An Optimal Agnostic PAC Algorithm

• URL: http://arxiv.org/abs/2608.06363v1
• Date: Thu, 06 Aug 2026 17:57:25 GMT
• Title: An Optimal Agnostic PAC Algorithm
• Abstract summary: We write $L$ for the binary risk and $L*=min_hin HL(h)$, we construct a learner achieving the statistically optimal risk bound.<n>This settles the agnostic sample complexity of PAC learning up to universal constants at every fixed $L*$.
• Score: 15.361702135159845
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Let $H\subseteq\{-1,+1\}^X$ be a class of finite VC dimension $d\ge1$. Writing $L$ for the binary risk and $L^*=\min_{h\in H}L(h)$, we construct a learner achieving the statistically optimal risk bound: from an i.i.d.\ sample of size $n$, for every $0<δ\le 1/2$, with probability at least $1-δ$, \[ L(\widehat h) \le L^*+ 7\cdot10^8\left( \sqrt{\frac{L^*(d+\log(1/δ))}{n}} +\frac{d+\log(1/δ)}{n} \right). \] This settles the sample complexity of agnostic PAC learning up to universal constants at every fixed $L^*$, matching the lower bounds of Devroye, Györfi, and Lugosi [A Probabilistic Theory of Pattern Recognition, Springer, 1996].

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