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arxiv: 2605.02147 · v1 · submitted 2026-05-04 · 💻 cs.RO · math.OC

Sampling-Based Control via Entropy-Regularized Optimal Transport

Vincent Pacelli , Akash Ratheesh , Evangelos A. Theodorou This is my paper

Pith reviewed 2026-05-08 18:23 UTC · model grok-4.3

classification 💻 cs.RO math.OC
keywords optimal transportmodel predictive controlsampling-based controlentropy regularizationrobotic systemsSinkhorn algorithmMPPICEM
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The pith

Entropy-regularized optimal transport refines control sequence candidates in sampling-based MPC to avoid mode-averaging while preserving solution space coverage.

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

The paper introduces OT-MPC, a new sampling-based model predictive control algorithm. Traditional methods like MPPI and CEM suffer from issues like mode-averaging in complex cost landscapes because they ignore the geometry of the control problem. OT-MPC uses an entropy-regularized optimal transport formulation to couple candidate sequences with low-cost proposals, refining them toward better solutions while coordinating updates to keep diversity. This leads to closed-form updates via the Sinkhorn algorithm that run in real time. Experiments on various robotic tasks show higher success rates compared to prior methods.

Core claim

By computing an optimal coupling between candidate control sequences and low-cost proposals through entropy-regularized optimal transport, OT-MPC refines candidates toward nearby promising samples while coordinating updates across the ensemble to maintain coverage of the solution space, with closed-form gradient-free updates derived via the Sinkhorn algorithm.

What carries the argument

The entropy-regularized optimal transport problem between control sequence candidates and low-cost proposals, solved using the Sinkhorn algorithm to produce the optimal coupling matrix for updates.

If this is right

  • Improved success rates over MPPI and CEM on navigation, manipulation, and locomotion tasks.
  • Real-time performance enabled by gradient-free closed-form updates.
  • Avoidance of pathological behaviors such as mode-averaging in complex cost landscapes.
  • Maintained coverage of the solution space during refinement of the ensemble.

Where Pith is reading between the lines

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

  • OT-MPC could extend to other sampling-based optimization problems beyond MPC in robotics.
  • The approach might reduce sensitivity to initial proposal distributions in highly multimodal environments.
  • Integration with learned dynamics models could further enhance performance in uncertain settings.

Load-bearing premise

The geometry induced by the entropy-regularized optimal transport cost reliably avoids new failure modes such as over-concentration or sensitivity to the proposal distribution while allowing real-time execution.

What would settle it

Demonstrating persistent mode-averaging or lower success rates than standard methods on tasks with complex, multimodal cost landscapes would falsify the claim that OT-MPC overcomes the limitations of prior sampling-based MPC methods.

Figures

Figures reproduced from arXiv: 2605.02147 by Akash Ratheesh, Evangelos A. Theodorou, Vincent Pacelli.

Figure 1
Figure 1. Figure 1: The proposed OT-MPC algorithm controlling a Unitree Go2 quadruped. Colored curves show planned foot trajectories from different candidate solutions at a key decision point where multiple gait strategies are viable. OT-MPC naturally maintains diverse candidates when the cost landscape is multimodal; when candidates converge to a single mode, local proposal sampling concentrates refinement where it is needed… view at source ↗
Figure 2
Figure 2. Figure 2: Overview of OT-MPC. (a) Proposals are sampled to explore the trajectory space. (b) The Sinkhorn algorithm computes a soft coupling between candidates (bold curves) and proposals factoring in both cost and proximity. Colors indicate which to which proposals the candidate is most strongly coupled. (c) Each candidate updates toward its coupled proposals via a barycentric projection (17)—refining locally while… view at source ↗
Figure 3
Figure 3. Figure 3: Robotics control experiments used to evaluate OT-MPC. (a) Kinematic Bicycle navigating through an obstacle field. (b) Quadrotor navigating from start to goal through cluttered environment. (c) Two-Quadrotor system cooperatively carrying a suspended load through an opening in the wall. (d) Push-T task using Franka where the manipulator has to push and align the T-Block to a goal location (e) Unitree Go2 box… view at source ↗
read the original abstract

Sampling-based model predictive control methods like MPPI and CEM are essential for real-time control of nonlinear robotic systems, particularly where discontinuous dynamics preclude gradient-based optimization. However, these methods derive from information-theoretic objectives that are agnostic to the geometry of the control problem, leading to pathological behaviors such as mode-averaging when the cost landscape is complex. We present OT-MPC, a sampling-based algorithm that overcomes these limitations through an entropy-regularized optimal transport formulation. By computing an optimal coupling between candidate control sequences and low-cost proposals, OT-MPC refines candidates toward nearby promising samples while coordinating updates across the ensemble to maintain coverage of the solution space. We derive closed-form, gradient-free updates via the Sinkhorn algorithm, enabling real-time performance. Experiments on navigation, manipulation, and locomotion tasks demonstrate improved success rates over existing methods.

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 manuscript proposes OT-MPC, a sampling-based model predictive control algorithm that reformulates the problem as an entropy-regularized optimal transport task. Candidate control sequences are coupled to low-cost proposals via the Sinkhorn algorithm, yielding closed-form gradient-free refinements that aim to avoid mode-averaging while preserving ensemble coverage; real-time execution is claimed, with supporting experiments on navigation, manipulation, and locomotion tasks showing higher success rates than MPPI and CEM baselines.

Significance. If the central claims hold, the work offers a geometrically informed alternative to information-theoretic sampling methods, potentially improving robustness in discontinuous or multimodal cost landscapes without sacrificing the real-time property essential for robotic hardware. The explicit use of Sinkhorn for closed-form updates and the emphasis on maintaining coverage are strengths that could influence subsequent sampling-based MPC research.

major comments (2)
  1. [§3] §3 (OT-MPC formulation): The claim that Sinkhorn iterations produce reliable real-time refinements without over-concentration depends on the specific cost metric between control sequences and the entropy weight; no convergence bounds, iteration limits, or sensitivity analysis to these choices are provided, which directly bears on whether the method escapes the mode-averaging pathology of MPPI while remaining computationally tractable.
  2. [Experiments] Experiments (navigation/manipulation/locomotion results): Reported success-rate gains are presented without ablations on proposal distribution, entropy regularization strength, or Sinkhorn iteration count; this leaves open whether the improvements are robust or sensitive to hyper-parameter tuning, undermining the assertion that the OT geometry reliably avoids new failure modes.
minor comments (2)
  1. [§3] Notation for the transport cost and coupling matrix should be introduced with explicit definitions before the Sinkhorn update equations to improve readability.
  2. [Abstract] The abstract states 'improved success rates' without numerical values or baseline comparisons; adding a brief quantitative summary would strengthen the high-level claim.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below and indicate planned revisions.

read point-by-point responses

  1. Referee: [§3] §3 (OT-MPC formulation): The claim that Sinkhorn iterations produce reliable real-time refinements without over-concentration depends on the specific cost metric between control sequences and the entropy weight; no convergence bounds, iteration limits, or sensitivity analysis to these choices are provided, which directly bears on whether the method escapes the mode-averaging pathology of MPPI while remaining computationally tractable.

    Authors: The referee correctly identifies that the manuscript provides no theoretical convergence bounds, explicit iteration limits, or sensitivity analysis for the Sinkhorn updates under our control-sequence cost metric and entropy weight. While Sinkhorn is known to converge for entropy-regularized OT, specific rates depend on these choices and were not analyzed. In the revision we will add empirical convergence plots, recommended iteration counts that preserve real-time execution, and sensitivity analysis to the entropy weight and cost metric, supporting that the coupling avoids over-concentration while escaping mode-averaging. We cannot supply general theoretical bounds for arbitrary cost metrics in this setting. revision: partial

  2. Referee: [Experiments] Experiments (navigation/manipulation/locomotion results): Reported success-rate gains are presented without ablations on proposal distribution, entropy regularization strength, or Sinkhorn iteration count; this leaves open whether the improvements are robust or sensitive to hyper-parameter tuning, undermining the assertion that the OT geometry reliably avoids new failure modes.

    Authors: We agree that the absence of ablations leaves the robustness of the reported gains open to question. The original experiments demonstrate higher success rates than MPPI and CEM across tasks, but do not vary proposal distribution, entropy regularization strength, or Sinkhorn iteration count. In the revised manuscript we will add these ablation studies for all three tasks, showing that performance improvements remain consistent within practical parameter ranges and that the OT coupling does not introduce new sensitivities or failure modes. revision: yes

standing simulated objections not resolved
  • Deriving theoretical convergence bounds for Sinkhorn iterations with the specific control-sequence cost metric and entropy weight in OT-MPC.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper formulates sampling-based MPC as an entropy-regularized optimal transport problem between control sequences and low-cost proposals, then applies the standard Sinkhorn algorithm to obtain updates. This is an application of an established algorithm (Sinkhorn) to a new problem framing rather than a self-referential derivation. No load-bearing step reduces by construction to fitted parameters, self-citations, or renamed inputs; the cost function is defined directly from the control objective, and the updates follow from known OT properties without circular redefinition. The central claim (improved coordination and coverage over MPPI/CEM) rests on the formulation and empirical results, not on tautological reduction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The method rests on the standard mathematical properties of entropy-regularized optimal transport and the Sinkhorn algorithm; the only new modeling choice is the definition of the transport cost between control sequences and proposals.

free parameters (1)
  • entropy regularization weight
    Controls the softness of the coupling; must be chosen for each task to balance refinement and diversity.
axioms (1)
  • domain assumption The control cost landscape admits a useful geometric interpretation under the chosen transport cost.
    Invoked to claim that the optimal coupling avoids mode averaging.

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discussion (0)

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Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages · 1 internal anchor

  1. [1]

    Unlike other sampling- based methods, SCD updates particles based on both cost and geometric proximity—enabling local refinement while preserving diversity

    We propose SCD, a gradient-free optimization algorithm, and its MPC instantiation, OT-MPC. Unlike other sampling- based methods, SCD updates particles based on both cost and geometric proximity—enabling local refinement while preserving diversity

    work page

  2. [2]

    We establish theoretical properties of SCD, including mono- tone descent, convergence guarantees, and closed-form up- dates for quadratic costs

    work page

  3. [3]

    We demonstrate empirically that OT-MPC achieves higher success rates than MPPI, CEM, and SV-MPC on challenging navigation, manipulation, and locomotion tasks. II. RE L AT E DWO R K The OT-MPC algorithm inherits the control-as-inference formulation common to sampling-based MPC but replaces the information-theoretic objective with an optimal transport one. ...

    work page 2000

  4. [4]

    Rehg, Byron Boots, and Evangelos A

    Grady Williams, Nolan Wagener, Brian Goldfain, Paul Drews, James M. Rehg, Byron Boots, and Evangelos A. Theodorou. Model predictive path integral control: From theory to parallel computation.Journal of Guidance, Control, and Dynamics, 40(2):344–357, 2017

    work page 2017

  5. [5]

    Rehg, Byron Boots, and Evangelos A

    Grady Williams, Nolan Wagener, Brian Goldfain, Paul Drews, James M. Rehg, Byron Boots, and Evangelos A. Theodorou. Information-theoretic MPC for model-based reinforcement learning.International Conference on Robotics and Automation, pages 1714–1721, 2017

    work page 2017

  6. [6]

    Rubinstein

    Reuven Y . Rubinstein. The cross-entropy method for combinatorial and continuous optimization.Methodology and Computing in Applied Probability, 1(2):127–190, 1999

    work page 1999

  7. [7]

    Cross-entropy randomized motion planning

    Marin Kobilarov. Cross-entropy randomized motion planning. InRobotics: Science and Systems, 2011

    work page 2011

  8. [8]

    Sample-efficient cross-entropy method for real- time planning

    Cristina Pinneri, Shambhuraj Sawant, Sebastian Blaes, Jan Achterhold, Joerg Stueckler, Michal Rolinek, and Georg Martius. Sample-efficient cross-entropy method for real- time planning. InConference on Robot Learning, pages 1049–1065. PMLR, 2020

    work page 2020

  9. [9]

    Murphey, and Dieter Fox

    Ian Abraham, Ankur Handa, Nathan Ratliff, Kendall Lowrey, Todd D. Murphey, and Dieter Fox. Model-based generalization under parameter uncertainty using path integral control.IEEE Robotics and Automation Letters, 5(2):2864–2871, 2020

    work page 2020

  10. [10]

    Available: https://arxiv.org/abs/2409.15610

    Haoru Xue, Chaoyi Pan, Zeji Yi, Guannan Qu, and Guanya Shi. Full-order sampling-based MPC for torque- level locomotion control via diffusion-style annealing. arXiv preprint arXiv:2409.15610, 2024

    work page arXiv 2024

  11. [11]

    TD-MPC2: Scalable, robust world models for continuous control

    Nicklas Hansen, Hao Su, and Xiaolong Wang. TD-MPC2: Scalable, robust world models for continuous control. InThe Twelfth International Conference on Learning Representations, 2024

    work page 2024

  12. [12]

    Zhang, Sofia Kwok, John M

    Juan Alvarez-Padilla, John Z. Zhang, Sofia Kwok, John M. Dolan, and Zachary Manchester. Real-time whole-body control of legged robots with model-predictive path in- tegral control. InInternational Conference on Robotics and Automation, pages 14721–14727. IEEE, 2025

    work page 2025

  13. [13]

    Control of legged robots using model predic- tive optimized path integral

    Hossein Keshavarz, Alejandro Ramirez-Serrano, and Ma- jid Khadiv. Control of legged robots using model predic- tive optimized path integral. InInternational Conference on Humanoid Robots, pages 1–8. IEEE, 2025

    work page 2025

  14. [14]

    Ratliff, Dieter Fox, Fabio Ramos, and Byron Boots

    Mohak Bhardwaj, Balakumar Sundaralingam, Arsalan Mousavian, Nathan D. Ratliff, Dieter Fox, Fabio Ramos, and Byron Boots. STORM: An integrated framework for fast joint-space model-predictive control for reactive manipulation. InConference on Robot Learning, pages 750–759. PMLR, 2022

    work page 2022

  15. [15]

    CoVO-MPC: Theoretical analysis of sampling-based MPC and optimal covariance design

    Zeji Yi, Chaoyi Pan, Guanqi He, Guannan Qu, and Guanya Shi. CoVO-MPC: Theoretical analysis of sampling-based MPC and optimal covariance design. InLearning for Dynamics and Control Conference, pages 1122–1135. PMLR, 2024

    work page 2024

  16. [16]

    Variational inference MPC for Bayesian model-based reinforcement learning

    Masashi Okada and Tadahiro Taniguchi. Variational inference MPC for Bayesian model-based reinforcement learning. InConference on robot learning, pages 258–272. PMLR, 2020

    work page 2020

  17. [17]

    Sinkhorn distances: Lightspeed com- putation of optimal transport

    Marco Cuturi. Sinkhorn distances: Lightspeed com- putation of optimal transport. InAdvances in Neural Information Processing Systems, volume 26, 2013

    work page 2013

  18. [18]

    Computational optimal transport: With applications to data science.Foundations and Trends in Machine Learning, 11(5-6):355–607, 2019

    Gabriel Peyr ´e and Marco Cuturi. Computational optimal transport: With applications to data science.Foundations and Trends in Machine Learning, 11(5-6):355–607, 2019

    work page 2019

  19. [19]

    Planning by probabilistic inference

    Hagai Attias. Planning by probabilistic inference. In International Workshop on Artificial Intelligence and Statistics, pages 9–16. PMLR, 2003

    work page 2003

  20. [20]

    Stein variational model pre- dictive control

    Alexander Lambert, Fabio Ramos, Byron Boots, Dieter Fox, and Adam Fishman. Stein variational model pre- dictive control. InConference on Robot Learning, pages 1278–1297. PMLR, 2021

    work page 2021

  21. [21]

    Variational inference MPC using Tsallis divergence

    Ziyi Wang, Oswin So, Jason Gibson, Bogdan Vlahov, Manan S Gandhi, Guan-Horng Liu, and Evangelos A Theodorou. Variational inference MPC using Tsallis divergence. InRobotics: Science and Systems, 2021

    work page 2021

  22. [22]

    Stein variational guided model predictive path integral control: Proposal and experiments with fast maneuvering vehicles

    Kohei Honda, Naoki Akai, Kosuke Suzuki, Mizuho Aoki, Hirotaka Hosogaya, Hiroyuki Okuda, and Tatsuya Suzuki. Stein variational guided model predictive path integral control: Proposal and experiments with fast maneuvering vehicles. InIEEE International Conference on Robotics and Automation, pages 8604–8610, 2024

    work page 2024

  23. [23]

    Theodorou

    Yuichiro Aoyama and Evangelos A. Theodorou. General- ized maximum entropy differential dynamic programming. InConference on Decision and Control, pages 8825–8831. IEEE, 2024

    work page 2024

  24. [24]

    Stein variational gradient descent without gradient

    Jun Han and Qiang Liu. Stein variational gradient descent without gradient. InInternational Conference on Machine Learning, pages 1900–1908. PMLR, 2018

    work page 1900

  25. [25]

    Entropic model predictive optimal transport over dynamical systems.Automatica, 152:110980, 2023

    Kaito Ito and Kenji Kashima. Entropic model predictive optimal transport over dynamical systems.Automatica, 152:110980, 2023

    work page 2023

  26. [26]

    Optimal stochastic vehicle path planning using covariance steering

    Kazuhide Okamoto and Panagiotis Tsiotras. Optimal stochastic vehicle path planning using covariance steering. Robotics and Automation Letters, 4(3):2276–2281, 2019

    work page 2019

  27. [27]

    Trajectory distribution control for model predictive path integral control using covariance steering

    Ji Yin, Zhiyuan Zhang, Evangelos Theodorou, and Pana- giotis Tsiotras. Trajectory distribution control for model predictive path integral control using covariance steering. InInternational Conference on Robotics and Automation, pages 1478–1484. IEEE, 2022

    work page 2022

  28. [28]

    Saravanos, Isin M

    Augustinos D. Saravanos, Isin M. Balci, Efstathios Bako- las, and Evangelos A. Theodorou. Distributed model predictive covariance steering. InInternational Confer- ence on Intelligent Robots and Systems, pages 5740–5747. IEEE, 2024

    work page 2024

  29. [29]

    Sara- vanos, and Evangelos A

    Akash Ratheesh, Vincent Pacelli, Augustinos D. Sara- vanos, and Evangelos A. Theodorou. Operator splitting covariance steering for safe stochastic nonlinear control. InConference on Decision and Control, pages 3552–3559. IEEE, 2025

    work page 2025

  30. [30]

    Primal Wasserstein imitation learning

    Robert Dadashi, L ´eonard Hussenot, Matthieu Geist, and Olivier Pietquin. Primal Wasserstein imitation learning. InInternational Conference on Learning Representations, 2021

    work page 2021

  31. [31]

    Imitation learning with sinkhorn distances

    Georgios Papagiannis and Yunpeng Li. Imitation learning with sinkhorn distances. InJoint European Confer- ence on Machine Learning and Knowledge Discovery in Databases, pages 116–131, 2022

    work page 2022

  32. [32]

    Denoising diffusion probabilistic models.Advances in Neural Information Processing Systems, 33:6840–6851, 2020

    Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models.Advances in Neural Information Processing Systems, 33:6840–6851, 2020

    work page 2020

  33. [33]

    Planning with diffusion for flexible behavior synthesis

    Michael Janner, Yilun Du, Joshua Tenenbaum, and Sergey Levine. Planning with diffusion for flexible behavior synthesis. InInternational Conference on Machine Learning, pages 9902–9915. PMLR, 2022

    work page 2022

  34. [34]

    Diffusion policy: Visuomotor policy learning via action diffusion.International Journal of Robotics Research, 44(10-11):1684–1704, 2025

    Cheng Chi, Zhenjia Xu, Siyuan Feng, Eric Cousineau, Yilun Du, Benjamin Burchfiel, Russ Tedrake, and Shuran Song. Diffusion policy: Visuomotor policy learning via action diffusion.International Journal of Robotics Research, 44(10-11):1684–1704, 2025

    work page 2025

  35. [35]

    Accelerating motion planning via optimal transport

    An T Le, Georgia Chalvatzaki, Armin Biess, and Jan Pe- ters. Accelerating motion planning via optimal transport. InAdvances in Neural Information Processing Systems, volume 36, 2023

    work page 2023

  36. [36]

    C´edric Villani.Optimal transport: old and new, volume

    work page

  37. [37]

    Envelope theorems for arbitrary choice sets.Econometrica, 70(2):583–601, 2002

    Paul Milgrom and Ilya Segal. Envelope theorems for arbitrary choice sets.Econometrica, 70(2):583–601, 2002

    work page 2002

  38. [38]

    Fan, Patrick Spieler, Ali-akbar Agha-mohammadi, and Evangelos A

    Bogdan Vlahov, Jason Gibson, David D. Fan, Patrick Spieler, Ali-akbar Agha-mohammadi, and Evangelos A. Theodorou. Low frequency sampling in model predictive path integral control.Robotics and Automation Letters, 9(5):4543–4550, 2024

    work page 2024

  39. [39]

    Reference-Free Sampling-Based Model Predictive Control

    Fabian Schramm, Pierre Fabre, Nicolas Perrin-Gilbert, and Justin Carpentier. Reference-free sampling- based model predictive control.arXiv preprint arXiv:2511.19204, 2025

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  40. [40]

    Autonomous navigation of agvs in unknown cluttered environments: log-mppi control strategy.IEEE Robotics and Automation Letters, 7(4):10240–10247, 2022

    Ihab S Mohamed, Kai Yin, and Lantao Liu. Autonomous navigation of agvs in unknown cluttered environments: log-mppi control strategy.IEEE Robotics and Automation Letters, 7(4):10240–10247, 2022

    work page 2022

  41. [41]

    Optuna: A next-generation hyperparameter optimization framework

    Takuya Akiba, Shotaro Sano, Toshihiko Yanase, Takeru Ohta, and Masanori Koyama. Optuna: A next-generation hyperparameter optimization framework. InProceedings of the 25th ACM SIGKDD international conference on knowledge discovery & data mining, pages 2623–2631, 2019. AP P E N D I XA PRO O F S O FAL G O R I T H MPRO P E RT I E S Proposition 3(Biconvexity)....

    work page 2019

  42. [42]

    Compared to standard locomotion task we have an extra 13 states corresponding to the position, orientation and velocities of the box

    Box Pushing:We have further extended the locomotion task to a box-pushing task, where the quadruped has to move a box from a starting location to the goal location. Compared to standard locomotion task we have an extra 13 states corresponding to the position, orientation and velocities of the box. The MPC cost is similar to the locomotion cost; however th...

    work page