Information-Geometric Forward Policy Training in GFlowNets

Generative Flow Networks (GFlowNets) have emerged as a flexible framework for amortized inference over discrete and mixed discrete-continuous objects, requiring only an unnormalized target density specified through a reward. In this work, we formulate forward-policy training in GFlowNets through the information geometry of the induced trajectory sampler. Treating the forward policy as an induced trajectory sampler, we show that its intrinsic first-order geometry is given by the Fisher-Rao metric of the trajectory family, and that the associated natural gradient provides the canonical local update whenever the corresponding Fisher information is computable or accurately approximable. We derive an exact decomposition of the trajectory Fisher into per-step conditional second moments, which clarifies when temporal score interactions vanish and when dense couplings remain under shared parameterization. This leads to three computational regimes: settings with tractable exact Fisher information, settings where Monte Carlo estimators of the expected Fisher are sufficient, and structure-exploitable settings in which target locality or factorization yields accurate approximations of the Fisher expectation. In the latter case, graphical-model tools such as exact marginalization, separator methods, and belief propagation provide principled surrogates for natural-gradient updates. The resulting framework turns target structure into optimization geometry and yields a tractable route to structure-aware forward-policy training in GFlowNets. We illustrate the framework empirically through examples comparing convergence and exploration behavior under Riemannian and Euclidean optimization.