Bregman Centroid Guided Cross-Entropy Method

Hongpeng Cao; Marco Caccamo; Naira Hovakimyan; Yuliang Gu

REVIEW 4 major objections 5 minor 24 references

Bregman Centroid Guided Cross-Entropy Method

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read This paper claims that a performance-weighted Bregman centroid, used as a trust-region anchor for resampling, lets ensemble CEM converge faster and find lower-cost trajectories at negligible extra computation.

desk verdict Clean theory and a genuinely new ensemble-CEM coupling; empirical support is thinner than the claims, and the stress-test failure mode is real but untested. read the letter →

arxiv 2506.02205 v2 pith:3GCZLYXW submitted 2025-06-02 cs.LG cs.AIcs.SYeess.SY

Yuliang Gu , Hongpeng Cao , Marco Caccamo , Naira Hovakimyan This is my paper · ORCID

classification cs.LGcs.AIcs.SYeess.SY

keywords cross-entropymethodBregmancentroidensembleoptimizationdiversitycontrolmodel-basedreinforcementlearningtrajectoryexponentialfamilytrustregionsampling

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

This paper is trying to establish that the main weakness of the Cross-Entropy Method (CEM) in model-based reinforcement learning, premature convergence caused by its unimodal sampling, can be fixed without changing the CEM update itself. The fix is an ensemble-level bookkeeping rule: compute a performance-weighted Bregman centroid of all CEM workers, rank each worker by its weighted divergence to that centroid, and replace the least-contributing worker with a sample from a Bregman ball around the centroid. Because CEM workers are exponential-family distributions, this aggregation costs almost nothing: the centroid is just the weighted average of the sufficient statistics CEM already computes, and the trust-region samples have closed form for the diagonal-Gaussian planners used in model predictive control. The paper reports faster convergence and better solution quality than vanilla and decentralized CEM on synthetic multimodal functions, a cluttered navigation task, and model-based RL benchmarks. If the claim holds, any CEM-based planner can be upgraded by wrapping it in this centroid-guided evolution strategy.

What carries the argument

The central object is the performance-weighted Bregman centroid of the ensemble, together with the dual trust region around it. For a regular exponential family in natural form $p_\theta(x)\propto\exp(\theta^\top T(x)-\Psi(\theta))$, the right-sided Bregman divergence $D_\Psi$ is dual to the mean parameterization $\eta=\nabla\Psi(\theta)$, and the centroid in mean coordinates is exactly $\eta_c=\sum_i w_i\eta_i$, the weighted average of the empirical sufficient statistics CEM already computes. The replacement rule uses a Bregman ball around $\theta_c$ as the trust region, and the paper proposes an exact radial sampler plus a second-order proxy sampler that becomes an axis-aligned box for diagonal Gaussian planners. The machinery does the work of making aggregation essentially free and making diversity injection a simple, geometry-adapted resampling step.

What would settle it

Take a two-basin cost landscape in which six of eight workers sit in a poor but broad basin, the other two sit in a narrow global-optimum basin, and the performance weights are close to equal; compute the Bregman centroid and let BC-EvoCEM respawn the lowest-scoring workers inside the trust region. If replacing a worker with a sample near this centroid consistently produces higher final cost than simply respawning from the best worker's neighborhood, the central claim of improved solution quality is false.

Watch

Extended reading notes

Core claim

BC-EvoCEM claims that the failure mode of ensemble CEM, where workers independently collapse onto the same or different local optima, can be corrected by a single information-geometric summary. At each iteration it assigns each worker a weight $w_i \propto \exp(-\mathbb{E}_{p_{\theta_i}}[J])$, computes the weighted Bregman centroid $\theta_c = \arg\min_\theta \sum_i w_i D_F(\theta_i\|\theta)$, scores every worker by its weighted divergence $\gamma_i = w_i D_F(\theta_i\|\theta_c)$ to the centroid, and replaces the lowest-scoring worker with a sample from the Bregman ball $B_\Delta(\theta_c)$. In the mean parameterization of an exponential family the centroid is just the weighted arithmetic mean of the workers' sufficient statistics, so no extra optimization is needed; the relevance score is a cheap log-likelihood evaluation, and the trust-region sampler is a box for the diagonal-Gaussian planners used in MPC. The paper argues that this coupling keeps exploration alive exactly where it is cheap, and reports that it improves convergence and solution quality over vanilla and decentralized CEM on multimodal synthetic functions, a cluttered navigation task, and model-based RL benchmarks.

Load-bearing premise

The trust region around the Bregman centroid is a good place to put new workers: the centroid is merely an average of all workers' information, not a guarantee of a good or feasible solution, so if that average lies in a poor region, respawning around it can pull the whole ensemble away from better solutions.

Editorial extensions

If this is right

On multimodal synthetic functions, BC-EvoCEM converges faster in both average and best cost than vanilla and decentralized CEM while keeping higher information radius, a proxy for diversity, at iteration 25.
On the cluttered navigation task, BC-EvoCEM keeps trajectories clustered around the centroid path and cuts normalized average cost by 82.4 percent and best cost by 45.3 percent versus decentralized CEM.
In model-based RL with a deterministic dynamics model, BC-EvoCEM attains faster learning and higher asymptotic return on most of the six benchmark tasks tested.
In model-based RL with a probabilistic ensemble dynamics model, the three optimizers are statistically indistinguishable, because the model's intrinsic stochasticity already supplies trajectory dispersion.
The extra computation is at most linear in parameter dimension and negligible compared with environment rollouts, so the method can serve as a drop-in warm-start wrapper for MPC loops.

Reading between the lines

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

Beyond the paper: the authors' deterministic-versus-probabilistic comparison suggests a testable prescription, namely to enable centroid-guided exploration when the dynamics model is deterministic or near its capacity limit and to leave it off when a stochastic model already generates trajectory diversity.
Beyond the paper: the likelihood-based relevance score is not CEM-specific, so the same centroid-anchored resampling transfers to any sampling-based optimizer whose proposal family is exponential, with the sufficient statistics swapped.
Beyond the paper: because the centroid is an unconstrained average, safety-critical planning would likely need the trust-region samples projected onto the feasible set or a constrained centroid; otherwise obstacle avoidance is not guaranteed.
Beyond the paper: the trust-region radius $\Delta$ is fixed; scheduling it downward across iterations would interpolate between exploration and consensus, an untested variant that follows naturally from the method's design.

Share X Bluesky LinkedIn Reddit HN

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

Quick take: BC-EvoCEM is a genuinely new and theoretically tidy upgrade to ensemble CEM. The Bregman-centroid coupling, the trust-region sampler, and the likelihood-based relevance ranking are derived cleanly, and the method costs almost nothing on top of standard CEM. But the empirical evidence is under-powered—three seeds, no error bars on the MBRL figures, no code, unspecified delta and replacement schedule—and the paper never tests the one case where its core heuristic is most fragile: when the weighted centroid sits on a high-cost saddle between modes. That gap is worth naming before anyone adopts this as a drop-in planner.

The theory is the strongest part. Proposition 1 and the mean-coordinate centroid are correct; the relevance score reducing to a weighted log-likelihood is a nice observation, and Theorem 1's uniformity argument holds. The proxy sampler for high dimensions is sensible and clearly derived. This is real work, not a repackaging.

The soft spots are primarily empirical. Three seeds per task is thin for MBRL, and the headline training curves don't show error bars, so the claimed "significant improvements" in Section 6.2 and the deterministic-model results in Figure 5 are suggestive rather than established. The paper honestly flags missing convergence guarantees and simulation-only evaluation. The stress-test concern about the centroid lying on a saddle is legitimate and untreated: the synthetic and navigation tasks don't include a bimodal landscape with separated good modes, and the relevance score can indeed mark a high-quality worker near the centroid as replaceable. That's not a fatal flaw—it might be fixable with a modified score or a check on the centroid's cost—but it is exactly the kind of failure a serious revision should probe. The fact that the method's advantage disappears under probabilistic ensemble dynamics is also worth more discussion than a hypothesis about intrinsic stochasticity.

Who is this for? Anyone using CEM in MPC loops who wants a cheap diversity mechanism. I'd read it as a solid workshop-plus paper; with stronger experiments it could be a proper journal contribution. It deserves a serious referee despite my reservations.

Referee Report

4 major / 5 minor

Summary. The paper proposes BC-EvoCEM, a lightweight augmentation of ensemble CEM. At each iteration, after independent CEM updates, it assigns each worker a performance weight, computes the weighted Bregman centroid of the workers' distributions, scores workers by their weighted Bregman divergence to the centroid, and respawns the lowest-scoring worker by sampling from a Bregman ball around the centroid. Using exponential-family duality, the centroid is shown to be the weighted arithmetic mean in mean coordinates, the score is a likelihood-based ranking, and trust-region sampling can be done with linear-cost operations. Experiments on a synthetic multimodal function, a cluttered 2D navigation task, and PETS-based MBRL benchmarks report faster convergence and lower cost relative to vanilla and decentralized CEM.

Significance. If the central claim holds, BC-EvoCEM is an attractive drop-in wrapper: the extra operations are linear in dimension, the information-geometric derivations (Proposition 1, Section 4.1, Appendix C) are clean and internally consistent, and the method addresses a recognized weakness of CEM (premature convergence) without changing the underlying optimizer. The paper is honest about its limitations, including the absence of global optimality guarantees and the fact that the centroid need not avoid obstacles. However, the empirical support is narrow: 3 seeds, no statistical tests, and no stress test of the trust-region heuristic in the regime where it can concentrate workers on poor solutions. The central claim is plausible but not yet established at the level required for publication.

major comments (4)

[Section 3 (Definition 2) and Algorithm 1] The trust region around the centroid can fail on multimodal landscapes with separated modes, and no experiment stresses this failure mode. Proposition 1 shows the centroid is the weighted arithmetic mean of worker means; with two separated good modes, this mean can lie on a high-cost saddle. Since Algorithm 1 replaces the lowest-scoring worker by sampling from B_Delta(theta_c), the ensemble can be pulled toward the saddle. Section 6.2 concedes that the centroid is not guaranteed to avoid obstacles, yet none of the experiments (Section 6) includes a case where the centroid is in a high-cost region; the synthetic benchmark in Section 6.1 has a symmetric cost with quadratic penalty, and the navigation task reports only average/best cost, not centroid cost. The paper should add an experiment with separated modes or otherwise show that the method degrades gracefully when the centroid is poor; without this, the claimed improvement in solution quality is untested in the method's most dangerous regime.
[Section 3, Definition 1 and Algorithm 1 line 6] The relevance score gamma_i = w_i D(theta_i || theta_c) conflates poor quality with redundancy. A worker with high performance (large w_i) but close to the centroid has small divergence and hence low gamma_i, making it a replacement candidate. The interpretation in Section 3 states that such workers contribute minimally to both exploitation and exploration, but a high-quality worker near the centroid contributes to exploitation; replacing it can discard the best current solution. The paper should either separate the two mechanisms, e.g., by protecting high-weight workers from replacement, or provide an ablation showing that this failure mode does not occur.
[Section 6.3 and Figures 5-8] The MBRL results lack statistical rigor. All curves are means over 3 seeds, with no error bars or significance tests; the statement that performance differences are 'statistically indistinguishable' for probabilistic ensembles is unsupported. Additionally, the 'negligible overhead' claim is not backed by wall-clock data: Appendix D.1 reports normalized costs (Table 2), not computation time. The paper should add error bars or confidence intervals, statistical tests, and at least one overhead measurement (e.g., per-iteration wall-clock time) to substantiate the central empirical claims.
[Section 4.2 and Appendix B, Remark 1] There is a theory-practice gap in the trust-region sampling. Theorem 1 guarantees uniformity for the exact sampler (Algorithm 2), and uniformity of theta_new in B_Delta(theta_c) only when Psi is quadratic. The experiments in Sections 6.2 and 6.3 use the PROXY SAMPLER (Algorithm 3) with the fixed-variance trick for high-dimensional planning. For fixed-variance diagonal Gaussians the proxy is exact, but for the general exponential-family setting in which the theory is presented, the proxy samples from an approximate ellipsoidal trust region, not the exact Bregman ball. The paper should either state clearly that the theoretical guarantees apply only to the exact sampler or provide a bound on the approximation error.

minor comments (5)

[Section 3, after Algorithm 1] There is a typo: 'condidate' should be 'candidate'.
[Theorem 1 statement] The phrase 'TURST-R EGION SAMPLER' contains a typo; it should read 'TRUST-REGION SAMPLER'.
[Section 6.2] The description of Figure 4 says decentralized CEM trajectories 'disperse widely' while BC-EvoCEM produces 'a more diverse and goal-directed planning'; these two uses of 'diverse' are confusing and should be clarified (e.g., diversity in the relevant region versus global dispersion).
[Table 4] The abbreviations 'DE' and 'PE' in the table caption are not defined; they should be spelled out in the caption or in the surrounding text.
[Appendix D.1] The text says a cost summary is provided in Appendix D.1, but Table 2 reports only normalized costs and percentage drops, not computational overhead; the wording should be corrected to avoid implying wall-clock measurements are included.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the centroid and trust-region sampler restate standard, externally cited convex-analysis facts, and the claimed improvements are benchmarked against external CEM baselines with honest null results.

full rationale

The paper's derivation chain is self-contained, and none of its reported improvements reduces to an input by construction. Proposition 1 (the weighted Bregman centroid is the weighted arithmetic mean in mean coordinates) is a standard mean-as-minimizer property of right-sided Bregman divergences, cited to external references [14, 18] and proved in one line from the optimality condition; the reference list contains no self-citations by the present authors. Section 4.1's 'scoring as likelihood-based ranking' (gamma_i proportional to -w_i times the per-sample log-likelihood at the centroid mean) is a pure algebraic rearrangement of the Bregman divergence in Definition 1 via Appendix A, presented as an interpretation of how well a worker explains the aggregated information, not as a fitted parameter renamed as a prediction. Theorem 1 (uniform trust-region sampling) is proven from scratch in Appendix C using polar-coordinate measure facts, with no imported uniqueness claim. The central empirical claims, faster convergence and higher solution quality than vanilla and decentralized CEM on the deterministic-model tasks, are evaluated against external benchmarks (PETS [11] and DecentCEM [12]) under matched hyperparameters, and the paper reports statistically indistinguishable results in the probabilistic-ensemble setting (Fig. 6), which would not occur if the improvement were forced by construction. The one self-referential element, Algorithm 1's replacement of the lowest-scoring worker with a sample from a ball around the workers' own centroid, is a heuristic mechanism rather than a derivation of the claimed result, and the paper explicitly flags its failure mode in Section 6.2 ('the centroid itself is not guaranteed to avoid obstacles') and in the Limitations section (no global optimality guarantees). No equation in the paper equates a reported result to a fitted value or to a self-citation.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The paper adds no new physical or invented entities. Its central algorithm introduces one main hyperparameter (the trust region radius delta), an unspecified replacement schedule, and several heuristic choices (variance freezing, proxy sampler, performance-weight estimation). The mathematical results rely on standard exponential-family duality and on the heuristic assumption that the centroid neighborhood is a promising region for resampling.

free parameters (3)

Trust region radius delta = not reported
Controls how far respawned workers can be from the Bregman centroid; no value or sensitivity analysis is given, yet it determines the exploration-exploitation balance in Algorithms 1 and 4.
Replacement schedule = unspecified, described as periodic
In Algorithm 4 the least-contributing workers are replaced periodically, but the period is not given; it affects diversity and computational cost.
CEM hyperparameters (ensemble size 3, population 100, elite 10%, initial variance 0.1, internal iterations 5) = fixed across benchmarks
These standard CEM settings are inputs shared with baselines, not fitted to outcomes, but the method's relative gains could vary with them.

assumptions (5)

domain assumption Workers belong to a regular, minimal exponential family in natural form with strictly convex cumulant Psi.
Used in Section 4 for the closed-form centroid, likelihood scoring, and trust-region sampling; excludes non-exponential CEM distributions.
domain assumption Performance weights w_i proportional to exp(-E_{p_theta_i}[J]) can be reliably estimated from the CEM population and rank worker quality.
Used in Algorithm 1 line 3 and centroid computation; estimation of E_{p_theta_i}[J] is not detailed in the main text.
ad hoc to paper The performance-weighted Bregman centroid is a meaningful aggregate summary of worker quality, and low relevance score gamma_i marks workers safe to replace.
Core heuristic of Algorithm 1; no formal justification that replacing low gamma_i workers preserves the global optimum.
ad hoc to paper The trust region around the centroid is a promising region for resampling.
Relies on the centroid neighborhood being useful; the paper itself notes in Section 6.2 that the centroid is not guaranteed to avoid obstacles.
ad hoc to paper The fixed-variance trick and second-order proxy sampler are faithful in high-dimensional planning.
Remark 1 freezes variance after a few iterations and uses a Hessian-based ellipsoid instead of the exact Bregman ball; approximation error is not quantified.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Bregman Centroid Guided Cross-Entropy Method." pith.science (2026). https://pith.science/paper/3GCZLYXW

@misc{pith2026250602205,
  author       = {Pith},
  title        = {Pith review of: Bregman Centroid Guided Cross-Entropy Method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3GCZLYXW}},
  note         = {Machine review of arXiv:2506.02205}
}

abstract

The Cross-Entropy Method (CEM) is a widely adopted trajectory optimizer in model-based reinforcement learning (MBRL), but its unimodal sampling strategy often leads to premature convergence in multimodal landscapes. In this work, we propose Bregman Centroid Guided CEM ($\mathcal{BC}$-EvoCEM), a lightweight enhancement to ensemble CEM that leverages $\textit{Bregman centroids}$ for principled information aggregation and diversity control. $\textbf{$\mathcal{BC}$-EvoCEM}$ computes a performance-weighted Bregman centroid across CEM workers and updates the least contributing ones by sampling within a trust region around the centroid. Leveraging the duality between Bregman divergences and exponential family distributions, we show that $\textbf{$\mathcal{BC}$-EvoCEM}$ integrates seamlessly into standard CEM pipelines with negligible overhead. Empirical results on synthetic benchmarks, a cluttered navigation task, and full MBRL pipelines demonstrate that $\textbf{$\mathcal{BC}$-EvoCEM}$ enhances both convergence and solution quality, providing a simple yet effective upgrade for CEM.

Figures

Figures reproduced from arXiv: 2506.02205 by the authors.

**Figure 1.** Illustration of BC-EvoCEM (only means are shown). • Active CEM workers. • Worst workers identified due to redundancy(left) and poor quality(right). ⋆ Bregman Centroid as a geometric average of active workers. × Potential Candidates sampled from the trust-region. (EM) steps. Decentralized ensembles run multiple CEM instances in parallel, keep them independent, and output the best solution at termination [12]. This … view at source ↗

**Figure 2.** Illustration of the Bregman centroid of two Gaussians. Bregman Centroid & Information Radius. Given a collection of points {xi} n i=1 ⊂ S, the Bregman centroid (right-sided) is the solution to the following minimization problem [14]: xc = arg min x∈S 1 n Xn i=1 DF (xi∥x). The corresponding minimized value is known as the Information Radius (IR) [19] (Bregman Information in [18]), which characterizes the diversity… view at source ↗

**Figure 3.** Performance comparison for vanilla, decentralized, and our CEM methods. Solid/dashed lines show the mean/best cost, shaded bands ±1 std. Information-radius (IR) at iter 25 is shown. We first demonstrate our method on a multimodal optimization problem with the cost function ( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Trajectory distributions from decentralized CEM (left) and Bregman–centroid guided [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: Training return curves across six control tasks using PETS with different CEM-based [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: Training return curves across 3 control tasks using PETS with different CEM-based opti [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: Testing return curves across six control tasks using PETS with different CEM-based opti [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

**Figure 8.** Figure 8: Testing return curves across 3 control tasks using PETS with different CEM-based opti [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

24 extracted references · 18 canonical work pages

[1]

R. Y . Rubinstein and D. P. Kroese. The cross-entropy method: a unified approach to com- binatorial optimization, Monte-Carlo simulation and machine learning . Springer Science & Business Media, 2004

work page 2004
[2]

De Boer, D

P.-T. De Boer, D. P. Kroese, S. Mannor, and R. Y . Rubinstein. A tutorial on the cross-entropy method. Annals of operations research, 134:19–67, 2005

work page 2005
[3]

Pinneri, S

C. Pinneri, S. Sawant, S. Blaes, J. Achterhold, J. Stueckler, M. Rolinek, and G. Martius. Sample-efficient cross-entropy method for real-time planning. In Conference on Robot Learn- ing, pages 1049–1065. PMLR, 2021

work page 2021
[4]

Kobilarov

M. Kobilarov. Cross-entropy motion planning. The International Journal of Robotics Re- search, 31(7):855–871, 2012

work page 2012
[5]

Banks, S

C. Banks, S. Wilson, S. Coogan, and M. Egerstedt. Multi-agent task allocation using cross- entropy temporal logic optimization. In 2020 IEEE International Conference on Robotics and Automation (ICRA), pages 7712–7718. IEEE, 2020

work page 2020
[6]

Ha and J

D. Ha and J. Schmidhuber. World models. arXiv preprint arXiv:1803.10122, 2018

arXiv 2018
[7]

Nagabandi, G

A. Nagabandi, G. Kahn, R. S. Fearing, and S. Levine. Neural network dynamics for model- based deep reinforcement learning with model-free fine-tuning. In 2018 IEEE international conference on robotics and automation (ICRA), pages 7559–7566. IEEE, 2018

work page 2018
[8]

Silver, T

D. Silver, T. Hubert, J. Schrittwieser, I. Antonoglou, M. Lai, A. Guez, M. Lanctot, L. Sifre, D. Kumaran, T. Graepel, et al. Mastering chess and shogi by self-play with a general reinforce- ment learning algorithm. arXiv preprint arXiv:1712.01815, 2017. 9

arXiv 2017

Show all 24 references

[9]

Williams, P

G. Williams, P. Drews, B. Goldfain, J. M. Rehg, and E. A. Theodorou. Aggressive driving with model predictive path integral control. In 2016 IEEE international conference on robotics and automation (ICRA), pages 1433–1440. IEEE, 2016

2016
[10]

Okada and T

M. Okada and T. Taniguchi. Variational inference mpc for bayesian model-based reinforcement learning. In Conference on robot learning, pages 258–272. PMLR, 2020

2020
[11]

K. Chua, R. Calandra, R. McAllister, and S. Levine. Deep reinforcement learning in a handful of trials using probabilistic dynamics models. Advances in neural information processing systems, 31, 2018

2018
[12]

Zhang, J

Z. Zhang, J. Jin, M. Jagersand, J. Luo, and D. Schuurmans. A simple decentralized cross- entropy method. Advances in Neural Information Processing Systems, 35:36495–36506, 2022

2022
[13]

Deisenroth and C

M. Deisenroth and C. E. Rasmussen. Pilco: A model-based and data-efficient approach to pol- icy search. In Proceedings of the 28th International Conference on machine learning (ICML- 11), pages 465–472, 2011

2011
[14]

Nielsen and R

F. Nielsen and R. Nock. Sided and symmetrized bregman centroids. IEEE transactions on Information Theory, 55(6):2882–2904, 2009

2009
[15]

L. M. Bregman. The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR computational mathe- matics and mathematical physics, 7(3):200–217, 1967

1967
[16]

Snell, K

J. Snell, K. Swersky, and R. Zemel. Prototypical networks for few-shot learning. Advances in neural information processing systems, 30, 2017

2017
[17]

Ahn and S

K. Ahn and S. Chewi. Efficient constrained sampling via the mirror-langevin algorithm. Ad- vances in Neural Information Processing Systems, 34:28405–28418, 2021

2021
[18]

Banerjee, S

A. Banerjee, S. Merugu, I. S. Dhillon, and J. Ghosh. Clustering with bregman divergences. Journal of machine learning research, 6(Oct):1705–1749, 2005

2005
[19]

Csisz ´ar, P

I. Csisz ´ar, P. C. Shields, et al. Information theory and statistics: A tutorial. Foundations and Trends® in Communications and Information Theory, 1(4):417–528, 2004

2004
[20]

Barndorff-Nielsen

O. Barndorff-Nielsen. Information and exponential families: in statistical theory. John Wiley & Sons, 2014

2014
[21]

S.-i. Amari. Information geometry of the em and em algorithms for neural networks. Neural networks, 8(9):1379–1408, 1995

1995
[22]

Villani et al

C. Villani et al. Optimal transport: old and new, volume 338. Springer, 2008

2008
[23]

Schneider

R. Schneider. Convex bodies: the Brunn–Minkowski theory, volume 151. Cambridge university press, 2013

2013
[24]

T. Wang, X. Bao, I. Clavera, J. Hoang, Y . Wen, E. Langlois, S. Zhang, G. Zhang, P. Abbeel, and J. Ba. Benchmarking model-based reinforcement learning. arXiv preprint arXiv:1907.02057, 2019. 10 A Relevance Score as Likelihood Evaluation Recall that the Bregman divergence induc...

1907 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

R. Y . Rubinstein and D. P. Kroese. The cross-entropy method: a unified approach to com- binatorial optimization, Monte-Carlo simulation and machine learning . Springer Science & Business Media, 2004

work page 2004

[2] [2]

De Boer, D

P.-T. De Boer, D. P. Kroese, S. Mannor, and R. Y . Rubinstein. A tutorial on the cross-entropy method. Annals of operations research, 134:19–67, 2005

work page 2005

[3] [3]

Pinneri, S

C. Pinneri, S. Sawant, S. Blaes, J. Achterhold, J. Stueckler, M. Rolinek, and G. Martius. Sample-efficient cross-entropy method for real-time planning. In Conference on Robot Learn- ing, pages 1049–1065. PMLR, 2021

work page 2021

[4] [4]

Kobilarov

M. Kobilarov. Cross-entropy motion planning. The International Journal of Robotics Re- search, 31(7):855–871, 2012

work page 2012

[5] [5]

Banks, S

C. Banks, S. Wilson, S. Coogan, and M. Egerstedt. Multi-agent task allocation using cross- entropy temporal logic optimization. In 2020 IEEE International Conference on Robotics and Automation (ICRA), pages 7712–7718. IEEE, 2020

work page 2020

[6] [6]

Ha and J

D. Ha and J. Schmidhuber. World models. arXiv preprint arXiv:1803.10122, 2018

arXiv 2018

[7] [7]

Nagabandi, G

A. Nagabandi, G. Kahn, R. S. Fearing, and S. Levine. Neural network dynamics for model- based deep reinforcement learning with model-free fine-tuning. In 2018 IEEE international conference on robotics and automation (ICRA), pages 7559–7566. IEEE, 2018

work page 2018

[8] [8]

Silver, T

D. Silver, T. Hubert, J. Schrittwieser, I. Antonoglou, M. Lai, A. Guez, M. Lanctot, L. Sifre, D. Kumaran, T. Graepel, et al. Mastering chess and shogi by self-play with a general reinforce- ment learning algorithm. arXiv preprint arXiv:1712.01815, 2017. 9

arXiv 2017