It is well-known that for classical PAC learning, $\Omega(d/\varepsilon)$ examples are necessary in order to acheive an error bound of $\varepsilon$ w.h.p., where $d$ is the VC-dimension of the concept class.

Is it known that $\Omega(d/\varepsilon^2)$ examples are needed in the agnostic case?

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    $\begingroup$ I am not sure what the lower bound looks like, one should exist if the Hoefding bound is tight (and I think it is). This bound states that for 1 fn, if the probability of error is p, then you need at most $m = O(1/\epsilon^2)$ samples to estimate p to within error +- $\epsilon$ w.h.p. So consider any concept class with 2 concepts, $f_1$ and $f_2$ and VC-dimension 2. Take a distribution over examples so that $p_1 = p_2 + \epsilon$ (or vice versa) -- this is possible because VC-dimension is 2. It seems that an algorithm using only $O(1/\epsilon)$ examples would imply an improved Hoefding bound. $\endgroup$
    – Aaron Roth
    Jun 14, 2011 at 20:31
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    $\begingroup$ Namely, I think the Hoeffding bound is tight at $p=1/2$ for $O(1/\epsilon^2)$. I think the reasoning above is generally known... $\endgroup$
    – Lev Reyzin
    Jun 14, 2011 at 23:09
  • $\begingroup$ OK -- looks like I've got myself another exercise for the ML course... :) Thanks for the input, Aaron and Lev! $\endgroup$
    – Aryeh
    Jun 15, 2011 at 5:42
  • $\begingroup$ @Aaron, maybe this should have been an answer. $\endgroup$ Jun 17, 2011 at 17:54

1 Answer 1


I now realize that a lower bound has indeed been established by Anthony and Bartlett (see the presentation here).

Edit 24-Sep-2018. This question has kept me occupied all of these years, and recently, I. Pinelis and I have obtained the exact optimal constant in the agnostic PAC lower bound to appear in Ann. Stat.

  • $\begingroup$ In your paper you do not cite this work (jmlr.org/papers/volume17/15-389/15-389.pdf). Does optimal sample complexity upperbounds in the realizable case not of any connection to your work? Are these corresponding optimal sample complexity upperbounds known for the agnostic case? $\endgroup$ Feb 18, 2019 at 17:36
  • $\begingroup$ I don't think the realizable case is all that related. In the realizable case, ERM does not guarantee optimal rates -- hence all the hard work Hanneke and others had to expend in order to remove the log factor, and it's still unknown whether a proper learner can achieve the optimal rate. Contrariwise, in the agnostic case, it has been known for a long time that ERM achieves the optimal rate. $\endgroup$
    – Aryeh
    Feb 18, 2019 at 19:21
  • $\begingroup$ It's now known that proper learning cannot always guarantee the optimal rate for realizable PAC. $\endgroup$
    – Aryeh
    Jan 14, 2021 at 22:30

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