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Learning a Mixture of GFlowNets

Tiago da Silva, Amauri H. Souza, Salem Lahlou

arXiv:2610.07562v1cs.LGcs.AIstat.ML

TL;DR

GFlowNets and their ensembles can struggle with exploration, convergence, and instability, while existing ensemble methods can add substantial computational overhead and lack a shared conceptual foundation. This paper develops a mixture framework, including CI and DI samplers and the SC method. Across benchmark tasks, SC improves convergence and state-space exploration without significant extra computation.

  • Problem

    GFlowNets can have insufficient state-space exploration, slow convergence, and learning instability, while ensemble approaches may add substantial overhead and lack a shared conceptual foundation.

    Source: §1 · p. 2

  • Method

    The paper develops a general mixture framework, specializes it into CI and DI GFlowNets, and proposes SC GFlowNets that partition outputs into strata using a prescribed modular function.

    Source: §1 · p. 2

  • Results

    SC GFlowNets substantially improve convergence and state-space exploration across benchmark tasks, with no significant runtime difference from an unpartitioned sampler.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The framework unifies prior mixture approaches and supports centralized or component-wise parallel training; SC provides an empirically beneficial mixture at negligible computational cost.

    Source: §1 · p. 2, §7 · p. 11

    Determining whether a stratum is reachable from a state is NP-complete in general, although length-based partitioning is an easily solved example.

    Source: §5.1 · p. 7

Abstract

from arXiv · show

Learning an ensemble of GFlowNets to sample from a discrete target distribution has become a common approach for achieving better state space exploration and convergence than that of a monolithic sampler. However, these methods often add a substantial runtime overhead to the base model, and their conceptual connection remains elusive. To address this, we first propose a general-purpose theoretical framework for describing a mixture of GFlowNets, which we specialize into continuously (CI) and discretely indexed (DI) collections. On the one hand, we show CI GFlowNets can be interpreted through the lens of a random features expansion, provably boosting the sampler's expressivity in graph-structured tasks and reducing learning instability via spectral shifting. On the other hand, we demonstrate DI GFlowNets encompass prior approaches for GFlowNet training and provide the foundation for the newly proposed Stratum-Conditioned (SC) GFlowNets. This method, which is inspired by the Doob's h-transform of Markov chains, decomposes the state space according to a prescribed modular function and restricts each component to sample from a distinct subset of it. Importantly, SC GFlowNets support centralized and component-wise embarrassingly parallel training, and we show both of them significantly speed up learning convergence and mode coverage without introducing any non-negligible extra computation.

1 Introduction

The paper develops a framework for GFlowNet mixtures and specializes it into continuously and discretely indexed samplers. Its SC method improves convergence and state-space exploration across benchmark tasks without significant extra runtime.

1 Introduction

  • GFlowNets face insufficient state-space exploration, slow convergence, and instability, while sampling from large discrete targets remains challenging.The introduction notes that GFlowNets can fail to simulate simple multimodal distributions over relatively small spaces.Source: §1 · pp. 1–2
  • The authors introduce a general mixture framework to unify prior ensemble methods, which can require several separately trained networks and lack a shared conceptual foundation.The framework is used to develop algorithms intended to stabilize and accelerate learning.Source: §1 · p. 2
  • CI GFlowNets use real-valued inputs as random features, boosting expressivity in graph tasks and stabilizing learning through spectral shifting.The paper says this stabilization reduces dependence on exogenous factors such as pseudorandom-number-generator initialization.Source: §1 · p. 2
  • DI GFlowNets include centralized shared-model training and embarrassingly parallel component-wise training, and encompass two prior methods.The named prior methods are Subgraph Asynchronous Learning and Boosted GFlowNets.Source: §1 · p. 2
  • SC GFlowNets partition outputs into strata by binning a prescribed modular real-valued function, relating the resulting samplers through Doob’s h-transform.Examples of the function include trajectory length and distance to a reference point; conditioning on arbitrary subsets is NP-complete in general.Source: §1 · p. 2
  • SC GFlowNets substantially improve convergence and state-space exploration across benchmark tasks, with no significant runtime difference from an unpartitioned sampler.The paper attributes the small runtime difference to partition-sampling and mask-computation costs being dwarfed by policy and reward evaluation.Source: §1 · pp. 2–3

2 Preliminaries

The preliminaries define the state-space structure and the GFlowNet sampling objective. They then introduce trajectory balance as a consistency condition used to train the sampler.

2 Preliminaries

  • A state graph is a directed acyclic graph with an initial state, terminal outputs, and paths from the initial state to every output.The text illustrates this with constructing size-K subsets from the empty set through smaller subsets.Source: §2 · p. 3
  • A GFlowNet stochastically constructs an output from the initial state, aiming for its marginal distribution over outputs to match the unnormalized target.The state space includes partial realizations and terminal outputs.Source: §2 · p. 3
  • Trajectory balance trains forward and backward policies to satisfy a consistency condition, using a learned normalizing constant and stochastic gradient descent.The forward policy is parameterized as a neural network; the backward policy represents the probability of a trajectory given its sampled output.Source: §2 · p. 3

3 Learning a Mixture of GFlowNets

This section formulates mixtures of GFlowNets as weighted collections of samplers whose aggregate matches the target distribution. It develops a learning objective for mixtures and describes how the framework relates to existing balance conditions.

3 Learning a Mixture of GFlowNets

  • Mixtures address GFlowNet objective under-determination and sample inefficiency, including the need for extensive state-graph exploration.The paper focuses on sample inefficiency in the main text and discusses under-determination in the supplement.Source: §3 · p. 3
  • A mixture framework seeks to make the weighted average of indexed samplers match the target distribution.The index space may be discrete or continuous, with common examples using counting or Lebesgue measure.Source: §3 · p. 3
  • A posterior mixing distribution rewrites the mixture-marginal objective, whose components are otherwise intractable, as a learning objective using tractable policies.The posterior mixing distribution represents the probability that a given output came from an indexed sampler, analogous to responsibilities in expectation-maximization.Source: §3 · p. 4
  • The mixture framework also supports balance conditions beyond trajectory balance, including Detailed Balance and SubTB.The paper notes that related learning objectives can be extended to the mixture setting.Source: §3 · pp. 4–5

4 Continuously Indexed GFlowNets

Continuously indexed GFlowNets can be viewed as random-feature-augmented samplers. The paper establishes greater expressivity for this formulation and links it to reduced training instability.

4 Continuously Indexed GFlowNets

  • CI GFlowNets augment a sampler with random features by giving each component an additional real-valued input, and can arbitrarily approximate graph distributions.The paper describes the random-feature approach as computationally simple and commonly used for graph-structured tasks.Source: §4 · p. 5
  • CI GFlowNets are strictly more expressive than single GFlowNets and are universal approximators for graph tasks with edge-additive generative processes.The universal-approximation result is stated for this graph-structured setting.Source: §4 · p. 5
  • CI GFlowNets can smooth the loss surface and significantly reduce training instability.The paper relates this effect to shifting the loss-function Hessian spectrum.Source: §4 · p. 5

5 Discretely Indexed GFlowNets

The discrete-mixture objective can collapse to a sampler whose target does not depend on its component. The paper treats that as a design choice and proposes finite component sets with sparse mixing distributions.

Discretely Indexed GFlowNets.

  • The TB loss permits a collapsed sampler whose target-to-normalizer ratio is independent of the component.A component-independent target may have benefits such as greater expressivity and a smoother loss surface, but should be chosen deliberately.Source: §5 · p. 5
  • Learning a non-collapsed mixture requires constraining the mixing distributions through soft regularization or hard constraints.Source: §5 · p. 5
  • The paper fixes the component index set as finite and the mixing distribution as sparse, assigning zero probability to most components.It also briefly discusses countably infinite component sets through Boosted GFlowNets.Source: §5 · p. 5

Discretely Indexed GFlowNets.

The paper defines discrete GFlowNet mixtures and introduces Stratum-Conditioned GFlowNets, which restrict components to subsets formed by a tractable cover. It characterizes stratum-conditioned policies through Doob’s h-transform and describes centralized and parallel training.

Discretely Indexed GFlowNets.

  • DI GFlowNets use a finite or countable mixture whose components learn reweighted targets, with centralized shared-model or embarrassingly parallel training options.SC GFlowNets set the component mixture from a hand-crafted cover of the state space.Source: Discretely Indexed GFlowNets · p. 6, §5.1 · p. 6, §5.3 · p. 9

5.1 Stratum-Conditioned GFlowNets

  • SC GFlowNets define strata as sets in a tractable cover, with each state's component probabilities uniform over the strata containing it.The cover must be exact, and the number of strata containing each state must be known to compute those probabilities efficiently.Source: §5.1 · p. 6
  • In Lines, SC GFlowNets converge to the target in over three times fewer training steps than conventional GFlowNets; Set Generation needs less than half the iterations for equal accuracy, and Hypergrid shows similar benefits across two partition strategies.Source: Figure 2 · p. 7, §5.1 · p. 7, Figure 3 · p. 10
  • Reachability is NP-complete for arbitrary polynomial-time modular functions, while symmetric junta-based covers permit polynomial-time reachability decisions.The paper presents this complexity result as a design principle for choosing covers.Source: §5.1 · pp. 7–8

5.2 Complementary Characterization of SC GFlowNets

  • For a disjoint cover, each SC policy is a Doob h-transform of the base policy, with its harmonic function encoding the probability of reaching that stratum first.The transformed kernel samples terminal states in the stratum in proportion to the target distribution.Source: §5.2 · pp. 8–9

5.3 Sampling Partitions & Parallel Learning for SC GFlowNets

  • Centralized training samples partitions using estimated component mass, while parallel training learns independent stratum samplers and selects them at inference by partition function.Average wallclock times for the two samplers are reported as similar.Source: §5.3 · p. 9, Figure 4 · p. 10

6 Experiments

The experiments evaluate SC GFlowNets on Set Generation, Hypergrid, and Ancestral Graphs. They report faster convergence, improved exploration, and similar runtime to unpartitioned samplers in the stated settings.

Set Generation

  • Length-based SC GFlowNets in Set Generation achieve the same accuracy in less than half the iterations, with no significant runtime difference from the unpartitioned sampler.The reported runtime difference is small because partition sampling and mask computation are outweighed by policy and reward evaluation.Source: §6 · p. 10

Hypergrid

  • SC GFlowNets improve convergence in Hypergrid even with imperfectly balanced length-based partitions; length- and octant-based partitions both show benefits.Source: §6 · p. 10

Ancestral Graphs

  • In Ancestral Graphs, partitioning by the number of non-absent edges improves both exploration and convergence.The task uses a BIC-based target and also considers an Erdős-Rényi model for tractable edge-marginal evaluation.Source: §6 · pp. 10–11

7 Conclusions

The paper concludes that its mixture framework explains why and how to learn mixtures of GFlowNets, with SC GFlowNets providing empirical benefits at negligible computational cost. Its background reviews GFlowNet challenges and the sampling and training setup.

7 Conclusions

  • SC GFlowNets restrict ensemble samplers to state-space subsets and provide substantial empirical benefits at negligible computational cost.The paper calls for future work on automatic, tractable state-space partitioning.Source: §7 · p. 11

A GFlowNets: Background & Challenges

A.1 Background

  • A GFlowNet constructs terminal states through a directed acyclic state graph and trains forward and backward policies to satisfy trajectory balance.The forward-policy marginal over terminal states is intended to match the target up to a normalizing constant.Source: §A.1 · p. 17

A.2 Challenges in training GFlowNets

GFlowNet training faces underdetermination, instability in the learned backward policy, model expressivity limits, and insufficient exploration. Mixture formulations offer a way to address these challenges, while the paper also relates mixture training to existing GFlowNet losses.

A.2 Challenges in training GFlowNets

  • TB permits infinitely many forward–backward policy pairs whose forward marginal matches the target, leaving the choice of solution unclear.The family can be continuous, except in settings such as autoregressive generation where each target has a single trajectory.Source: §A.2 · p. 18
  • On Hypergrid, the learned backward policy varies substantially across runs that differ only in the pseudorandom sampling seed.The passage reports five runs with the same parameter initialization and optimizer.Source: §A.2 · p. 18, Figure 8 · p. 18

B Mixture of GFlowNets

  • Mixtures address underdetermination by learning a family of backward policies rather than committing to one potentially unknown policy.The supplement extends SubTB to mixtures and states that familiar GFlowNet losses follow from the resulting framework.Source: §A.2 · p. 18, B Mixture of GFlowNets · p. 19

A.2 Challenges in training GFlowNets

  • Near state aliasing occurs when a solution to the balance objective cannot be accurately represented by the chosen parametric model.For graph-structured domains, a 1-WL GNN can fail to distinguish states that require different probabilities.Source: §A.2 · p. 18
  • Accurate learning requires visiting diverse, sparse high-probability regions, while prior exploration approaches can multiply per-step computation by requiring additional models.The paper presents mixtures as an alternative that improves exploration without costly, independently parameterized extra models.Source: §A.2 · p. 19

C Continuously Indexed GFlowNets

Continuously indexed mixtures augment GFlowNets with random features, connecting them to graph random-feature methods. The paper establishes greater expressivity and reports a stabilizing effect, then describes discrete stratum-based partitions and their tractable implementations.

C Continuously Indexed GFlowNets

  • CI GFlowNets use a collapsed mixing posterior, qM(dγ|x)=q(dγ), because the relevant measures are generally intractable when the index space is uncountable.The paper identifies this choice as random features for GFlowNets.Source: C Continuously Indexed GFlowNets · p. 20
  • If the flow function is flexible enough to mimic the target, the optimal mixing posterior can collapse to the prior, so distinct components require deliberate structural constraints.The paper notes that such a collapse is not necessarily undesirable, but should be a design choice.Source: C Continuously Indexed GFlowNets · p. 20
  • Random-feature-augmented GNN GFlowNets can approximate any target distribution on edge-additive graph-generation processes with arbitrarily small total variation error, with high probability.The guarantee is over the draw of the random features.Source: C Continuously Indexed GFlowNets · p. 21
  • The paper leaves a thorough investigation of random-feature benefits and the useful definition of the mixing prior pM for future research.It particularly identifies how to define pM usefully as important.Source: C Continuously Indexed GFlowNets · p. 21
  • Random features implicitly regularize training by increasing loss-Hessian eigenvalues, and five-run Hypergrid experiments report reduced variability in avgKL(pB||pU).The experiments vary the exploratory-policy seed while holding the underlying architecture and parameter initialization fixed.Source: C Continuously Indexed GFlowNets · pp. 21–22

D Discretely Indexed GFlowNets

  • Discrete stratum-conditioned partitions can use description length or Hamming weight, with transitions masked or pruned to keep samples within the assigned strata.The paper relates both examples to a symmetric-junta function class and says their implementations are tractable.Source: D Discretely Indexed GFlowNets · p. 23

E Revisiting Subgraph Asynchronous Learning (SAL)

The section interprets SAL within the DI GFlowNet framework using fixed-horizon coverings and a Doob h-transform. This interpretation connects SAL’s balance-equation solution to mixture weights, while SAL remains inherently distributed because its mixing functions are generally intractable to compute.

E Revisiting Subgraph Asynchronous Learning (SAL)

  • A fixed-horizon covering assigns disjoint state subsets to sets S_k, with each S_k within a fixed distance d of the initial state.States reachable from S_k belong to X_k, while X_o contains terminal states closer than d to the initial state.Source: E Revisiting Subgraph Asynchronous… · p. 24
  • SAL’s partition components enforce amortized trajectory balance with a shared uniform backward policy before their learned flows inform the centralized model.The centralized model uses component flow functions as targets for its learning objective.Source: E Revisiting Subgraph Asynchronous… · p. 24
  • A Doob h-transform yields policies that pass through each S_k, and the associated hitting probabilities h_k and g_k are harmonic outside S_k.The section uses these functions to establish SAL’s representation in the mixture framework.Source: E Revisiting Subgraph Asynchronous… · p. 24
  • SAL can be represented as a DI GFlowNet by mapping its balance-equation solution to specific prior and posterior mixing distributions.The construction uses h_k(s_o) as the prior and g_k(x) as the posterior mixing distribution.Source: E Revisiting Subgraph Asynchronous… · p. 25
  • SAL is inherently distributed because h_k and g_k cannot generally be computed tractably.The framework interpretation also recovers SAL’s correctness statements from Proposition E.4.Source: E Revisiting Subgraph Asynchronous… · p. 25

F.1 Boosted GFlowNets satisfy the Mixed Trajectory Balance

The section characterizes Boosted GFlowNets as DI GFlowNets under stated assumptions on underallocation and residual convergence. Their sequential residual-based training yields a countably infinite mixture representation, but does not support near-embarrassingly parallel training.

F.1 Boosted GFlowNets satisfy the Mixed Trajectory Balance

  • Assumption F.1 requires each trained booster’s trajectory-wise mass estimate not to exceed the target, while F.3 requires residuals to converge to the target.F.3 also assumes residuals collapse to a state-subset-dependent form.Source: §F.1 · p. 26
  • Each booster trains sequentially on the residual target left by earlier samplers, which mostly excludes regions where earlier samplers concentrate.This residual-based design makes the approach adaptive and sequential rather than amortized or parallel over fixed structural primitives.Source: §F.1 · pp. 25–26
  • Under Assumptions F.1 and F.3, Boosted GFlowNets satisfy the mixed trajectory balance equations as a countably infinite DI GFlowNet mixture.The representation has normalized posterior mixing weights and component normalizers that sum to Z.Source: §F.1 · pp. 26–27
  • Boosted GFlowNets cannot use near-embarrassingly parallel training because their mixture has unbounded support and each model depends on predecessor residuals.The experiments typically truncate the prior mixing distribution to a chosen K; Dall’Antonia et al. (2026b) use K=2 or 3.Source: §F.1 · pp. 26–27
  • SC GFlowNets are described as more foundational because stratum-conditioning can train SAL samplers or Boosted GFlowNet boosters, but not vice versa without losing monolithicness.Monolithicness means one model can learn the entire mixture.Source: §F.1 · p. 27

F.2 Experiments

Across the considered tasks, SC GFlowNets are more compute-efficient than Boosted GFlowNets and often converge faster. The section attributes the cost difference to amortized component policies versus separately trained sequential networks, and reports faster convergence with larger gains for larger state spaces in an additional experiment.

F.2 Experiments

  • SC GFlowNets are up to roughly 1.6× faster per step for Set Generation and use up to 50% fewer FLOPs per training step than Boosted GFlowNets.The comparison covers Set Generation, Hypergrid, and Lazy Random Walk.Source: §F.2 · p. 27, Figure 13 · p. 28
  • SC GFlowNets often converge faster than Boosted GFlowNets across the considered tasks.The section reports this pattern for Set Generation and Lazy Random Walk comparisons.Source: Figure 14 · p. 28, Figure 15 · p. 28, §F.2 · p. 28
  • Boosted GFlowNet experiments allocate a prescribed iteration budget equally across a predefined number of boosters, while automatic sampler elicitation remains unresolved.The paper notes this as an unresolved issue for Boosted GFlowNets.Source: §F.2 · p. 28
  • SC’s stable partitioning supports consistent, non-random exploration of subsets, whereas boosting relies on accurately training several independent networks and can be biased unless Assumptions F.1 and F.3 hold.Both methods are described as improving discovery of high-probability regions through enhanced state-space exploration.Source: §F.2 · p. 28

G Experimental Details & Further Experiments

  • In the additional generative-process experiment, convergence gains are greater for larger state spaces; supplemental analyses also cover additional domains, target distributions, and TC constructions.The authors provide JAX code for training and evaluating the models on the considered tasks.Source: G Experimental Details & Further… · p. 28, Figure 17 · p. 29

G.1 More Experiments

This section provides additional experiments on SC and ASC GFlowNets, alongside implementation details and proofs concerning GFlowNet properties, component reachability, and balance conditions. The experiments examine task- and partition-dependent effects on learning convergence or fit, while the proofs include an implication from SAL to the MTB condition.

G.1 More Experiments

  • For sequence generation, SC significantly improved learning convergence, and parallel ASC training on distinct subsets also sped training.ASC used K = 3 separate samplers; Figure 18 compares it against the best of K independently trained GFlowNets at matched computational cost.Source: §G.1 · p. 29, Figure 18 · p. 30
  • Partition-count effects differed: one unspecified domain showed no significant difference across K ∈ {2, 3, 4, 5, 10}, while Lazy Random Walk improved over a monolithic sampler for all evaluated K ≥ 2.In Lazy Random Walk, accuracy varied noticeably with K. The authors suggest K = 3 when compatible with the chosen TC, or short training runs to select K, and advise against too large a value.Source: §G.1 · pp. 30, 32, 33
  • For Lazy Random Walk, SC improved convergence and fit despite imperfect partitioning, while ASC achieved a better fit, especially on Rings.The proposed distance-based partition uses K = 3; the ASC advantage is particularly apparent when components receive connected high-probability regions.Source: §G.1 · p. 30, Figure 20 · p. 31, Figure 24 · p. 32
  • SC GFlowNets improved both learning and the rate of discovering modes in Bayesian variable selection.The samplers were evaluated using predictive posterior and average unnormalized log-posterior metrics for the 10 most likely variable sets.Source: §G.1 · p. 33, Figure 25 · p. 33

G.2 Experimental Details

  • Experiments generally used two 64-dimensional MLP layers, Muon for network parameters, Adam for log-partition parameters, and dγ = 8 for mixture representations.Results and error bars were averages and standard deviations over three independent runs.Source: §G.2 · p. 34

H.1 Proof of Proposition 3.3

  • The proof of Proposition 3.3 obtains its equalities by marginalizing trajectory probabilities and summing over terminal states.Source: §H.1 · pp. 36–37

H.2 Proof of Corollary 5.2

  • The proof of Corollary 5.2 establishes its equalities using the component-region indicator and the definition of Zk.Source: §H.2 · p. 37

H.3 Proof of Proposition 5.6

  • The proof of Proposition 5.6 reduces component reachability to SubsetSum, showing that deciding whether Xk is reachable from a given state is NP-complete in general.Source: §H.3 · p. 37

H.4 Proof of Proposition 5.7

  • For symmetric juntas, the decision problem is polynomial-time solvable, including for bounded multisubsets when the size bound is polynomial in |C|.The multisubset argument reduces verification to checking whether some m from 0 to B − #(s∩r) yields ξ(#(s∩r) + m) ∈ [a_k, b_k).Source: §H.4 · p. 38

H.5 Proof of Proposition 5.9

  • Under a balance condition, the Markov chain defined by κ_F yields the target distribution R at its hitting state, as asserted by Proposition 5.9.The proof also uses the finite maximum trajectory length to establish finite hitting times.Source: §H.5 · pp. 38–39

H.6–H.7 Proofs of Lemma 5.10 and Proposition 5.11

  • The functions h_k are harmonic, equal 1 on X_k and 0 outside it, and h_k(s_o) is proportional to Z_k under a balance condition.For the transformed kernel, trajectories reach X_k before any other X_m, and its marginal matches R restricted to X_k up to normalization.Source: §H.6 · p. 39, §H.7 · pp. 39–40

H.8–H.9 Proofs of Proposition B.1 and Corollary C.2

  • A mixture satisfying SubTB samples from the target, while the prior and posterior mixing distributions are related when both satisfy SubTB.The target claim follows because SubTB entails trajectory balance on complete trajectories; the passages do not state the mixing-distribution formulas.Source: §H.8 · p. 41, §H.9 · p. 41

H.10 Proof of Proposition C.5

  • Random-feature GNNs can approximate any graph distribution with high probability under the described edge-additive process.The proof bounds the marginal-distribution error through a policy-approximation error and a union bound over states.Source: §H.10 · pp. 41, 43

H.11–H.12 Proofs of Lemma C.6 and Proposition C.7

  • The CI GFlowNet population loss has a local expansion whose random features act as spectral regularization, shifting eigenvalues by σ² eig(Hreg).The local-loss characterization is given for the learning objective, and the spectral shift follows from the Hessian relation.Source: §H.11 · p. 43, §H.12 · p. 43

H.13–H.14 Proofs of Lemmas E.2 and E.3

  • Both h_k and g_k are harmonic outside S_k, while h_k(s_o) and g_k(x) form probability distributions over k ∈ {0, . . . , K}.The latter distributions account for reaching one of the S_k or, for the residual category, reaching X_o without hitting any S_k.Source: §H.13 · p. 44, §H.14 · p. 44

H.15 Proof of Proposition E.4

  • Policies satisfying the ATB condition from Silva et al. (2025b) yield policies satisfying the MTB condition.The proof establishes this implication for the policies denoted p_B^(k) and their transformed counterparts.Source: §H.15 · p. 44
  • The proof establishes that every solution to SAL also satisfies the MTB condition in Definition 3.2.Source: §H.15 · p. 45

H.16 Proof of Proposition F.4

The proof establishes that the learned policies satisfy the MTB condition under Assumptions F.1 and F.3, and identifies Boosted GFlowNets as DI GFlowNets with countably infinite components. Related work places GFlowNets among models used across diverse applications and connects them to mixture approaches.

H.16 Proof of Proposition F.4

  • The learned policies satisfy the MTB condition under Assumptions F.1 and F.3.Source: §H.16 · p. 45
  • The proof verifies that q(k|x) is a probability distribution over k ≥ 1.Source: §H.16 · p. 46
  • Boosted GFlowNets can be interpreted as DI GFlowNets with countably infinite components.Source: §H.16 · p. 46

I Related works

  • Related work includes GFlowNet state-space modifications focused on exploration or strictly distributed training, alongside mixture methods in inference and normalizing flows.The cited state-space approaches include Boussif et al. (2024), Yu et al. (2026b), and Silva et al. (2025b).Source: I Related works · p. 46

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