We continue the study of doubly-efficient proof systems for verifying agnostic PAC learning, for which we obtain the following results. - We construct an interactive protocol for learning the $t$ largest Fourier characters of a given function $f \colon \{0,1\}^n \to \{0,1\}$ up to an arbitrarily small error, wherein the verifier uses $\mathsf{poly}(t)$ random examples. This improves upon the Interactive Goldreich-Levin protocol of Goldwasser, Rothblum, Shafer, and Yehudayoff (ITCS 2021) whose sample complexity is $\mathsf{poly}(t,n)$. - For agnostically learning the class $\mathsf{AC}^0[2]$ under the uniform distribution, we build on the work of Carmosino, Impagliazzo, Kabanets, and Kolokolova (APPROX/RANDOM 2017) and design an interactive protocol, where given a function $f \colon \{0,1\}^n \to \{0,1\}$, the verifier learns the closest hypothesis up to $\mathsf{polylog}(n)$ multiplicative factor, using quasi-polynomially many random examples. In contrast, this class has been notoriously resistant even for constructing realisable learners (without a prover) using random examples. - For agnostically learning $k$-juntas under the uniform distribution, we obtain an interactive protocol, where the verifier uses $O(2^k)$ random examples to a given function $f \colon \{0,1\}^n \to \{0,1\}$. Crucially, the sample complexity of the verifier is independent of $n$. We also show that if we do not insist on doubly-efficient proof systems, then the model becomes trivial. Specifically, we show a protocol for an arbitrary class $\mathcal{C}$ of Boolean functions in the distribution-free setting, where the verifier uses $O(1)$ labeled examples to learn $f$.

翻译：我们继续研究用于验证agnostic PAC学习的双高效证明系统，并获得了以下结果： - 我们构建了一个交互式协议，用于学习给定函数$f \colon \{0,1\}^n \to \{0,1\}$的前$t$个最大傅里叶特征，误差可任意小，其中验证者使用$\mathsf{poly}(t)$个随机样本。这改进了Goldwasser、Rothblum、Shafer和Yehudayoff（ITCS 2021）的交互式Goldreich-Levin协议，其样本复杂度为$\mathsf{poly}(t,n)$。 - 对于均匀分布下agnostically学习$\mathsf{AC}^0[2]$类的问题，我们基于Carmosino、Impagliazzo、Kabanets和Kolokolova（APPROX/RANDOM 2017）的工作，设计了一个交互式协议。在该协议中，给定函数$f \colon \{0,1\}^n \to \{0,1\}$，验证者使用拟多项式数量的随机样本，学习到最接近假设的$\mathsf{polylog}(n)$乘法因子。相比之下，该类在仅使用随机样本构建可实现学习器（无需证明者）时一直极具挑战性。 - 对于均匀分布下agnostically学习$k$-junta的问题，我们获得了一个交互式协议，其中验证者使用$O(2^k)$个随机样本学习给定函数$f \colon \{0,1\}^n \to \{0,1\}$。关键在于，验证者的样本复杂度与$n$无关。我们还证明，如果不坚持双高效证明系统，则该模型变得平凡。具体而言，我们展示了在分布无关设定下任意布尔函数类$\mathcal{C}$的一个协议，其中验证者仅需$O(1)$个带标签样本即可学习$f$。