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A Theory of Non-acyclic Generative Flow Networks
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Extends GFlowNet theory to measurable (including continuous) state spaces with cycles, defines 0-flows as generalized cycles, shows FM/DB/TB losses are unstable with cycles, and proposes stable loss variants tested on hypergrids, Cayley graphs, and a continuous Point-Robot task.
Problem
GFlowNet training assumes acyclic state graphs; with cycles, flows diverge along loops, sampling time explodes, and the usual FM, DB and TB losses are unstable. Continuous state spaces also lack a cycle notion.
Approach
Generalize GFlowNets to measurable spaces with finite measures as flows, define 0-flows as a generalization of cycles, prove that FM/DB/TB-style divergence losses are unstable under adding 0-flows, and propose stable losses (difference-based with a nonnegative f and g) and regularizations that discourage cycles.
Key results
- On the Cayley graph of S_20, stable losses keep flow bounded while unstable FM flow grows uncontrollably, with paths staying near the optimal length of 5.
- On Hypergrids, the expected path length under unstable FM explodes as paths wander, while the stable FM loss yields relatively straight paths to rewarding regions.
- On Point-Robot-Sparse, Stable-CFlowNets obtains higher reward and lower variance than CFlowNets and the RL baselines DDPG, TD3, SAC and PPO.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Stable FM loss instead of FM loss | GFlowNets · 2DW20 Hypergrid | sampled path length | Helped |
| Use FM loss instead of Stable FM loss | GFlowNets · 2DW20 Hypergrid | sampled path length | Hurt |
| Use TB loss instead of Stable FM loss | GFlowNets · Hypergrids | expected sampling time | Mixed |
| Use Stable FM loss instead of FM loss | GFlowNets · Cayley graphs | edgeflow magnitude | Helped |
| Use Stable-CFlowNets loss instead of CFlowNets loss | CFlowNets · Point-Robot-Sparse | valid-distinctive trajectories explored | No clear effect |
| Use Stable-CFlowNets loss instead of CFlowNets loss | CFlowNets · Point-Robot-Sparse | reward | Helped |
Surprising findings
- TB loss does not show exploding sampling time in some experiments, stabilizing at a higher value, so instability is only partial for TB.
- Irrational rotations on the unit circle give perpetually wandering acyclic trajectories, so cycles alone do not explain exploding sampling time in continuous spaces.
Limitations
- Trajectory balance instability is only partially analyzed and no stable TB variant is proposed.
- Experiments are limited; stable detailed balance is not tested in the continuous setting.
- Cayley graph experiments use simple feedforward networks and fail to reach the theoretical initial flow value.
Experimental setup
Datasets: Hypergrid 2DW20 Cayley graph S_20 Point-Robot-Sparse (Gym)
Benchmarks: Hypergrid Cayley graph of S_20 Point-Robot-Sparse
Metrics: average reward expected path length number of valid-distinctive trajectories
Baselines: FM loss TB loss CFlowNets DDPG TD3 SAC PPO Metropolis-Hastings
Builds on: GFlowNets Flow Matching loss Detailed Balance loss Trajectory Balance loss CFlowNets
Closest papers by meaning
- Revisiting Non-Acyclic GFlowNets in Discrete Environments (ICML 2025 · similarity 0.94)
- Flow Factorization for Efficient Generative Flow Networks (AAAI 2025 · similarity 0.90)
- Random Policy Evaluation Uncovers Policies of Generative Flow Networks (ICML 2025 · similarity 0.90)
- Pessimistic Backward Policy for GFlowNets (NeurIPS 2024 · similarity 0.89)
- Pre-Training and Fine-Tuning Generative Flow Networks (ICLR 2024 · similarity 0.89)
- Expected flow networks in stochastic environments and two-player zero-sum games (ICLR 2024 · similarity 0.89)
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