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A set of points $X = \{x\}$ is $\gamma$-shattered by a set of functions $\mathcal{F}$ if there are real numbers $r_x$ indexed by $x$ such that for any binary vector $b$ defining labeling of points from $X$ we can find a function $f \in \mathcal{F}$ such that $f(x) \geq r_x + \gamma$ if $x$ has label 1 and $f(x) \leq r_x - \gamma$ if $x$ has label -1.

In this definition, what is the role of numbers $r_x$? If we don't use $r_x$ in the definition, what will change?

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I was also looking for an explanation and this is the best one I got (Section 4.1.2). Apparently fat-shattering is a restrictive form of P-shattering that says for some fixed $r_x$ there is some $f$ that has a margin of atleast $\gamma$.

Check out the figure in reference for clearer explanation.

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