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arxiv: 2605.22507 · v1 · pith:T7J6U6BWnew · submitted 2026-05-21 · 💻 cs.LG · stat.ML

Generative Modeling by Value-Driven Transport

Pablo Moreno-Mu\~noz , Adrian M\"uller , Gergely Neu This is my paper

Pith reviewed 2026-05-22 07:55 UTC · model grok-4.3

classification 💻 cs.LG stat.ML
keywords generative modelingvalue-driven transportmeasure transportstochastic controlprimal-dual algorithmstraight pathsoptimal transportdiffusion models
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0 comments X

The pith

Generative modeling can be recast as optimal control for measure transport, yielding straight-path policies from value functions.

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

The paper proposes a new framework for generative modeling by formulating measure transport as a discrete-time stochastic control problem. Adapting classic control theory results, they pose the problem as a linear program whose dual variables correspond to the optimal value function that directly encodes the optimal control policy. A simulation-free primal-dual algorithm then computes approximate value functions and the associated value-driven transport policies. Well-trained VDT policies produce straight transport paths that support fast and robust simulation while allowing the same enhancements as diffusion and flow models, such as conditional generation and classifier-free guidance. Experiments indicate competitive performance with potential for scalability.

Core claim

By adapting results from control theory, the measure transport problem is posed as a linear program whose dual variables correspond to the optimal value function of the control problem, which directly encodes the optimal control policy. An efficient simulation-free primal-dual algorithm computes approximately optimal value functions and the resulting value-driven transport policies that approximate the true optimal policy for generative modeling.

What carries the argument

The primal-dual algorithm approximating the optimal value function of the stochastic control formulation of measure transport, which directly defines the value-driven transport policy.

If this is right

  • Transport occurs along straight paths, enabling quick and robust simulation of the generative process.
  • VDT policies can incorporate conditional generation, classifier-free guidance, and unpaired data-to-data translation.
  • The simulation-free training supports scalability to larger problems.
  • Performance remains competitive with flows, diffusions, and Schrödinger bridges in experiments.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This control formulation may reduce training instability by avoiding the need for simulation steps during optimization.
  • Straight paths could lower sampling variance compared to the curved trajectories common in diffusion models.
  • The approach might extend naturally to other measure transport tasks outside generative modeling, such as domain adaptation.

Load-bearing premise

That policies from the approximated value functions are sufficiently close to the true optimal control policy to produce straight paths and practical robustness.

What would settle it

Measuring the average deviation from straight lines in trajectories sampled from a trained VDT policy; large curvature or non-linear paths would indicate the approximation fails to deliver the claimed transport properties.

Figures

Figures reproduced from arXiv: 2605.22507 by Adrian M\"uller, Gergely Neu, Pablo Moreno-Mu\~noz.

Figure 1
Figure 1. Figure 1: Continuous- and discrete-time transport plans. Continuous-time plans are defined as smooth [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Value functions and value-driven transport policies in a two-dimensional example, plotted [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Few-step generation with a learned VDT model. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Experiments on MNIST: conditional generation and data translation. [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Downscaled images and their deblurred counterparts produced by VDT, [PITH_FULL_IMAGE:figures/full_fig_p027_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Downscaled images and their deblurred counterparts produced by VDT, [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Downscaled images and their deblurred counterparts produced by VDT, [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Forward sampling from a VDT policy from EMNIST to MNIST. [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Reverse sampling from a VDT policy from MNIST to EMNIST. [PITH_FULL_IMAGE:figures/full_fig_p028_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: MNIST digits generated by CFG with various guidance scales: [PITH_FULL_IMAGE:figures/full_fig_p029_10.png] view at source ↗
read the original abstract

We propose a new framework for generative modeling based on a discrete-time stochastic control formulation of measure transport. Adapting classic results from control theory, we formulate our problem as a linear program whose dual variables correspond to the \emph{optimal value function} of the control problem, which directly encodes the optimal control policy. Exploiting this LP formulation, we develop an efficient simulation-free primal-dual algorithm for computing approximately optimal value functions and the associated \emph{value-driven transport} (VDT) policies which approximate the true optimal policy. We show that well-trained VDT policies enjoy numerous favorable properties in comparison with other state-of-the-art methods based on flows, diffusions, or Schr\"odinger bridges: they lead to straight transport paths which can be simulated quickly and robustly, and can be enhanced in all the same ways as diffusion and flow-based models (e.g., conditional generation, classifier-free guidance, unpaired data-to-data translation are all easy to incorporate). We evaluate our methodology in a range of experiments, with results that indicate strong performance and good potential for scalability.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces a generative modeling framework called Value-Driven Transport (VDT) that reformulates measure transport as a discrete-time stochastic control problem. It casts this as a linear program whose dual variables correspond to the optimal value function, which encodes the optimal control policy. A simulation-free primal-dual algorithm is proposed to compute approximate value functions and the resulting VDT policies. The central claims are that well-trained VDT policies produce straight transport paths that can be simulated quickly and robustly, and that these policies support the same enhancements as diffusion and flow models (conditional generation, classifier-free guidance, unpaired translation). Experiments are reported to indicate competitive performance and scalability potential.

Significance. If the approximation guarantees and empirical claims hold, the work would provide a useful alternative to flow-, diffusion-, and Schrödinger-bridge-based generative models by enabling straight-line paths with reduced simulation cost and improved robustness. The LP-dual construction and simulation-free training are clear strengths that adapt classic stochastic control results to this setting. The ease of incorporating conditional and guidance mechanisms is a practical advantage. These elements could influence future work on efficient transport-based generation if the policy approximation quality is rigorously established.

major comments (2)
  1. [§4] §4 (primal-dual algorithm): no explicit convergence rates, discretization error bounds, or approximation guarantees are provided for how closely the learned value functions and induced policies approach the true optimal control policy. The central claim that VDT policies yield straight transport paths and simulation robustness rests on this approximation being sufficiently accurate; without quantitative bounds on step size, iteration count, or function-class capacity, it is possible for deviations to produce curved paths or require corrective simulation, undermining the stated advantages over flows and diffusions.
  2. [§5] §5 (experiments): the reported results compare VDT to baselines but do not include ablation studies or quantitative metrics (e.g., path straightness measured by integrated curvature or simulation variance) that directly test whether the learned policies achieve the claimed straight paths and robustness. This weakens the link between the algorithmic construction and the favorable properties asserted in the abstract.
minor comments (2)
  1. [§3] Notation for the discrete-time control problem and the LP dual could be clarified with an explicit statement of the continuous-time limit and how the policy is recovered from the value function.
  2. [§1] The abstract and introduction would benefit from a brief comparison table or paragraph situating VDT relative to recent optimal-transport and Schrödinger-bridge generative models.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their constructive and insightful comments on our manuscript. We provide point-by-point responses to the major comments below.

read point-by-point responses

  1. Referee: [§4] §4 (primal-dual algorithm): no explicit convergence rates, discretization error bounds, or approximation guarantees are provided for how closely the learned value functions and induced policies approach the true optimal control policy. The central claim that VDT policies yield straight transport paths and simulation robustness rests on this approximation being sufficiently accurate; without quantitative bounds on step size, iteration count, or function-class capacity, it is possible for deviations to produce curved paths or require corrective simulation, undermining the stated advantages over flows and diffusions.

    Authors: The manuscript builds on the exact equivalence between the linear program and the optimal control problem, which guarantees straight transport paths for the true optimal value function. For the approximate primal-dual algorithm with neural network parameterization, we do not provide explicit convergence rates or error bounds in the current version. This is a valid observation, and we will revise the paper to include a discussion section addressing approximation quality, potential sources of error, and their implications for path straightness, drawing on related literature in approximate dynamic programming and stochastic control. However, establishing rigorous quantitative bounds for this specific setting would constitute a significant extension of the theoretical analysis. revision: partial

  2. Referee: [§5] §5 (experiments): the reported results compare VDT to baselines but do not include ablation studies or quantitative metrics (e.g., path straightness measured by integrated curvature or simulation variance) that directly test whether the learned policies achieve the claimed straight paths and robustness. This weakens the link between the algorithmic construction and the favorable properties asserted in the abstract.

    Authors: We agree that incorporating quantitative metrics for path straightness and simulation robustness, as well as ablation studies, would provide stronger empirical support for the claimed advantages. The current experiments emphasize generative quality and comparisons to baselines, with qualitative evidence of straight paths. In the revised manuscript, we will add these quantitative evaluations and ablations to directly validate the straight-path and robustness properties. revision: yes

standing simulated objections not resolved
  • Deriving explicit convergence rates, discretization error bounds, or approximation guarantees for the primal-dual algorithm with neural network function approximation.

Circularity Check

0 steps flagged

No significant circularity; derivation adapts external control theory

full rationale

The paper formulates measure transport as a linear program whose dual encodes the optimal value function from stochastic control theory, then introduces a primal-dual algorithm to approximate the associated policies. The claimed straight transport paths and simulation robustness are presented as consequences of approximating the optimal control policy in this LP setting, drawing on classic external results rather than any fitted parameter renamed as a prediction or any self-referential definition. No load-bearing step reduces by construction to the paper's own inputs or prior self-citations; the central claims rest on the LP-dual construction and the new algorithm, which remain independent of the target generative properties.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review based on abstract only; limited visibility into specific assumptions or parameters.

axioms (1)
  • standard math Classic results from control theory on stochastic control problems and their linear programming formulations hold and can be adapted to measure transport.
    The paper states it adapts these results to formulate the generative modeling problem as an LP.
invented entities (1)
  • Value-driven transport (VDT) policies no independent evidence
    purpose: Approximate optimal control policies obtained from the dual of the transport LP.
    Introduced as the output of the primal-dual algorithm that directly encodes the optimal policy via the value function.

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