(Failure of OOD detection under invariant classifier) Consider an out-of-distribution input which contains the environmental feature: ? out ( x ) = M inv z out + M e z e , where z out ? ? inv . Given the invariant classifier (cf. Lemma 2), the posterior probability for the OOD input is p ( y = 1 ? ? out ) = ? ( 2 p ? z e ? + log ? / ( 1 ? ? ) ) , where ? is the logistic function. Thus for arbitrary confidence 0 < c : = P ( y = 1 ? ? out ) < 1 , there exists ? out ( x ) with z e such that p ? z e = 1 2 ? log c ( 1 ? ? ) ? ( 1 ? c ) .
Facts. Thought an away-of-delivery enter in x out having Yards inv = [ I s ? s 0 step one ? s ] , and you will M e = [ 0 s ? age p ? ] , then the ability image was ? age ( x ) = [ z away p ? z age ] , in which p ‘s the product-standard vector outlined inside Lemma 2 .
Then we have P ( y = 1 ? ? out ) = P ( y = 1 ? z out , p ? z e ) = ? ( 2 p ? z e ? + log ? / ( 1 ? ? ) ) , where ? is the logistic function. Thus for arbitrary confidence 0 < c : = P ( y = 1 ? ? out ) < 1 , there exists ? out ( x ) with z e such that p ? z e = 1 2 ? log c ( 1 ? ? ) ? ( 1 ? c ) . ?
Remark: During the a very standard circumstances, z aside can be modeled while the an arbitrary vector that’s independent of the in the-delivery brands y = 1 and you will y https://datingranking.net/pl/transgenderdate-recenzja/ = ? step one and you will environmental features: z out ? ? y and you can z out ? ? z elizabeth . Thus inside Eq. 5 i’ve P ( z aside ? y = 1 ) = P ( z out ? y = ? step 1 ) = P ( z out ) . Next P ( y = step one ? ? aside ) = ? ( dos p ? z elizabeth ? + log ? / ( step one ? ? ) ) , just like for the Eq. 7 . Thus our very own fundamental theorem nevertheless keeps below far more general situation.
Appendix B Extension: Colour Spurious Relationship
To advance examine our very own findings beyond record and you will intercourse spurious (environmental) enjoys, we offer even more fresh show into the ColorMNIST dataset, given that revealed within the Contour 5 .
Comparison Task 3: ColorMNIST.
[ lecun1998gradient ] , which composes colored backgrounds on digit images. In this dataset, E = < red>denotes the background color and we use Y = < 0>as in-distribution classes. The correlation between the background color e and the digit y is explicitly controlled, with r ? < 0.25>. That is, r denotes the probability of P ( e = red ? y = 0 ) = P ( e = purple ? y = 0 ) = P ( e = green ? y = 1 ) = P ( e = pink ? y = 1 ) , while 0.5 ? r = P ( e = green ? y = 0 ) = P ( e = pink ? y = 0 ) = P ( e = red ? y = 1 ) = P ( e = purple ? y = 1 ) . Note that the maximum correlation r (reported in Table 4 ) is 0.45 . As ColorMNIST is relatively simpler compared to Waterbirds and CelebA, further increasing the correlation results in less interesting environments where the learner can easily pick up the contextual information. For spurious OOD, we use digits < 5>with background color red and green , which contain overlapping environmental features as the training data. For non-spurious OOD, following common practice [ MSP ] , we use the Textures [ cimpoi2014describing ] , LSUN [ lsun ] and iSUN [ xu2015turkergaze ] datasets. We train on ResNet-18 [ he2016deep ] , which achieves 99.9 % accuracy on the in-distribution test set. The OOD detection performance is shown in Table 4 .