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arXiv 2404.08158cs.CCcs.DScs.LG
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论交互式证明在学习中的能力

On the Power of Interactive Proofs for Learning

University of Cambridge（剑桥大学）；Simon Fraser University（西蒙菲莎大学）；Qualcomm（高通公司）；University of Warwick（华威大学）

Tom Gur, Mohammad Mahdi Jahanara, Mohammad Mahdi Khodabandeh, Ninad Rajgopal, Bahar Salamatian, Igor Shinkar

备注：
58 pages, To appear in STOC 2024
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AI总结：

该研究针对不可知PAC学习的双高效交互式证明系统展开，在傅里叶特征学习、AC⁰[2]类与k-juntas学习上取得样本复杂度优化成果，并证明非双高效系统的平凡性。

摘要：

我们继续研究用于验证不可知PAC学习的双高效证明系统，并得到以下结果。\n- 我们构建了一种交互式协议，用于学习给定函数$f \colon \{0,1\}^n \to \{0,1\}$的$t$个最大傅里叶特征，误差可任意小，其中验证器使用$\mathsf{poly}(t)$个随机样本。这改进了Goldwasser、Rothblum、Shafer和Yehudayoff提出的Interactive Goldreich-Levin协议（发表于ITCS 2021），该协议的样本复杂度为$\mathsf{poly}(t,n)$。\n- 针对均匀分布下不可知学习$\mathsf{AC}^0[2]$类的问题，我们基于Carmosino、Impagliazzo、Kabanets和Kolokolova的研究（发表于APPROX/RANDOM 2017）设计了一种交互式协议：给定函数$f \colon \{0,1\}^n \to \{0,1\}$，验证器使用准多项式数量的随机样本，学习到最接近的假设，误差为$\mathsf{polylog}(n)$的乘法因子。相比之下，该类函数即使在构建（无证明者的）可实现学习器时，也一直因难以用随机样本求解而闻名。\n- 针对均匀分布下不可知学习$k$-juntas的问题，我们得到了一种交互式协议：对于给定函数$f \colon \{0,1\}^n \to \{0,1\}$，验证器使用$O(2^k)$个随机样本。关键在于，验证器的样本复杂度与$n$无关。\n我们还证明，如果不要求双高效证明系统，那么该模型会变得平凡。具体而言，我们提出了一种适用于无分布设置下任意布尔函数类$\mathcal{C}$的协议，其中验证器使用$O(1)$个带标记样本即可学习$f$。

英文摘要：

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 $f$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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