We establish a fundamental connection between optimal structure learning and optimal conditional independence testing by showing that the minimax optimal rate for structure learning problems is determined by the minimax rate for conditional independence testing in these problems. This is accomplished by establishing a general reduction between these two problems in the case of poly-forests, and demonstrated by deriving optimal rates for several examples, including Bernoulli, Gaussian and nonparametric models. Furthermore, we show that the optimal algorithm in these settings is a suitable modification of the PC algorithm. This theoretical finding provides a unified framework for analyzing the statistical complexity of structure learning through the lens of minimax testing.

翻译：我们在最优结构学习与最优条件独立性检验之间建立了一个基本联系，证明结构学习问题的极小化最优速率由这些问题中条件独立性检验的极小化速率决定。这一结果通过建立多森林情况下这两个问题之间的通用归约实现，并借助多个示例（包括伯努利、高斯及非参数模型）推导出最优速率来加以验证。此外，我们证明在这些场景下，最优算法是对PC算法的一种适当修正。这一理论发现为通过极小化检验的视角分析结构学习的统计复杂性提供了一个统一框架。