www-ai.cs.tu-dortmund.de/LEHRE/SEMINARE/WS2122/TrustworthyAIMachineLearning/yurochkin2020a.pdf
dx(x1, x2) = (x1 − x2)TΣ∗(x1 − x2) 1 2 ,
then the error in the transportation cost function is at most |c((x1, y), (x2, y))− c∗((x1, y), (x2, y))|
= |(x1 − x2)TΣ(x1 − x2)− (x1 − x2)TΣ∗(x1 − x2)|
≤ D2 ‖Σ−Σ∗‖2 [...] ε) 1 2 dε
≤ 12√ n
∫ ∞ 0
( logN(L, ‖ · ‖∞, ε2 ) +N
( [0, L√
ε ], | · |, ε
2D2
)) 1 2 dε
≤ 12√ n
(∫ ∞ 0
logN(L, ‖ · ‖∞, ε2 ) 1 2 dε+
∫ ∞ 0
N ( [0, L√
ε ], | · |, ε
2D2
) 1 2 dε
) ≤ 24C(L)√
n +
24LD2
√ nε [...] `cλn(z, θ)|
=
∣∣∣∣ sup x2∈X
`((x2, y), θ)− λnc∗((x, y), (x2, y))− sup x2∈X
`((x2, y), θ)− λnc((x, y), (x2, y))
∣∣∣∣ ≤ sup x2∈X
λn|c∗((x, y), (x2, y))− c((x, y), (x2, y))|
≤ λnδc ·D2.
This implies sup
P :W∗(P …
