Graph-Informed Adversarial Modeling: Infimal Subadditivity of Interpolative Divergences
Authors: Panagiota Birmpa, Eric Joseph Hall
Organizations: Heriot--Watt University · Maxwell Institute for Mathematical Sciences · School of Mathematical and Computer Sciences Heriot–Watt University Edinburgh EH14 4AS, UK · Maxwell Institute for Mathematical Sciences Edinburgh EH8 9BT
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.
Figures & tables
Figure 1: The linear-Gaussian Bayesian network given by the Hasse diagram of the subset lattice for the 8 subsets of a three-element set. The edge weights from node j to i are the parent coefficients ϕji∼N(μϕ=0.6,σϕ2=0.052) in equation 31 , which are fixed once across all experiments.
Figure 2: Objective-level and structural diagnostics for the structured linear-Gaussian model equation 31 under graph-agnostic and graph-informed adversarial training; see Figure 1 for the underlying Hasse network. (a) Best achieved validation variational objective, i.e., the minimum variational loss attained over training epochs for the validation set. Values closer to zero indicate a closer fit, and gray line segments pair runs with the same random seed. In the interpolative setting, the graph-informed multi-discriminator surrogate attains values close to those of the graph-agnostic monolithic objective, consistent with the interpretation of Theorem 5.3 . (b) Mean L2 error in the recovered parent coefficients, averaged over non-root nodes, with lower values indicating better recovery of the underlying conditional structure. Here, the graph-informed construction aligned with the true families outperforms both the graph-agnostic baseline and the misaligned multi-discriminator construction. Boxplots summarize 13 paired runs per model.
Figure 3: Additional diagnostics for the structured linear-Gaussian Hasse network, complementing the objective-level comparison in Figure 2 . (a) Energy distance (lower is better) provides a global measure of distributional discrepancy across the three discriminator constructs under the Lipschitz-restricted interpolative divergence (cf. parent-coefficient recovery in Figure 2(b) ); boxplots summarize 13 paired runs per model. (b) Variational loss for the GiGAN trained using an interpolative divergence at several Lipschitz constants L , with the graph-agnostic GAN at L=1 for reference; mean with a min-max envelope based on 3 runs per model.
Figure 4: Fifty ball-throwing trajectory time series from various distributions. (a) Samples from the target distribution with v0∼N(μv=4,σv2=32) . (b) Fifty generated trajectories from a GiGAN with 14 localized discriminators trained using an interpolative divergence with a Lipschitz constraint. (c) Fifty generated trajectories from a graph-agnostic GAN with a monolithic discriminator trained using an interpolative divergence with a Lipschitz constraint.
Figure 5: Training diagnostics for the ball-trajectory experiment from a representative run; the GiGAN with 14 localized discriminators aligned to child–parent families in the Bayesian network is compared to the graph-agnostic GAN with a monolithic discriminator. (a) and (b) show the evolution of the KL variational loss equation 37 on the training and validation sets, for an interpolative ( 20 runs in total) and classical divergence ( 10 runs in total), i.e., with and without Lipschitz constraints enforced via spectral normalization. (c) and (d) show the corresponding energy-distance diagnostics, both for the full trajectory distribution and for the recovered physics parameters (g^,v^0) , again for an interpolative and classical divergence.
Figure 6: The two real-world discrete Bayesian networks used in the experiments. (a) Child , a medium-sized, discrete Bayesian network with 20 nodes and 25 arcs ( 230 parameters) describing a possible diagnostic tool for birth asphyxia for newborn babies with congenital heart disease, from Spiegelhalter (1992) . (b) Earthquake , a small-sized, discrete Bayesian network consisting of 5 nodes and 4 arcs (10 parameters), from Pearl (1988) ; Korb and Nicholson (2010) .
Figure 7: Training and validation variational loss (interpolative JS-Wasserstein) for the real-world discrete Bayesian network experiments. (a) Compares the graph-agnostic GAN model M with the GiGAN models M1 (child–parent families) and M3 (ancestor depth- 3 families) on Child over 15 runs. (b) Compares the graph-agnostic GAN and the GiGAN on Earthquake over 85 runs.
Figure 8: Validation diagnostics for the real-world discrete Bayesian network experiments (single run highlighted). The Child experiment compares M , M1 , and M3 , corresponding to graph-agnostic GAN and two GiGANs with child–parent and ancestor depth-3 families over 15 runs. The Earthquake experiment compares graph-agnostic and GiGANs over 85 runs. (a) and (b) show global validation metrics, namely energy distance and average log-likelihood. (c) and (d) show representative discriminator AUC values.
Figure 9: Total variation error between generated marginals and the corresponding target marginals from the known CPT. (a) The mean error by node by depth, across 15 runs for Child for graph-agnostic model M and graph-informed models M1 and M3 . (b) The mean error, averaged over all nodes by depth, across 15 runs for Child , highlighting variables such as CO2 with high error relative to other nodes. (c) The mean error by depth by node, across 85 runs for Earthquake ; note the outliers in the graph-agnostic predictions.
Generative Adversarial Networks (GANs) train competing generators and discriminators to synthesize data, translate images, and model complex distributions.
College of Computer Science, Sichuan University, China · Department of Computer and Information Sciences, Northumbria University, UK · School of Artificial Intelligence, Sichuan University, China