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If the VC dimension of a fixed hypothesis class is $d$, then showing the gap between the sample and generalization error for any hypothesis in the usual agnostic computational learning problem is $O(\sqrt{d/m}\log^{1/2}\frac{d}{m})$ with high probability is a common problem addressed by elementary statistical learning theory classes.

This reference mentions the above in Equation 3.32, and also claims that "a finer analysis" may remove the log factor, but the linked document does not cite any source for this.

Who has shown this in the past, or, more directly, how can we get rid of the log factor?

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It seems that using a "chaining" technique from Dudley 1978 one can get rid of the logarithmic factor, as derived here.

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