ŷ<latexit sha1_base64="jPZmvWu31gXEHv19NR427Bm/VUk=">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</latexit>• g(·) (nonlinear) feature transformation<latexit sha1_base64="ngzDfbmlbWO/58BVMXnQ6g73ABM=">AAAEAXicdZPNbhMxEMfdLNASvlo4clkRVSqXKGmR6KVSBVQKnIpo2kjZKPJ6ZxMTf2xtb5Ng7Yk34AovwA1x5Uk48yJ400VNHDHSSqP5/WfGO/bEGaPatFq/N2rBrdt3Nrfu1u/df/Dw0fbO43Mtc0WgSySTqhdjDYwK6BpqGPQyBZjHDC7iyeuSX1yB0lSKMzPPYMDxSNCUEmxcqDfai0gizfPhdqPVbC0sXHfaldNAlZ0Od2p/okSSnIMwhGGt++1WZgYWK0MJg6Ie5RoyTCZ4BH3nCsxBD+ziwEW46yJJmErlPmHCRXQ5w2Ku9ZzHTsmxGWuflcH/MTPmhR8ru+giXNgKc2IjJdOrGWXd9XA/N+nhwFKR5QYEuf6LNGehkWE52jChCohhc+dgoqgbREjGWGFi3AXUd0O/RYY/ydUOHE9AScndoLAgwBwVMCWScywSG1GRFDYqU+PYvi08OgU6Gpt/gtROfcFsdgNnPtSZqCjBzH5Yx0YBLCnOfEWns0Q7PgWl3MXb6AoryDRlUniCxL31pQJv/AIJpHBZFiDli3MPRhz5LS6dprBHSxJPcTLLbuZ3stbhoy5sMrSRgZmx79wICrcWbX8J1p3z/Wb7oLn//kXj+FW1IFvoKXqG9lAbvUTHqINOURcRxNAX9BV9Cz4H34Mfwc9raW2jynmCViz49RcdVmTw</latexit>• h(·) 2 {0, 1} (randomized) classification function<latexit sha1_base64="IBDm9C+usnKs1MYLBrtS5HzA07Q=">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</latexit>

16

An Information-Theoretic Lower Bound
Can we extend the counter-example to more general cases?
Consider the general adaptation scenario:gh

X ! Z ! Ŷ<latexit sha1_base64="jPZmvWu31gXEHv19NR427Bm/VUk=">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</latexit>• g(·) (nonlinear) feature transformation<latexit sha1_base64="ngzDfbmlbWO/58BVMXnQ6g73ABM=">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</latexit>• h(·) 2 {0, 1} (randomized) classification function<latexit sha1_base64="IBDm9C+usnKs1MYLBrtS5HzA07Q=">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</latexit>

Theorem: suppose the Markov chain holds and dJS (DSY , DTY ) dJS (DSZ , DTZ ) ,
then
1
Y
Y
Z
Z 2
"S (h g) + "T (h g)dJS (DS , DT ) dJS (DS , DT )2ZZ
• DS /DT : induced distributions by g(·) from DS /DT• dJS (·, ·) : Jensen-Shannon distance<latexit sha1_base64="1n4GSkFYerjriQZwOPSaKwy1IGo=">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</latexit><latexit sha1_base64="o0vVP/pKHqoVvwfwz2GoZY7xVeA=">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</latexit><latexit sha1_base64="ngzDfbmlbWO/58BVMXnQ6g73ABM=">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</latexit><latexit sha1_base64="amjjdqfEL7qFuQ3pqlE9TRRyFlc=">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</latexit><latexit sha1_base64="cq98tzt2ECQpWdCP8l++oWRbuvg=">AAAECnicdVPLbtNAFJ3GPEp4NIUlG4uoEquQFCTYVKpaKgVWRU3aSnEUjcfXyZB5uDPjJmHkP+AP2MIPsENs+QnW/AjjJKjJRFzJ8tU5595j35kbZ4xq02z+3qoEt27fubt9r3r/wcNHO7Xdx+da5opAl0gm1WWMNTAqoGuoYXCZKcA8ZnARj49L/uIalKZSdMwsgz7HQ0FTSrBx0KC2EyXOZHD2YvHuDGr1ZqM5j3AzaS2TOlrG6WC38idKJMk5CEMY1rrXamamb7EylDAoqlGuIcNkjIfQc6nAHHTfzr+8CPcckoSpVO4RJpyjqxUWc61nPHZKjs1I+1wJ/o8zI174WOmii3Aea5wTGymZXq8o+27Cvdykb/qWiiw3IMjiL9KchUaG5YzDhCoghs1cgomibhAhGWGFiXEnUd0LfYsMf5LrDhyPQUnJ3aCwIMAcK2BCJOdYJDaiIilsVJbGsX1XeOwE6HBk/glSO/EF0+kNOfVJnYklSzCzZ5u0UQArio6vaLdX2LbPglLu4G10jRVkmjIpPEF5D1cavPUbJJDCVdmAlDfOXRhx4FtcOU1hD1YknuJkmt3M72TD4aMubDKwkYGpse/dCAq3Fi1/CTaT8/1G62Vj/8Or+uHRckG20VP0DD1HLfQaHaI2OkVdRFCOvqCv6FvwOfge/Ah+LqSVrWXNE7QWwa+/o9loXQ==</latexit><latexit sha1_base64="dozgRgP3BIWmfq3VN7kE7ijJptU=">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</latexit>

d2JS (D, D0 ) =<latexit sha1_base64="UQmM0VBIc0ChmbUtbFYJ0R13kk4=">AAAEsHicdZNtT9swEMdd6BhjD8D2ctJkDSFAQ1XbTdreIKENpO5JYhpPEukqx7lQU8cOtgNlJnyIfbt9k72ckxaRusJSksv9f3enXO7ClDNtms2/tZnZ+oO5h/OPFh4/efpscWn5+aGWmaJwQCWX6jgkGjgTcGCY4XCcKiBJyOEoHHwq9KMLUJpJsW+uUugm5FSwmFFinKu39CeIzvSv9noQuVqb5X1tA2/hIFaE2lZu23kw4BG7GBE319c3pdH7voHf3EetVbHN4DwjER6/uifEcF6NHMUU2UbVe0srzUazPHjaaI2NFTQ+e73l2bkgkjRLQBjKidYnrWZqupYowyiHfCHINKSEDsgpnDhTkAR015bNy/Gq80Q4lspdwuDSW42wJNH6KgkdmRDT175WOO/TTD/JfV9RRee4PBOag42UXE9GFHmn3SeZiT90LRNpZkDQ0VfEGcdG4uI344gpoIZfOYNQxVwjMO0T13PjhmFhFfslUvJbTlZIyACUlIlrFBEUuFMFXFKZJERENmAiym1QhIah/Zx76iWw0765BWJ76QPD4Z049EWdirFKCbc/p2WjACrEvk90OhW146uglPvxNrggClLNuBQeUMxhJcGOn6Ac4SIBLSbODYzY8kucOya3WxXEI3aH6V3/dqcqnOncRj0bGBga+8W1wAPKbcvtzi3y9ZtD3Oa0/D2ZNg7bjdbbRvvHu5Xtj+Mdmkcv0Wu0jlroPdpGHbSHDhBF/2qvamu19Xq7flzv1ckInamNY16giVM/+w9ouaKE</latexit>

11
DKL (D || DM ) + DKL (D0 || DM ),2
2

DM : =

1(D + D0 )2

16

An Information-Theoretic Lower Bound
Theorem: suppose the Markov chain holds and dJS (DSY , DTY ) dJS (DSZ , DTZ ) ,
then
1
Y
Y
Z
Z 2
"S (h g) + "T (h g)dJS (DS , DT ) dJS (DS , DT )2ZZ
• DS /DT : induced distributions by g(·) from DS /DT• dJS (·, ·) : Jensen-Shannon distance<latexit sha1_base64="1n4GSkFYerjriQZwOPSaKwy1IGo=">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</latexit><latexit sha1_base64="o0vVP/pKHqoVvwfwz2GoZY7xVeA=">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</latexit><latexit sha1_base64="ngzDfbmlbWO/58BVMXnQ6g73ABM=">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</latexit><latexit sha1_base64="amjjdqfEL7qFuQ3pqlE9TRRyFlc=">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</latexit><latexit sha1_base64="cq98tzt2ECQpWdCP8l++oWRbuvg=">AAAECnicdVPLbtNAFJ3GPEp4NIUlG4uoEquQFCTYVKpaKgVWRU3aSnEUjcfXyZB5uDPjJmHkP+AP2MIPsENs+QnW/AjjJKjJRFzJ8tU5595j35kbZ4xq02z+3qoEt27fubt9r3r/wcNHO7Xdx+da5opAl0gm1WWMNTAqoGuoYXCZKcA8ZnARj49L/uIalKZSdMwsgz7HQ0FTSrBx0KC2EyXOZHD2YvHuDGr1ZqM5j3AzaS2TOlrG6WC38idKJMk5CEMY1rrXamamb7EylDAoqlGuIcNkjIfQc6nAHHTfzr+8CPcckoSpVO4RJpyjqxUWc61nPHZKjs1I+1wJ/o8zI174WOmii3Aea5wTGymZXq8o+27Cvdykb/qWiiw3IMjiL9KchUaG5YzDhCoghs1cgomibhAhGWGFiXEnUd0LfYsMf5LrDhyPQUnJ3aCwIMAcK2BCJOdYJDaiIilsVJbGsX1XeOwE6HBk/glSO/EF0+kNOfVJnYklSzCzZ5u0UQArio6vaLdX2LbPglLu4G10jRVkmjIpPEF5D1cavPUbJJDCVdmAlDfOXRhx4FtcOU1hD1YknuJkmt3M72TD4aMubDKwkYGpse/dCAq3Fi1/CTaT8/1G62Vj/8Or+uHRckG20VP0DD1HLfQaHaI2OkVdRFCOvqCv6FvwOfge/Ah+LqSVrWXNE7QWwa+/o9loXQ==</latexit><latexit sha1_base64="dozgRgP3BIWmfq3VN7kE7ijJptU=">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</latexit>

17

An Information-Theoretic Lower Bound
Theorem: suppose the Markov chain holds and dJS (DSY , DTY ) dJS (DSZ , DTZ ) ,
then
1
Y
Y
Z
Z 2
"S (h g) + "T (h g)dJS (DS , DT ) dJS (DS , DT )2ZZ
• DS /DT : induced distributions by g(·) from DS /DT• dJS (·, ·) : Jensen-Shannon distance<latexit sha1_base64="1n4GSkFYerjriQZwOPSaKwy1IGo=">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</latexit><latexit sha1_base64="o0vVP/pKHqoVvwfwz2GoZY7xVeA=">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</latexit><latexit sha1_base64="ngzDfbmlbWO/58BVMXnQ6g73ABM=">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</latexit><latexit sha1_base64="amjjdqfEL7qFuQ3pqlE9TRRyFlc=">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</latexit><latexit sha1_base64="cq98tzt2ECQpWdCP8l++oWRbuvg=">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</latexit><latexit sha1_base64="dozgRgP3BIWmfq3VN7kE7ijJptU=">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</latexit>•

For domain-invariant representation, dJS (DSZ , DTZ ) = 0 , lower boundachieves maximum value<latexit sha1_base64="wJcwlVlzkobq3fRTi5Aou7zKpWI=">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</latexit>

17

An Information-Theoretic Lower Bound
Theorem: suppose the Markov chain holds and dJS (DSY , DTY ) dJS (DSZ , DTZ ) ,
then
1
Y
Y
Z
Z 2
"S (h g) + "T (h g)dJS (DS , DT ) dJS (DS , DT )2ZZ
• DS /DT : induced distributions by g(·) from DS /DT• dJS (·, ·) : Jensen-Shannon distance<latexit sha1_base64="1n4GSkFYerjriQZwOPSaKwy1IGo=">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</latexit><latexit sha1_base64="o0vVP/pKHqoVvwfwz2GoZY7xVeA=">AAAEenicdVNLbxMxEHbaACXl0fI4cbFaVWpUGiWhElwqVUClwKko6TObRl7vbOLWa29tp0mxVvxOzoh/wQFvEtTEESPtavR938zsziNMOdOmWv1ZWFouPnj4aOVxafXJ02fP19ZfnGg5UBSOqeRSnYVEA2cCjg0zHM5SBSQJOZyG159y/vQWlGZStMxdCp2E9ASLGSXGQd21HwEoJVW3ud0PKFMU98p4B0/B1gwY9OAGB7Ei1NYyW88CDrHZDqIr7V7uO7vNy/O3eOK2Ls/LeBfPkRf35EU5UKzXN+XLeqm7tlmtVMeGF53a1NlEUzvqri/9CiJJBwkIQznRul2rpqZjiTKMcshKwUBDSug16UHbuYIkoDt23KkMbzkkwrFU7hEGj9HZCEsSre+S0CkTYvra53Lwf5zpJ5mP5VV0hsc2xzmxkZLr+Yg87yLcHpj4Q8cykQ4MCDr5i3jAsZE4nymOmAJq+J1zCFXMNQLTPnGjMm7ypS3sl0jJdzlfISHXoKRMXKOIoMAdK2BIZZIQEdmAiSizQR4ahvZL5rFDyKf5TxDboS8Yje7JkU/qVExZSrhtLtJGAcwoWr6i0ZhhGz473mQnuCUKUs24FJ4gX8qZBJ/9BBHEcJMnoPnGuYUR+36JG6fJ7P6MxFMcjtL7/h0uVLjSmY26NjAwMvara0HmzqLmH8Gic1Kv1N5V6t/2Ng8+Tg9kBb1BG2gb1dB7dIAa6AgdI4p+F1YLrwqvl/8UN4rl4s5EulSYxrxEc1bc+wsCyYu/</latexit><latexit sha1_base64="ngzDfbmlbWO/58BVMXnQ6g73ABM=">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</latexit><latexit sha1_base64="amjjdqfEL7qFuQ3pqlE9TRRyFlc=">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</latexit><latexit sha1_base64="cq98tzt2ECQpWdCP8l++oWRbuvg=">AAAECnicdVPLbtNAFJ3GPEp4NIUlG4uoEquQFCTYVKpaKgVWRU3aSnEUjcfXyZB5uDPjJmHkP+AP2MIPsENs+QnW/AjjJKjJRFzJ8tU5595j35kbZ4xq02z+3qoEt27fubt9r3r/wcNHO7Xdx+da5opAl0gm1WWMNTAqoGuoYXCZKcA8ZnARj49L/uIalKZSdMwsgz7HQ0FTSrBx0KC2EyXOZHD2YvHuDGr1ZqM5j3AzaS2TOlrG6WC38idKJMk5CEMY1rrXamamb7EylDAoqlGuIcNkjIfQc6nAHHTfzr+8CPcckoSpVO4RJpyjqxUWc61nPHZKjs1I+1wJ/o8zI174WOmii3Aea5wTGymZXq8o+27Cvdykb/qWiiw3IMjiL9KchUaG5YzDhCoghs1cgomibhAhGWGFiXEnUd0LfYsMf5LrDhyPQUnJ3aCwIMAcK2BCJOdYJDaiIilsVJbGsX1XeOwE6HBk/glSO/EF0+kNOfVJnYklSzCzZ5u0UQArio6vaLdX2LbPglLu4G10jRVkmjIpPEF5D1cavPUbJJDCVdmAlDfOXRhx4FtcOU1hD1YknuJkmt3M72TD4aMubDKwkYGpse/dCAq3Fi1/CTaT8/1G62Vj/8Or+uHRckG20VP0DD1HLfQaHaI2OkVdRFCOvqCv6FvwOfge/Ah+LqSVrWXNE7QWwa+/o9loXQ==</latexit><latexit sha1_base64="dozgRgP3BIWmfq3VN7kE7ijJptU=">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</latexit>•

For domain-invariant representation, dJS (DSZ , DTZ ) = 0 , lower boundachieves maximum value<latexit sha1_base64="wJcwlVlzkobq3fRTi5Aou7zKpWI=">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</latexit>•

After a given threshold, minimizing source error and domain
discrepancy leads to larger target error

17

An Information-Theoretic Lower Bound
Theorem: suppose the Markov chain holds and dJS (DSY , DTY ) dJS (DSZ , DTZ ) ,
then
1
Y
Y
Z
Z 2
"S (h g) + "T (h g)dJS (DS , DT ) dJS (DS , DT )2
Proof Sketch:<latexit sha1_base64="1n4GSkFYerjriQZwOPSaKwy1IGo=">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</latexit><latexit sha1_base64="o0vVP/pKHqoVvwfwz2GoZY7xVeA=">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</latexit>

18

An Information-Theoretic Lower Bound
Theorem: suppose the Markov chain holds and dJS (DSY , DTY ) dJS (DSZ , DTZ ) ,
then
1
Y
Y
Z
Z 2
"S (h g) + "T (h g)dJS (DS , DT ) dJS (DS , DT )2
Proof Sketch:<latexit sha1_base64="1n4GSkFYerjriQZwOPSaKwy1IGo=">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</latexit><latexit sha1_base64="o0vVP/pKHqoVvwfwz2GoZY7xVeA=">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</latexit>

JS distance is a metric:dJS (DSY , DTY )  dJS (DSY , DSŶ ) + dJS (DSŶ , DTŶ ) + dJS (DTŶ , DTY )<latexit sha1_base64="uk58GrQxfTIvxUdtn8HA6wrrckc=">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</latexit>

18

An Information-Theoretic Lower Bound
Theorem: suppose the Markov chain holds and dJS (DSY , DTY ) dJS (DSZ , DTZ ) ,
then
1
Y
Y
Z
Z 2
"S (h g) + "T (h g)dJS (DS , DT ) dJS (DS , DT )2
Proof Sketch:<latexit sha1_base64="1n4GSkFYerjriQZwOPSaKwy1IGo=">AAAENHicdZPNbhMxEMfdho8SvlI4clkRVWolFCUtEnCoVAGVAqeiJv3KppHXO5u48dob29skWPssvAVvwBXOSNwqrjwD3iQoiSNGsjSa339m5Bk7SBhVulr9ubZeuHX7zt2Ne8X7Dx4+elzafHKiRCoJNIlgQp4FWAGjHJqaagZniQQcBwxOg/67nJ9eg1RU8IYeJ9COcZfTiBKsbahTeuOHV2rbD22nzvHl+Qtv6jYuz3f8Lgy8JXwxxxc7nVK5WqlOzFt1ajOnjGZ21Nlcv/FDQdIYuCYMK9WqVRPdNlhqShhkRT9VkGDSx11oWZfjGFTbTO6YeVs2EnqRkPZw7U2iixkGx0qN48AqY6x7ymV58H9M9+LMjeVdVOZNbIlZsRaCqeWMvO5quJXq6HXbUJ6kGjiZ3iJKmaeFl2/DC6kEotnYOphIagfhkR6WmGi7s+KW57ZI8Gex3CHGfZBCxHZQmBNglnIYEhHHmIfGpzzMjJ+nBoH5kDl0CLTb0/8EkRm6gtFoDkcuVAmfUYKZOV7FWgIsKBquol5foHWXgpR28ca/xhISRZngjiB/igsF3rsFQohgkBcg+YuzD4bvuy0GVpOZ/QWJozgcJfP5Ha50uFKZCTvG1zDS5qMdQWa/Rc39BKvOyW6ltlfZ/fSyfPB29kE20DP0HG2jGnqFDlAdHaEmIugL+oa+ox+Fr4VfhZvC76l0fW2W8xQtWeHPX7zAeGI=</latexit><latexit sha1_base64="o0vVP/pKHqoVvwfwz2GoZY7xVeA=">AAAEenicdVNLbxMxEHbaACXl0fI4cbFaVWpUGiWhElwqVUClwKko6TObRl7vbOLWa29tp0mxVvxOzoh/wQFvEtTEESPtavR938zsziNMOdOmWv1ZWFouPnj4aOVxafXJ02fP19ZfnGg5UBSOqeRSnYVEA2cCjg0zHM5SBSQJOZyG159y/vQWlGZStMxdCp2E9ASLGSXGQd21HwEoJVW3ud0PKFMU98p4B0/B1gwY9OAGB7Ei1NYyW88CDrHZDqIr7V7uO7vNy/O3eOK2Ls/LeBfPkRf35EU5UKzXN+XLeqm7tlmtVMeGF53a1NlEUzvqri/9CiJJBwkIQznRul2rpqZjiTKMcshKwUBDSug16UHbuYIkoDt23KkMbzkkwrFU7hEGj9HZCEsSre+S0CkTYvra53Lwf5zpJ5mP5VV0hsc2xzmxkZLr+Yg87yLcHpj4Q8cykQ4MCDr5i3jAsZE4nymOmAJq+J1zCFXMNQLTPnGjMm7ypS3sl0jJdzlfISHXoKRMXKOIoMAdK2BIZZIQEdmAiSizQR4ahvZL5rFDyKf5TxDboS8Yje7JkU/qVExZSrhtLtJGAcwoWr6i0ZhhGz473mQnuCUKUs24FJ4gX8qZBJ/9BBHEcJMnoPnGuYUR+36JG6fJ7P6MxFMcjtL7/h0uVLjSmY26NjAwMvara0HmzqLmH8Gic1Kv1N5V6t/2Ng8+Tg9kBb1BG2gb1dB7dIAa6AgdI4p+F1YLrwqvl/8UN4rl4s5EulSYxrxEc1bc+wsCyYu/</latexit>

JS distance is a metric:dJS (DSY , DTY )  dJS (DSY , DSŶ ) + dJS (DSŶ , DTŶ ) + dJS (DTŶ , DTY )<latexit sha1_base64="uk58GrQxfTIvxUdtn8HA6wrrckc=">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</latexit>

Data-processing Principle:dJS (DSŶ , DTŶ )  dJS (DSZ , DTZ )<latexit sha1_base64="Vh+n9HhF2vK5y4KswRtsTc+Ln/s=">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</latexit>

18

An Information-Theoretic Lower Bound
Theorem: suppose the Markov chain holds and dJS (DSY , DTY ) dJS (DSZ , DTZ ) ,
then
1
Y
Y
Z
Z 2
"S (h g) + "T (h g)dJS (DS , DT ) dJS (DS , DT )2
Proof Sketch:<latexit sha1_base64="1n4GSkFYerjriQZwOPSaKwy1IGo=">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</latexit><latexit sha1_base64="o0vVP/pKHqoVvwfwz2GoZY7xVeA=">AAAEenicdVNLbxMxEHbaACXl0fI4cbFaVWpUGiWhElwqVUClwKko6TObRl7vbOLWa29tp0mxVvxOzoh/wQFvEtTEESPtavR938zsziNMOdOmWv1ZWFouPnj4aOVxafXJ02fP19ZfnGg5UBSOqeRSnYVEA2cCjg0zHM5SBSQJOZyG159y/vQWlGZStMxdCp2E9ASLGSXGQd21HwEoJVW3ud0PKFMU98p4B0/B1gwY9OAGB7Ei1NYyW88CDrHZDqIr7V7uO7vNy/O3eOK2Ls/LeBfPkRf35EU5UKzXN+XLeqm7tlmtVMeGF53a1NlEUzvqri/9CiJJBwkIQznRul2rpqZjiTKMcshKwUBDSug16UHbuYIkoDt23KkMbzkkwrFU7hEGj9HZCEsSre+S0CkTYvra53Lwf5zpJ5mP5VV0hsc2xzmxkZLr+Yg87yLcHpj4Q8cykQ4MCDr5i3jAsZE4nymOmAJq+J1zCFXMNQLTPnGjMm7ypS3sl0jJdzlfISHXoKRMXKOIoMAdK2BIZZIQEdmAiSizQR4ahvZL5rFDyKf5TxDboS8Yje7JkU/qVExZSrhtLtJGAcwoWr6i0ZhhGz473mQnuCUKUs24FJ4gX8qZBJ/9BBHEcJMnoPnGuYUR+36JG6fJ7P6MxFMcjtL7/h0uVLjSmY26NjAwMvara0HmzqLmH8Gic1Kv1N5V6t/2Ng8+Tg9kBb1BG2gb1dB7dIAa6AgdI4p+F1YLrwqvl/8UN4rl4s5EulSYxrxEc1bc+wsCyYu/</latexit>

JS distance is a metric:dJS (DSY , DTY )  dJS (DSY , DSŶ ) + dJS (DSŶ , DTŶ ) + dJS (DTŶ , DTY )<latexit sha1_base64="uk58GrQxfTIvxUdtn8HA6wrrckc=">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</latexit>

Data-processing Principle:dJS (DSŶ , DTŶ )  dJS (DSZ , DTZ )<latexit sha1_base64="Vh+n9HhF2vK5y4KswRtsTc+Ln/s=">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</latexit>

Lin’s Lemma (1991):

YŶ

dJS (D , D ) <latexit sha1_base64="M4qTacHpxfVgICVJIrQ36jJIxKM=">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</latexit>p"(h g)

18

An Information-Theoretic Lower Bound
Theorem: suppose the Markov chain holds and dJS (DSY , DTY ) dJS (DSZ , DTZ ) ,
then
1
Y
Y
Z
Z 2
"S (h g) + "T (h g)dJS (DS , DT ) dJS (DS , DT )2
Proof Sketch:<latexit sha1_base64="1n4GSkFYerjriQZwOPSaKwy1IGo=">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</latexit><latexit sha1_base64="o0vVP/pKHqoVvwfwz2GoZY7xVeA=">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</latexit>

JS distance is a metric:dJS (DSY , DTY )  dJS (DSY , DSŶ ) + dJS (DSŶ , DTŶ ) + dJS (DTŶ , DTY )<latexit sha1_base64="uk58GrQxfTIvxUdtn8HA6wrrckc=">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</latexit>

Data-processing Principle:dJS (DSŶ , DTŶ )  dJS (DSZ , DTZ )<latexit sha1_base64="Vh+n9HhF2vK5y4KswRtsTc+Ln/s=">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</latexit>

Lin’s Lemma (1991):

YŶ

dJS (D , D ) <latexit sha1_base64="M4qTacHpxfVgICVJIrQ36jJIxKM=">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</latexit>p"(h g)

Combining all three inequalities with AM-GM inequality finishes the proof.

18

An Information-Theoretic Lower Bound
Theorem: suppose the Markov chain holds and dJS (DSY , DTY ) dJS (DSZ , DTZ ) ,
then
1
Y
Y
Z
Z 2
"S (h g) + "T (h g)dJS (DS , DT ) dJS (DS , DT )2<latexit sha1_base64="1n4GSkFYerjriQZwOPSaKwy1IGo=">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</latexit><latexit sha1_base64="o0vVP/pKHqoVvwfwz2GoZY7xVeA=">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</latexit>

19

An Information-Theoretic Lower Bound
Theorem: suppose the Markov chain holds and dJS (DSY , DTY ) dJS (DSZ , DTZ ) ,
then
1
Y
Y
Z
Z 2
"S (h g) + "T (h g)dJS (DS , DT ) dJS (DS , DT )2<latexit sha1_base64="1n4GSkFYerjriQZwOPSaKwy1IGo=">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</latexit><latexit sha1_base64="o0vVP/pKHqoVvwfwz2GoZY7xVeA=">AAAEenicdVNLbxMxEHbaACXl0fI4cbFaVWpUGiWhElwqVUClwKko6TObRl7vbOLWa29tp0mxVvxOzoh/wQFvEtTEESPtavR938zsziNMOdOmWv1ZWFouPnj4aOVxafXJ02fP19ZfnGg5UBSOqeRSnYVEA2cCjg0zHM5SBSQJOZyG159y/vQWlGZStMxdCp2E9ASLGSXGQd21HwEoJVW3ud0PKFMU98p4B0/B1gwY9OAGB7Ei1NYyW88CDrHZDqIr7V7uO7vNy/O3eOK2Ls/LeBfPkRf35EU5UKzXN+XLeqm7tlmtVMeGF53a1NlEUzvqri/9CiJJBwkIQznRul2rpqZjiTKMcshKwUBDSug16UHbuYIkoDt23KkMbzkkwrFU7hEGj9HZCEsSre+S0CkTYvra53Lwf5zpJ5mP5VV0hsc2xzmxkZLr+Yg87yLcHpj4Q8cykQ4MCDr5i3jAsZE4nymOmAJq+J1zCFXMNQLTPnGjMm7ypS3sl0jJdzlfISHXoKRMXKOIoMAdK2BIZZIQEdmAiSizQR4ahvZL5rFDyKf5TxDboS8Yje7JkU/qVExZSrhtLtJGAcwoWr6i0ZhhGz473mQnuCUKUs24FJ4gX8qZBJ/9BBHEcJMnoPnGuYUR+36JG6fJ7P6MxFMcjtL7/h0uVLjSmY26NjAwMvara0HmzqLmH8Gic1Kv1N5V6t/2Ng8+Tg9kBb1BG2gb1dB7dIAa6AgdI4p+F1YLrwqvl/8UN4rl4s5EulSYxrxEc1bc+wsCyYu/</latexit>•

Lower bound still holds when separate transformations are used insource/target domains

19

An Information-Theoretic Lower Bound
Theorem: suppose the Markov chain holds and dJS (DSY , DTY ) dJS (DSZ , DTZ ) ,
then
1
Y
Y
Z
Z 2
"S (h g) + "T (h g)dJS (DS , DT ) dJS (DS , DT )2<latexit sha1_base64="1n4GSkFYerjriQZwOPSaKwy1IGo=">AAAENHicdZPNbhMxEMfdho8SvlI4clkRVWolFCUtEnCoVAGVAqeiJv3KppHXO5u48dob29skWPssvAVvwBXOSNwqrjwD3iQoiSNGsjSa339m5Bk7SBhVulr9ubZeuHX7zt2Ne8X7Dx4+elzafHKiRCoJNIlgQp4FWAGjHJqaagZniQQcBwxOg/67nJ9eg1RU8IYeJ9COcZfTiBKsbahTeuOHV2rbD22nzvHl+Qtv6jYuz3f8Lgy8JXwxxxc7nVK5WqlOzFt1ajOnjGZ21Nlcv/FDQdIYuCYMK9WqVRPdNlhqShhkRT9VkGDSx11oWZfjGFTbTO6YeVs2EnqRkPZw7U2iixkGx0qN48AqY6x7ymV58H9M9+LMjeVdVOZNbIlZsRaCqeWMvO5quJXq6HXbUJ6kGjiZ3iJKmaeFl2/DC6kEotnYOphIagfhkR6WmGi7s+KW57ZI8Gex3CHGfZBCxHZQmBNglnIYEhHHmIfGpzzMjJ+nBoH5kDl0CLTb0/8EkRm6gtFoDkcuVAmfUYKZOV7FWgIsKBquol5foHWXgpR28ca/xhISRZngjiB/igsF3rsFQohgkBcg+YuzD4bvuy0GVpOZ/QWJozgcJfP5Ha50uFKZCTvG1zDS5qMdQWa/Rc39BKvOyW6ltlfZ/fSyfPB29kE20DP0HG2jGnqFDlAdHaEmIugL+oa+ox+Fr4VfhZvC76l0fW2W8xQtWeHPX7zAeGI=</latexit><latexit sha1_base64="o0vVP/pKHqoVvwfwz2GoZY7xVeA=">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</latexit>••

Lower bound still holds when separate transformations are used insource/target domains
The proof can be extended using other distance metrics as well19

Experiments
Digit classification:
MNIST

SVHN

USPS

Marginal label distribution:

20

USPS - MNISTUSPS - SVHN

SVHN - USPS

SourceTargetSVHN - MNIST

Experiments

21

USPS - MNISTUSPS - SVHN

SVHN - USPS

SourceTargetSVHN - MNIST

Experiments

21

Summary•Suﬃcient condition: matching conditional distribution"S (h)<latexit sha1_base64="7DBUCBE0/4XbImNI3BbAdTJUWv8=">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</latexit>•"T (h)  dH̃ (DS , DT ) + min{EDS [|fS

fT |], EDT [|fS

1
dJS (DSY , DTY )2ZZ 2dJS (DS , DT )Necessary condition: matching marginal label distribution"S (h g) + "T (h g)<latexit sha1_base64="o0vVP/pKHqoVvwfwz2GoZY7xVeA=">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</latexit>fT |]}

Collaborators:

Remi Tachet des Combes

Kun Zhang

Geoﬀrey J. Gordon

Poster: #71, Pacific Ballroom, Today
22