In the previous post I mentioned I did not understand the

The keys to understanding are the following:

*relative distance*metric they used for analyzing the security simple learning problem. However, in the paper they refer to a Master thesis that explains the metric in more detail.The keys to understanding are the following:

- To move the decision boundary as much as possible, each data point equals the previous mean added with the radius:
t__X__*-1 + R* - This is done alpha times at each iteration:
*(*__X__t-1 + R) * at - The next mean will be
where n is the number of previous data points__X__t =__[X__t-1 * (n + a1+ ... +at-1) +__(X__t-1 + R) * at ]/(n + a1+ ... +at-1 +at) - If we simplify the expression we will get:
__X__t =__X__t-1 + R * at /(n + a1+ ... +at) - Assuming that the attacker has complete control of the learning process then n=0 thus:
__X__t =__X__t-1 + R * at /(a1+ ... +at) - Using
*Mt = (a1+ ... +at)*as the effort of the attacker and by noticing that the recursion leads to:__X__t =__X__0 + R * [1 + a2/M2+ ... +at /Mt] - Thus:
*(*__X__t -__X__0)/R = [1 + a2/M2+ ... +at /Mt] - Noticing that
*ai/Mi = (Mi - Mi-1)/Mi = 1 - Mi-1/Mi* - Thus we have the relative displacement
*(*__X__t -__X__0)/R =*1 + a2/M2+ ... +at /Mt = 1 + 1 - M1/M2 + ... +1 - Mt-1/Mt = t - [M1/M2**+ ... + M**t-1/Mt]* - The relative distance (or displacement) is then for
*t = T*as in the paper:

*D({Mi}) = T − [M1/M2 + ... + MT-1/MT]*

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