Pith. sign in

REVIEW 4 major objections 6 minor 2 cited by

Consensus-based optimization—a particle swarm pulled toward its cost-weighted consensus—provably reaches global optima and finds lower-cost robot trajectories than MPPI, CEM, and CMA-ES on three demanding benchmarks.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

On long-horizon, underactuated, and high-dimensional simulated robot planning tasks, consensus-based optimization finds lower-cost trajectories than MPPI, CEM, and CMA-ES.

T0 review reviewed 2026-08-03 challenge →

load-bearing objection The empirical case for CBO on three robotics benchmarks is worth taking seriously; the paper's own global-convergence proof has a real gap in the finite-sample step, so the guarantee as stated does not follow. the 4 major comments →

arxiv 2602.06868 v2 pith:BO2OFIJG submitted 2026-02-06 cs.RO

Consensus-based optimization (CBO): Towards Global Optimality in Robotics

Xudong Sun , Armand Jordana , Massimo Fornasier , Jalal Etesami , Majid Khadiv This is my paper
classification cs.RO
keywords consensus-based optimizationglobal optimizationzero-order optimizationtrajectory optimizationparticle dynamicssoftmax weightingrobot control
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

This paper introduces consensus-based optimization (CBO) to robot trajectory optimization. CBO keeps a population of candidate control sequences, weights each by the softmax of its cost, and drives every particle toward the resulting consensus point with a diffusion term scaled by distance to that point. Unlike local sampling methods, the dynamics are not gradient-driven, so particles are not trapped by individual local minima; the authors prove under certain regularity conditions that the population converges to a global optimum. On three challenging benchmarks—long-horizon navigation with a 200-dimensional decision space, a double cartpole under a 0.5 N force limit, and a 23-DoF humanoid with only a terminal cost—CBO reports lower costs than MPPI, CEM, and CMA-ES.

Core claim

CBO's particle dynamics, du = -λ(u-ū)dr + σ‖u-ū‖dW, with ū the softmax-weighted consensus (Eq. 22), drive the surrogate cost down and the consensus point to a global optimizer when the cost function satisfies inverse-continuity and relaxed-Lipschitz conditions over a neighborhood of the optimum. The paper argues that this is the first demonstration that CBO scales to robotic trajectory optimization, and that it outperforms standard zero-order methods by producing a population whose members are consistently low-cost rather than a few good samples among many poor ones.

What carries the argument

The central mechanism is the consensus point ū^r, the softmax-weighted average of particle costs, and the associated SDE (Eq. 21) combining a linear drift toward ū with Brownian noise scaled by the particle's distance to ū. The distance-scaled noise lets the empirical distribution develop irregular, asymmetric tails that concentrate on promising regions, avoiding the shape constraints of Gaussian parameterizations. Proposition 2, resting on Lemmas 2–3, shows exponential decay of the Lyapunov function and convergence of the consensus to the global optimizer.

Load-bearing premise

The global-convergence guarantee rests on the cost function satisfying inverse-continuity and relaxed-Lipschitz conditions (Eqs. 25–27) in a neighborhood of the optimum, and on the noise satisfying 2λ > n_a T σ²; the benchmark cost functions (with indicator penalties and contact-rich dynamics) are not shown to satisfy these, and the experiments use decaying σ rather than the constant σ assumed in the theory.

What would settle it

Compute the constants η and L in Eqs. (25)–(26) for the long-horizon cost (32) near a tunnel-boundary optimum: if the inverse-continuity ratio is zero or unbounded, the assumptions of Proposition 2 fail, and any success of CBO there is empirical rather than guaranteed. Alternatively, run CBO with constant σ and 2λ ≤ n_a T σ² on a smooth two-minimum cost; if the consensus still converges to the global minimum reliably, then the stated rate in Eq. (53) is not a necessary condition for global convergence.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • CBO offers a zero-order trajectory optimization method that avoids being trapped by local minima in the way MPPI, CEM, and CMA-ES are, because particle updates are driven by distance to the consensus rather than by local gradient estimates.
  • Under the stated regularity assumptions, the theory guarantees a lower surrogate cost after a bounded number of iterations and convergence of the consensus point to a global optimizer with high probability.
  • On the three tested benchmarks, CBO achieves lower costs than the baselines, suggesting it can handle long horizons, severe underactuation, and high-dimensional terminal-cost problems.
  • Because CBO's empirical distribution is non-parametric, it circumvents the curse of distribution parametrization that limits Gaussian-based methods in high-dimensional decision spaces.
  • The framework provides a unifying lens for interpreting existing zero-order methods—path integral, CEM, CMA-ES—as instances of surrogate minimization, making their local nature explicit.

Where Pith is reading between the lines

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

  • Readers should expect the practical success of CBO to depend on an annealing schedule: the paper's experiments use an exponential decay of σ, while the theory assumes constant σ; formalizing this schedule could extend the global-convergence guarantee to time-varying noise.
  • CBO's history-carrying particles suggest a natural fit for model-predictive control: rather than resampling from a fixed covariance, the population could be reused across control cycles, potentially enabling real-time global replanning.
  • A testable prediction of the paper's mechanism is that an MPPI variant with a shared, history-carrying particle population and distance-proportional exploration should close part of the reported performance gap, isolating which design choice matters.
  • The high variance in the humanoid results hints that CBO can discover qualitatively different locomotion solutions; a multi-consensus extension might exploit this multimodality rather than averaging it away.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. This paper introduces consensus-based optimization (CBO) to robotic trajectory optimization. It casts MPPI, CEM, and CMA-ES as local zero-order methods based on Gaussian smoothing / surrogate minimization, then proposes CBO, where particles evolve by the SDE (21) toward a softmax-weighted consensus point. The main theoretical claim (Prop. 2) is that under inverse-continuity and local Lipschitz assumptions (25)-(27), CBO's surrogate cost decreases and the consensus converges to a global optimizer with high probability; the proof, in Appendix B, restates Lemmas 2-3 from [21]. The paper then reports experiments on long-horizon 200-D navigation, a double cartpole with a 0.5 N force limit, and a 23-DoF humanoid with terminal cost only, where CBO achieves lower cost than MPPI, CMA, and CMA-ES.

Significance. If the theoretical claim were fully established, the paper would be a solid contribution: it brings a global zero-order method to robotics and supplies a unified interpretation of existing local methods, with encouraging empirical results on challenging problems. The paper is transparent that Lemmas 2-3 are restatements of peer-reviewed results [21], and the experiments address problems of real interest. However, the proof of Prop. 2 has internal gaps and the assumptions are not met by the benchmarks, so the advertised 'guaranteed global convergence' is not supported as stated. The empirical comparisons may still be valid as an engineering contribution, but the central theoretical claim needs substantial repair.

major comments (4)
  1. [Appendix B, finite-sample step] The finite-sample argument is invalid. Lemma 3 (Eq. 54) bounds only the squared distance of the empirical mean (1/N)Σ u^(r,i) to u*, yet the proof states that choosing ε_e small ensures 'all particles reside in Bκ(u*)'. An average near u* does not control the maximum distance; a single particle at distance √(N ε_e) satisfies the mean bound. Moreover, the quantity to decrease in (29) is the average cost (1/N)Σ J(u^(r,i)); controlling the mean position does not control this average unless every particle is in the Lipschitz region and J is locally Lipschitz. No concentration bound on the empirical measure or on the cost tail is supplied. The stated reformulation of Theorem 3.8 in [21] appears to be misstated as a mean bound rather than a Wasserstein/empirical-measure bound; as written it cannot support the conclusion.
  2. [Section III-C / IV] The assumptions underlying Lemma 2/3 and Prop. 2 are not verified for any benchmark and are plausibly violated. The long-horizon cost (32) contains discontinuous indicator penalties I_not_in_tunnel and I_obstacle, so J is neither locally Lipschitz (25) nor inverse continuous (26) in Bκ(u*). The double-cartpole and humanoid costs are evaluated through contact-rich MuJoCo dynamics, which are non-smooth. Since (25)-(27) are load-bearing for the exponential decay of V in Lemma 2, the guarantee of Prop. 2 does not transfer to the experiments. The reported lower costs are an empirical claim, not a consequence of the theory. The paper should either verify the assumptions for these problems (unlikely) or explicitly present the experiments as heuristics without the global-convergence guarantee.
  3. [Appendix B, Eq. (64)-(68)] In the mean-field part, inequality (26) is applied to the initial measure U^(0) over the full space, although (26) is stated only for u∈Bκ(u*). The step from ∫||u-u*||² dU^(0) to (1/η²)∫(J(u)-J(u*))² dU^(0) requires supp(U^(0))⊂Bκ(u*), which is not true for the random initial populations used in the experiments. Consequently the definition of ζ(J,U^(0)) in (67)-(68) is not justified, and the strict decrease in (69) is not established even in the mean-field setting unless additional global inverse-continuity assumptions are imposed.
  4. [Section III-C / Eq. (28)] The experiments replace the constant-σ SDE in (21) with an exponential decay schedule for σ, but Lemma 2/3 and the bound r* ∝ 1/(2λ - n_a T σ²) are derived for constant σ. No argument is given that annealing preserves the exponential decay or the positivity of the denominator. Since the choice of λ is said to follow Remark 13, the paper should report the actual λ and σ schedules and prove or cite a result covering time-dependent σ. As written, the theory and experiments are mismatched.
minor comments (6)
  1. [Appendix B, Lemma 3] The notation in (54) and (56) uses U both as a measure and as a dummy variable; the displayed bound appears to be for the empirical mean, not the squared 2-Wasserstein distance claimed in the lemma title. Please correct the statement and align it with [21].
  2. [Remark 11, Eq. (24)] The reduction λΔr=1, σ=√Δr leads to σ√Δr = Δr in the discretized update, not σ² as printed. Please clarify the intended scaling.
  3. [Appendix A, proof of Prop. 1] The proof contains an uncontrolled approximation in (48) ('reasonable due to the KL-divergence constraint'). If Proposition 1 is a formal statement, the approximation should be quantified or removed; otherwise label it as heuristic.
  4. [Abstract] The phrase 'guaranteed to converge to a global optimum under mild assumptions' should be qualified by quoting assumptions (25)-(27) and noting that they are not verified in the experiments.
  5. [Section IV-A] Hyperparameters (λ, σ schedule, number of iterations) are not reported per experiment; please include them for reproducibility.
  6. [Section IV-B, Eq. (32)] The indicator penalties are discontinuous; if the authors intend the cost to satisfy (25)-(27), consider replacing them with smooth approximations or explicitly stating that the theory does not cover this cost.

Circularity Check

0 steps flagged

No significant circularity: the global-convergence guarantee is imported from prior peer-reviewed CBO theorems with assumptions independent of the robotics benchmarks, and the benchmark results are new empirical measurements rather than fitted predictions.

full rationale

The paper's central theoretical claim, Proposition 2, is explicitly built on Lemmas 2 and 3, which are verbatim restatements of Theorem 3.7 and Theorem 3.8 of [21] (Fornasier, Klock, Riedl). Although one of the current authors (Fornasier) is also an author of [21], the cited theorems are parameter-free mathematical statements with assumptions (inverse continuity, relaxed Lipschitz conditions on the cost, noise regime 2λ > n_a T σ²) that do not include the robotics results. They are not fitted to the benchmark data, and they are externally checkable results from a peer-reviewed journal; therefore this self-citation qualifies as independent support rather than circularity. The paper does not rename a known result as its own: it transfers an existing CBO convergence framework to trajectory optimization and supplies new experiments. The benchmark claims are empirical comparisons against MPPI, CEM, and CMA-ES, and are not derived from the assumptions of Proposition 2; they are new measurements, so they cannot be circular. The skeptical observation that Lemma 3's finite-sample bound on the empirical mean does not by itself imply that all particles enter Bκ(u*), and the use of exponential σ-annealing beyond the constant-σ theory, are proof-completeness concerns about the strength of the guarantee, not instances where a prediction is equivalent to an input by construction. No equation is defined in terms of the predicted quantity, no fitted parameter is renamed as a prediction, and no load-bearing uniqueness theorem is invoked solely from the authors' own prior work. Thus the score is 0.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

The paper's genuine contribution is architectural (application + framework), not ontological: it postulates no new entities. Its free parameters are the standard CBO knob set (ρ, λ, σ, N) plus a hand-added exponential σ decay that is outside the theory. The central untested premise is that the benchmark cost functions satisfy the smoothness-type conditions required by the imported convergence theorems — which the discontinuous indicators and contact dynamics in the experiments likely violate.

free parameters (4)
  • ρ (softmax temperature, Eq. 9) = not reported per task
    Controls weight concentration on low-cost particles. Prop 2 requires ρ 'large enough'; in the benchmarks ρ is a free knob whose numerical values are absent.
  • λ (drift/decay rate, Eq. 21) = not reported
    Set per Remark 13 using the imported rate (28); its numerical value is absent, and for n_a·T = 138 (humanoid) the recommended regime implies a very large λ.
  • σ and its exponential decay schedule = initial σ = 10 (long horizon); decay rate not reported
    Initial σ is matched to baselines, but the exponential annealing ('we use an exponential decay of the σ... ensures the convergence behavior', §IV-B) is an ad hoc addition outside the constant-σ theory of Lemmas 2-3.
  • population size N = 1000 / 5000 / 10,000
    Chosen per experiment; the finite-sample guarantee (Lemma 3) requires N, Δr, ϵ_e, δ to be instantiated, which the paper does not do.
axioms (5)
  • domain assumption J satisfies inverse continuity and relaxed Lipschitz conditions (25)-(27) in a neighborhood of u*.
    Needed by Prop 2/Lemma 2 for exponential Lyapunov decay. Unverified for the benchmarks; the long-horizon cost (32) has discontinuous indicator terms, so the assumption is likely false there.
  • domain assumption The global optimum u* lies in the support of the initial particle distribution U(0).
    Condition for Lemma 2's decay and Prop 2's conclusion; plausible for unbounded Gaussian initialization but never stated or checked for the benchmarks.
  • standard math Constant-σ mean-field CBO dynamics with 2λ > n_a·T·σ² (Eq. 28) as analyzed in Theorems 3.7-3.8 of [21].
    Lemmas 2-3 are restatements of [21]; the paper's experiments use discrete Euler-Maruyama updates, finite N, and exponentially decaying σ, which is outside this theorem's stated regime.
  • domain assumption Hydrax/MuJoCo MJX simulations faithfully represent the double cartpole and G1 humanoid dynamics.
    All benchmark cost claims are simulator outputs; the paper provides no validation of the sims against the physical systems.
  • domain assumption The finite-sample softmax consensus (22) approximates the mean-field weighted mean well enough for Lemma 3's probability bounds to apply at the chosen N and Δr.
    Lemma 3 gives δ(ϵ_e, N, Δr, n_a×T, V), but the paper never instantiates these quantities for the experiments.

reviewed 2026-08-03 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Consensus-based optimization (CBO): Towards Global Optimality in Robotics." pith.science (2026). https://pith.science/paper/BO2OFIJG

@misc{pith2026260206868,
  author       = {Pith},
  title        = {Pith review of: Consensus-based optimization (CBO): Towards Global Optimality in Robotics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BO2OFIJG}},
  note         = {Machine review of arXiv:2602.06868}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

Zero-order optimization has recently received significant attention for designing optimal trajectories and policies for robotic systems. However, most existing methods (e.g., MPPI, CEM, and CMA-ES) are local in nature, as they rely on gradient estimation. In this paper, we introduce consensus-based optimization (CBO) to robotics, which is guaranteed to converge to a global optimum under mild assumptions. We provide theoretical analysis and illustrative examples that give intuition into the fundamental differences between CBO and existing methods. To demonstrate the scalability of CBO for robotics problems, we consider three challenging trajectory optimization scenarios: (1) a long-horizon problem for a simple system, (2) a dynamic balance problem for a highly underactuated system, and (3) a high-dimensional problem with only a terminal cost. Our results show that CBO is able to achieve lower costs with respect to existing methods on all three challenging settings. This opens a new framework to study global trajectory optimization in robotics.

Figures

Figures reproduced from arXiv: 2602.06868 by Armand Jordana, Jalal Etesami, Majid Khadiv, Massimo Fornasier, Xudong Sun.

Figure 1
Figure 1. Figure 1: An irregular, non-parametric distribution can focus [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A cost function landscape in a contour plot constructed [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Long horizon planning: comparison of the best loss [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: MPPI and CMA get stuck at a local minimum and [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Double cartpole with cart mass mc = 1.0 kg. Pole masses: m1 = m2 = 0.1 kg. Pole lengths: l1 = l2 = 1.0m. we set friction loss to 0 for both revolute joints. The running cost is set to be the bound violation cost of control according to [31] and the terminal cost is calculated as the summation of all the following terms with equal weighting: • distance to upright pole angles, i.e., for i = 1, 2, minimize (c… view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of the best loss among the population of [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: CBO optimized trajectory: Three snapshots (start, mid [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Exploiting Structure with Anisotropic Consensus-Based Optimization

    math.OC 2026-07 accept novelty 6.0

    Anisotropic CBO's computational complexity depends exponentially only on the intrinsic dimension of an additively separable objective, not the ambient dimension, under aligned anisotropic noise.

  2. Convergence of Consensus-Based Particle Methods for Nonconvex Bi-Level Optimization

    math.OC 2026-05 unverdicted novelty 6.0

    Establishes exponential convergence in Wasserstein distance for the mean-field limit and finite-particle approximation of a consensus-based method solving nonconvex bi-level optimization problems.

Reference graph

Works this paper leans on

30 extracted references · 5 linked inside Pith · cited by 2 Pith papers

  1. [1]

    Optimization-based control for dynamic legged robots

    Patrick M Wensing, Michael Posa, Yue Hu, Adrien Escande, Nicolas Mansard, and Andrea Del Prete. Optimization-based control for dynamic legged robots. IEEE Transactions on Robotics, 40:43–63, 2023

    2023

  2. [2]

    Learning- based legged locomotion: State of the art and future perspectives.The International Journal of Robotics Research, 44(8):1396–1427, 2025

    Sehoon Ha, Joonho Lee, Michiel van de Panne, Zhaom- ing Xie, Wenhao Yu, and Majid Khadiv. Learning- based legged locomotion: State of the art and future perspectives.The International Journal of Robotics Research, 44(8):1396–1427, 2025

    2025

  3. [3]

    Visual imitation enables contextual humanoid control.arXiv preprint arXiv:2505.03729, 2025

    Arthur Allshire, Hongsuk Choi, Junyi Zhang, David McAllister, Anthony Zhang, Chung Min Kim, Trevor Darrell, Pieter Abbeel, Jitendra Malik, and Angjoo Kanazawa. Visual imitation enables contextual humanoid control.arXiv preprint arXiv:2505.03729, 2025

    Pith/arXiv arXiv 2025

  4. [4]

    Omniretarget: Interaction- preserving data generation for humanoid whole-body loco-manipulation and scene interaction.arXiv preprint arXiv:2509.26633, 2025

    Lujie Yang, Xiaoyu Huang, Zhen Wu, Angjoo Kanazawa, Pieter Abbeel, Carmelo Sferrazza, C Karen Liu, Rocky Duan, and Guanya Shi. Omniretarget: Interaction- preserving data generation for humanoid whole-body loco-manipulation and scene interaction.arXiv preprint arXiv:2509.26633, 2025

    Pith/arXiv arXiv 2025

  5. [5]

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

    2025

  6. [6]

    π0.5: A vision-language-action model with open-world generalization.arXiv preprint arXiv:2504.16054, 2025

    Physical Intelligence, Kevin Black, Noah Brown, James Darpinian, Karan Dhabalia, Danny Driess, Adnan Es- mail, Michael Equi, Chelsea Finn, Niccolo Fusai, et al. π0.5: A vision-language-action model with open-world generalization.arXiv preprint arXiv:2504.16054, 2025

    Pith/arXiv arXiv 2025

  7. [8]

    Isaac gym: High performance gpu-based physics simulation for robot learning.arXiv preprint arXiv:2108.10470, 2021

    Viktor Makoviychuk, Lukasz Wawrzyniak, Yunrong Guo, Michelle Lu, Kier Storey, Miles Macklin, David Hoeller, Nikita Rudin, Arthur Allshire, Ankur Handa, et al. Isaac gym: High performance gpu-based physics simulation for robot learning.arXiv preprint arXiv:2108.10470, 2021

    Pith/arXiv arXiv 2021

  8. [9]

    An introduction to zero-order op- timization techniques for robotics.arXiv preprint arXiv:2506.22087, 2025

    Armand Jordana, Jianghan Zhang, Joseph Amigo, and Ludovic Righetti. An introduction to zero-order op- timization techniques for robotics.arXiv preprint arXiv:2506.22087, 2025

    arXiv 2025

  9. [10]

    Sampling-based methods for optimal control: Theory, algorithms, and applications

    Chaoyi Pan, Zeji Yi, Guanya Shi, and Guannan Qu. Sampling-based methods for optimal control: Theory, algorithms, and applications. In2025 IEEE 64th Confer- ence on Decision and Control (CDC), pages 3775–3793. IEEE, 2025

    2025

  10. [11]

    Aggressive driving with model predictive path integral control

    Grady Williams, Paul Drews, Brian Goldfain, James M Rehg, and Evangelos A Theodorou. Aggressive driving with model predictive path integral control. In2016 IEEE international conference on robotics and automation (ICRA), pages 1433–1440. IEEE, 2016

    2016

  11. [12]

    Springer Science & Business Media, 2004

    Reuven Y Rubinstein and Dirk P Kroese.The cross- entropy method: a unified approach to combinatorial op- timization, Monte-Carlo simulation and machine learn- ing. Springer Science & Business Media, 2004

    2004

  12. [13]

    Completely derandomized self-adaptation in evolution strategies

    Nikolaus Hansen and Andreas Ostermeier. Completely derandomized self-adaptation in evolution strategies. Evolutionary computation, 9(2):159–195, 2001

    2001

  13. [14]

    Full-order sampling-based mpc for torque- level locomotion control via diffusion-style annealing

    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. In 2025 IEEE International Conference on Robotics and Automation (ICRA), pages 4974–4981. IEEE, 2025

    2025

  14. [15]

    Control of legged robots using model predictive optimized path integral

    Hossein Keshavarz, Alejandro Ramirez-Serrano, and Ma- jid Khadiv. Control of legged robots using model predictive optimized path integral. In2025 IEEE-RAS 24th International Conference on Humanoid Robots (Hu- manoids), pages 1–8. IEEE, 2025

    2025

  15. [16]

    Sampling-based model predictive control leverag- ing parallelizable physics simulations.IEEE Robotics and Automation Letters, 2025

    Corrado Pezzato, Chadi Salmi, Elia Trevisan, Max Spahn, Javier Alonso-Mora, and Carlos Hern ´andez Cor- bato. Sampling-based model predictive control leverag- ing parallelizable physics simulations.IEEE Robotics and Automation Letters, 2025

    2025

  16. [17]

    Td-cd- mppi: Temporal-difference constraint-discounted model predictive path integral control.IEEE Robotics and Automation Letters, 11(1):498–505, 2025

    Pietro Noah Crestaz, Ludovic De Matteis, Elliot Chane- Sane, Nicolas Mansard, and Andrea Del Prete. Td-cd- mppi: Temporal-difference constraint-discounted model predictive path integral control.IEEE Robotics and Automation Letters, 11(1):498–505, 2025

    2025

  17. [18]

    Information-geometric optimization al- gorithms: A unifying picture via invariance principles

    Yann Ollivier, Ludovic Arnold, Anne Auger, and Niko- laus Hansen. Information-geometric optimization al- gorithms: A unifying picture via invariance principles. Journal of Machine Learning Research, 18(18):1–65, 2017

    2017

  18. [19]

    A consensus-based model for global optimization and its mean-field limit.Mathematical Models and Methods in Applied Sciences, 27(01):183–204, 2017

    Ren ´e Pinnau, Claudia Totzeck, Oliver Tse, and Stephan Martin. A consensus-based model for global optimization and its mean-field limit.Mathematical Models and Methods in Applied Sciences, 27(01):183–204, 2017

    2017

  19. [20]

    An analytical framework for consensus-based global optimization method.Mathematical Models and Methods in Applied Sciences, 28(06):1037–1066, 2018

    Jos ´e A Carrillo, Young-Pil Choi, Claudia Totzeck, and Oliver Tse. An analytical framework for consensus-based global optimization method.Mathematical Models and Methods in Applied Sciences, 28(06):1037–1066, 2018

    2018

  20. [21]

    Consensus-based optimization methods converge glob- ally.SIAM Journal on Optimization, 34(3):2973–3004, 2024

    Massimo Fornasier, Timo Klock, and Konstantin Riedl. Consensus-based optimization methods converge glob- ally.SIAM Journal on Optimization, 34(3):2973–3004, 2024

    2024

  21. [22]

    World Scientific Publishing Company, 1991

    Yuval Davidor.Genetic Algorithms and Robotics: A heuristic strategy for optimization, volume 1. World Scientific Publishing Company, 1991

    1991

  22. [23]

    Analytical integral global optimization

    Sebastien Labbe and Andrea Del Prete. Analytical integral global optimization. InProceedings of the 7th Annual Learning for Dynamics & Control Confer- ence, volume 283 ofProceedings of Machine Learning Research, pages 711–722. PMLR, 04–06 Jun 2025

    2025

  23. [24]

    Springer Science & Business Media, 2001

    Pedro Larra ˜naga and Jose A Lozano.Estimation of distribution algorithms: A new tool for evolutionary computation, volume 2. Springer Science & Business Media, 2001

    2001

  24. [25]

    Random gradient-free minimization of convex functions.Foun- dations of Computational Mathematics, 17(2):527–566, April 2017

    Yurii Nesterov and Vladimir Spokoiny. Random gradient-free minimization of convex functions.Foun- dations of Computational Mathematics, 17(2):527–566, April 2017

    2017

  25. [26]

    A generalized path integral control approach to rein- forcement learning.The Journal of Machine Learning Research, 11:3137–3181, 2010

    Evangelos Theodorou, Jonas Buchli, and Stefan Schaal. A generalized path integral control approach to rein- forcement learning.The Journal of Machine Learning Research, 11:3137–3181, 2010

    2010

  26. [27]

    Model predictive path integral control: From theory to parallel computation.Journal of Guidance, Control, and Dynamics, 40(2):344–357, 2017

    Grady Williams, Andrew Aldrich, 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

    2017

  27. [28]

    Bidirectional relation between cma evolution strategies and natural evolution strategies

    Youhei Akimoto, Yuichi Nagata, Isao Ono, and Shigenobu Kobayashi. Bidirectional relation between cma evolution strategies and natural evolution strategies. InParallel Problem Solving from Nature, PPSN XI: 11th International Conference, Krak ´ow, Poland, September 11-15, 2010, Proceedings, Part I 11, pages 154–163. Springer, 2010

    2010

  28. [29]

    Hydrax: Sampling-based model predic- tive control on gpu with jax and mujoco mjx, 2024

    Vince Kurtz. Hydrax: Sampling-based model predic- tive control on gpu with jax and mujoco mjx, 2024. https://github.com/vincekurtz/hydrax

    2024

  29. [30]

    evosax: Jax-based evolution strate- gies

    Robert Tjarko Lange. evosax: Jax-based evolution strate- gies. InProceedings of the Companion Conference on Genetic and Evolutionary Computation, pages 659–662, 2023

    2023

  30. [31]

    Spline-interpolated model predictive path in- tegral control with stein variational inference for re- active navigation

    Takato Miura, Naoki Akai, Kohei Honda, and Susumu Hara. Spline-interpolated model predictive path in- tegral control with stein variational inference for re- active navigation. (arXiv:2404.10395), April 2024. arXiv:2404.10395 [cs]. APPENDIX To ensure consistent cross-referencing with the main text, the equation numbering is continued in this appendix. A. ...

    Pith/arXiv arXiv 2024

This paper was first reviewed by deepseek-v4-flash on August 3, 2026.