(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 ) .
Evidence. Think an out-of-distribution input x out that have M inv = [ I s ? s 0 step one ? s ] , and you may Yards e = [ 0 s ? e p ? ] , then the function symbol was ? elizabeth ( x ) = [ z away p ? z elizabeth ] , where p is the equipment-standard vector laid out inside the Lemma dos .
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 a more standard case, z aside will be modeled once the a random vector that is independent of the from inside the-shipment labels y = 1 and you can y = ? 1 and you will ecological has actually: z away ? ? y and z aside ? ? z elizabeth . Therefore into the Eq. 5 i’ve P ( z aside ? y = 1 ) = P ( z out ? y = ? step 1 ) = P ( z aside ) . Upcoming P ( y = step 1 ? ? away ) = ? ( 2 p ? z e ? + log ? / ( step one ? ? ) ) , identical to when you look at the Eq. seven . Hence our very own head theorem still retains less than a great deal more general situation.
Appendix B Extension: Color Spurious Correlation
To further confirm our results past background and you may sex spurious (environmental) has, we provide extra fresh performance towards ColorMNIST dataset, just like the shown from kinkyads inside the Profile 5 .
Review Task 3: ColorMNIST.
[ lecun1998gradient ] , which composes colored backgrounds on digit images. In this dataset, E = denotes the background color and we use Y = as in-distribution classes. The correlation between the background color e and the digit y is explicitly controlled, with r ? . 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 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 .