Signal from Structure: Exploiting Submodular Upper Bounds in Generative Flow Networks
Authors: Alexandre Larouche, Audrey Durand
Organizations: D´epartement d’informatique et de g´enie logiciel, Universit´e Laval, Qu´ebec, Canada · Mila - Quebec AI Institute, Montr´eal, Canada · Canada CIFAR AI Chair
Generative Flow Networks (GFlowNets; GFNs) are a class of generative models that learn to sample compositional objects proportional to their a priori unknown value, their reward. We focus on the case where the reward has a specified, actionable structure, namely that it is submodular. We show submodularity can be harnessed to retrieve upper bounds on the reward of compositional objects that have not yet been observed. We provide in-depth analyses of the probability of such bounds occurring, as well as how many unobserved compositional objects can be covered by a bound. Following the Optimism in the Face of Uncertainty principle, we then introduce SUBo-GFN , which uses the submodular upper bounds to train a GFN. We show that SUBo-GFN generates orders of magnitude more learning signal than classical GFNs for the same number of queries to the reward function. We demonstrate the effectiveness of SUBo-GFN in terms of distribution matching and high-quality candidate generation on graph submodular tasks.
Figures & tables
Figure 1: An example DAG with submodular rewards. Submodularity constrains the reward of states, ensuring that there is a diminishing return over time for the reward of later states.
Figure 2: An example DAG, showing a pair of trajectories for {\color[rgb]{1,0,0}\mathcal{T}(p_{x,c})} and {\color[rgb]{0,0,1}\mathcal{T}(c)} for x={a,b,c} . This pair of trajectory yields the bound UB({a,b,c}∣∅,c)=R({c})−R(∅)+R({a,b}) . The ellipsis ( … ) indicates any number of valid transitions are taken. Bold edges and vertices highlight key elements in trajectory.
Figure 3: Normalized coverage of unobserved states (Cor. 4.1 ) w.r.t. the cardinality-actions ratio and the number of sampled trajectories (Assumption 4.1 ) ( m∈[500,10000] ) for fixed N=1000 . The dashed line shows the 1:1 ratio between coverage and query.
Figure 4: Normalized coverage of unobserved terminating states for SUBo-GFN with trajectories collected with PF and the uniform policy (Assumption 4.1 ), against the lower bound (Cor. 4.1 ) for random graphs with C=15 .
Figure 5: Performance metrics of FL/TB-GFNs and SUBo-GFN variants on random graphs with C=15 . Strategies are trained online (dashed line) until the , then trained offline (full line) without further queries to R until the end of the experiment indicated by the ■ .
Appendix figures & tables11 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 6: Normalized coverage of unobserved terminating states for SUBo-GFN with trajectories collected with PF and the uniform policy (Assumption 4.1 ), against the lower bound (Cor. 4.1 ) for ER graphs.
Figure 7: Normalized coverage of unobserved terminating states for SUBo-GFN with trajectories collected with PF and the uniform policy (Assumption 4.1 ), against the lower bound (Cor. 4.1 ) for BA graphs.
Figure 8: Normalized coverage of unobserved terminating states for SUBo-GFN with trajectories collected with PF and the uniform policy (Assumption 4.1 ), against the lower bound (Cor. 4.1 ) for CM-BI graphs.
Figure 9: Normalized coverage of unobserved terminating states for SUBo-GFN with trajectories collected with PF and the uniform policy (Assumption 4.1 ), against the lower bound (Cor. 4.1 ) for CM-POW graphs.
Figure 10: Normalized coverage of unobserved terminating states for SUBo-GFN with trajectories collected with PF and the uniform policy (Assumption 4.1 ), against the lower bound (Cor. 4.1 ) for CM-REG graphs.
Figure 11: FCS, Top-100 Average Rewards, Loss, and Coverage for SUBo-GFN and FL/TB-GFNs on ER graphs.
Figure 12: FCS, Top-100 Average Rewards, Loss, and Coverage for SUBo-GFN and FL/TB-GFNs on BA graphs.
Figure 13: FCS, Top-100 Average Rewards, Loss, and Coverage for SUBo-GFN and FL/TB-GFNs on CM-BI graphs.
Figure 14: FCS, Top-100 Average Rewards, Loss, and Coverage for SUBo-GFN and FL/TB-GFNs on CM-POW graphs.
Figure 15: FCS, Top-100 Average Rewards, Loss, and Coverage for SUBo-GFN and FL/TB-GFNs on CM-REG graphs.
Generative Flow Networks (GFlowNets) enable fine-tuning large language models to approximate reward-proportional posteriors, but they remain prone to mode collapse, manifesting as prefix collapse and length bias. We attribute this to two factors: (i) weak credit assignment to early prefixes, and (ii) biased replay that induces a shifted, non-representative training flow distribution. We propose Rooted absorbed prefix Trajectory Balance RapTB, an objective that anchors subtrajectory supervision at the root and propagates terminal rewards to intermediate prefixes via absorbed suffix-based backups, providing dense prefix-level learning signals. To mitigate replay-induced distribution shift, we further introduce SubM, a submodular replay refresh strategy that promotes both high reward and diversity. Empirically, on tasks such as molecule generation with LLM using SMILES strings, RapTB combined with SubM consistently improves optimization performance and molecular diversity while preserving high validity. The code is released on https://github.com/ComDec/ChemGFN.
Xi Wang, Wenbo Lu, Shengjie Wang
Courant Institute School of Mathematics, Computing, and Data Science, New York University
Generative Flow Networks (GFlowNets) learn to sample diverse candidates in proportion to a reward function, making them well-suited for scientific discovery, where exploring multiple promising solutions is crucial. Further extending GFlowNets to multi-objective settings has attracted growing interest as real-world applications often involve multiple, conflicting objectives. However, existing approaches require joint training for each combination of objectives, meaning that any change in the objective set necessitates retraining from scratch. We propose a framework that composes pre-trained GFlowNets at inference time, enabling rapid adaptation without fine-tuning or retraining. Importantly, our framework is flexible, capable of handling diverse reward combinations ranging from linear scalarization to complex nonlinear operators, which are often handled separately in previous literature. We prove that our method exactly recovers the target distribution for linear scalarization, and quantify the approximation quality for nonlinear operators through a distortion factor. Experiments on a synthetic 2D grid and real-world molecule generation tasks demonstrate that our approach achieves performance comparable to baselines.
Seokwon Yoon, Youngbin Choi, Seunghyuk Cho +3
Department of Computer Science & Engineering, POSTECH, South Korea · Graduate School of Artificial Intelligence, POSTECH, South Korea
We present GFlowState, a visual analytics system designed to illuminate the training process of Generative Flow Networks (GFlowNets or GFNs). GFlowNets are a probabilistic framework for generating samples proportionally to a reward function. While GFlowNets have proved to be powerful tools in applications such as molecule and material discovery, their training dynamics remain difficult to interpret. Standard machine learning tools allow metric tracking but do not reveal how models explore the sample space, construct sample trajectories, or shift sampling probabilities during training. Our solution, GFlowState, allows users to analyze sampling trajectories, compare the sample space relative to reference datasets, and analyze the training dynamics. To this end, we introduce multiple views, including a chart of candidate rankings, a state projection, a node-link diagram of the trajectory network, and a transition heatmap. These visualizations enable GFlowNet developers and users to investigate sampling behavior and policy evolution, and to identify underexplored regions and sources of training failure. Case studies demonstrate how the system supports debugging and assessing the quality of GFlowNets across application domains. By making the structural dynamics of GFlowNets observable, our work enhances their interpretability and can accelerate GFlowNet development in practice.
Florian Holeczek, Andreas Hinterreiter, Alex Hernandez-Garcia +2
Johannes Kepler University Linz · Mila – Quebec AI Institute · Eidgenössische Technische Hochschule Zürich