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An Optimal Agnostic PAC Algorithm

Research Learning Theory

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TL;DR - A theoretical paper constructing a learner for agnostic PAC learning over hypothesis classes of finite VC dimension d that attains the statistically optimal excess-risk bound, closing the long-standing gap between upper and lower bounds up to universal constants.

  • For a class $H\subseteq{-1,+1}^X$ with VC dimension $d\ge1$, the learner achieves, with probability $\ge 1-\delta$, $L(\widehat h)\le L^+7\cdot10^8\big(\sqrt{L^(d+\log(1/\delta))/n}+(d+\log(1/\delta))/n\big)$ from an i.i.d. sample of size $n$.
  • The bound is "first-order"/optimistic: it interpolates between the fast $O((d+\log(1/\delta))/n)$ rate in the realizable case ($L^=0$) and the slow $\sqrt{\cdot/n}$ rate as $L^$ grows.
  • The result settles agnostic PAC sample complexity up to universal constants at every fixed $L^*$, matching the lower bounds of Devroye, Györfi, and Lugosi (1996).
  • The constant $7\cdot10^8$ is explicitly large, so the contribution is asymptotic/constant-factor optimality rather than practical tightness; no empirical results are claimed in the provided abstract.

Sources (1)

An Optimal Agnostic PAC Algorithm

arXiv cs.LG Markus Engelund Mathiasen, Jian Qian, Nikita Zhivotovskiy 2026-08-06 arXiv:2608.06363
Public signals Semantic Scholar citations 0 · Semantic Scholar influential citations 0
Providers: Hugging Face · N/A OpenAlex · N/A Publisher · N/A Semantic Scholar · Citations 0 · Influential citations 0 X · N/A Fetched 2026-08-26 14:34:57.466173 UTC

TL;DR - A theoretical paper constructing a learner for agnostic PAC learning over hypothesis classes of finite VC dimension d that attains the statistically optimal excess-risk bound, closing the long-standing gap between upper and lower bounds up to universal constants.

  • For a class $H\subseteq{-1,+1}^X$ with VC dimension $d\ge1$, the learner achieves, with probability $\ge 1-\delta$, $L(\widehat h)\le L^+7\cdot10^8\big(\sqrt{L^(d+\log(1/\delta))/n}+(d+\log(1/\delta))/n\big)$ from an i.i.d. sample of size $n$.
  • The bound is "first-order"/optimistic: it interpolates between the fast $O((d+\log(1/\delta))/n)$ rate in the realizable case ($L^=0$) and the slow $\sqrt{\cdot/n}$ rate as $L^$ grows.
  • The result settles agnostic PAC sample complexity up to universal constants at every fixed $L^*$, matching the lower bounds of Devroye, Györfi, and Lugosi (1996).
  • The constant $7\cdot10^8$ is explicitly large, so the contribution is asymptotic/constant-factor optimality rather than practical tightness; no empirical results are claimed in the provided abstract.
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