In Section 3. The paper uses a measure ρ
that is essentially the fraction of examples at which local monotonicity (in any of the prescribed directions in M
) is violated and then show that this measure decreases when using the paper's method over the baselines. However, I'm not certain that this measure corresponds to the global monotonicity requirment that is often desired in practice: namely, the one that appears in Definition 1. For example, consider a 1
-D function over [ 1 , 99.99 ]
whose graph is a piecewise linear curve connecting the points ( 0 , 100 ) , ( 0.99 , 100.99 ) , ( 1 , 99 ) , ( 1.99 , 99.99 ) , ( 2 , 98 ) , ( 2.99 , 98.99 ) , . . . , ( 99 , 1 ) , ( 99.99 , 1.99 )
. This function has nonnegative derivative at about 99
of its domain, yet if one chooses two points x 1 , x 2
uniformaly and independently from the domain, then there's at least a 97
chance that f ( m i n x 1 , x 2 ) > f ( m a x x 1 , x 2 )
. I think, therefore, that it would be good to complement the local ρ
with an estimate of the probability that Definition 1 would not hold over the distribution in question (training, test or random).
Section 4.1 The authors introduce the notion of group monotonicity, but it's unclear how the regularizer introduced in equation 3 helps to encourage that property. Specifically, 1) Only the sum of the gradient is taken into account (so it could be that a component a_w_{i,j} has a very negative gradient, but still the sum will be positive), and 2) the softmax in equation 3 seems to encourage that the total gradient of S y
is larger than the total gradient of all the other S k
's, not that it's positive. Perhaps I'm missing something?
Section 4.2 The paper claims that the fact that a good performance of the "total activation classifier" shows evidence that the original classifier satisfies group monotonicity. But that claim is not clear to me. The total activation classifier does not depend on the part of the network that computes the output from the intermediate layer which is critical for the satisfaction of group monotonicity.
Section 4.2.2 The paper doesn't compare their methods to other methods for detecting noisey/adverserial test examples.
1) Only the sum of the gradient is taken into account (so it could be that a component a_w_{i,j} has a very negative gradient, but still the sum will be positive), and