vainaijr:

does this mean, that instead of parameterization of a line, we could parameterize a curve also, like a sigmoid curve, or a sinusoidal curve, or a circle and that would be a different attribution method.


So there are several things to be considered here:

The line parametrization (x’ + α(x-x’)) is of constant speed along the path. This enables multiplying with (x_i-x’_i) at the last step rather than having to do this separately for each “time step” in the integral. But done properly such a change of in the parametrization of the path would not change the result.
As the figure illustrates, if you took a path different than the straight line, then you would get a different attribution. While the “sum of the two attribution parts” is again the difference in function values by the fundamental theorem of calculus, you can get a wildly different split by coordinate. e.g. if you take F = min(x_1, x_2) as your function and consider paths between (0, 1) and (1, 0), where the function is both 0. You could go along the coordinate axes to get (depending on which you do first) an attribution of (0, 0) or (1, -1). You can also get anything in between by interleaving directions.
One of the bad parts of all this (with having to arbitrarily choose the line) is that it will no compose well, i.e. if you declare the last conv layer of your network the features and now you want to attribute the change in features to the change in input and the change in output to the change in features, you’ll get something completely different between the two, because it’s unlikely that the straight line in input space corresponds to a straight line in feature-space.
In the end, it is some sort of non-definedness issue around the choice of path. The next question could be how to get around this. The straight line is straightforward to pick in Euclidean space, as would be geodesics (but even then it might not be unique if you have several) for manifolds. You could also try to get avoid the choice of a single path by introducing some probability measure on the paths and then integrating over that. But by then you’re in a much more complex setting (and I’m not sure anyone has done this).

Hmhm. Now the last post might have been peak happiness. 
Best regards
Thomas