Path-dependent Discrete Amortized Inference

Esmeralda S. Whitammer; Salem Lahlou; Tiago da Silva

REVIEW 2 major objections 6 minor 300 references

Path-dependent Discrete Amortized Inference

T0 review · 2 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A deterministic latent memory added to a GFlowNet's state space lets the policy depend on the whole past trajectory, and the paper proves the standard balance losses still yield the correct terminal distribution.

desk verdict Solid central contribution (balance losses on a lifted DAG for path-dependent GFlowNets) undermined by a flawed expressivity proof in Proposition 3.1 that should be fixed or downgraded before peer review. read the letter →

arxiv 2608.08644 v1 pith:CH2YKO6U submitted 2026-08-09 cs.LG

Tiago da Silva , Esmeralda S. Whitammer , Salem Lahlou This is my paper · ORCID

classification cs.LG

keywords GFlowNetspath-dependentpolicieslatentdynamicalsystemsdiscreteamortizedsamplingtrajectorybalancestatealiasingself-referentialweightmatrixcompositionalobject

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

GFlowNets learn to sample compositional discrete objects---graphs, sequences, subsets---in proportion to an unnormalized target by training a Markovian policy that builds objects step by step. This paper argues that the Markovian restriction is the bottleneck: identical states reached by different histories are aliased, so the policy cannot tell them apart, and credit from the final reward propagates poorly. The proposed fix is to lift the state space with a learnable latent dynamical system $W_t = W_{t-1} + \phi(s_t, W_{t-1})$, making the policy depend on the whole trajectory while remaining trainable with the same trajectory-balance, subtrajectory-balance, and contrastive-balance losses. The paper proves these lifted losses still make the terminal marginal proportional to the target, and reports on benchmark discrete-sampling tasks that the recurrent policy often converges faster and fits the target better than a Markovian counterpart.

What carries the argument

The central object is the lifted pointed DAG $S^\star = S \times \Omega$ with deterministic latent dynamics $W_t = W_{t-1} + \phi(s_t, W_{t-1})$: it expands the state graph so that each distinct history receives its own latent context, while keeping the lifted space finite. The paper implements $\phi$ with a modified self-referential weight matrix (a fast-weight update rule that rotates the weight matrix by a fixed ergodic rotation and self-adjusts its update strength), so the memory both encodes the past and actively reduces state aliasing. The subtrajectory balance, trajectory balance, and contrastive balance conditions are then evaluated on this lifted graph, and the balance identity is what transfers the correctness guarantee from the Markovian setting to the path-dependent one.

What would settle it

Train the LINES environment from Section 3 with the same recurrent policy but a stochastic latent update $W_t = W_{t-1} + \phi(s_t, W_{t-1}) + \varepsilon_t$ and minimize the TB loss; if the terminal marginal's distance to $R$ does not approach the deterministic version's level while all else is held equal, the finiteness assumption behind Proposition 4.5 is the reason.

Watch

Extended reading notes

Core claim

On a lifted pointed DAG $S^\star = S \times \Omega$, where each state carries a latent memory updated by a deterministic, state-conditional rule, the central result (Proposition 4.5) is that if the subtrajectory balance condition $F(s^\star_i)p_F(s^\star_{i+1:j} \mid s^\star_i) = p_B(s^\star_{i:j-1} \mid s^\star_j)F(s^\star_j)$ holds for every slice of every trajectory, then the marginal of the forward policy over terminal states is proportional to the flow $F$, and therefore to the target $R$ on terminal objects. Trajectory balance and contrastive balance follow as corollaries. Because the latent update is deterministic, each discrete path determines a unique latent sequence, keeping the number of paths through each lifted state finite, which is what carries the proof. The same lifting is shown to strictly enlarge the representable class of distributions for linear policies and for GNN-based policies, and a two-terminal toy graph exhibits a uniform Markovian output where the path-dependent model can fit any target. A companion result shows that a Markovian backward policy forces the optimal forward policy to be Markovian as well, so the lift buys expressivity only when the backward model is also allowed to be path-dependent.

Load-bearing premise

The load-bearing premise is that the latent update is deterministic, $W_t = W_{t-1} + \phi(s_t, W_{t-1})$, so each discrete path determines a unique latent trajectory and the lifted graph stays finite; a stochastic latent update breaks that finiteness and the paper leaves that case open.

Editorial extensions

If this is right

Existing GFlowNet training pipelines can swap a Markovian policy head for a recurrent policy and keep the same balance-loss guarantees that the terminal marginal is proportional to the target.
Path-dependent parameterizations can represent distributions that Markovian parameterizations of the same architecture cannot, so state aliasing is an expressivity ceiling, not just an optimization difficulty.
A Markovian backward policy drives the optimal lifted forward policy back to being Markovian; to benefit from path-dependence, the backward model must also be path-dependent, which the paper's experimental designs do.
The vector field $\phi$ and the policy are learned jointly from a single loss, so the method adds memory without introducing a separate training objective.
Distributed and streaming amortized inference schemes transfer to the lifted setting, as shown in the supplement, because the balance conditions remain valid there.

Reading between the lines

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

A natural testable extension is stochastic latent dynamics: the paper sketches a measure-theoretic formulation but leaves the learning guarantees open, so an open problem is whether any SubTB-type condition can be proven when each history branches over infinitely many latent continuations.
The expressivity results suggest that symmetry-rich generation tasks, where GNN-based GFlowNets are known to collapse to uniform or near-uniform outputs, are prime candidates for this memory-based lifting.
Because the lifted policy is effectively a recurrent policy, techniques from partially observable MDPs and recurrent reinforcement learning---belief-state compression, memory regularization, action abstraction---could be imported directly into discrete amortized sampling.
The ablation study singles out the rotation matrix as the component that most accelerates convergence, which suggests a cheap improvement test for other recurrent samplers: swap the learned memory update for a rotation-based fast-weight rule before scaling the network.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Hi [Colleague],

Here's my read of arXiv:2608.08644.

The one thing to know: this paper has a solid core and a shaky headline. The core is the lifted pointed DAG construction (Definition 4.1) and the proof that trajectory balance, subtrajectory balance, and contrastive balance remain valid on it (Propositions 4.5–4.7). The proof is correct: the lifted DAG is still a pointed DAG, and with deterministic latent dynamics the number of trajectories per terminal state is finite, so the standard flow-matching argument goes through. That's a real contribution.

What's also good: the empirical work is broad and honest. The authors test on set generation, sequence design, grid world, Ising, lazy random walk, plus ablations on the rotation matrix, link function, and a comparison with LSTM/GRU. The path-dependent SRWM consistently beats the Markovian baseline. The code is in the supplement, though a public repository would be better.

Now the soft spots. The expressivity claims are the weak link. Proposition 3.1 claims that path-dependent linear policies strictly expand the set of learnable distributions, but the proof in Appendix A.1 is not valid. It equates the dimension of the set of achievable terminal distributions with the rank of the state embedding matrix, ignoring that the map from policy log-odds to terminal distributions is softmax-based and nonlinear. A one-dimensional subspace of log-odds can easily produce terminal distributions whose linear span is two-dimensional (the N=2, d=1 example from the stress test shows this). So the inequality dim span(WM) ≤ dim span(WP) is not proven. Proposition 3.2 has a reversed set inclusion in its proof (RP ⊆ RM instead of RM ⊆ RP), which is probably a typo but should be fixed. And Proposition 4.8's proof drops R(x) factors when marginalizing; it's not convincing as written.

None of these soft spots break the central correctness result. But the paper's own framing leans on these expressivity propositions, so they deserve serious repair. As it stands, the expressivity gain is better treated as an empirical observation than a proven theorem.

Who should read this: GFlowNet researchers working on structured discrete spaces where state aliasing is a practical issue. It might be a useful drop-in improvement. It won't reshape the field, but it's a legitimate step forward.

My recommendation: send it to peer review. The main method and its theoretical underpinning are sound enough to warrant referee time, and the authors should be pushed to fix or downgrade the expressivity claims before publication.

Best, [You]

Referee Report

2 major / 6 minor

Summary. The paper proposes to lift the pointed DAG of a GFlowNet by attaching a deterministic, learnable latent dynamical system W_t = W_{t-1} + phi(s_t, W_{t-1}), so that the forward policy can condition on the whole trajectory through W_t. It argues that Markovian policies suffer from state aliasing and limited expressivity, illustrates this with linear LINES and GNN-based graph examples, and proves (Propositions 4.5-4.7) that trajectory balance, subtrajectory balance, and contrastive balance remain valid on the lifted DAG. It also proves (Proposition 4.8) that a Markovian backward policy forces the optimal forward policy to be Markovian. Experiments on set generation, sequence design, grid worlds, preference learning, bit sequences, lazy random walks, and Ising models report improved goodness-of-fit and faster convergence relative to Markovian baselines with comparable parameter counts.

Significance. If the main theoretical claims held as stated, the paper would make a useful contribution: a principled recipe for recurrent/path-dependent GFlowNet policies with the same training objectives as standard GFlowNets, plus a formal demonstration that lifting can strictly increase the representable family in some settings. The proof strategy for Propositions 4.5-4.7 is sound: the lifted graph is a finite pointed DAG for fixed deterministic phi, and the standard balance-condition argument applies. The experimental section is broad, code is promised in the supplement, and the ablations (rotation matrix, link function, RNN baselines) support the claimed mechanism. The main reservation is that one of the headline expressivity results (Proposition 3.1) is not established by its proof, and the appendix proof of Proposition 4.8 contains a nontrivial omission. The central balance-loss extension is nonetheless credible and worth publishing after revision.

major comments (2)

[Section 3.1, Proposition 3.1; Appendix A.1] The proof equates the expressive power of the Markovian sampler with the dimension of the set of achievable policy log-odds, writing o = Psi k (Eq. 13) and concluding dim W_M = rank Psi. This is not valid: the map from log-odds o to the terminal distribution r is nonlinear, since for M = 1, r_i = (1 - sigma(o_i)) prod_{j < i} sigma(o_j). For N = 2, d = 1 and psi(p_i) = i, the three achievable normalized terminal vectors r(0) = (1/2, 1/4, 1/4), r(-ln 2) = (2/3, 4/15, 1/15), and r(ln 2) = (1/3, 2/15, 8/15) are linearly independent (determinant -1/60), while rank Psi = 1. Hence the linear span of the achievable terminal distributions can have dimension 3 even though the log-odds lie on a line; the claimed inequality dim span(W_M) <= dim span(W_P) and the conclusion dim W_M <= d are not established. The proposition should be reformulated (e.g., in terms of the intrinsic dimension of the achievable distribution manifold) and proved for that statement.
[Appendix A.4, proof of Proposition 4.8] In the marginalization step, the proof uses Eq. (16) to replace products of pF with products of pB-tilde, but Eq. (16) contains the factor R(s_T) on the right-hand side. When computing pS(s_l | xi*) as a ratio of sums over continuations tau: s_l -> X and tau': s_{l-1} -> X, the factors R(x_tau) and R(x_tau') appear inside the sums and do not cancel with the displayed pB-tilde factors. They are dropped in the manuscript's derivation. The conclusion is recoverable: define A(s_l) = sum_{tau: s_l -> X} R(x_tau) pB-tilde(tau | s_l) and B(s_{l-1}) = sum_{tau': s_{l-1} -> X} R(x_tau') pB-tilde(tau' | s_{l-1}); then the conditional is pB-tilde(s_{l-1} | s_l) A(s_l) / B(s_{l-1}), which still depends only on s_{l-1} and s_l. The proof should be rewritten accordingly, and the notation pB(xi* | s_T) for an incomplete prefix xi* should be clarified or replaced.

minor comments (6)

[Appendix A.2, proof of Proposition 3.2] The first inclusion is written R_P(S) subset of R_M(S); it should be R_M(S) subset of R_P(S). The same reversal appears in the sentence beginning 'Consequently, R_P(S) subset of R_M(S)'.
[Appendix A.3] The sentence 'Corollaries 4.6, 4.7, B.1 follow directly from Proposition 3.2' should refer to Proposition 4.5, not Proposition 3.2.
[Appendix A.1, Eq. (14)] The definition c_i = sum_{0 <= j <= i} psi(p_i) uses the wrong index; it should be psi(p_j).
[Appendix A.3, proof of Proposition 4.5] In the line 'P_{tau: s_o* -> s*} pF(tau | s*) = 1', the sum should be over the backward policy pB(tau | s*), not pF.
[Section 5, Table 1 and surrounding text] The text says K is in {16, 32}, but Table 1 reports K = 16 and K = 24; the cited cardinalities 10^14 and 10^18 correspond to K = 16 and K = 32, respectively. Please align the text and table.
[Section D and Conclusions] The main text and Section D appear to claim that a stochastic latent dynamical system can be developed, while the Conclusions state that the feasibility of a stochastic dynamics remains open; please clarify which parts of Section D are formal results and which are preliminary.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the core balance-correctness theorem is proved from stated assumptions and the self-citations are not load-bearing.

full rationale

The paper's central formal claim, Proposition 4.5, is a self-contained proof: assuming the SubTB condition (Eq. 10) on the lifted DAG, it derives p^T(s*) proportional to F(s*) by summing the backward-policy decomposition and using finiteness of trajectories, which follows from the deterministic latent update. No fitted parameter is renamed as a prediction, and no theorem is assumed into its own conclusion. The expressivity propositions are stated as parameter-free comparisons of representable sets; although the proof of Proposition 3.1 contains a gap (it identifies the dimension of policy log-odds with the dimension of the terminal-distribution span), that is a correctness issue rather than a circular reduction. Several cited prior works are by overlapping authors (Silva et al. 2025a for GNN expressivity, da Silva et al. 2024a for CB-implies-TB), but those citations are supporting steps, not the definition of the target result, and the manuscript also gives its own construction in Appendix A.2. The FCS metric is cited from the same authors but is explicitly defined in Eq. (18) and used only for evaluation. Thus the derivation chain is not circular; the modest score reflects only the presence of minor non-load-bearing self-citations.

Assumptions & free parameters 3 free parameters · 5 assumptions · 2 invented entities

The central theoretical claims rest on viewing the lifted DAG as a pointed DAG; the learned latent dynamics parameters (W, b, W0) are optimized by SGD and are not fitted constants in the derivations. The hand-chosen latent dimension and the fixed rotation matrix are the only non-learned knobs besides standard hyperparameters.

free parameters (3)

Latent dimension d = d=32 in experiments
The dimension of the latent dynamical system and the SRWM is chosen by hand; the ablation in Figure 10 varies it. Central to the architecture but not a fitted constant in a prediction.
Rotation matrix R = fixed ergodic rotation matrix
Introduced ad hoc in the SRWM update rule to reduce state aliasing; not learned, but a design choice that the ablation (Figure 13) shows is important.
Learning rates = 1e-3 for policy, 1e-1 for logZ
Standard hyperparameters from torchgfn, not fitted to a target. Listed for completeness.

assumptions (5)

domain assumption The lifted DAG SG* (Definition 4.1) with deterministic phi and fixed W0 is a pointed DAG with finitely many trajectories per terminal state.
This makes the standard flow-matching proof applicable; it fails if phi is stochastic (Section D) or the state space is non-acyclic.
standard math The balance conditions (TB, CB, SubTB) are sufficient and their unrestricted optima are achieved by the corresponding objectives.
This is the standard GFlowNet theory from Bengio et al. 2023 and Malkin et al. 2022; the paper re-states it for the lifted DAG.
domain assumption In Proposition 3.1, the state embeddings are fixed (e.g., psi(p_i)=i for the natural embedding), and the comparison holds within the restricted linear policy class.
The comparison between linear Markovian and linear path-dependent models is within a restricted class; with sufficiently wide deep networks the representability gap can vanish.
standard math In Proposition 3.2, GNN expressivity is bounded by the Weisfeiler-Lehman hierarchy.
This is a known limitation of GNNs applied to the GFlowNet policy parameterization; temporal features break the symmetry.
domain assumption The target distribution is positive: R(x)>0 for all terminal x.
Flow definitions and balance losses assume positive rewards; this matches the standard GFlowNet setup.

invented entities (2)

Latent dynamical system {W_t}
purpose: Makes the policy path-dependent, providing a memory of the trajectory and mitigating state aliasing.
A model component introduced by the paper; its effectiveness is demonstrated on benchmarks inside the paper, but there is no external falsifiable prediction that would differ in an independently observable way.
Ergodic rotation matrix R inside the SRWM
purpose: Rotates the eigenvectors of the fast weight matrix to promote representation diversity and reduce aliasing.
An architectural invention imported from prior fast-weight-programming work; no independent evidence outside the reported ablations.

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Cite this review

Pith. "Pith review of Path-dependent Discrete Amortized Inference." pith.science (2026). https://pith.science/paper/CH2YKO6U

@misc{pith2026260808644,
  author       = {Pith},
  title        = {Pith review of: Path-dependent Discrete Amortized Inference},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CH2YKO6U}},
  note         = {Machine review of arXiv:2608.08644}
}

read the original abstract

We consider the problem of sampling compositional and discrete objects from a given unnormalized posterior distribution. Notably, recent studies have shown that this problem can be efficiently solved by learning a deterministic Markov Decision Process (MDP) that progressively builds each object in proportion to the posterior. In this work, however, we demonstrate that the Markovian assumption can both hamper signal propagation during training and catastrophically reduce the learned sampler's expressivity due to state aliasing. To address these issues, we propose lifting the MDP with a learnable latent dynamical system that allows the underlying policy to depend on the entire past trajectory---and not only on the current state. In view of this, we refer to the resulting method as path-dependent discrete amortized inference. Importantly, we provably extend existing learning algorithms for discrete amortized samplers to our setting. In experiments on standard benchmark problems, we also show that our approach often leads to faster learning convergence and improved state space exploration relatively to prior techniques.

Figures

Figures reproduced from arXiv: 2608.08644 by the authors.

**Figure 1.** b) is Markovian. As we argue in Section 3, however, the introduction of path-dependent state-transitions can strictly improve the expressivity of the hypothesis class defined by the space of parametric policies. To show this, we present illustrative environments and distributions that can be solved through path-dependent parameterizations, but not through their Markovian counterparts. Additionally, traditional learn… view at source ↗

**Figure 2.** Illustration of the LINES environment with length N and step size M = 2. Starting at po, an agent can either move up to M steps forward or transition to a terminal state (q) and stop. 3. On the limitations of Markovian samplers This section lays out the empirical and provable limitations of Markovian samplers. In Sections 3.1 and 3.2, we demonstrate that the introduction of path-dependence strictly improves the mo… view at source ↗

**Figure 3.** Illustration of Proposition 3.1. While a linear Markovian model (left) completely misses the high-probability regions of the target (right; see Equation (5)), our path-dependent parameterization (middle) provides a near-perfect distributional fit. in which kj ∈ {1, . . . , N} for 1 ≤ j ≤ 4 and wj ∈ [0, 1] with P4 j=1 wj = 1 as weighting factors; see [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: Illustration of Proposition 3.2. We present an example in which a 1-WL GNN-based Markovian amortized sampler (Equation (6)) cannot sample from an non-uniform target over {q1, q2}, while a corresponding path-dependent model can (right). for α ∈ {0.1, 0.2, . . . , 0.9} a…

**Figure 5.** Figure 5: Markovian samplers struggle to distinguish similar states (Section 3.3). (a) LINES environment with a sparse target; the optimal action distribution at p2 puts a large weight on moving up (due to the largeness of R(q2)), while the one at its adjacent state, p3, is bias…

**Figure 6.** Figure 6: illustrates Proposition 4.8 for the LINES environment (recall [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: Goodness-of-fit to the target in the grid-world . 8 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: Multi-gated flow network interpretation for path-dependent (a) and Markovian (b) amortized samplers. Under our framework, each node sends a variable amount of flow to its children, depending on the flow’s past trajectory (e.g., either green or red for x1’s parent). In …

**Figure 9.** Figure 9: Illustration of Definition D.1. We endow each node in the unlifted DAG (recall Definition 2.1) with a latent and stochastic random variable (in red). This section builds on the framework of Lahlou et al. (2023) to expand our pathdependent algorithm in Figure 1b to the…

**Figure 10.** Figure 10: TV distance to the target for varying model sizes for the sequence environment. We show that our path-dependent parameterization based on the modified SRWM architecture consistently outperforms its Markovian counterpart (based on a 3-layer ReLU network). In each case,…

**Figure 11.** Figure 11: (left) Our path-dependent parameterization improves goodness-of-fit for the standard 16×16 hypergrid task. (right) By learning a latent dynamical system, we also improve exploration of high-probability regions, as measured by the average R(x) of the 20 most probable e…

**Figure 12.** Figure 12: Results for Lazy Random Walk. A path-dependent parameterization yields a better distributional approximation than a Markovian one for both the RINGS (left) and SINUSOIDAL (right) targets, which we describe in Equations (21) and (22). 0 5000 10000 15000 Training steps …

**Figure 13.** Figure 13: The rotation matrix plays a key role in accelerating learning convergence. While both the chosen link function σ (Equation (23)) and the rotation matrix R are beneficial for training, removing R effectively hinders training. Lazy Random Walk. We evaluate our algorithm…

**Figure 14.** Figure 14: Our SRWM leads to faster convergence than gated RNNs. In contrast to LSTM and GRU, our SRWM-based policy network promotes the diversity of the model’s hidden representations—via the rotation matrix—reducing aliasing and accelerating convergence. v ∼ U({v ∈ R d : ∥v∥2 …

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Pith tools

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