Proofs of the Main Results of Chapter 6

Appendix 4 Proofs of the Main Results of Chapter 6

T HEOREM 6.1 ([GER 13]).– Let H be a hypothesis class. We have ∀ρ on H, E RT01 (h) ≤ E RS01 (h) + h∼ρ

h∼ρ

1 disρ (SX , TX ) + λρ , 2

where λρ is the deviation between the expected joint errors between pairs for voters on the target and source domains, which is defined as     [6.3] λρ =  eT (ρ) − eS (ρ) .

P ROOF .– First, from equation [6.2], we recall that, given a domain D on X × Y and a distribution ρ over H, we have E RD01 (h) =

h∼ρ

1 dD (ρ) + eD (ρ). 2 X

Therefore, we have    1 dTX (ρ) − dSX (ρ) + eT (ρ) − eS (ρ) 2    1     ≤ dTX (ρ) − dSX (ρ) + eT (ρ) − eS (ρ) 2 1 = disρ (SX , TX ) + λρ . 2

E RT01 (h) − E RS01 (h) =

h∼ρ

h∼ρ



172

Advances in Domain Adaptation Theory

T HEOREM 6.3 ([GER 16]).– Let H be a hypothesis space, let S and T , respectively, be the source and the target domains on X × Y and let q > 0 be a constant. We have for all posterior distributions ρ on H, " #1− q1 1 dTX (ρ) + βq × eS (ρ) + ηT \S , 2

E RT01 (h) ≤

h∼ρ

where ηT \S =

Pr

(x,y)∼T

 (x, y) ∈ /

 SUPP(S)

sup RT \S (h)

h∈H

with T \S the distribution of (x, y)∼T conditional an (x, y) ∈ SUPP(T )\SUPP(S). P ROOF .– From equation [6.2], we know that E RT01 (h) =

h∼ρ

1 2

dTX (ρ) + eT (ρ).

Let us split eT (ρ) into two parts: eT (ρ) = =

E

E

(x,y)∼T (h,h )∼ρ2

E

(x,y)∼S

+

E

(x,y)∼T

01 (h(x), y) 01 (h (x), y)

T (x, y) E 01 (h(x), y) 01 (h (x), y) S(x, y) (h,h )∼ρ2 I [(x, y) ∈ / SUPP(S)]

E

(h,h )∼ρ2

[A4.1]

01 (h(x), y) 01 (h (x), y). [A4.2]

(i) On the one hand, we upper bound the first part (line A4.1) using Hölder’s inequality, with p such that p1 = 1− 1q : T (x, y) E 01 (h(x), y) 01 (h (x), y) S(x, y) (h,h )∼ρ2 q q1 

p1 T (x, y) p  E [01 (h(x), y) 01 (h (x), y)] E ≤ E S(x, y) (x,y)∼S (h,h )∼ρ2 (x,y)∼S " # p1 = βq × eS (ρ) , E

(x,y)∼S

where we have removed the exponent from expression [01 (h(x), y)01 (h (x), y)]p without affecting its value, which is either 1 or 0.

Appendix 4

173

(ii) On the other hand, we upper bound the second part (line A4.2) by the term ηT \S ;   I [(x, y) ∈ / SUPP(S)] E  (h(x), y)  (h (x), y) E 01 01  2 (x,y)∼T

(h,h )∼ρ





E

=

(x,y)∼T

I [(x, y)∈ / SUPP(S)]



E

=  =

(x,y)∼T

I [(x, y)∈ / SUPP(S)]

E

01 (h(x), y) 01 (h (x), y)

eT \S (ρ)  E R01 (h) h∼ρ T \S



I [(x, y)∈ / SUPP(S)]

E

I [(x, y)∈ / SUPP(S)] sup RT01\S (h) = ηT \S .

(x,y)∼T

−

1 2 dT \S (ρ)

E

(x,y)∼T

 ≤

E

(x,y)∼T \S (h,h )∼ρ2

h∈H