(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. Imagine an out-of-shipments input x out that have Meters inv = [ We s ? s 0 step 1 ? s ] , and Meters e = [ 0 s ? e p ? ] , then your element representation is ? e ( x ) = [ z out p ? z age ] , in which p is the product-norm vector laid out in 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: From inside the an even more standard case, z out would be modeled once the an arbitrary vector which is in addition to the inside-shipment brands y = 1 and you may y = ? step 1 and ecological features: z aside ? ? y and you will z out ? ? z age https://www.datingranking.net/pl/jpeoplemeet-recenzja/ . Thus during the Eq. 5 you will find P ( z out ? y = 1 ) = P ( z aside ? y = ? step one ) = P ( z out ) . Next P ( y = step one ? ? out ) = ? ( dos p ? z elizabeth ? + log ? / ( step one ? ? ) ) , identical to inside Eq. seven . Hence our very own main theorem nonetheless keeps less than significantly more standard instance.
Appendix B Extension: Colour Spurious Relationship
To advance verify all of our conclusions beyond records and you may gender spurious (environmental) has, we provide additional fresh efficiency towards the ColorMNIST dataset, because the revealed inside the Profile 5 .
Evaluation Activity 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 .