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：我们通过证明结构学习问题的最小最大最优速率由这些问题中条件独立性检验的最小最大速率所决定，从而建立了最优结构学习与最优条件独立性检验之间的基本联系。这一结论是通过在多森林情形下建立这两个问题之间的广义归约关系实现的，并通过推导若干示例（包括伯努利模型、高斯模型及非参数模型）的最优速率加以验证。此外，我们证明了这些场景下的最优算法是PC算法的适当修正版本。这一理论发现为通过最小最大检验视角分析结构学习的统计复杂性提供了统一框架。