We study adversarial learning when the target distribution factorizes according to a known Bayesian network. For interpolative divergences, including $(f,Γ)$-divergences, we prove a new infimal subadditivity principle showing that, under suitable conditions, a global variational discrepancy is controlled by an average of family-level discrepancies aligned with the graph. In an additive regime, the surrogate is exact. This closes a theoretical gap in the literature; existing subadditivity results justify graph-informed adversarial learning for classical discrepancies, but not for interpolative divergences, where the usual factorization argument breaks down. In turn, we provide a justification for replacing a standard, graph-agnostic GAN with a monolithic discriminator by a graph-informed GAN (GiGAN) with localized family-level discriminators, without requiring the optimizer itself to factorize according to the graph. We also obtain parallel results for integral probability metrics and proximal optimal transport divergences, identify natural discriminator classes for which the theory applies, and present experiments showing improved stability and structural recovery relative to graph-agnostic baselines.

翻译：我们研究当目标分布根据已知贝叶斯网络进行因子化时的对抗学习。对于插值散度（包括$(f,Γ)$-散度），我们证明了新的下确界次可加性原理，表明在适当条件下，全局变分差异可由沿图对齐的族级差异平均值控制。在加性框架中，该替代量是精确的。这填补了文献中的理论空白：已有的次可加性结果仅能证明经典差异下基于图信息的对抗学习，但无法适用于插值散度（其中常规因子化论证失效）。因此，我们为使用具有局部族级判别器的图信息生成对抗网络（GiGAN）替代标准无图感知生成对抗网络（包含单一整体判别器）提供了理论依据，且不要求优化器本身根据图进行因子化。我们还获得了积分概率度量和近端最优传输散度的平行结论，识别了适用于该理论的自然判别器类别，并通过实验表明相较于无图感知基线方法在稳定性和结构恢复方面的改进。