Meaning of this (convoluted) sentence from the book "Probably Approximately Correct"

As a non native english speaker sometimes, I'm having hard time understanding not conventional sentence as the following. Can you rephrase the sentence (in Bold) in another way?

This upper bound is equal to M/m^2, where M is the square of the distance of the furthest data point in the training set from the point (0,0) and m is the margin. The margin has a trickier definition: It is the minimum distance of any data point from the separating line for the line for which this distance is the largest

From the book "Probably Approximately Correct" by Leslie Valiant

• I think most native English speakers would struggle with that. One would expect it to be followed by an example, or further explanation - does that exist? Commented Jan 14, 2019 at 13:42

The sentence definitely could be written better. I would approach the interpretation in stages. First:

It is the minimum distance of any data point from the separating line

There are multiple data points available. This is implied here and made explicit earlier in the full quotation you provided. Given the separating line, we look at the distance of each data point to that line and then we take the smallest value.

The big confusion comes from the second part:

for the line for which this distance is the largest.

This implies that there are multiple possible separating lines. Which means that there are multiple possible minimum distances, but we want only one. So we look at the minimum distance for each possible separating line, and then we select the separating line that gives us the largest value for the minimum distance.

The sentence is tough because it refers twice to the line that is part of the definition.

The margin is the minimum distance of a specific point from this line. The specific point in question must have the largest distance from this line.

The two conditions seem to contradict. The point is the closest for which it is the greatest distance.

Max is right; an example is needed. That or the next paragraph.