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Revisiting Non-Acyclic GFlowNets in Discrete Environments

T0 review · 1 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

Pith's one-line read GFlowNets can train on cyclic graphs if the backward path is always available

desk verdict Solid, checkable theory for non-acyclic GFlowNets with one overbroad corollary that should be qualified before publication. read the letter →

arxiv 2502.07735 v3 pith:HGHOL5HH submitted 2025-02-11 cs.LG stat.ML

classification cs.LGstat.ML
keywords GFlowNetsnon-acyclicenvironmentsdiscretestatespacesbackwardpolicyexpectedvisitcountsdetailedbalancelossstabilityentropy-regularizedreinforcementlearning
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that GFlowNets do not actually need acyclic graphs in discrete settings, provided every edge has a positive backward probability. It constructs flows as expected numbers of visits of a reversed random walk, which stay finite and satisfy detailed balance and flow matching even with cycles. The consequence is that with a fixed backward policy, every standard acyclic GFlowNet loss learns the correct forward policy, so special stable losses are only needed when the backward policy is trained. The paper also proves that expected trajectory length equals normalized total flow, proposes state flow regularization to control that length, and extends the known equivalence between GFlowNets and entropy-regularized RL to cyclic environments.

What carries the argument

The reversed random walk with a strictly positive backward policy $P_B$: starting at the sink $s_f$ and stepping backward through edges with probabilities $P_B(s \mid s')$, with $s_0$ absorbing. Its expected edge and state visit counts define the flows; Lemma 3.4 guarantees that the walk is absorbed almost surely with finite expected length. This makes flows finite, yields detailed balance $F(s \to s') = F(s') P_B(s \mid s')$, and turns standard GFlowNet losses into valid training objectives for cyclic graphs.

What would settle it

Take a finite cyclic graph with edges $s_0 \to a$, $a \to a$, and $a \to s_f$. Set $P_B(a \mid a) = 1$, $P_B(a \mid s_f) = 1$, and $P_B(s_0 \mid a) = 0$. Then the only finite trajectory from $s_0$ to $s_f$ has induced probability zero and the reversed walk is trapped in the self-loop, so the sum of induced trajectory probabilities is $0$ instead of $1$, violating Lemma 3.4 exactly when strict positivity is dropped.

Watch

Extended reading notes

Core claim

The central claim is Corollary 3.11: when a backward policy $P_B > 0$ is fixed, any loss from the acyclic GFlowNet literature can be used to learn the corresponding forward policy in a non-acyclic environment. The mechanism is to define flows not as probabilities of visiting an edge but as expected numbers of visits under the reversed random walk. Those expected visit counts are finite because strict positivity makes the reversed chain absorbing at $s_0$, and they satisfy detailed balance and flow matching. Consequently reward matching and uniqueness of the forward policy hold, and the positive backward distribution automatically keeps expected trajectory length finite. The paper additionally proves an exact identity linking expected trajectory length to total state flow, proposes state flow regularization as a practical way to control trajectory length when the backward policy is trained, and generalizes the GFlowNet/entropy-regularized-RL equivalence to cyclic graphs.

Load-bearing premise

Everything rests on the backward policy assigning strictly positive probability to every edge of the cyclic graph: if any edge has zero backward probability, the reversed random walk can fail to be absorbed, the induced trajectory measure can stop being a probability distribution, and the flows can diverge.

Editorial extensions

If this is right

  • Any acyclic GFlowNet loss, such as flow matching, detailed balance, and trajectory balance, can train a correct sampler on a cyclic graph as long as the backward policy is fixed and strictly positive.
  • Expected trajectory length equals the normalized total state flow exactly, so minimizing expected sampling length and minimizing total flow are the same optimization target.
  • State flow regularization allows log-flow-scale losses to keep trajectory lengths bounded while preserving accurate reward matching, whereas flow-scale losses bias toward shorter trajectories but can degrade sampling fidelity.
  • The equivalence between GFlowNets and entropy-regularized RL carries over to cyclic environments, so RL algorithms and insights transfer to non-acyclic GFlowNet training.
  • When the backward policy is trained, log-flow-scale losses without regularization can let trajectory length grow without bound; regularization is needed to control it.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The strict positivity condition suggests that any practical implementation must initialize the backward policy with full support and clip it away from zero during training; the paper's experiments indeed use a small epsilon for this purpose.
  • Loss stability is best understood as a property of training dynamics when the backward policy is learnable, not as a requirement for the existence of reward-matching solutions, since a fixed positive backward policy makes all standard losses valid.
  • The exact identity between expected trajectory length and normalized total flow gives a practical diagnostic: monitor normalized total flow during training as a proxy for sampling cost.
  • The scaling hypothesis could be tested directly on other GFlowNet losses, such as subtrajectory balance or flow matching, by comparing the same loss in flow scale versus log-flow scale and measuring trajectory length and distribution error.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The paper develops a theory of GFlowNets on discrete graphs that may contain cycles. Starting from a strictly positive backward policy P_B, the authors define state and edge flows as expected visit counts under the induced trajectory distribution (Definition 3.5), prove that these flows satisfy flow matching and detailed balance (Proposition 3.6), establish a one-to-one correspondence between flows and pairs (P_B, F(s_f)) (Proposition 3.7), and show that every fixed P_B determines a unique forward policy P_F with the same trajectory distribution (Proposition 3.8). Proposition 3.10 gives a reward-matching condition, and Corollary 3.11 claims that any loss from the acyclic GFlowNet literature can be used to learn this P_F in the non-acyclic case. The paper further proposes a state-flow regularizer for training with a learnable backward policy, connects non-acyclic GFlowNets to entropy-regularized RL (Theorem 3.13), and reports experiments on cyclic hypergrids and permutation generation.

Significance. If the results are correct, the paper provides a substantially simpler and more intuitive foundation for non-acyclic GFlowNets than the measure-theoretic treatment of Brunswic et al. (2024), clarifies the central role of the backward policy, and supplies a practical regularizer that can control expected trajectory length. The appendices contain detailed proofs, and the paper includes code, exact computations on small environments, and a careful experimental comparison of loss scales, which are all strengths. The main practical corollary, however, is overstated as written, and this affects a load-bearing claim of the paper.

major comments (1)
  1. [Section 3.5, Corollary 3.11] Corollary 3.11 states that with a fixed P_B > 0, 'any loss from the acyclic GFlowNet literature (Bengio et al., 2021; Malkin et al., 2022; Bengio et al., 2023; Madan et al., 2023)' can be used to learn the corresponding forward policy P_F in the non-acyclic case. This is not true for the Flow Matching loss (Bengio et al., 2021), which is explicitly listed. FM is defined solely on edge/state flows and never involves P_B. In a cyclic graph, any edge flow satisfying the flow matching conditions (3) and the terminal condition F(x → s_f) = R(x) is a global FM optimum; adding a circulation around a directed cycle preserves these constraints and yields a continuum of distinct forward policies P_F(s'|s) = F(s → s')/F(s). Only one of these is the expected-visit-count flow of Definition 3.5 that Proposition 3.8 associates with the fixed P_B, and FM contains no term that can select it. The paper's own Appendix C.1 concedes that FM 'does not admit explicit parameterization of a backward policy, as well as training with fixed backward policies,' which contradicts the breadth of the corollary. The corollary should be restricted to losses that actually involve P_B (e.g., DB, TB, SubTB), or the claim should be reformulated to specify which losses are covered; as stated, the main practical corollary of the paper is internally inconsistent.
minor comments (5)
  1. [Section 3.5, Eq. (12)] In Eq. (12) the regularizer is written as λ F_θ(s), but since the network parameterizes the log-flow, the authors should state explicitly that this means λ·exp(log F_θ(s)) to avoid ambiguity about the scale of the regularization.
  2. [Section 3.5, Eq. (11)] The constrained optimization problem in (11) is over strictly positive backward policies; because the positivity constraint is an open condition, the infimum of the total flow may not be attained. The authors should clarify whether (11) is meant as an infimum or whether additional compactness assumptions are intended.
  3. [Appendix C.1] The statement that FM 'does not admit explicit parameterization of a backward policy, as well as training with fixed backward policies' should be reconciled with Corollary 3.11; at present the appendix explicitly excludes the FM loss from the setting in which the corollary is claimed to hold.
  4. [Table 1] For n = 20, the rows for DB ΔF and SDB ΔF report ΔlogZ ≈ 42.98 with zero standard error, indicating that these runs completely fail to estimate the normalizing constant. The authors should add a note that these configurations are included primarily for the trajectory-length comparison and are not competitive for reward matching.
  5. [Appendix B.2] The scaling hypothesis is supported by a heuristic gradient argument about the derivative of losses in ΔF scale. The authors should state clearly in the main text that this is an explanatory hypothesis rather than a proven statement, since the paper does not provide a theoretical guarantee for the stability of ΔF-scale losses.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the core theory is derived from explicit expected-visit-count definitions, and the main overreach in Corollary 3.11 is a correctness issue rather than a circular reduction.

full rationale

The paper's claimed derivation chain is self-contained. Definition 3.5 defines flows as F(sf) times expected visit counts under the reversed Markov chain induced by PB. Proposition 3.6 derives detailed balance and flow matching from this definition — F(s -> s') = F(s')PB(s | s') follows by Fubini and the Markov property — rather than assuming them. Proposition 3.8 constructs the unique forward policy PF(s' | s) = F(s')PB(s | s')/F(s) and verifies equality of trajectory probabilities, so uniqueness is a mathematical consequence of the definitions, not a fitted input. Proposition 3.10 obtains reward matching from F(x -> sf) = R(x) together with F(sf) = sum_x R(x) = Z. Theorem 3.13 generalizes Tiapkin et al. (2024), but Appendix A.7 supplies an independent proof via occupancy measures and soft Bellman equations instead of resting on the cited acyclic theorem. The main overreach is Corollary 3.11, which says that 'any loss from the acyclic GFlowNet literature' learns the PF corresponding to fixed PB, whereas Appendix C.1 concedes that the flow matching loss 'does not admit explicit parameterization of a backward policy, as well as training with fixed backward policies.' This is an internal inconsistency and overgeneralization, not a circular reduction: no equation is assumed true and then recovered as a prediction. The fixed-PB DB/TB-style losses do have global optima that recover the induced PF by Propositions 3.8–3.9, so the non-circular core of the paper stands. Self-citations to Tiapkin et al. (2024) and Morozov et al. (2024) are used as context or external baselines, and the central results are proved in the present paper, so they are not load-bearing. The score of 1 reflects a minor breadth-of-claim issue, not circularity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central theory introduces no fitted parameters and no new physical entities. The only hand-chosen numbers are experimental hyperparameters (λ, ε, η) for the loss and regularization. The axioms are the standard environment assumptions in GFlowNet work, plus the strict positivity of the backward policy, which is the load-bearing condition for finite flows in cyclic graphs.

free parameters (2)
  • λ (state flow regularization strength) = 10^-3 and 10^-5 in experiments
    Controls the trade-off between expected trajectory length and reward matching accuracy in the regularized DB loss (12). The theoretical equivalence in Proposition 3.12 does not depend on λ, so this is an experimental hyperparameter, not a fitted model parameter.
  • SDB hyperparameters ε and η = ε=1.0, η=10^-3
    Hyperparameters of the stable detailed balance loss (5) from Brunswic et al. 2024, used as a baseline and in scale comparisons. Chosen by the authors; the paper notes that larger η reduces fidelity.
assumptions (4)
  • domain assumption Assumption 3.1: G is finite, s0 has no incoming edges, sf has no outgoing edges, and every state lies on some path from s0 to sf.
    Used throughout Section 3 to ensure trajectories are finite and flows are well-defined; stated as an assumption at the start of Section 3.1.
  • domain assumption PB(s|s') > 0 for every edge s→s' ∈ E whenever a backward policy is used.
    Required by Lemma 3.4 and Definition 3.5 to guarantee the reversed Markov chain is absorbing and expected visit counts are finite. Stated explicitly before Definition 3.5.
  • domain assumption Reward R(x) > 0 for all terminal states x.
    Used in Proposition 3.10 and in the RL equivalence to ensure log R is defined and Z > 0. Standard in GFlowNet literature.
  • standard math Standard results for finite absorbing Markov chains (e.g., fundamental matrix (I-Q)^{-1} exists).
    Used in the proofs of Lemma 3.4 and in Appendix B.4 to solve the flow system exactly.

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Cite this review

Pith. "Pith review of Revisiting Non-Acyclic GFlowNets in Discrete Environments." pith.science (2026). https://pith.science/paper/HGHOL5HH

@misc{pith2026250207735,
  author       = {Pith},
  title        = {Pith review of: Revisiting Non-Acyclic GFlowNets in Discrete Environments},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HGHOL5HH}},
  note         = {Machine review of arXiv:2502.07735}
}
read the original abstract

Generative Flow Networks (GFlowNets) are a family of generative models that learn to sample objects from a given probability distribution, potentially known up to a normalizing constant. Instead of working in the object space, GFlowNets proceed by sampling trajectories in an appropriately constructed directed acyclic graph environment, greatly relying on the acyclicity of the graph. In our paper, we revisit the theory that relaxes the acyclicity assumption and present a simpler theoretical framework for non-acyclic GFlowNets in discrete environments. Moreover, we provide various novel theoretical insights related to training with fixed backward policies, the nature of flow functions, and connections between entropy-regularized RL and non-acyclic GFlowNets, which naturally generalize the respective concepts and theoretical results from the acyclic setting. In addition, we experimentally re-examine the concept of loss stability in non-acyclic GFlowNet training, as well as validate our own theoretical findings.

Figures

Figures reproduced from arXiv: 2502.07735 by the authors.

Figure 1
Figure 1. Comparison of non-acyclic GFlowNet training losses on a small hypergrid environment. We use DB or SDB to specify the utilized loss, ∆ log F or ∆F to specify the flow scale used to compute the error in the loss, and use λ = C to specify the usage of the proposed state flow regularization. Top: evolution of L 1 distance between an empirical distribution of samples and target distribution. Bottom: evolution of mean len… view at source ↗
Figure 2
Figure 2. Comparison of non-acyclic GFlowNet training losses on a larger hypergrid environment. We use DB or SDB to specify the utilized loss, ∆ log F or ∆F to specify the flow scale used to compute the error in the loss, and use λ = C to specify the usage of the proposed state flow regularization. Left: evolution of L 1 distance between an empirical distribution of samples and target distribution. Middle: evolution of mean l… view at source ↗
Figure 3
Figure 3. Plots for DB and SDB losses in ∆F and ∆ log F scales with fixed predicted log backward flow = 1 and varying predicted log forward flow. More specifically, green curve is y = (x − 1)2 , red curve is y = [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Left: evolution of L 1 distance between empirical distribution of samples and target distribution. Right: evolution of mean length of sampled trajectories. Here we note that when ∆ log F scale losses are employed without state flow regularization, mean trajectory lengt…
Figure 5
Figure 5. Figure 5: Left: evolution of L 1 distance between empirical distribution of samples and target distribution. Right: evolution of mean length of sampled trajectories. Here, we see the effects of state flow regularization of different strength λ. Larger values of λ lead to smaller…
Figure 6
Figure 6. Figure 6: Comparison of non-acyclic GFlowNet training losses on a small permutation environment. Left: evolution of L1 distance between true and empirical distribution of fixed point probabilities C(k). Right: evolution of mean length of sampled trajectories. The results are sim…

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.