We prove a general structural theorem for a wide family of local algorithms, which includes property testers, local decoders, and PCPs of proximity. Namely, we show that the structure of every algorithm that makes $q$ adaptive queries and satisfies a natural robustness condition admits a sample-based algorithm with $n^{1- 1/O(q^2 \log^2 q)}$ sample complexity, following the definition of Goldreich and Ron (TOCT 2016). We prove that this transformation is nearly optimal. Our theorem also admits a scheme for constructing privacy-preserving local algorithms. Using the unified view that our structural theorem provides, we obtain results regarding various types of local algorithms, including the following. - We strengthen the state-of-the-art lower bound for relaxed locally decodable codes, obtaining an exponential improvement on the dependency in query complexity; this resolves an open problem raised by Gur and Lachish (SICOMP 2021). - We show that any (constant-query) testable property admits a sample-based tester with sublinear sample complexity; this resolves a problem left open in a work of Fischer, Lachish, and Vasudev (FOCS 2015) by extending their main result to adaptive testers. - We prove that the known separation between proofs of proximity and testers is essentially maximal; this resolves a problem left open by Gur and Rothblum (ECCC 2013, Computational Complexity 2018) regarding sublinear-time delegation of computation. Our techniques strongly rely on relaxed sunflower lemmas and the Hajnal-Szemer\'edi theorem.

翻译：我们证明了一类广泛局部算法的通用结构定理，该类算法包含性质测试器、局部解码器和邻近概率可检验证明（PCP）。具体而言，我们证明了：在满足自然鲁棒性条件且进行$q$次自适应查询的算法中，其结构可转化为基于采样的算法，根据Goldreich和Ron（TOCT 2016）的定义，该算法的样本复杂度为$n^{1- 1/O(q^2 \log^2 q)}$，并证明了该转化近乎最优。该定理还提出了一种用于构建隐私保护局部算法的方案。通过结构定理提供的统一视角，我们获得了关于各类局部算法的以下成果： - 我们改进了松弛局部可解码码的最新下界，在查询复杂度的依赖关系上实现指数级优化，解决了Gur和Lachish（SICOMP 2021）提出的开放问题。 - 我们证明任何（常数查询）可测试性质均存在样本复杂度为次线性的基于采样的测试器，通过将Fischer、Lachish和Vasudev（FOCS 2015）的主要结论扩展到自适应测试器，解决了该研究遗留的问题。 - 我们证明邻近证明与测试器之间的已知分离本质上是最大化的，解决了Gur和Rothblum（ECCC 2013, Computational Complexity 2018）提出的关于次线性时间计算委托的开放问题。 我们的技术方法强烈依赖于松弛太阳花引理和Hajnal-Szemerédi定理。